the fundamental theorem of asset pricing for unbounded stochastic processes the fundamental theorem of asset pricing states roughly speaking that the absence of arbitrage possibilities for a stochastic process s is equivalent to the existence of an equivalent martingale measure for s it turns out that it is quite hard to give precise and sharp versions of this theorem in proper generality if one insists on modifying the concept of no arbitrage as little as possible it was shown in ds94 that for a locally bounded r d valued semi martingale s the condition of no free lunch with vanishing risk is equivalent to the existence of an equivalent local martingale measure for the process s it was asked whether the local boundedness assumption on s may be dropped in the present paper we show that if we drop in this theorem the local boundedness assumption on s the theorem remains true if we replace the term equivalent local martingale measure by the term equivalent sigma martingale measure the concept of sigma martingales was introduced by chou and emery under the name of semimartingales de la classe sigma m we provide an example which shows that for the validity of the theorem in the non locally bounded case it is indeed necessary to pass to the concept of sigma martingales on the other hand we also observe that for the applications in mathematical finance the notion of sigma martingales provides a natural framework when working with non locally bounded processes s the duality results which we obtained earlier are also extended to the non locally bounded case as an application we characterize the hedgeable elements author s abstract

optimal and better transport plans we consider the monge kantorovich transport problem in a purely measure theoretic setting i e without imposing continuity assumptions on the cost function it is known that transport plans which are concentrated on c monotone sets are optimal provided the cost function c is either lower semi continuous and nite or continuous and may possibly attain the value1 we show that this is true in a more general setting in particular for merely borel measurable cost functions provided thatfc 1g is the union of a closed set and a negligible set in a previous paper schachermayer and teichmann considered strongly c monotone transport plans and proved that every strongly c monotone transport plan is optimal we establish that transport plans are strongly c monotone if and only if they satisfy a better notion of optimality called robust optimality

optimal transport and skorokhod embedding the skorokhod embedding problem is to represent a given probability as the distribution of brownian motion at a chosen stopping time over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties these constructions employ a variety of techniques ranging from excursion theory to potential and pde theory and have been used in many different branches of pure and applied probability we develop a new approach to skorokhod embedding based on ideas and concepts from optimal mass transport in analogy to the celebrated article of gangbo and mccann on the geometry of optimal transport we establish a geometric characterization of skorokhod embeddings with desired optimality properties this leads to a systematic method to construct optimal embeddings it allows us for the first time to derive all known optimal skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings while previous constructions typically used particular properties of brownian motion our approach applies to all sufficiently regular markov processes

sasa satsuma complex modified korteweg de vries ii and the complex sine gordon ii equation revisited recursion operators nonlocal symmetries and more we present a new symplectic structure and a hereditary recursion operator for the sasa satsuma equation which is widely used in nonlinear optics using an integrodifferential substitution relating this equation to a third order symmetry flow of the complex sine gordon ii equation enabled us to find a hereditary recursion operator and higher hamiltonian structures for the latter equation we also show that both the sasa satsuma equation and the third order symmetry flow for the complex sine gordon ii equation are bi hamiltonian systems and we construct several hierarchies of local and nonlocal symmetries for these systems

on the monotonicity principle of optimal skorokhod embedding problem this is a continuation of our accompanying paper siam j control optim 54 2016 pp 2174 2201 we provide an alternative proof of the monotonicity principle for the optimal skorokhod embedding problem established in beiglbock cox and huesmann optimal transport and skorokhod embedding preprint 2013 our proof is based on the adaptation of the monge kantorovich duality in our context a delicate application of the optional cross section theorem and a clever conditioning argument introduced in beiglbock cox and huesmann optimal transport and skorokhod embedding preprint 2013

integration of the soliton hierarchy with self consistent sources in contrast with the soliton equations the evolution of the eigenfunctions in the lax representation of soliton equation with self consistent sources sescs possesses singularity we present a general method to treat the singularity to determine the evolution of scattering data the akns hierarchy with self consistent sources the mkdv hierarchy with self consistent sources the nonlinear schrodinger equation hierarchy with self consistent sources the kaup newell hierarchy with self consistent sources and the derivative nonlinear schrodinger equation hierarchy with self consistent sources are integrated directly by using the inverse scattering method the n soliton solutions for some sescs are presented it is shown that the insertion of a source may cause the variation of the velocity of soliton this approach can be applied to all other 1 1 dimensional soliton hierarchies

geometry of distribution constrained optimal stopping problems we adapt ideas and concepts developed in optimal transport and its martingale variant to give a geometric description of optimal stopping times tau of brownian motion subject to the constraint that the distribution of tau is a given probability mu the methods work for a large class of cost processes at a minimum we need the cost process to be measurable and mathcal f 0 t t ge 0 adapted continuity assumptions can be used to guarantee existence of solutions we find that for many of the cost processes one can come up with the solution is given by the first hitting time of a barrier in a suitable phase space as a by product we recover classical solutions of the inverse first passage time problem shiryaev s problem

semistable sheaves on projective varieties and their restriction to curves let x be a nonsingular projective variety of dimension n over an algebraically closed field k let h be a very ample line bundle on x if v is a torsion free coherent sheaf on x we define deg v to be cl v c h and v deg v rk v we call v sernistable resp stable if for all proper subsheaves w of v we have w v resp w v cf 7 14 in this paper we prove that if v is semistable on x then its restriction to a general complete intersection curve of sufficiently high degree is semistable theorem 6 1 to give an idea of the proof assume x is a surface and v a vector bundle of rank 2 the restriction of v to a general curve c of degree m is not semistable if and only if it is not semistable on the generic curve ym defined over the function field of iph h let s be the line bundle on y contradicting the semistability of vi y cf sects 4 1 and 4 2 first we show that l extends uniquely to a line bundle l m on x proposition 2 1 if we can get l as a subbundle of v we are through for then l would contradict the semistability of v so we would like the restriction map h hom lm v h hom l m v to be surjective now for fixed l it follows from the lemma of enriques severi proposition 3 2 6 corollary 7 8 that h horn l v h horn l v is surjective for large m therefore it is enough if the l remain the same line bundle l for infinitely many m to prove that l l we construct a degenerating family of curves d f s x x s 3 d p x such that the generic fibre is a curve ct 1 of degree 2 1 and the special fibre is a reduced curve with two nonsingular components ci of degree 2 cf sect 5 let m denote 2 extending the subbundle lt ct to a subsheaf of p v on d and restricting the extension to ci gives a lower bound for the maximal degree of a line subbundle of v cim in terms of that for v ctm i proposition 4 3 this implies that degl is bounded lemma 6 5 1 so that for an infinite subsequence of m degl is constant if degltm r deglt by refining the above argument with the degenerating family one can prove that lt ci

martingale optimal transport duality we obtain a dual representation of the kantorovich functional defined for functions on the skorokhod space using quotient sets our representation takes the form of a choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient sets these sets contain stochastic integrals defined pathwise and two such definitions starting with simple integrands are given another important ingredient of our analysis is a regularized version of jakubowski s s topology on the skorokhod space

direct perturbation theory for solitons of the derivative nonlinear schrodinger equation and the modified nonlinear schrodinger equation a direct perturbation theory for solitons of the derivative nonlinear schrodinger dnls equation is developed based on a closure of eigenfunctions of the linearized dnls equation around a one soliton solution the slow evolution of soliton parameters and the perturbation induced radiation are obtained under the known simple gaugelike transformation these results are transformed into those for the perturbed modified nonlinear schrodinger mnls equation describing propagation of femtosecond pulses in optical fibers a calculation of the perturbation induced radiation fields for the perturbed dnls and mnls equations is also made our results for the perturbed mnls equation can be reduced perfectly to those for the perturbed nonlinear schrodinger equation in the small nonlinear dispersion limit

the root solution to the multi marginal embedding problem an optimal stopping and time reversal approach we provide a complete characterisation of the root solution to the skorokhod embedding problem sep by means of an optimal stopping formulation our methods are purely probabilistic and the analysis relies on a tailored time reversal argument this approach allows us to address the long standing question of a multiple marginals extension of the root solution of the sep our main result establishes a complete solution to the n marginal sep using first hitting times of barrier sets by the time space process the barriers are characterised by means of a recursive sequence of optimal stopping problems moreover we prove that our solution enjoys a global optimality property extending the one marginal root case our results hold for general one dimensional martingale diffusions

complete duality for martingale optimal transport on the line we study the optimal transport between two probability measures on the real line where the transport plans are laws of one step martingales a quasi sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward cost functions absence of a duality gap and existence of dual optimizers both properties are shown to fail in the classical formulation as a consequence of the duality result we obtain a general principle of cyclical monotonicity describing the geometry of optimal transports

the perturbed korteweg de vries equation considered anew the perturbed korteweg de vries equation is studied in a new way by a green s function formalism without use of inverse scattering methods the green s function is determined by employing the backlund transformation and green s theorem after a thorough analysis of the exact first order solution with regard to secular terms a two time scale expansion leads to the adiabatic approximation and the first order correction in accordance with the results of karpman and maslov contrary to statements in the literature the term tanh2 z in the expression for the modified phase of the perturbed soliton arises as a consequence of the systematically conducted first order perturbation theory

all adapted topologies are equal a number of researchers have introduced topological structures on the set of laws of stochastic processes a unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes aldous defines an extended weak topology based on the weak convergence of prediction processes in the economic literature hellwig introduced the information topology to study the stability of equilibrium problems bion nadal and talay introduce a version of the wasserstein distance between the laws of diffusion processes pflug and pichler consider the nested distance and the weak nested topology to obtain continuity of stochastic multistage programming problems these distances can be seen as a symmetrization of lassalle s causal transport problem but there are also further natural ways to derive a topology from causal transport our main result is that all of these seemingly independent approaches define the same topology in finite discrete time moreover we show that this weak adapted topology is characterized as the coarsest topology that guarantees continuity of optimal stopping problems for continuous bounded reward functions

the distribution of brownian motion in rn at a natural stopping time abstract for a measure on rn let bt p be brownian motion in rn with initial distribution let d be an open subset of rn with exit time inf t 0 bt d in the case where d is a green region with green function g and is a measure in d such that g is not identically infinite on any component of d we have given necessary and sufficient conditions for a measure in d to be of the form dx p bt dx t

linear integral transformations and hierarchies of integrable nonlinear evolution equations abstract integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations these are riemann hilbert type of integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an indefinite parameter k the potential matrices in these equations obey a set of nonlinear recursion relations leading to a heirarchy of nonlinear partial differential equations in decreasing order the same equations give rise to a reciprocal hierarchy associated with heisenberg ferromagnet type of equations central in the treatment is an embedding of the hierarchy into an infinite matrix structure which is constructed on the basis of the integral equations in terms of this infinite matrix structure the equations governing the hierarchies become quite simple furthermore it leads in a straightforward way to various generalizations such as to other types of linear spectral problems multicomponent system and lattice equations generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment finally some results on conserved densities and the hamiltonian structure are briefly discussed

completeness of the eigenfunctions for the caudrey beals coifman system the direct and the inverse scattering problem for the first order linear systems of the type l x i d dx q x j x 0 j h q x gj which generalizes the zakharov shabat system and the systems studied by caudrey beals and coifman cbc is analyzed herein here j is a regular complex constant element of the cartan subalgebra h g of the simple lie algebra g and the potential q x vanishes fast enough for x taking values in the image gj of adj the cbc results are generalized and the fundamental analytic solution m x for any choice of the irreducible finite dimensional representation v of g is constructed four pairwise equivalent minimal sets of scattering data for l invariant with respect to the choice of the representation of g are extracted from the asymptotics of m x for x from m x the resolvent of l is constructed in the adjoint representation vad and the completeness relation is proven for the eigenfunctions of l in vad it is also proven that the discrete spectrum

complete eigenfunctions of linearized integrable equations expanded around a soliton solution complete eigenfunctions for an integrable equation linearized around a soliton solution are the key to the development of a direct soliton perturbation theory in this article we explicitly construct such eigenfunctions for a large class of integrable equations including the kdv nls and mkdv hierarchies we establish the striking result that the linearization operators of all equations in the same integrable hierarchy share the same complete set of eigenfunctions furthermore these eigenfunctions are precisely the squared eigenfunctions of the associated eigenvalue problem the key step in our derivation is to show that the linearization operator of an integrable equation can be factored into a function of the integro differential operator which generates the integrable equation and the linearization operator of the lowest order integrable equation in the same hierarchy we also obtain similar results for the adjoint linearization operator of an integrable equation even though our analysis is conducted only for the kdv nls and mkdv hierarchies similar results are expected for other integrable hierarchies as well we further explicitly present the complete eigenfunctions for the kdv nls and mkdv hierarchy equations and give their inner products thus they can be readily used to develop a direct soliton perturbation theory for any of those hierarchy equations

martingale optimal transport and robust hedging in continuous time the duality between the robust or equivalently model independent hedging of path dependent european options and a martingale optimal transport problem is proved the financial market is modeled through a risky asset whose price is only assumed to be a continuous function of time the hedging problem is to construct a minimal super hedging portfolio that consists of dynamically trading the underlying risky asset and a static position of vanilla options which can be exercised at the given fixed maturity the dual is a monge kantorovich type martingale transport problem of maximizing the expected value of the option over all martingale measures that have a given marginal at maturity in addition to duality a family of simple piecewise constant super replication portfolios that asymptotically achieve the minimal super replication cost is constructed

the squared eigenstates of the sine gordon eigenvalue problem an analysis of the squared sine gordon eigenvalue problem in laboratory coordinates is presented it is shown that unlike the unsquared laboratory coordinate eigenvalue problem the squared laboratory coordinate eigenvalue problem may be cast into the form of a standard eigenvalue problem wherein an eigenvalue independent operator operating on an eigenfunction generates the eigenvalue with this form it becomes rather elementary to obtain the squared eigenfunction expansion of the sine gordon potentials as well as to demonstrate closure

skorokhod embeddings minimality and non centred target distributions in this paper we consider the skorokhod embedding problem for target distributions with non zero mean in the zero mean case uniform integrability provides a natural restriction on the class of embeddings but this is no longer suitable when the target distribution is not centred instead we restrict our class of stopping times to those which are minimal and we find conditions on the stopping times which are equivalent to minimality

the geometry of multi marginal skorokhod embedding the skorokhod embedding problem is one of the classical problems in the theory of stochastic processes with applications in many different fields cf the surveys hobson in paris princeton lectures on mathematical finance 2010 volume 2003 of lecture notes in mathematics springer berlin 2011 obloj in probab surv 1 321 390 2004 many of these applications have natural multi marginal extensions leading to the optimal multi marginal skorokhod problem some of the first papers to consider this problem are brown et al probab theory relat fields 119 4 558 578 2001 hobson seminaire de probabilites xxxii volume 1686 of lecture notes in mathematics springer berlin 1998 madan and yor bernoulli 8 4 509 536 2002 however this turns out to be difficult using existing techniques only recently a complete solution was be obtained in cox et al probab theory relat fields 173 211 259 2018 establishing an extension of the root construction while other instances are only partially answered or remain wide open in this paper we extend the theory developed in beiglbock et al invent math 208 2 327 400 2017 to the multi marginal setup which is comparable to the extension of the optimal transport problem to the multi marginal optimal transport problem as for the one marginal case this viewpoint turns out to be very powerful in particular we are able to show that all classical optimal embeddings have natural multi marginal counterparts notably these different constructions are linked through a joint geometric structure and the classical solutions are recovered as particular cases moreover our results also have consequences for the study of the martingale transport problem as well as the peacock problem

a simple treatment of the sine gordon equation to avoid difficulties due to the complexity of the lax pair of the sine gordon equation the orthogonality relations of the squared jost solutions is derived simply using the 1 1 dimensional green s theorem the direct perturbation theory for sine gordon in the laboratory reference is re developed correcting some numerical coefficients in published orthogonality relationships

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

unbiased shifts of brownian motion let b b t t r be a two sided standard brownian motion an unbiased shift of b is a random time t which is a measurable function of b such that b t t b t t r is a brownian motion independent of b t we characterise unbiased shifts in terms of allocation rules balancing mixtures of local times of b for any probability distribution on r we construct a stopping time t 0 with the above properties such that b t has distribution we also study moment and minimality properties of unbiased shifts a crucial ingredient of our approach is a new theorem on the existence of allocation rules balancing stationary diffuse random measures on r another new result is an analogue for diffuse random measures on r of the cycle stationarity characterisation of palm versions of stationary simple point processes

lax pair operators for squared sum and squared difference eigenfunctions an interrelationship between various representations of the inverse scattering transformation is established by examining eigenfunctions of lax pair operators of the sine gordon equation and the modified korteweg de vries equation in particular it is shown explicitly that there exist lax pair operators for the squared sum and squared difference eigenfunctions of the ablowitz kaup newell segur inverse scattering transformation

embedding laws in diffusions by functions of time we present a constructive probabilistic proof of the fact that if b bt t 0 is standard brownian motion started at 0 and is a given probability measure on r such that 0 0 then there exists a unique left continuous increasing function b 0 r and a unique left continuous decreasing function c 0 r such that b stopped at b c inf t 0 bt b t or bt c t has the law the method of proof relies upon weak convergence arguments arising from helly s selection theorem and makes use of the levy metric which appears to be novel in the context of embedding theorems we show that b c is minimal in the sense of monroe so that the stopped process b b c bt b c t 0 satisfies natural uniform integrability conditions expressed in terms of we also show that b c has the smallest truncated expectation among all stopping times that embed into b the main results extend from standard brownian motion to all recurrent diffusion processes on the real line

direct perturbation theory for a nearly integrable nonlinear equation with application to dark soliton solutions since the linearized equation for a nearly integrable nonlinear equation is essentially a linear equation the direct perturbation theory is now developed in the frame of green s function theory of linear differential equations the perturbed nls equation with normal dispersion is treated as an example some obscure points in previous works about this equation are clarified

the squared eigenfunctions of the massive thirring model in laboratory coordinates we demonstrate a simple procedure for finding the squared eigenfunctions of an integrable system and illustrate the method by deriving the squared eigenfunctions of the massive thirring model we give the formula for the inner products between the direct squared eigenfunction problem and its adjoint we also demonstrate closure and express the variations of the potentials in terms of variations of the scattering data

optimal skorokhod embedding under finitely many marginal constraints the skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of brownian motion stopped at a chosen stopping time in this paper we consider an extension of the weak formulation of the optimal skorokhod embedding problem in beiglbock cox and huesmann optimal transport and skorokhod embedding preprint 2013 to the case of finitely many marginal constraints using the classical convex duality approach together with the optimal stopping theory we establish some duality results under more general conditions than beiglbock cox and huesmann we also relate these results to the problem of martingale optimal transport under multiple marginal constraints

systematic perturbation theory for sine gordon solitons without use of inverse scattering methods the perturbed sine gordon soliton solutions kinks breathers in laboratory coordinates are derived by a green function formalism in a novel way dispensing with any inverse scattering methods besides the backlund transformation a simple approach based on green s theorem is employed for determining the green functions a systematic treatment allows the adiabatic approximation to be derived in a new manner the results are compared to those of the relevant papers in this field and thereby further insight is provided in a simple example a kink scattered by an impurity the calculation of energy radiation is demonstrated

a direct approach to studying soliton perturbations starting with an integrable nonlinear evolution equation the author investigates perturbations about a one soliton solution through the inversion of a linear equation for the first order correction to the soliton solution this inversion differs from past methods as the proposed method takes place in coordinate space not spectral space while it employs some of the tools of inverse scattering theory the method is applied to the korteweg de vries nonlinear schrodinger and sine gordon equations the first order corrections are then obtained

green s function method for perturbed dnls equation with the squared spectrum parameter abstract a general direct perturbation theory for the perturbed dnls equation is developed the completeness relation is given by the green s function method similar to that used for the linear equations the effect of damping on a single soliton is calculated as an example and the result is good with the squared spectrum parameter

squared eigenfunctions for the sasa satsuma equation squared eigenfunctions are quadratic combinations of jost functions and adjoint jost functions which satisfy the linearized equation of an integrable equation they are needed for various studies related to integrable equations such as the development of its soliton perturbation theory in this article squared eigenfunctions are derived for the sasa satsuma equation whose spectral operator is a 3 3 system while its linearized operator is a 2 2 system it is shown that these squared eigenfunctions are sums of two terms where each term is a product of a jost function and an adjoint jost function the procedure of this derivation consists of two steps first is to calculate the variations of the potentials via variations of the scattering data by the riemann hilbert method the second one is to calculate the variations of the scattering data via the variations of the potentials through elementary calculations while this procedure has been used before on other integrable equations it is shown here for

travelling waves in a singularly perturbed sine gordon equation we determine the linearised stability of travelling front solutions of a perturbed sine gordon equation this equation models the long josephson junction using the rcsj model for currents across the junction and includes surface resistance for currents along the junction the travelling waves correspond to the so called fluxons and their linear stability is determined by calculating the evans function surface resistance corresponds to a singular perturbation term in the governing equation which specifically complicates the computation of the corresponding evans function both the flow of quasi particles across and along the junction stabilise the waves

robust expected utility maximization with medial limits abstract in this paper we study a robust expected utility maximization problem with random endowment in discrete time we give conditions under which an optimal strategy exists and derive a dual representation for the optimal utility our approach is based on a general representation result for monotone convex functionals a functional version of choquet s capacitability theorem and medial limits the novelty is that it works under nondominated model uncertainty without any assumptions of time consistency as applications we discuss robust utility maximization problems with moment constraints wasserstein constraints and wasserstein penalties

constrained optimal transport the classical duality theory of kantorovich c r doklady acad sci urss ns 37 199 201 1942 and kellerer z wahrsch verw gebiete 67 4 399 432 1984 for classical optimal transport is generalized to an abstract framework and a characterization of the dual elements is provided this abstract generalization is set in a banach lattice mathcal x with an order unit the problem is given as the supremum over a convex subset of the positive unit sphere of the topological dual of mathcal x and the dual problem is defined on the bi dual of mathcal x these results are then applied to several extensions of the classical optimal transport

green s functions for nonlinear klein gordon kink perturbation theory a green s function is defined for nonlinear klein gordon theories in terms of the solutions to the eigenvalue equation obtained by linearizing the nonlinear wave equation about a static kink waveform analytic forms in terms of modified lommel functions of two variables are derived for the sine gordon phi 4 and double quadratic potentials asymptotic forms for the green s functions are obtained by investigating the asymptotic behavior of the modified lommel functions methods for calculating the lommel functions are also outlined

duality theorems for marginal problems given topological spaces x1 xn with product space x probability measures i on xi together with a real function h on x define a marginal problem as well as a dual problem using an extended version of choquet s theorem on capacities an analogue of the classical duality theorem of linear programming is established imposing only weak conditions on the topology of the spaces xi and the measurability resp boundedness of the function h applications concern among others measures with given support stochastic order and general marginal problems

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159