weakly gibbsian measures for lattice spin systems we present a weaker notion of gibbs measures by requiring that only an almost everywhere absolutely summable potential is given this has recently appeared in the context gibbsian versus non gibbsian measures we give a first exploration of the main features of this weaker notion we concentrate on the questions of where do weakly gibbsian measures appear and what remains of their thermodynamic description

a note on the projecton of gibbs measures we give an example of a projection which maps two gibbs measures for the same interaction into gibbs measures for different interactions as a corollary we find a case where by decimation a non gibbsian measure is transformed into a gibbs measure

gauge optimization and duality gauge functions significantly generalize the notion of a norm and gauge optimization as defined by r m freund math programming 38 1987 pp 47 67 seeks the element of a convex set that is minimal with respect to a gauge function this conceptually simple problem can be used to model a remarkable array of useful problems including a special case of conic optimization and related problems that arise in machine learning and signal processing the gauge structure of these problems allows for a special kind of duality framework this paper explores the duality framework proposed by freund and proposes a particular form of the problem that exposes some useful properties of the gauge optimization framework such as the variational properties of its value function and yet maintains most of the generality of the abstract form of gauge optimization

a note on the strong polynomiality of convex quadratic programming we prove that a general convex quadratic program qp can be reduced to the problem of finding the nearest point on a simplicial cone ino n3 n logl steps wheren andl are respectively the dimension and the encoding length of qp the proof is quite simple and uses duality and repeated perturbation the implication however is nontrivial since the problem of finding the nearest point on a simplicial cone has been considered a simpler problem to solve in the practical sense due to its special structure also we show that theoretically this reduction implies that i if an algorithm solves qp in a polynomial number of elementary arithmetic operations that is independent of the encoding length of data in the objective function then it can be used to solve qp in strongly polynomial time and ii ifl is bounded by a first order exponential function ofn then i can be stated even in stronger terms to solve qp in strongly polynomial time it suffices to find an algorithm running in polynomial time that is independent of the encoding length of the quadratic term matrix or constraint matrix finally based on these results we propose a conjecture

relative entropy and variational properties of generalized gibbsian measures we study the relative entropy density for generalized gibbs measures we first show its existence and obtain a familiar expression in terms of entropy and relative energy for a class of almost gibbsian measures almost sure continuity of conditional probabilities for quasilocal measures we obtain a full variational principle for the joint measures of the random field ising model we show that the weak gibbs property holds with an almost surely rapidly decaying translation invariant potential for these measures we show that the variational principle fails as soon as the measures lose the almost gibbs property these examples suggest that the class of weakly gibbsian measures is too broad from the perspective of a reasonable thermodynamic formalism

generalized goal programming polynomial methods and applications in this paper we address a general goal programming problem with linear objectives convex constraints and an arbitrary componentwise nondecreasing norm to aggregate deviations with respect to targets in particular classical linear goal programming problems as well as several models in location and regression analysis are modeled within this framework

approximating semidefinite packing programs in this paper we define semidefinite packing programs and describe an algorithm to approximately solve these problems semidefinite packing programs arise in many applications such as semidefinite programming relaxations for combinatorial optimization problems sparse principal component analysis and sparse variance unfolding techniques for dimension reduction our algorithm exploits the structural similarity between semidefinite packing programs and linear packing programs

almost gibbsianness and parsimonious description of the decimated 2d ising model in this paper we complete and provide details for the existing characterizations of the decimation of the ising model on mathbb z 2 in the generalized gibbs context we first recall a few features of the dobrushin program of restoration of gibbsianness and present the construction of global specifications consistent with the extremal decimated measures we use them to prove that these renormalized measures are almost gibbsian at any temperature and to analyse in detail its convex set of dlr measures we also recall the weakly gibbsian description and complete it using a potential that admits a quenched correlation decay i e a well defined configuration dependent length beyond which this potential decays exponentially we use these results to incorporate these decimated measures in the new framework of parsimonious random fields that has been recently developed to investigate probability aspects related to neurosciences

extended neuro fuzzy models of multilayer perceptrons in this paper the famous neural model the multilayer perceptron is extended to a new neural model that is called the additive takagi sugeno type multilayer perceptron the present study proves that this new model can also act as a universal approximator and thus it can be used in many fields such as system modeling and identification the concept of f duality and the fuzzy operator interactive or are used to prove that the proposed neural model is functionally equal to a kind of fuzzy inference system further this paper presents another new neuro fuzzy model that is called the sigmoid adaptive network based fuzzy inference system simulation studies show that our proposed models both have stronger approximation capability than multilayer perceptrons

ill defined block spin transformations at arbitrarily high temperatures examples are presented of block spin transformations which map the gibbs measures of the ising model in two or more dimensions at temperature intervals extending to arbitrarily high temperatures onto non gibbsian measures in this way we provide the first example of this kind of pathology for very high temperatures and as a corollary also the first example of such a pathology happening at a critical point

variational principle for some renormalized measures we show that some measures suffering from the so called renormalization group pathologies satisfy a variational principle and that the corresponding limit of the pressure with boundary conditions in a set of measure 1 is equal to the pressure of the ising model module a scale factor

global specifications and nonquasilocality of projections of gibbs measures reference gr pf article 1997 001doi 10 1214 aop 1024404514view record in web of science record created on 2008 11 27 modified on 2017 05 12

robustness of the non gibbsian property some examples we discuss some examples of measures on lattice systems which lack the property of being a gibbs measure in a rather strong sense

projections of gibbs measures may be non gibbsian consider the phase of the two dimensional nearest neighbor ferromagnetic ising model at a temperature belowt c let be the restriction of this measure to a coordinate axis we prove that there is no one dimensional translation invariant summable interaction for which is a gibbs measure this is proven by showing that if such an interaction existed would have large deviation properties different from those it actually has percolation methods are used in the proof

variational principle and almost quasilocality for renormalized measures we study the variational principle for some non gibbsian measures we give a necessary and sufficient condition for the validity of the implication zero relative entropy density implies common version of conditional probabilities so called second part of the variational principle applying this to noisy decimations of the low temperature phases of the ising model we obtain almost sure quasilocality for these measures and the second part of the variational principle for the projection of low temperature ising phases on a one dimensional layer we also obtain the second part of the variational principle

optimal inequalities in probability theory a convex optimization approach we propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for p x in s for a set s defined by polynomial inequalities and a random vector x defined on omega subseteq cal r n that has a given collection of up to kth order moments in the univariate case we provide optimal bounds on p x in s when the first k moments of x are given as the solution of a semidefinite optimization problem in k 1 dimensions in the multivariate case if the sets s and omega are given by polynomial inequalities we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n for fixed k we characterize the complexity of the problem of deriving tight moment inequalities we show that it is np hard to find tight bounds for k geq 4 and omega cal r n and for k geq 2 and omega cal r n when the data in the problem is rational for k 1 and omega cal r n we show that we can find tight upper bounds by solving n convex optimization problems when the set s is convex and we provide a polynomial time algorithm when s and omega are unions of convex sets over which linear functions can be optimized efficiently for the case k 2 and omega cal r n we present an efficient algorithm for finding tight bounds when s is a union of convex sets over which convex quadratic functions can be optimized efficiently

large deviations and variational principle for harmonic crystals we consider massless gaussian fields with covariance related to the green function of a long range random walk on d these are viewed as gibbs measures for a linear quadratic interaction we establish thermodynamic identities and prove a version of gibbs variational principle showing that translation invariant gibbs measures are characterized as minimizers of the relative entropy density we then study the large deviations of the empirical field of a gibbs measure we show that a weak large deviation principle holds at the volume order with rate given by the relative entropy density

pathological behavior of renormalization group maps at high fields and above the transition temperature we show that decimation transformations applied to high q potts models result in non gibbsian measures even for temperatures higher than the transition temperature we also show that majority transformations applied to the ising model in a very strong field at low temperatures produce non gibbsian measures this shows that pathological behavior of renormalization group transformations is even more widespread than previous examples already suggested

some remarks on pathologies of renormalization group transformations for the ising model the results recently obtained by van enter fernandez and sokal on non gibbsianness of the measurev t b h arising from the application of a single decimation transformationt b with spacingb to the gibbs measure h of the ising model for suitably chosen large inverse temperature and nonzero external fieldh are critically analyzed in particular we show that if keeping fixed the same values of h andb one iterates a sufficiently large number of timesn the transformationt b one obtains a new measurev t b n h which is gibbsian and moreover very weakly coupled

a recursive algorithm for finding the minimum norm point in a polytope and a pair of closest points in two polytopes for a given pair of finite point setsp andq in some euclidean space we consider two problems problem 1 of finding the minimum euclidean norm point in the convex hull ofp and problem 2 of finding a minimum euclidean distance pair of points in the convex hulls ofp andq we propose a finite recursive algorithm for these problems the algorithm is not based on the simplicial decomposition of convex sets and does not require to solve systems of linear equations

basic concepts for a theory of evaluation the aggregative operator abstract starting with an explication of the aggregative concept and deducing a general structure which satisfies a number of minimal requirements properties of clustering the main features of a new mathematical theory called theory of evaluation are developed the theory sheds new light on such well known concepts as membership conjunction and disjunction and seems to be a very promising tool to handle representation problems as they grow from the fields of theory of fuzzy set and its many applications of human decision making and of multicriteria analysis

dual gauge programs with applications to quadratic programming and the minimum norm problem a gauge functionf is a nonnegative convex function that is positively homogeneous and satisfiesf 0 0 norms and pseudonorms are specific instances of a gauge function this paper presents a gauge duality theory for a gauge program which is the problem of minimizing the value of a gauge functionf over a convex set the gauge dual program is also a gauge program unlike the standard lagrange dual we present sufficient conditions onf that ensure the existence of optimal solutions to the gauge program and its dual with no duality gap these sufficient conditions are relatively weak and are easy to verify and are independent of any qualifications on the constraints the theory is applied to a class of convex quadratic programs and to the minimuml p norm problem the gauge dual program is shown to provide a smaller duality than the standard dual in a certain sense discussed in the text

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

renormalization group transformations under strong mixing conditions gibbsianness and convergence of renormalized interactions in this paper we study a renormalization group map the block averaging transformation applied to gibbs measures relative to a class of finite range lattice gases when suitable strong mixing conditions are satisfied using a block decimation procedure cluster expansion and detailed comparison between statistical ensembles we are able to prove gibbsianness and convergence to a trivial i e gaussian and product fixed point our results apply to the 2d standard ising model at any temperature above the critical one and arbitrary magnetic field