genetic algorithm approach on multi criteria minimum spanning tree problem minimum spanning tree mst problem is of high importance in network optimization the multi criteria mst mc mst is a more realistic representation of the practical problem in the real world but it is difficult for the traditional network optimization technique to deal with in this paper a genetic algorithm ga approach is developed to deal with this problem without neglecting its network topology the proposed method adopts the prufer number as the tree encoding and applies the multiple criteria decision making mcdm technique and nondominated sorting technique to make the ga search give out all pareto optimal solutions either focused on the region near the ideal point or distributed all along the pareto frontier compared with the enumeration method of pareto optimal solution the numerical analysis shows the efficiency and effectiveness of the ga approach on the mc mst problem

on the vectoral ekeland s variational principle and minimal points in product spaces abstract phelps 13 noticed that the scalar ekeland s variational principle evp is equivalent to the existence of a minimal point of the epigraph of the corresponding function with respect to an appropriate order attouch and riahi 1 showed that evp is equivalent to the existence of maximal points with respect to cones satisfying some additional conditions taking these into account gopfert and tammer 6 7 established a maximal point theorem in a product space the aim of this paper is to obtain several minimal point theorems in product spaces and the corresponding variants of the vectorial evp

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

the poisson kernel for drinfeld modular curves in an earlier work t 1 the author described a technique for constructing rigid analytic modular forms on the p adic upper half plane by means of an integral transform a poisson kernel in this paper we apply these methods to the study of rigid analytic modular forms on the upper half plane over a complete local field k of characteristic p arising out of the theory of drinfeld modules d these modular forms first studied by goss see gol go2 are in many ways analogous to the classical modular forms for sl2 z for a brief but effective introduction to the theory of these characteristic p modular forms see the introduction to gekeler s paper gekl also useful is gekeler s book gek2 we establish three main results first of all we show that if k is the function field of a complete geometrically irreducible curve c over the finite field lfq with q elements k is a completion of k f is an arithmetic subgroup of gl2 k and u is a measure on pi coming from a harmonic cocycle of weight n for i see definition 1 below then

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

heegner points on mumford tate curves 1 shimura curves 417 2 heegner points 429 3 the regulator term 441 4 the conjecture 445 5 applications and re nements 447

stickelberger elements and modular parametrizations of elliptic curves in the present paper we shall give evidence to support the claim conjecture i below and 1 3 that every elliptic curve a o which can be parametrized by modular functions admits a canonical modular parametrization whose properties can be related to intrinsic properties of a in particular we will see how such a parametrizat ion can be used to prove some rather pleasant integrality properties of stickelberger elements ad p adic l functions attached to a in addition if conjecture i is true then we can give an intrinsic characterization of the isomorphism class of a special elliptic curve in the q isogeny class of a distinguished by modular considerations for most of the paper we have opted for the concrete approach and defined modular parametrizations in terms of x i n definition 1 1 however to justify our view of these parametrizations as being canonical we begin here with a more intrinsic definition recall that shimura 19 chap 6 see w 1 of this paper has defined a compatible system of canonical models of modular curves xs ses where 5 p is a certain collection of open subgroups of the group gl 2 az over the finite adeles a i of q we define the adelic upper half plane to be the pro variety ll m xs and give the q structure induced by the s field of modular functions whose q expansions at the 0 cusp have coefficients in q a modular parametrization of a is a q morphism a which sends

compact gravity inversion we present a new criterion for the inversion of gravity data the principle employed is to minimize the volume of the causative body which is equivalent to maximizing its compactness the anomalous density distribution is obtained using an iterative technique which is numerically stable and rapidly convergent the principle can also be adapted to include modeling of gravity anomalies by single density sources the advantage of this approach is that desirable geologic characteristics are automatically incorporated into the model with a minimum of subjective judgments on the part of the interpreter the treatment of noise in the data fits naturally into the formulation of the inversion procedure the method is illustrated by the inversion of noise free and noisy data generated from a two dimensional model consisting of a regular array of identical rectangular blocks whose densities can be individually specified in every case the algorithm successfully recovers the correct density distribution from the dat

three dimensional gravity inversion using graph theory to delineate the skeleton of homogeneous sources abstractwe developed a new strategy based on graph theory concepts to invert gravity data using an ensemble of simple point masses our method consisted of a genetic algorithm with elitism to generate a set of possible solutions each estimate was associated to a graph to solve the minimum spanning tree mst problem to produce unique and stable estimates we restricted the position of the point masses by minimizing the statistical variance of the distances of an mst jointly with the data misfit function during the iterations of the genetic algorithm hence the 3d spatial distribution of the point masses identified the skeleton of homogeneous gravity sources in addition our method also gave an estimation of the anomalous mass of the source so together with the anomalous mass the skeleton could aid other 3d methods with promising geometric a priori parameters several tests with different values of regularizing parameter were made to bespeak this new regularizing strategy the inversion results ap

adaptive learning 3d gravity inversion for salt body imaging we have developed an iterative scheme for inverting gravity data produced by salt bodies with density contrasts relative to the sediments varying from positive to negative crossing in this way the nil zone our inversion method estimates a 3d density contrast distribution through a piecewise constant function defined on a user specified grid of cells it consists of two nested iterative loops the outer loop uses an adaptive learning strategy that starts with a coarse grid of cells a set of first guess geometric elements axes and points and the corresponding assigned density contrasts from the second iteration on this strategy refines the grid and automatically creates a new set of geometric elements points only and associated density contrasts each geometric element operates as the first guess skeletal outline of a section of the salt body to be imaged the inner loop estimates the 3d density contrast distribution for the grid of cells and for the set of geometric elements defined in the oute

vector valued perturbed equilibrium problems the goal of this paper is to prove some general vector valued perturbed equilibrium principles and some existence results of vector equilibrium points for bifunctions satisfying a new natural notion of lower semi continuity we obtain these results by going through a new concept of approximative equilibrium point

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

derived heights and generalized mazur tate regulators 2 algebraic preliminaries 4 2 1 notations and conventions 4 2 2 duality 5 2 2 1 local duality 5 2 2 2 global duality 6 2 3 assumptions 8 2 4 preliminary calculations 10

integration on cal h _p times cal h and arithmetic applications this article describes a conjectural p adic analytic construction of global points on modular elliptic curves points which are defined over the ring class fields of real quadratic fields the resulting conjectures suggest that the classical heegner point construction and the theory of complex multiplication on which it is based should extend to a variety of contexts in which the underlying field is not a cm field

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

3 d inversion of gravity data we present two methods for inverting surface gravity data to recover a 3 d distribution of density contrast in the first method we transform the gravity data into pseudomagnetic data via poisson s relation and carry out the inversion using a 3 d magnetic inversion algorithm in the second we invert the gravity data directly to recover a minimum structure model in both approaches the earth is modeled by using a large number of rectangular cells of constant density and the final density distribution is obtained by minimizing a model objective function subject to fitting the observed data the model objective function has the flexibility to incorporate prior information and thus the constructed model not only fits the data but also agrees with additional geophysical and geological constraints we apply a depth weighting in the objective function to counteract the natural decay of the kernels so that the inversion yields depth information applications of the algorithms to synthetic and field data produce density models representative of true structures our results have shown that the inversion of gravity data with a properly designed objective function can yield geologically meaningful information

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

traveltime inversion for 2 d anomaly structures this work presents a traveltime inversion method that uses parametric functions to represent 2 d anomaly structures these functions are described by a small set of unknown parameters which in turn are obtained after solving a highly nonlinear optimization problem via simulated annealing sa the procedure favors neither smooth nor high contrasting anomalies and keeps the number of unknowns very small so as to make the problem tractable using sa yet the strategy allows one to accommodate a large class of velocity models results indicate that this new approach typically yields better images than a standard linearized inversion based on a cell parameterization scheme

ekelandʼs variational principle for an l 0 valued function on a complete random metric space abstract motivated by the recent work on conditional risk measures this paper studies the ekeland s variational principle for a proper lower semicontinuous and lower bounded l 0 valued function where l 0 is the set of equivalence classes of extended real valued random variables on a probability space first we prove a general form of ekeland s variational principle for such a function defined on a complete random metric space then we give a more precise form of ekeland s variational principle for such a local function on a complete random normed module finally as applications we establish the bishop phelps theorem in a complete random normed module under the framework of random conjugate spaces

the solution of nonlinear inverse problems and the levenberg marquardt method although the levenberg marquardt damped least squares method is an extremely powerful tool for the iterative solution of nonlinear problems its theoretical basis has not been described adequately in the literature this is unfortunate because levenberg and marquardt approached the solution of nonlinear problems in different ways and presented results that go far beyond the simple equation that characterizes the method the idea of damping the solution was introduced by levenberg who also showed that it is possible to do that while at the same time reducing the value of a function that must be minimized iteratively this result is not obvious although it is taken for granted moreover levenberg derived a solution more general than the one currently used marquardt started with the current equation and showed that it interpolates between the ordinary least squares method and the steepest descent method in this tutorial the two papers are combined into a unified presentation which will help the reader gain a better understanding of what happens when solving nonlinear problems because the damped least squares and steepest descent methods are intimately related the latter is also discussed in particular in its relation to the gradient when the inversion parameters have the same dimensions and units the direction of steepest descent is equal to the direction of minus the gradient in other cases it is necessary to introduce a metric i e a definition of distance in the parameter space to establish a relation between the two directions although neither levenberg nor marquardt discussed these matters their results imply the introduction of a metric some of the concepts presented here are illustrated with the inversion of synthetic gravity data corresponding to a buried sphere of unknown radius and depth finally the work done by early researchers that rediscovered the damped least squares method is put into a historical context

conjugate convex operators convex mappings from a locally convex space x into f f are considered where f is an ordered topological vector space and an arbitrary greatest element adjoined to f in view of applications to the polarity theory of convex operators the possibility is investigated of representing a convex mapping taking values in f as a supremum of continuous affine mappings

existence theorems for variational inclusion problems and the set valued vector ekeland variational principle in a complete metric space abstract in this paper we apply an existence theorem for the variational inclusion problem to study the existence results for the variational intersection problems in ekeland s sense and the existence results for some variants of set valued vector ekeland variational principles in a complete metric space our results contain ekeland s variational principle as a special case and our approaches are different to those for any existence theorems for such problems

a generalized ekeland vector variational principle and its applications in optimization abstract in this paper we give a generalized ekeland vector variational principle by using the principle we extend and improve the related results in sharp efficiency in the framework of locally convex spaces we introduce two kinds of generalized sharp efficiencies and prove that they are equivalent in particular we show that a sharp efficient solution with respect to an interior point of the ordering cone is also one with respect to every interior point moreover we introduce the generalized takahashi s condition and the generalized hamel s condition for vector valued functions from the generalized ekeland principle we deduce that the two conditions are equivalent from this we discuss the relationship between the distance of f x from e f x and the distance of x from e f where e f x denotes the efficient point set of f x and e f denotes the efficient solution set

generalized stark formulae over function fields we establish formulae of stark type for the stickelberger elements in the function field setting our result generalizes a work of hayes and a conjecture of gross it is used to deduce a p adic version of rubin stark conjecture and burns conjecture

p adic periods and modular symbols of elliptic curves of prime conductor under certain assumptions we prove a conjecture of mazur and tate describing a relation between the modular symbol attached to an elliptic curve with split multiplicative reduction atp and itsp adic period we generalize this relation to modular forms of weight 2 with coefficients not necessarily in

vector valued variational principles in the context of vector valued extensions of variational principles we are dealing with functions taking values in a banach space partially ordered by a closed convex pointed cone we introduce and study a new notion of semi continuity connected with the order and we improve the vector valued extensions of deville godefroy zizler perturbed minimization principle

a refined beilinson bloch conjecture for motives of modular forms we propose a refined version of the beilinson bloch conjecture for the motive associated with a modular form of even weight this conjecture relates the dimension of the image of the relevant p adic abel jacobi map to certain combinations of heegner cycles on kuga sato varieties we prove theorems in the direction of the conjecture and in doing so obtain higher weight analogues of results for elliptic curves due to darmon

sur la nature non cyclotomique des points d ordre fini des courbes elliptiques appendice par e kowalski et p michel

the equivariant tamagawa number conjecture and modular symbols we show that for each elliptic curve e mathbb q with l e mathbb q 1 ne 0 there are infinitely many odd primes l and for each such l infinitely many abelian l extensions k mathbb q such that the l part of the equivariant tamagawa number conjecture holds for the pair h 1 e k 1 mathbb z text gal k mathbb q

iwasawa theory for elliptic curves at supersingular primes we give a new formulation in iwasawa theory for elliptic curves at good supersingular primes this formulation is similar to mazur s at good ordinary primes namely we define a new selmer group and show that it is of cotorsion then we formulate the iwasawa main conjecture as that the characteristic ideal is generated by pollack s p adic l function we show that this main conjecture is equivalent to kato s and perrin riou s main conjectures we also prove an inequality in the main conjecture by using kato s euler system in terms of the and the invariants of our selmer group we specify the numbers and in the asymptotic formula for the order of the tate shafarevich group by kurihara and perrin riou

a birch and swinnerton dyer conjecture for the mazur tate circle pairing let e be an elliptic curve over q attached to a newform f of weight two on 0 n and let k be a real quadratic field in which all the primes dividing n are split this note relates the canonical r z valued circle pairing on e k defined by mazur and tate mt1 to a period integral i f k defined in terms of f and k the resulting conjecture can be viewed as an analogue of the classical birch and swinnerton dyer conjecture in which i f k replaces the derivative of the complex l series l f k s and the circle pairing replaces the neron tate height it emerges naturally as an archimedean fragment of the theory of anticyclotomic p adic l functions developed in bd1 and has been tested numerically in a variety of situations the last section formulates a conjectural variant of a formula of gross kohnen and zagier gkz for the mazur tate circle pairing

gravity and magnetic inversion with minimization of a specific functional the nonuniqueness of gravity or magnetic data inversion is well known in order to remove ambiguity some authors have sought solutions minimizing a functional describing geometrical or physical properties last and kubik 1983 in particular developed a method explaining the observed anomaly by structures of minimum volume in this method the domain where anomalous sources are searched is divided into elementary prisms of a constant density or susceptibility contrast each elementary contrast is allowed to vary individually thus a contrast distribution is computed the search for this kind of solution leads in general to geologically more appropriate bodies but exceptions do occur in this paper the technique is broadened to include the search for solutions minimizing the moment of inertia with respect to the center of gravity or with respect to a given dip line passing through it the resulting structures are both deeper and more compact precisely as is required in specific cases theoretical and

ekeland s principle for vector equilibrium problems abstract in this paper the authors deal with bifunctions defined on complete metric spaces and with values in locally convex spaces ordered by closed convex cones the aim is to provide a vector version of ekeland s theorem related to equilibrium problems to prove this principle a weak notion of continuity of a vector valued function is considered and some of its properties are presented via the vector ekeland s principle existence results for vector equilibria are proved in both compact and noncompact domains

une base explicite de symboles modulaires sur les corps de fonctions modular symbols for the congruence subgroup gamma 0 mathfrak n of gl 2 mathbf f q t have been defined by teitelbaum they have a presentation given by a finite number of generators and relations in a formalism similar to manin s for classical modular symbols we completely solve the relations and get an explicit basis of generators when mathfrak n is a prime ideal of odd degree as an application we give a non vanishing statement for l functions of certain automorphic cusp forms for mathbf f q t the main statement also provides a key step for a result towards the uniform boundedness conjecture for drinfeld modules of rank 2

linear inversion of gravity data for 3 d density distributions we have developed an improved levenburg marquart technique to rapidly invert bouguer gravity data for a 3 d density distribution as a source of the observed field this technique is designed to replace tedious forward modeling with an automatic solver that determines density models constrained by geologic information supplied by the user where such information is not available objective models are generated the technique estimates the density distribution within the source volume using a least squares inverse solution that is obtained iteratively by singular value decomposition using orthogonal decomposition of matrices with sequential householder transformations the source volume is subdivided into a series of right rectangular prisms of specified size but of unknown density this discretization allows the construction of a system of linear equations relating the observed gravity field to the unknown density distribution convergence of the solution to the system is tightly controlled by a damping pa