two and three spin triangular ising model variational approximations the implications of the known hard hexagon lattice gas results for the triangular ising model with both pair and triplet interactions are pointed out employing an appropriate generalization of the variational method of baxter we determine using the lowest order approximation the phase boundaries for this model when the pair interactions are ferromagnetic higher approximations are presented for the case of pure triplet interactions and the resulting phase diagrams are in excellent agreement with all exactly known results

on the lifetime behavior of a discrete time shock model in this article we study a shock model in which the shocks occur according to a binomial process i e the interarrival times between successive shocks follow a geometric distribution with mean 1 p according to the model the system fails when the time between two consecutive shocks is less than a prespecified level this is the discrete time version of the so called d shock model which has been previously studied for the continuous case we obtain the probability mass function and probability generating function of the system s lifetime we also present an extension of the results to the case where the shock occurrences are dependent in a markovian fashion

fft based homogenization for microstructures discretized by linear hexahedral elements summary the fft based homogenization method of moulinec suquet has recently emerged as a powerful tool for computing the macroscopic response of complex microstructures for elastic and inelastic problems in this work we generalize the method to problems discretized by trilinear hexahedral elements on cartesian grids and physically nonlinear elasticity problems we present an implementation of the basic scheme that reduces the memory requirements by a factor of four and of the conjugate gradient scheme that reduces the storage necessary by a factor of nine compared with a naive implementation for benchmark problems in linear elasticity the solver exhibits mesh and contrast independent convergence behavior and enables the computational homogenization of complex structures for instance arising from computed tomography computed tomography ct imaging techniques there exist 3d microstructures involving pores and defects for which the original fft based homogenization scheme does not converge in contrast for the proposed scheme convergence is ensured also the solution fields are devoid of the spurious oscillations and checkerboarding artifacts associated to conventional schemes we demonstrate the power of the approach by computing the elasto plastic response of a long fiber reinforced thermoplastic material with 172 106 displacement degrees of freedom copyright 2016 john wiley sons ltd

δ shock model based on polya process and its optimal replacement policy abstract shock models are of great interest in engineering reliability among the others the shock model has been widely studied in the literature in this model the system breaks down due to the arrivals of two successive shocks which are too close to each other that is the system fails when the time between two consecutive shocks falls below a fixed threshold in the literature the shock model has been mostly studied by assuming that shocks arrive according to a renewal process so that the interarrival times between shocks are independent and identically distributed in the current paper we consider the case when the shock arrival process is described by a polya process which has dependent interarrival times in particular we obtain survival function and mean lifetime of the system and study the optimal replacement policy for the shock model based on polya process

on the application of the critical minimum energy subspace method to disordered systems abstract we discuss the recent application to strongly disordered systems of the critical minimum energy subspace crmes method used to limit the energy subspace of the wang landau sampling we compare with our results on the 3d random field ising model obtained by a multi range wang landau simulation over the whole energy range we point out some problems that may arise when applying the crmes scheme to models having a complex free energy landscape

universal features and tail analysis of the order parameter distribution of the two dimensional ising model an entropic sampling monte carlo study we present a numerical study of the order parameter probability density function pdf of the square ising model for lattices with linear sizes l 80 140 a recent efficient entropic sampling scheme combining the wang landau and broad histogram methods and based on the high levels of the wang landau process in dominant energy subspaces is employed we find that for large lattices there exists a stable window of the scaled order parameter in which the full ansatz including the pre exponential factor for the tail regime of the universal pdf is well obeyed this window is used to estimate the equation of state exponent and to observe the behavior of the universal constants implicit in the functional form of the universal pdf the probability densities are used to estimate the universal privman fisher coefficient and to investigate whether one could obtain reliable estimates of the universal constants controlling the asymptotic behavior of the tail regime

the random field ising model with an asymmetric trimodal probability distribution the ising model in the presence of a random field is investigated within the mean field approximation based on landau expansion the random field is drawn from the trimodal probability distribution p hi p hi h0 q hi h0 r hi where the probabilities p q r take on values within the interval 0 1 consistent with the constraint p q r 1 asymmetric distribution hi is the random field variable and h0 the respective strength this probability distribution is an extension of the bimodal one allowing for the existence in the lattice of non magnetic particles or vacant sites the current random field ising system displays second order phase transitions which for some values of p q and h0 are followed by first order phase transitions thus confirming the existence of a tricritical point and in some cases two tricritical points also reentrance can be seen for appropriate ranges of the aforementioned variables using the variational principle we determine the equilibrium equation for magnetization solve it for both transitions and at the tricritical point in order to determine the magnetization profile with respect to h0

monte carlo analysis of the critical properties of the two dimensional randomly bond diluted ising model via wang landau algorithm abstract the influence of random bond dilution on the critical properties of the two dimensional ising model on a square lattice with periodic boundary conditions is presented using the monte carlo process with wang landau sampling the lattice linear size is l 20 140 and the concentration of the diluted bonds spans a wide range from weak to strong dilution namely q 0 05 0 1 0 2 0 3 0 4 the respective percolation limit for the square lattice is q c p e r c 0 5 its pure version q 0 has a second order phase transition with vanishing specific heat critical exponent an example of inapplicability of the harris criterion the main effort is focused on the temperature dependence of the specific heat and magnetic susceptibility to estimate the respective maximum values and subsequent pseudocritical temperatures for extracting the relative critical exponents we have also looked at the probability distribution of the susceptibility pseudocritical temperature and specific heat for assessing self averaging the study is carried out in the appropriately restricted but dominant energy subspaces by applying the finite size scaling analysis the critical exponents are estimated the specific heat exponent vanishes the correlation length exponent is equal to one and the critical exponents ratio retains its pure ising model value thus supporting the strong universality hypothesis

fast finite element calculation of effective conductivity of random continuum microstructures the recursive poincare steklov operator method abstract the effective conductivity of a block of composite can be extracted from the dirichlet to neumann poincare steklov operator pso for that block in this paper a domain decomposition method for computing the pso over a finite element mesh is discussed a new numerical strategy is introduced to accelerate the computation of this operator using the schur complement to calculate the pso for the smallest subdomains then recursively merging subdomain psos up to the full domain at each step of the algorithm information extraneous to the pso is discarded the effective conductivity values computed by this method are identical to those obtained from a basic finite element method an order of magnitude faster and with much less computer memory consumed as proof of concept effective conductivity measurements are presented for a percolating random fractal like microstructure across a range of phase fractions

determining the density of states for classical statistical models a random walk algorithm to produce a flat histogram we describe an efficient monte carlo algorithm using a random walk in energy space to obtain a very accurate estimate of the density of states for classical statistical models the density of states is modified at each step when the energy level is visited to produce a flat histogram by carefully controlling the modification factor we allow the density of states to converge to the true value very quickly even for large systems from the density of states at the end of the random walk we can estimate thermodynamic quantities such as internal energy and specific heat capacity by calculating canonical averages at any temperature using this method we not only can avoid repeating simulations at multiple temperatures but we can also estimate the free energy and entropy quantities that are not directly accessible by conventional monte carlo simulations this algorithm is especially useful for complex systems with a rough landscape since all possible energy levels are visited with the same probability as with the multicanonical monte carlo technique our method overcomes the tunneling barrier between coexisting phases at first order phase transitions in this paper we apply our algorithm to both first and second order phase transitions to demonstrate its efficiency and accuracy we obtained direct simulational estimates for the density of states for two dimensional ten state potts models on lattices up to 200 x 200 and ising models on lattices up to 256 x 256 our simulational results are compared to both exact solutions and existing numerical data obtained using other methods applying this approach to a three dimensional j spin glass model we estimate the internal energy and entropy at zero temperature and using a two dimensional random walk in energy and order parameter space we obtain the rough canonical distribution and energy landscape in order parameter space preliminary data suggest that the glass transition temperature is about 1 2 and that better estimates can be obtained with more extensive application of the method this simulational method is not restricted to energy space and can be used to calculate the density of states for any parameter by a random walk in the corresponding space

assessment of a generalized discrete time mixed δ shock model for the multi state systems abstract in this paper the life behavior of a shock model is studied when the external shocks occur according to a binomial process whose interarrival times between successive shocks follow a geometric distribution the system transits into a lower partially working state upon the occurrence of each interarrival time between two successive shocks less than a critical threshold say the system fails when k out of interarrival times between two successive shocks are less than or the magnitude of the shock is larger than the other critical level say such a model creates a multi state system having a number of different states the probability mass functions of system s life time the time spent by the system in a perfectly functioning state and the total time spent by the system in partially working states are derived for the proposed model the corresponding probability generating functions are also derived the case of markov shock occurrences is also studied to illustrate the model studied in this paper some numerical examples are also considered finally an application in insurance is also provided

efficient multiple range random walk algorithm to calculate the density of states we present a new monte carlo algorithm that produces results of high accuracy with reduced simulational effort independent random walks are performed concurrently or serially in different restricted ranges of energy and the resultant density of states is modified continuously to produce locally flat histograms this method permits us to directly access the free energy and entropy is independent of temperature and is efficient for the study of both 1st order and 2nd order phase transitions it should also be useful for the study of complex systems with a rough energy landscape

electrostatics and heat conduction in high contrast composite materials we present a robust integral equation method for the calculation of the electrostatic and thermal properties of systems made of piecewise homogeneous high contrast materials by high contrast we mean that the electrical or thermal conductivity ratios are high our approach involves a modification of the standard integral representation using in electrostatic terms the polarization charge as an unknown it is related to the perturbation approach proposed by tausch and white j tausch j white capacitance extraction of 3 d conductor systems in dielectric media with high permittivity ratios ieee trans microwave theory tech 47 1999 18 26 but based on a simple form of projection rather than on the solution of auxiliary integral equations numerical implementation is straightforward in the multiple dielectric case

phase behavior under the averaging over disorder realizations abstract effects of the averaging over disorder realizations samples on the phase behavior are analyzed in terms of the mean field approximation for the random field ising model with infinite range interactions it is found that the averaging is equivalent to a drastic modification in the statistics of the quenched variables in its turn this lowers the critical temperature of a second order phase transition or depending on the sampling even suppresses the ordered phase possible first order transitions are shown to be softened by the sample averaging common issues and differences in the interpretation of these effects in the context of the simulation and experimental studies are discussed

the effective conductivity of random checkerboards an algorithm is presented for the fast and accurate solution of the electrostatic equation on multi component random checkerboards it relies on a particular choice of integral equation extended as to separate ill conditioning due to singular fields in corners from ill conditioning due to interaction of clusters of well conducting squares at large distances two separate preconditioners take care of the two separate phenomena in a series of numerical examples effective conductivities are computed for random checkerboards containing up to 104 squares with conductivity ratios of up to 106 the achievable relative precision in these examples is on the order of 10 11

continuous phase transitions with a convex dip in the microcanonical entropy the appearance of a convex dip in the microcanonical entropy of finite systems usually signals a first order transition however a convex dip also shows up in some systems with a continuous transition as for example in the baxter wu model and in the four state potts model in two dimensions we demonstrate that the appearance of a convex dip in those cases can be traced back to a finite size effect the properties of the dip are markedly different from those associated with a first order transition and can be understood within a microcanonical finite size scaling theory for continuous phase transitions results obtained from numerical simulations corroborate the predictions of the scaling theory

lack of self averaging of the specific heat in the three dimensional random field ising model we apply the recently developed critical minimum energy subspace scheme for the investigation of the random field ising model we point out that this method is well suited for the study of this model the density of states is obtained via the wang landau and broad histogram methods in a unified implementation by employing the n fold version of the wang landau scheme the random fields are obtained from a bimodal distribution h i ifmmode pm else textpm fi 2 and the scaling of the specific heat maxima is studied on cubic lattices with sizes ranging from l 4 to l 32 observing the finite size scaling behavior of the maxima of the specific heats we examine the question of saturation of the specific heat the lack of self averaging of this quantity is fully illustrated and it is shown that this property may be related to the question mentioned above

finite size scaling analysis of the critical behavior of the baxter wu model abstract we use the recently developed critical minimum energy subspace crmes approximation scheme to study the critical behavior of the baxter wu model this scheme uses only a properly determined part of the energy spectrum and allows us to obtain high accuracy for relatively large systems with considerably reduced computational effort the density of states is constructed by a multi range wang landau sampling and from this we obtain the critical properties of specific heat and of the reduced binder cummulant the good agreement of our results with existing exact solutions demonstrates the accuracy of our approximation technique

the random field ising model with an asymmetric and anisotropic bimodal probability distribution abstract the ising model in the presence of a random field drawn from the asymmetric and anisotropic trimodal probability distribution p h i p h i h 0 q h i h 0 r h i is investigated the partial probabilities p q r take on values within the interval 0 1 consistent with the constraint p q r 1 asymmetric distribution h i is the random field variable with basic absolute value h 0 strength is the competition parameter which is the ratio between the respective strength of the random magnetic field in the two principal directions z and z and is positive so that the random fields are competing anisotropic distribution this probability distribution is an extension of the bimodal one allowing for the existence in the lattice of non magnetic particles or vacant sites the current random field ising system displays mainly second order phase transitions which for some values of p q and h 0 are followed by first order phase transitions joined smoothly by a tricritical point occasionally two tricritical points appear implying another second order phase transition in addition to these points re entrant phenomena can be seen for appropriate ranges of the temperature and random field for specific values of p and q using the variational principle we write down the equilibrium equation for the magnetization and solve it for both phase transitions and at the tricritical point in order to determine the magnetization profile with respect to h 0 considered as an independent variable in addition to the temperature

the sherrington kirkpatrick spin glass model in the presence of a random field with a joint gaussian probability density function for the exchange interactions and random fields abstract the magnetic systems with disorder form an important class of systems which are under intensive studies since they reflect real systems such a class of systems is the spin glass one which combines randomness and frustration the sherrington kirkpatrick ising spin glass with random couplings in the presence of a random magnetic field is investigated in detail within the framework of the replica method the two random variables exchange integral interaction and random magnetic field are drawn from a joint gaussian probability density function characterized by a correlation coefficient the thermodynamic properties and phase diagrams are studied with respect to the natural parameters of both random components of the system contained in the probability density the de almeida thouless line is explored as a function of temperature and other system parameters the entropy for zero temperature as well as for non zero temperatures is partly negative or positive acquiring positive branches as h 0 increases

the effective conductivity of arrays of squares large random unit cells and extreme contrast ratios an integral equation based scheme is presented for the fast and accurate computation of effective conductivities of two component checkerboard like composites with complicated unit cells at very high contrast ratios the scheme extends recent work on multi component checkerboards at medium contrast ratios general improvement include the simplification of a long range preconditioner the use of a banded solver and a more efficient placement of quadrature points this together with a reduction in the number of unknowns allows for a substantial increase in achievable accuracy as well as in tractable system size results accurate to at least nine digits are obtained for random checkerboards with over a million squares in the unit cell at contrast ratio 106 furthermore the scheme is flexible enough to handle complex valued conductivities and using a homotopy method purely negative contrast ratios examples of the accurate computation of resonant spectra are given

a δ shock maintenance model for a deteriorating system abstract in this paper a shock maintenance model for a deteriorating system is studied assume that shocks arrive according to a renewal process the interarrival time of shocks has a weibull distribution or gamma distribution whenever an interarrival time of shocks is less than a threshold the system fails assume further the system is deteriorating so that the successive threshold values are geometrically nondecreasing and the consecutive repair times after failure form an increasing geometric process a replacement policy n is adopted by which the system will be replaced by an identical new one at the time following the n th failure then the long run average cost per unit time is evaluated afterwards an optimal policy n for minimizing the long run average cost per unit time could be determined numerically

new monte carlo algorithm entropic sampling we present a new monte carlo sampling algorithm with which one can obtain any desired distribution of the sampling in one monte carlo simulation the free energy and the entropy of a system can thus be obtained from a simple exercise of this algorithm the main idea is to sample directly the entropy of a system at infinite temperature importance sampling is shown to be a particular case of the new algorithm the algorithm is tested against the exact partition function of the ital l 4 simple cubic ising model a comparison with the multicanonical ensemble for the ital l 12 ital q 10 potts model shows that the new algorithm is more general and more efficient

estimation of critical behavior from the density of states in classical statistical models we present a simple and efficient approximation scheme which greatly facilitates the extension of wang landau sampling or similar techniques in large systems for the estimation of critical behavior the method presented in an algorithmic approach is based on a very simple idea familiar in statistical mechanics from the notion of thermodynamic equivalence of ensembles and the central limit theorem it is illustrated that we can predict with high accuracy the critical part of the energy space and by using this restricted part we can extend our simulations to larger systems and improve the accuracy of critical parameters it is proposed that the extensions of the finite size critical part of the energy space determining the specific heat satisfy a scaling law involving the thermal critical exponent the method is applied successfully for the estimation of the scaling behavior of specific heat of both square and simple cubic ising lattices the proposed scaling law is verified by estimating the thermal critical exponent from the finite size behavior of the critical part of the energy space the density of states of the zero field ising model on these lattices is obtained via a multirange wang landau sampling

wang landau study of the critical behavior of the bimodal 3d random field ising model we apply the wang landau method to the study of the critical behavior of the three dimensional random field ising model with a bimodal probability distribution for high values of the random field intensity we find that the energy probability distribution at the transition temperature is double peaked suggesting that the phase transition is first order on the other hand the transition looks continuous for low values of the field intensity in spite of the large sample to sample fluctuations observed the double peak in the probability distribution is always present for high fields

baxter wu model in the presence of an external magnetic field abstract in this work we study an unusual phase transition of the baxter wu model in the presence of an external magnetic field the model is pure baxter wu which means that only three spin interactions are taken into account we construct a phase diagram on the temperature field plane based mainly on the singularities of the specific heat these singularities are more clearly observed than those of the magnetic susceptibility which are used in existing works we establish a discontinuity in the critical exponents when the field is changed from zero to negative

critical endpoint behavior in an asymmetric ising model application of wang landau sampling to calculate the density of states using the wang landau sampling method with a two dimensional random walk we determine the density of states for an asymmetric ising model with two and three body interactions on a triangular lattice in the presence of an external field with an accurate density of states we were able to map out the phase diagram accurately and perform quantitative finite size analyses at and away from the critical endpoint we observe a clear divergence of the curvature of the spectator phase boundary and of the magnetization coexistence diameter derivative at the critical endpoint and the exponents for both divergences agree well with previous theoretical predictions

diffusion on percolation clusters at criticality the concept of fractal dimensionality is used to study the problem of diffusion on percolation clusters the authors find from monte carlo simulations that the fractal dimensionality of a random walk on a critical percolation cluster in three dimensional space is d 3 3 or 0 1 where the size of the cluster is restricted to be larger than the span of the walk and is d 3 9 or 0 1 for a walk on clusters not subject to this restriction for two dimensional space they find d approximately d approximately 2 7 or p0 1 the exponent d and d is related to the scaling of the average length r of n steps via rd varies as n the fracton dimensionality which is related to the density of states was found to be 1 26 or 0 1 these results are in good agreement with the predictions of alexander and orbach 1982

lack of self averaging in random systems liability or asset abstract the study of quenched random systems is facilitated by the idea that the ensemble averages describe the thermal averages for any specific realization of the couplings provided that the system is large enough careful examination suggests that this idea might have a flaw when the correlation length becomes of the order of the size of the system we find that certain bounded quantities are not self averaging when the correlation length becomes of the order of the size of the system this suggests that the lack of self averaging expressed in terms of properly chosen signal to noise ratios may serve to identify phase boundaries this is demonstrated by using such signal to noise ratios to identify the boundary of the ferromagnetic phase of the random field ising system and compare the findings with more traditional measures

critical energy distribution function of the baxter wu model abstract in this work we examine the critical finite size scaling behavior of the energy probability distribution function and its corresponding binder cumulant at critical point based on the results of monte carlo simulations at zero external magnetic field using the recently developed triangle cluster algorithm we calculate the energy distribution function s exponents and we derive the scaling relations for the energy binder cumulant finally we predict the exact form of the scaled energy distribution function most of our conclusions seem applicable not only to the baxter wu model but also to other ising like models

first order transition features of the triangular ising model with nearest and next nearest neighbor antiferromagnetic interactions abstract we implement a new and accurate numerical entropic scheme to investigate the first order transition features of the triangular ising model with nearest neighbor j nn and next nearest neighbor j nnn antiferromagnetic interactions in ratio r j nn j nnn 1 important aspects of the existing theories of first order transitions are briefly reviewed tested on this model and compared with previous work on the potts model using lattices with linear sizes l 30 40 100 120 140 160 200 240 360 and 480 we estimate the thermal characteristics of the present weak first order transition our results improve the original estimates of rastelli et al and verify all the generally accepted predictions of the finite size scaling theory of first order transitions including transition point shifts thermal and magnetic anomalies however two of our findings are not compatible with current phenomenological expectations the behavior of transition points derived from the number of phases parameter is not in accordance with the theoretically conjectured exponentially small shift behavior and the well known double gaussian approximation does not correctly describe higher correction terms of the energy cumulants it is argued that this discrepancy has its origin in the commonly neglected contributions from domain wall corrections

the random field ising model with asymmetric bimodal probability distribution abstract the ising model in the presence of a random field is investigated within the mean field approximation based on landau expansion the random field is drawn from the bimodal probability distribution p h p h h 0 1 p h h 0 where the probability p assumes any value within the interval 0 1 asymmetric distribution the prevailing transitions are of second order but for some values of p and h 0 first order phase transitions take place for smaller temperatures and higher h 0 thus confirming the existence of a tricritical point also the possible reentrant phenomena in the phase diagram t h 0 plane occur for appropriate values of p and h 0 using the variational principle we determine the equilibrium equation for magnetization and solve it for both transitions and at the tricritical point in order to determine the magnetization profile with respect to h 0

reliability analysis under marshall olkin run shock model abstract in this paper a new shock model called marshall olkin run shock model is defined and studied according to the model two components are subject to shocks that may arrive from three different sources and component i fails when it is subject to k consecutive critical shocks from source i or k consecutive critical shocks from source 3 i 1 2 reliability and mean residual life functions of such components are studied when the times between shocks follow phase type distribution

an extended optimal replacement model of systems subject to shocks abstract a system is subject to shocks that arrive according to a non homogeneous poisson process as shocks occur a system has two types of failures type i failure minor failure is rectified by a minimal repair whereas type ii failure catastrophic failure is removed by replacement the probability of a type ii failure is permitted to depend on the number of shocks since the last replacement this paper proposes a generalized replacement policy where a system is replaced at the nth type i failure or first type ii failure or at age t whichever occurs first the cost of the minimal repair of the system at age t depends on the random part c t and deterministic paper c t the expected cost rate is obtained the optimal n and optimal t which would minimize the cost rate are derived and discussed various special cases are considered and detailed

self averaging in many body quantum systems out of equilibrium time dependence of distributions in a disordered system a quantity is self averaging when the ratio between its variance for disorder realizations and the square of its mean decreases as the system size increases here we consider a chaotic disordered many body quantum system and search for a relationship between self averaging behavior and the properties of the distributions over disorder realizations of various quantities and at different timescales an exponential distribution as found for the survival probability at long times explains its lack of self averaging since the mean and the dispersion are equal gaussian distributions however are obtained for both self averaging and non self averaging quantities our studies show also that one can make conclusions about the self averaging behavior of one quantity based on the distribution of another related quantity this strategy allows for semianalytical results and thus circumvents the limitations of numerical scaling analysis which are restricted to few system sizes

on the mean and extreme distances between failures in markovian binary sequences this paper is concerned with the mean minimum and maximum distances between two successive failures in a binary sequence consisting of markov dependent elements these random variables are potentially useful for the analysis of the frequency of critical events occurring in certain stochastic processes exact distributions of these random variables are derived via combinatorial techniques and illustrative numerical results are presented

conformal invariance studies of the baxter wu model and a related site colouring problem the partition function of the baxter wu model is exactly related to the generating function of a site colouring problem on a hexagonal lattice we extend the original bethe ansatz solution of these models in order to obtain the eigenspectra of their transfer matrices in finite geometries and general toroidal boundary conditions the operator content of these models is studied by solving numerically the bethe ansatz equations and by exploring conformal invariance since the eigenspectra are calculated for large lattices the corrections to finite size scaling are also calculated

the ising model and critical behavior of transport in binary composite media we present a general theory for critical behavior of transport in binary composite media the theory holds for lattice and continuum percolation models in both the static case with real parameters and the quasi static case frequency dependent with complex parameters through a direct analytic correspondence between the magnetization of the ising model and the effective parameter problem of two phase random media we show that the critical exponents of the transport coefficients satisfy the standard scaling relations for phase transitions in statistical mechanics our work also shows that delta components form in the underlying spectral measures at the spectral endpoints precisely at the percolation threshold pc and at 1 pc this is analogous to the lee yang ruelle characterization of the ising model phase transition and identifies these transport transitions with the collapse of spectral gaps in these measures

bounds for effective parameters of heterogeneous media by analytic continuation we give a mathematical formulation of a method for obtaining bounds on effective parameters developed by d bergman and g w milton this method in contrast to others used before does not rely on a variational principle but exploits the properties of the effective parameter as an analytic function of the component parameters the method is at present restricted to two component media