solving set optimization problems based on the concept of null set abstract we introduce the concept of null set in the hyperspace that is consisting of all nonempty subsets of a given normed space based on the concepts of null set and convex cone we can define two partial orderings according to the algebraic difference and hukuhara difference between any two elements in the hyperspace which will be used to define the solution concepts of set optimization problems on the other hand we transform set optimization problems into a conventional vector optimization problem under these settings we can apply the technique of scalarization to solve this transformed vector optimization problem finally we show that the optimal solution of scalarized problem is also the optimal solution of original set optimization problem

energy eigenvalues for an arbitrary potential well with n minima for a general smooth one dimensional potential well with n minima containing n 1 barrier humps of arbitrary shape a phase integral quantization condition including terms is obtained from which the energy eigenvalues can be calculated with an accuracy relative to the chosen order of the froman phase integral approximation the quantization equation is derived from a condition for total transmission through a general system of potential barriers of arbitrary shape by means of a limiting procedure in which the two outermost barriers are made to grow large in a particular way finally the quantization condition is expressed by means of a concise recursion formula as an illustration the special case n 2 is treated in some detail for n 2 the quantization condition turns out to be identical to a previously well known phase integral quantization condition for the double minimum potential the accuracy of which has been examined in several papers by other authors their results concerning a symmetri

real trajectories in complex time method and barrier penetrationlike phenomena various applications of the real trajectories in the complex time method are discussed it is shown that the proper way of summation over semiclassical orbits removes existing doubts and leads to the more general formulas than the standard wkb type one in the energy eigenvalue problem this method is also used in the description of explicitly time dependent phenomena in the problem of escaping from a metastable state the estimation of a lifetime is in a good agreement with the inverse width of a false ground state energy

a pre order principle and set valued ekeland variational principle abstract we establish a pre order principle from the principle we obtain a very general set valued ekeland variational principle where the objective function is a set valued map taking values in a quasi ordered linear space and the perturbation contains a family of set valued maps satisfying certain property from this general set valued ekeland variational principle we deduce a number of particular versions of set valued ekeland variational principle which include many known ekeland variational principles their improvements and some new results

pseudo ϵ expansion and critical exponents of superfluid helium abstract the pseudo expansions series for critical exponents of the three dimensional x y model describing transition in liquid helium are derived up to the 6 terms numerical estimates extracted from the series obtained using the pade borel resummation technique scaling relations and the seven loop 7 estimate for the fisher exponent are presented including those for the exponents and measured in experiments with a record accuracy for the exponent the procedure argued to be most reliable gives 0 0117 the number that is very close to the most accurate experimental values it signals that the pseudo expansion approach is a powerful tool robust enough to evaluate critical exponents with small absolute errors the arguments in favor of such a robustness are presented

the lagrange multiplier rule in set valued optimization the known lagrange multiplier rule is extended to set valued constrained optimization problems using the contingent epiderivative as differentiability notion a necessary optimality condition for weak minimizers is derived which is also a sufficient condition under generalized convexity assumptions

pointwise well posedness in set optimization with cone proper sets abstract this paper deals with the well posedness property in the setting of set optimization problems by using a notion of well posed set optimization problem due to zhang et al 2009 18 and a scalarization process we characterize this property through the well posedness in the tykhonov sense of a family of scalar optimization problems and we show that certain quasiconvex set optimization problems are well posed our approach is based just on a weak boundedness assumption called cone properness that is unavoidable to obtain a meaningful set optimization problem

set relations and optimality conditions in set valued maps abstract problems in set valued optimization can be solved via set optimization in this paper properties of set optimization are investigated conditions for existence of solutions are established directional derivatives are studied for set valued mappings necessary and sufficient conditions in the existence of solutions are showed with directional derivatives

route to chaos and some properties in the boundary crisis of a generalized logistic mapping abstract a generalization of the logistic map is considered showing two control parameters and that can reproduce different logistic mappings including the traditional second degree logistic map cubic quartic and all other degrees we introduce a parametric perturbation such that the original logistic map control parameter r changes its value periodically according an additional parameter 2 q the value of q gives this period for this system an analytical expression is obtained for the first bifurcation that starts a period doubling cascade and using the feigenbaum universality we found numerically the accumulation point r c where the cascade finishes giving place to chaos in the second part of the paper we study the death of this chaotic behavior due to a boundary crisis at the boundary crisis orbits can reach a maximum value x x max 1 when it occurs the trajectory is mapped to a fixed point at x 0 we show that there exist a general recursive formula for initial conditions that lead to x x max

existence theorems for set optimization problems abstract in this paper we consider set optimization problems by using a partial ordering based on a set relation we establish several existence results of minimal elements more general than those existing in the literature as an application we present sufficient conditions for the existence of solutions of a set valued optimization problem under some semi continuity assumption

a unified approach and optimality conditions for approximate solutions of vector optimization problems this paper deals with approximate v efficient solutions of vector optimization problems we introduce a new v efficiency concept which extends and unifies different approximate solution notions introduced in the literature we obtain necessary and sufficient conditions via nonlinear scalarization which allow us to study this new class of approximate solutions in a general framework since any convexity hypothesis is required several examples are proposed to show the concepts introduced and the results attained

low energy levels and matrix elements in the large n small h cross asymptotic region the asymptotic form of the froman and froman solutions 1977 to the schrodinger equation has been used to calculate the low energy spectra and corresponding matrix elements for many well potentials it has been proved that the bohr sommerfeld quantisation condition is also valid in the low energy region independently for each well it has also been shown that the corresponding matrix elements are dominated by the energy levels belonging to the same well the general results have then been used to show that in n dimensions with spherically symmetric hamiltonians quantum systems approach their classical configurations with constraints when n to infinity

effective characteristic polynomials and two point pade approximants as summation techniques for the strongly divergent perturbation expansions of the ground state energies of anharmonic oscillators pad e approximants are able to sum effectively the rayleigh schr odinger perturbation series for the ground state energy of the quartic anharmonic oscillator as well as the corresponding renormalized perturbation expansion e j weniger j ifmmode check c else v c fi i ifmmode check z else v z fi ek and f vinette j math phys 34 571 1993 in the sextic case pad e approximants are still able to sum these perturbation series but convergence is so slow that they are computationally useless in the octic case pad e approximants are not powerful enough and fail on the other hand the inclusion of only a few additional data from the strong coupling domain e j weniger ann phys ny to be published greatly enhances the power of summation methods the summation techniques which we consider are two point pad e approximants and effective characteristic polynomials it is shown that these summation methods give good results for the quartic and sextic anharmonic oscillators and even in the case of the octic anharmonic oscillator which represents an extremely challenging summation problem two point pad e approximants give relatively good results textcopyright 1996 the american physical society

the pauli principle and the restricted primitive model the restricted primitive model is an electrically neutral classical model consisting of hard spheres charged either q or q we show that by appropriately selecting the diameter of the hard spheres the pressure when q 0 can be made equal to that for a fluid of maxwell boltzmann point ions and an ideal fermi gas of electrons we compare the series expansion of these classical and quantum systems and find that except for intermediate de broglie density and moderate to strong electrical interaction strength the restricted primitive model gives a reasonable representation of the pressure of the corresponding quantum system much of the current interest however has been focused on the above excepted region

well posedness for set optimization problems abstract in this paper three kinds of well posedness for set optimization are first introduced by virtue of a generalized gerstewitz s function the equivalent relations between the three kinds of well posedness and the well posedness of three kinds of scalar optimization problems are established respectively then sufficient and necessary conditions of well posedness for set optimization problems are obtained by using a generalized forcing function respectively finally various criteria and characterizations of well posedness are given for set optimization problems

on approximate solutions in set valued optimization problems in this paper we introduce several concepts of approximate solutions of set valued optimization problems with vector and set optimization we prove existence results and necessary and sufficient conditions by using limit sets

approximations of nondifferentiable functions and local topological equivalence a concept of local approximation of a function is introduced this concept is defined via directional derivatives in consequence the local approximation is carried out by a positively homogeneous mapping we obtain local approximations for functions that are not necessarily locally lipschitzian nor continuous this is the case of some large classes of functions such as stable functions or contingently epidifferentiable and directionally lipschitzian functions using the concept of topological equivalence we establish the existence of a local coordinate transformation between the original function and the positively homogeneous function this investigation is developed for contingently epidifferentiable functions around a noncritical point and for noncontingently epidifferentiable functions under particular conditions

relative pareto minimizers for multiobjective problems existence and optimality conditions in this paper we introduce and study enhanced notions of relative pareto minimizers for constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of pareto and weak pareto efficiency minimality using advanced tools of variational analysis and generalized differentiation we establish the existence of relative pareto minimizers for general multiobjective problems under a refined version of the subdifferential palais smale condition for set valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers as well as for conventional efficient and weak efficient counterparts that are new in both finite dimensional and infinite dimensional settings our proofs are based on variational and extremal principles of variational analysis in particular on new versions of the ekeland variational principle and the subdifferential variational principle for set valued and single valued mappings in infinite dimensional spaces

the ekeland variational principle for henig proper minimizers and super minimizers abstract in this paper we consider for the first time approximate henig proper minimizers and approximate super minimizers of a set valued map f with values in a partially ordered vector space and formulate two versions of the ekeland variational principle for these points involving coderivatives in the sense of ioffe clarke and mordukhovich as applications we obtain sufficient conditions for f to have a henig proper minimizer or a super minimizer under the palais smale type conditions the techniques are essentially based on the characterizations of henig proper efficient points and super efficient points by mean of the henig dilating cones and the hiriart urruty signed distance function

minimal element theorems revisited abstract starting with the brezis browder principle we give stronger versions of many variational principles and minimal element theorems which appeared in the recent literature relationships among the elements of different sets of assumptions are discussed and clarified i e assumptions to the metric structure of the underlying space and boundedness assumptions new results involving set valued maps and the increasingly popular set relations are obtained along the way

solution of the anharmonic quartic potential oscillator problem the problem of the one dimensional generalized anharmonic quartic potential oscillator is studied in full solutions of the classical equations of motion for the different types of potential are given in terms of the jacobi elliptic functions in both classically allowed and forbidden regions of each potential for physically interesting initial conditions the solutions and some of their properties are also given by means of bilinear transformations of the corresponding anharmonic symmetric oscillator solutions for similar regions or initial conditions

generalized e quasi solutions in multiobjective optimization problems existence results and optimality conditions abstract this paper is concerned with approximate solutions of multiobjective optimization problems whose order cone is not necessarily the nonnegative orthant we introduce the concept of generalized e quasi solution that extends other well known approximate efficiency notions of the literature and we study the limit behavior of these solutions as approximations to the efficient and weak efficient sets moreover we prove several existence results and a bound for the generalized e quasi solution set under convexity assumptions and by using asymptotic analysis tools finally we develop optimality conditions for a particular case of these kinds of e quasi solutions in nonsmooth convex and nonconvex problems

an ekeland s variational principle for set valued mappings with applications in this paper we obtain a general ekeland s variational principle for set valued mappings in complete metric space which is different from those in g y chen x x huang ekeland s e variational principle for set valued mapping mathematical methods of operations research 48 1998 181 186 g y chen x x huang s h hou general ekeland s variational principle for set valued mappings journal of optimization theory and applications 106 2000 151 164 s j li w y zhang on ekeland s variational principle for set valued mappings acta mathematicae application sinica english series 23 2007 141 148 by the result we prove some existence results for a general vector equilibrium problem under nonconvex and compact or noncompact assumptions of its domain respectively moreover we give some equivalent results to the variational principle

exactness of conventional and supersymmetric jwkb formulae and their relation to the shape invariance property of potentials a mechanism for the jwkb formulae in one dimensional quantum mechanics to quantize energy levels exactly is discussed this mechanism is most easily applied when considered potentials are represented by a suitable change of variable as periodic ones to be successful the mechanism demands potentials with no more than one second order pole and no more than two occasionally four turning points in the basic period strips it then selects 11 potentials which have been known in the literature for a long time it is also shown that the exactness of the supersymmetric susy jwkb formulae for these 11 potentials follows directly from the corresponding exactness of the conventional ones being a consequence of the singularity structures of the potentials a relation of these singularity structures to the shape invariance symmetry of the quantized potentials is shown to guarantee the conventional and susy jwkb formulae to be exact simultaneously the relevant two non perturbative theorems describing these facts are formulated and proved

epidifferentiability and hypodifferentiability of pseudoconvex maps in set optimization problems abstract in this paper we study set valued optimization problems with set optimization criteria with this aim we introduce two classes of pseudoconvex maps whose elements are epidifferentiable or hypodifferentiable we establish optimality conditions in terms of the epiderivatives and hypoderivatives of associated maps of infima or suprema under pseudoconvexity conditions finally by connecting set optimization criteria with vector criterion we apply these results in order to solve a vector optimization problem with constraints given by a set valued map

nonconvex scalarization in set optimization with set valued maps abstract the aim of this work is to obtain scalar representations of set valued optimization problems without any convexity assumption using a criterion of solution introduced by kuroiwa d kuroiwa some duality theorems of set valued optimization with natural criteria in proceedings of the international conference on nonlinear analysis and convex analysis world scientific river edge nj 1999 pp 221 228 which is based on ordered relations between sets we characterize this type of solutions by means of nonlinear scalarization the scalarizing function is a generalization of the gerstewitz s nonconvex separation function as applications of our results we give two existence theorems for set valued optimization problems

pade approximant for hard sphere square well and hard sphere square well square shoulder model fluids abstract the coupling parameter series expansion cpse is used to calculate coefficients a i i 1 2 3 7 of the high temperature series expansion htse for two hard sphere square well models and two hard sphere square well square shoulder models the four sets of coefficients are used to construct 28 pade approximant variants p n d for the four models based on the 28 pade variants six common thermodynamic functions for the four models are calculated and compared with corresponding results based on mc simulations reported in the literature and the htse truncated to seventh order in the present work respectively several conclusions are generalized from the comparisons i the commonly used rule of the minimum value for n d to choose the best pade approximation may not select the best one which depends on the model and relevant parameters values and has to be determined in advance by reference to simulations ii from the whole table of 28 pade approximants reliable throughout the text by reliable we mean the results are at least qualitatively correct variants always can be singled out which not only provides an improvement over the raw htse truncated at the optimal order but also is free of pole s and consequently applicable over reasonably wide phase space of interest particularly the improvement can be immense for most but not all of the models considered when the excess heat capacity is being considered iii among the reliable p n d approximations picked some have high values of n d indicating that the higher order htse coefficients calculated by the cpse provide important and reliable information for the unknown excess helmholtz free energy and the conventional pade summation indeed provides an effective method to utilize the information iv the pade summation can be a resort in statistical mechanics perturbation theories of simple potential fluids because of its simplicity in the actual application and possibility in improving the raw htse however to ensure the use reliably one comprehensive pre research should be carried out

borel summable solutions to one dimensional schrodinger equation it is shown that so called fundamental solutions the semiclassical expansions of which have been established earlier to be borel summable to the solutions themselves appear also to be the unique solutions to the one dimensional 1d schrodinger equation having this property namely it is shown in this paper that for the polynomial potentials the borel function defined by the fundamental solutions can be considered as the canonical one the latter means that any borel summable solution can be obtained by the borel transformation of this unique canonical borel function multiplied by some dependent and borel summable constant this justifies the exceptional role the fundamental solutions play in 1d quantum mechanics and completes the relevant semiclassical theory relied on the borel resummation technique and developed in our other papers

1 n expansion for the number of matchings on regular graphs and monomer dimer entropy using a 1 n expansion that is an expansion in descending powers of n for the number of matchings in regular graphs with 2n vertices we study the monomer dimer entropy for two classes of graphs we study the difference between the extensive monomer dimer entropy of a random r regular graph g bipartite or not with 2n vertices and the average extensive entropy of r regular graphs with 2n vertices in the limit n rightarrow infty we find a series expansion for it in the numbers of cycles with probability 1 it converges for dimer density p 1 and for g bipartite it diverges as mathrm ln 1 p for p rightarrow 1 in the case of regular lattices we similarly expand the difference between the specific monomer dimer entropy on a lattice and the one on the bethe lattice we write down its taylor expansion in powers of p through the order 10 expressed in terms of the number of totally reducible walks which are not tree like we prove through order 6 that its expansion coefficients in powers of p are non negative

the semiclassical small ℏ limit of loci of roots of subdominant solutions for polynomial potentials in this paper a description of the small limit of loci of zeros of subdominant solutions for polynomial potentials is given called as fundamental solutions in our earlier papers the considered potentials are those which provide us with the simple turning points only three types of stokes graphs sg s associated with the potentials are investigated the general non critical ones the general critical ones but with only single internal stokes line sl and the stokes graphs corresponding to arbitrary multiple well real even degree polynomial potentials with internal stokes lines distributed on the real axis only all these cases are considered in their both versions of the quantized and not quantized in particular due to the fact that the small limit is semiclassical it is shown that loci of roots of subdominant solutions in the cases considered are collected along stokes lines there are infinitely many roots of subdominant solutions on such lines escaping to infinity and a finite number of th

topological expansion and exponential asymptotics in 1d quantum mechanics borel summable semiclassical expansions in 1d quantum mechanics are considered these are the borel summable expansions of fundamental solutions and of quantities constructed with their help an expansion called topological is constructed for the corresponding borel functions this allows us to study the borel plane singularity structure in a systematic way examples of such structures are considered for linear harmonic and anharmonic potentials together with the best approximation provided by the semiclassical series the exponentially small contributions completing the approximation are considered a natural method of constructing such exponential asymptotics based on the borel plane singularity structures provided by the topological expansion is developed the method is used to form the semiclassical series including exponential contributions for the energy levels of the anharmonic oscillator

alternative phase integral approximations phase integral approximations of a new kind are obtained they are closely related to the phase integral approximations generated from an unspecified base function q z due to professor n froman and professor p o froman 1991 in the search for an optimal base function a sequence of successively improved base functions qn z are derived n 0 1 2 together with a concise recurrence formula for these functions that is suitable for computer calculations the function qn z is an approximate solution to the nonlinear q equation equivalent to the schrodinger equation the proposed approximation which is immediately obtained from qn z is expected to be somewhat more accurate for n or 2 than the corresponding phase integral approximation of order 2n 1 obtained from the function qn z given by qn z q z sigma n 0ny2n z a detailed comparison is made between qn z and qn z for n 0 1 2 and 3 the proposed approximations have the correct form required to combine them with the f matrix technique for solving connection problems

transmission coefficients in anharmonic symmetrical potentials barrier transmission in potentials of the type v x ax2 bx4 is studied using the phase integral method the same as the jwkb approximation in lower orders elliptic functions are used for the classical solutions the transmission coefficient is calculated for all signs and values of a and b that give a potential barrier

strict approximate solutions in set valued optimization with applications to the approximate ekeland variational principle this work deals with strict solutions of set valued optimization problems under the set optimality criterion in this context we introduce a new approximate solution concept and we obtain several properties of these solutions when the error is fixed and also for their limit behavior when the error tends to zero then we prove a general existence result which is applied to obtain approximate ekeland variational principles

a generalized distance and enhanced ekeland s variational principle for vector functions we propose a definition of lower closed transitive relations and prove the existence of minimal elements for such a relation this result is shown to contain probably a large part of existing versions of ekeland s variational principle evp we introduce the notion of a weak function p as a generalized distance and use it together with the above result on minimal elements to establish enhanced evp for various settings under relaxed lower semicontinuity assumptions these principles conclude the existence not only of p strict minimizers of p perturbations of the considered vector function but also p sharp and p strong minimizers our results are proved to be stronger than the classical evp and many generalizations in the literature even in the usual one dimensional case by numerous corollaries and examples we include equivalent formulations of our enhanced evp as well

optimality conditions for set valued maps with set optimization abstract problems in set valued optimization can be solved via set optimization in this paper optimality conditions are studied for set valued maps with set optimization optimality requirements are established for continuous selections using directional derivatives necessary and sufficient conditions for the existence of solutions are shown for set valued maps under generalized convexity assumptions and with the notion of the contingent derivative

detailed behavior of the phase integral approximations at zeros and singularities of the square of the base function approximate solutions to the one dimensional time independent wave equation called the phase integral approximations are analyzed in the vicinity of characteristic points the approximations are of arbitrary order and are generated from an unspecified base function the general theory is illustrated by examples involving the power and or the exponential behavior of the square of the base function in these cases simple estimates are derived for the integrals which define the accuracy of the phase integral approximation and the optimum approximation order is determined

perturbation theory calculation of the pressure for an electron ion system abstract by means of finite temperature many body perturbation theory we derive through order e4 the corrections to an ideal fermi gas plus an ideal maxwell boltzmann gas of ions this computation is carried out for general values of the de broglie density the behavior of these coefficients is reported and their implications for the ionization profile at low densities are described

the energy levels and the corresponding normalized wave functions for a model of a compressed atom in the model of a compressed atom or ion considered in the present paper the boundary condition associated with the corresponding uncompressed atom i e the condition that the radial wave function must vanish at r is replaced by the boundary condition that the radial wave function must have a node at the finite distance r a the treatment of the problem of obtaining the energy shift due to the compression is based on the phase integral method developed by froman and froman an essential feature of which is that one can use exact formulas in the calculations and make all approximations in the final stage the treatment of the problem of obtaining the relative change of the wave function due to the compression is based on the rigorous evaluation of the normalization integral developed by furry phys rev 71 360 1947 and yngve j math phys 13 324 1972 in which one also uses exact formulas in the calculations and makes all approximations in the final stage since compression of an atom giv

generalized exponential circular and hyperbolic functions for nonlinear wave equations wave functions are presented in the form of generalized exponentials that are solutions of some of the most usual linear and nonlinear wave equations the solutions are given in terms of the elliptic functions of jacobi and presented in a form as similar as possible to the usual circular functions some simple theorems are demonstrated to present the solutions as the simplest possible extension of the usual exponentials

a set valued ekeland s variational principle in vector optimization this paper deals with ekeland s variational principle for vector optimization problems by using a set valued metric a set valued perturbed map and a cone boundedness concept based on scalarization we introduce an original approach to extending the well known scalar ekeland s principle to vector valued maps as a consequence of this approach we obtain an ekeland s variational principle that does not depend on any approximate efficiency notion this result is related to other ekeland s principles proved in the literature and the finite dimensional case is developed via an varepsilon efficiency notion that we introduced in math methods oper res 64 2006 pp 165 185 siam j optim 17 2006 pp 688 710

transmission through a system of potential barriers i transmission coefficient the transmission coefficient for a one dimensional system of n potential barriers of general shapes unequal or equal is expressed exactly by means of the phase integral method of n froman and p o froman in terms of quantities characterizing the separate barriers and wells of the system the exact formula being of interest for further rigorous studies of transmission properties can for evaluation readily be converted into a useful approximate formula by insertion of available phase integral expressions of an arbitrary order for the characteristic quantities mentioned above error estimates are given the sub and superbarrier cases are treated in a unified way some key facts about the phase integral method are given in appendix a

change of variables as borel resummation of semiclassical series the 1d schr odinger equation with meromorphic potentials are considered it is shown that any fundamental solution constructed for such potentials has a property to depend in some particular way on an arbitrary meromorphic function f x a family of fundamental solutions created in this way can be reproduced by multiplying some fixed member of the family by appropriate dependent constants this freedom allows for a proper construction of fundamental solutions at simple and double poles of considered potentials it expresses the fact that any change of variablex ydy x keeping the form of 1d schr odinger equation keeps invariant the form of the fundamental solutions as well in fact there is one to one correspondence between the functions f x and the schwarzian ofy x since any fundamental solution is borel summable it means also that the effect of the change of variable in fundamental solutions can be recoverd by multiplying the initial fundamental solutions by suitably chosenh dependent constants and borel resumming the corresponding semiclassical expansions this explains why a change of variable can improve jwkb formulae however it is shown also that a change of variable itself cannot provide us with the exact jwkb formulae

transmission through a system of potential barriers ii necessary condition for complete transparency a maximum transmission problem a smooth one dimensional system of n potential barriers of arbitrary shapes unequal or equal is considered a general necessary condition for complete transparency is obtained that can be understood as a constraint on the reflection coefficients pertaining to the single barriers of the system a maximum transmission problem of a general kind is solved and the solution is used to give a physical interpretation of the necessary condition the sub and superbarrier cases are treated in a unified way the exact final formulas can readily be converted into accurate approximate ones by insertion of available phase integral expressions of an arbitrary order for certain characteristic quantities appearing in the formulas

quantum anharmonic symmetrical oscillators using elliptic functions the authors study in the jwkb approximation the energy levels of the symmetric anharmonic oscillators v x ax2 bx4 for different signs and values of a and b comparisons are made with published results for specific cases and with numerical calculations they given an additional example of exact value to add to the very rare catalogue of known examples