methods for multi objective optimization an analysis decomposition based methods are often cited as the solution to multi objective nonconvex optimization problems with an increased number of objectives these methods employ a scalarizing function to reduce the multi objective problem into a set of single objective problems which upon solution yield a good approximation of the set of optimal solutions this set is commonly referred to as pareto front in this work we explore the implications of using decomposition based methods over pareto based methods on algorithm convergence from a probabilistic point of view namely we investigate whether there is an advantage of using a decomposition based method for example using the chebyshev scalarizing function over pareto based methods we find that under mild conditions on the objective function the chebyshev scalarizing function has an almost identical effect to pareto dominance relations when we consider the probability of finding superior solutions for algorithms that follow a balanced trajectory we propose the hypothesis that this seemingly contradicting result compared with currently available empirical evidence signals that the disparity in performance between pareto based and decomposition based methods is due to the inability of the former class of algorithms to follow a balanced trajectory we also link generalized decomposition to the results in this work and show how to obtain optimal scalarizing functions for a given problem subject to prior assumptions on the pareto front geometry

motivic integral of k3 surfaces over a non archimedean field abstract we prove a formula expressing the motivic integral loeser and sebag 2003 34 of a k3 surface over c t with semi stable reduction in terms of the associated limit mixed hodge structure secondly for every smooth variety over a complete discrete valuation field we define an analogue of the monodromy pairing constructed by grothendieck in the case of abelian varieties and prove that our monodromy pairing is a birational invariant of the variety finally we propose a conjectural formula for the motivic integral of maximally degenerate k3 surfaces over an arbitrary complete discrete valuation field and prove this conjecture for kummer k3 surfaces

the moduli space of cubic fourfolds via the period map we characterize the image of the period map for cubic fourfolds with at worst simple singularities as the complement of an arrangement of hyperplanes in the period space it follows then that the geometric invariant theory git compactification of the moduli space of cubic fourfolds is isomorphic to the looijenga compactification associated to this arrangement this paper builds on and is a natural continuation of our previous work on the git compactification of the moduli space of cubic fourfolds

reference points and approximation algorithms in multicriteria discrete optimization abstract mathematical research on multicriteria optimization problems predominantly revolves around the set of pareto optimal solutions in practice on the other hand methods that output a single solution are more widespread reference point methods are a successful example of this approach and are widely used in real world multicriteria optimization a reference point solution is the solution closest to a given reference point in the objective space we study the connection between reference point methods and approximation algorithms for multicriteria optimization problems over discrete sets in particular we establish that in terms of computational complexity computing approximate reference point solutions is polynomially equivalent to approximating the pareto set complementing these results we show for a number of general algorithmic techniques in single criteria optimization how they can be lifted to reference point optimization in particular we lift the link between dynamic programming and fptas as well as certain lp rounding techniques the latter applies e g to set cover and several machine scheduling problems

lattices and k3 surfaces of degree 6 we relate a class of degenerations of k3 surfaces of degree 6 to certain lattices via hodge theory

the second homology group of the mapping class group of an orientable surface in i 7 mumford shows that the picard group p ic is isomorphic to h2 f 2 and conjectures the latter is rank one g 3 we prove this below for g 5 another interpretation of this theorem may be obtained by identifying h2 f as bordism classes of fiber bundles f w4 t where t is a closed oriented surface sect 0 when f is closed every such bundle is bordant to f w t a bundle admitting a section s t w the theorem then says that

degenerations of k3 surfaces of degree two we consider a semistable degeneration of k3 surfaces equipped with an effective divisor that defines a polarisation of degree two on a general fibre we show that the map to the relative log canonical model of the degeneration maps every fibre to either a sextic hypersurface in p 1 1 1 3 or a complete intersection of degree 2 6 in p 1 1 1 2 3 furthermore we find an explicit description of the hypersurfaces and complete intersections that can arise thereby giving a full classification of the possible singular fibres

on k3 surfaces with large complex structure abstract we discuss a notion of large complex structure for elliptic k3 surfaces with section inspired by the eight dimensional f theory heterotic duality in string theory this concept is naturally associated with the type ii mumford partial compactification of the moduli space of periods for these structures the paper provides an explicit hodge theoretic condition for the complex structure of an elliptic k3 surface with section to be large we also establish certain geometric consequences of this large complex structure condition in terms of the kodaira types of the singular fibers of the elliptic fibration

on the noether lefschetz theorem and some remarks on codimension two cycles here of general moduli means that there is a countable union v of subvarieties of the space pn of surfaces of degree d in p3 such that the statement pie s z holds for s p n v noether it would seem stated this theorem but never completely proved it instead he gave a plausibility argument based on the following construction 1 let g be the hilbert scheme of curves of degree n and genus 9 in p3 l gr3 d l ip n the space of surfaces of degree d in p3 and zc t the incidence correspondence s o a c s c z s

on monodromies of a degeneration of irreducible symplectic kahler manifolds we study the monodromy operators on the betti cohomologies associated to a good degeneration of irreducible symplectic manifold and we show that the unipotency of the monodromy operator on the middle cohomology is at least the half of the dimension this implies that the mildest singular fiber of a good degeneration with non trivial monodromy of irreducible symplectic manifolds is quite different from the generic degeneration of abelian varieties or calabi yau manifolds

combining equity and utilitarianism in a mathematical programming model we discuss the problem of combining the conflicting objectives of equity and utilitarianism for social policy making in a single mathematical programming model the definition of equity we use is the rawlsian one of maximizing the minimum utility over individuals or classes of individuals however when the disparity of utility becomes too great the objective becomes progressively utilitarian such a model is particularly applicable not only to health provision but to other areas as well building a mixed integer linear programming milp formulation of the problem raises technical issues because the objective function is nonconvex and the hypograph is not milp representable in its initial form we present a succinct formulation and show that it is sharp in the sense that its linear programming relaxation describes the convex hull of the feasible set before extra resource allocation or policy constraints are added we apply the formulation to a healthcare planning problem and show that instances of realistic size are easily solved by standard milp software this paper was accepted by dimitris bertsimas optimization

mathematics underlying the f theory heterotic string duality in eight dimensions we give an analytic description of the moduli space of classical vacua for f theory compactified on elliptic k3 surfaces on open tubular regions near the two type ii boundary divisors the structure of these open sets is related to the total spaces of certain holomorphic theta fibrations over the corresponding boundary divisors as the two type ii divisors can be naturally identified as moduli spaces of elliptic curves and flat g bundles with g e8 e8 or spin 32 one is led to an analytic isomorphism between these open domains and regions of the moduli spaces of heterotic string theory compactified on the two torus corresponding to large volumes of the torus this provides a proof for the classical version of f theory heterotic string duality in eight dimensions a description of the type ii boundary points in terms of elliptic stable k3 surfaces is also given

compromise programming non interactive calibration of utility based metrics utility functions have been used widely to support multi objective decision making expansion of a general additive utility function around the ideal results in a composite linear quadratic metric of a compromise programming problem determining the unknown parameters of the composite linear quadratic metric requires substantial interaction with the decision maker who might not always be available or capable to participate in such a process we propose a non interactive method that uses information on observed attribute levels to obtain the unknown parameters of the composite linear quadratic metric and enables forecasting and scenario analysis the method is illustrated with a small scale numerical example

a practical weight sensitivity algorithm for goal and multiple objective programming this paper presents a weight sensitivity algorithm that can be used to investigate a portion of weight space of interest to the decision maker in a goal or multiple objective programme the preferential information required from the decision maker is an initial estimate of their starting solution with an equal weights solution being used as a default if this is not available and preference information that will define the portion of weight space on which the sensitivity analysis is to be conducted the different types of preferential information and how they are incorporated by the algorithm are discussed the output of the algorithm is a set of distinct solutions that characterise the portion of weight space searched the possible different output requirements of decision makers are detailed in the context of the algorithm the methodology is demonstrated on two examples one hypothetical and the other relating to predicting cinema going behaviour conclusions and avenues for future research are given

degeneration of surfaces with trivial canonical bundle the object of this note is to prove the theorem stated below the main idea of the proof and much of the detail are already to be found in v i kulikov s important paper k our contribution to the proof and the differences with kulikov s paper are discussed later in this introduction theorem let wl x d be a semistable degeneration of surfaces such that

transcending classical invariant theory let sp2 r sp2 sp be the real symplectic group of rank n let sp denote the two fold cover of sp there is a unitary representation constructed by shale sh and weil wa of sp that is of considerable interest in the theory of automorphic forms we denote this representation by co and call it the oscillator representation the purpose of this paper is to establish about c a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co we begin by formulating the result let e be a reductive subgroup of sp and let e denote the inverse image of e in sp denote by r e the set of infinitesimal equivalence classes of continuous irreducible admissible representations of e on locally convex spaces let co be the smooth representation of sp associated to co let co be realized on a hilbert space and let the subspace of smooth vectors on which space co is defined be written denote by r9 e co the set of elements of e which are realized as quotients by co e invariant closed subspaces of t g consider a reductive dual pair g g c sp h2 it is not hard to show that g and g commute with one another hence we may regard co i g g as a representation of gx g it is well known f that r gx g s9 g xr9 g the identification associates to p e r g and p e i g the tensor product p x p we note that p x p is not defined as a topological vector space nevertheless the infinitesimal equivalence class of p op is well defined select p x p e g x g suppose that in fact p p e 9 g g c then clearly p e r g c and p e 9 g c hence g co defines the graph of a correspondence between certain subsets of r9 g cv and j g co in fact the situation is quite precise

a general structure of achievement function for a goal programming model abstract a key element of a goal programming model is the achievement function that is the function that measures the degree of minimisation of the unwanted deviation variables of the goals considered in the model the three oldest and still most widely used forms of achievement functions are the following archimedean lexicographic and minmax while they are the most common other newer versions can offer theoretical and practical advantages these newer forms will be analytically described and interpreted in utility terms throughout this paper the presentation will end with a very general structure which encompasses as particular cases basically all the achievement functions proposed in the literature

the tate conjecture for k3 surfaces in odd characteristic we show that the classical kuga satake construction gives rise away from characteristic 2 to an open immersion from the moduli of primitively polarized k3 surfaces of any fixed degree to a certain regular integral model for a shimura variety of orthogonal type this allows us to attach to every polarized k3 surface in odd characteristic an abelian variety such that divisors on the surface can be identified with certain endomorphisms of the attached abelian variety in turn this reduces the tate conjecture for k3 surfaces over finitely generated fields of odd characteristic to a version of the tate conjecture for certain endomorphisms on the attached kuga satake abelian variety which we prove as a by product of our methods we also show that the moduli stack of primitively polarized k3 surfaces of degree 2d is quasi projective and when d is not divisible by p 2 is geometrically irreducible in characteristic p we indicate how the same method applies to prove the tate conjecture for co dimension 2 cycles on cubic fourfolds

a non interactive elicitation method for non linear multiattribute utility functions theory and application to agricultural economics practical implementation of multiattribute utility theory is limited partly for the lack of operative methods to elicit the parameters of the multiattribute utility function particularly when this function is not linear as a consequence most studies are restricted to linear specifications which are easier to estimate and to interpret we propose an indirect method to elicit the parameters of a non linear utility function to be compatible with the observed behaviour of decision makers rather than with their answers to direct surveys the idea rests on approaching the parameter estimation problem as a dual of the decision problem by making the observed decisions to be compatible with a rational decision making process

the moduli of curves of genus six and k3 surfaces we prove that the coarse moduli space of curves of genus six is birational to an arithmetic quotient of a bounded symmetric domain of type iv by giving a period map to the moduli space of some lattice polarized k3 surfaces

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

moduli of nodal curves on k3 surfaces abstract we consider modular properties of nodal curves on general k3 surfaces let k p be the moduli space of primitively polarized k3 surfaces s l of genus p 3 and v p m k p be the universal severi variety of nodal irreducible curves in m l on s l k p we find conditions on p m for the existence of an irreducible component v of v p m on which the moduli map v m g with g m 2 p 1 1 has generically maximal rank differential our results which for any p leave only finitely many cases unsolved and are optimal for m 5 except for very low values of p are summarized in theorem 1 1 in the introduction

no mass drop for mean curvature flow of mean convex hypersurfaces a possible evolution of a compact hypersurface in rn 1 by mean curvature past singularities is defined via the level set flow in the case where the initial hypersurface has positive mean curvature we show that the brakke flow associated to the level set flow is actually a brakke flow with equality as a consequence we obtain the fact that no mass drop can occur along such a flow a further application of the techniques used above is to give a new variational formulation for mean curvature flow of mean convex hypersurfaces

semistable degenerations and period spaces for polarized k3 surfaces modular compactifications of moduli spaces for polarized k3 surfaces are constructed using the tools of logarithmic geometry in the sense of fontaine and illusie the relationship between these new moduli spaces and the classical minimal and toroidal compactifications of period spaces are discussed and it is explained how the techniques of this paper yield models for the latter spaces over number fields the paper also contains a discussion of picard functors for log schemes and a logarithmic version of artin s method for proving representability by an algebraic stack

selecting the cp metric a risk aversion approach abstract through a linkage between arrow s risk theory and compromise programming we obtain a reliable specification of the metric defining the compromise distance from a point to the ideal point in the n attribute space

l2 cohomology of arithmetic varieties abstract the l2 cohomology of arithmetic quotients of bounded symmetric domains is studied we establish the conjecture of zucker equating the l2 cohomology of these spaces to the intersection cohomology of their baily borel compactifications

the hodge conjecture and arithmetic quotients of complex balls let s be a closed shimura variety uniformized by the complex n ball associated with a standard unitary group the hodge conjecture predicts that every hodge class in h 2k s mathbb q k 0 dots n is algebraic we show that this holds for all degrees k away from the neighborhood bigl tfrac13n tfrac23n bigr of the middle degree we also prove the tate conjecture for the same degrees as the hodge conjecture and the generalized form of the hodge conjecture in degrees away from an interval depending on the codimension c of the subvariety centered at the middle dimension of s these results are derived from a general theorem that applies to all shimura varieties associated with standard unitary groups of any signature the proofs make use of arthur s endoscopic classification of automorphic representations of classical groups as such our results rely on the stabilization of the trace formula for the disconnected groups gl n rtimes langle theta rangle associated with base change

picard number of the generic fiber of an abelian fibered hyperkahler manifold we shall show that the picard number of the generic fiber of an abelian fibered hyperkahler manifold over the projective space is always one we then give a few applications for the mordell weil group in particular by deforming o grady s 10 dimensional manifold we construct an abelian fibered hyperkahler manifold of mordell weil rank 20 which is the maximum possible among all known ones

the noether lefschetz conjecture and generalizations we prove the noether lefschetz conjecture on the moduli space of quasi polarized k3 surfaces this is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal type are dual to the classes of special cycles i e sub arithmetic manifolds of the same type for compact manifolds this was proved in 3 here we extend the results of 3 to non compact manifolds this allows us to apply our results to the moduli spaces of quasi polarized k3 surfaces

compromise programming a utility based linear quadratic composite metric from the trade off between achievement and balanced non corner solutions abstract in this paper compromise programming cp is viewed as the maximization of the decision maker s additive utility function whose arguments are the criteria under consideration subject to an efficient frontier of criteria and the non negativity constraints in a deterministic context this is equivalent to minimizing the difference distance between utility at the ideal point and utility at a frontier point on the criteria map a meaningful statement as minimizing distances to the utopia is the ethos of compromise programming by taylor expansion of utility around the ideal point the distance to the utopia becomes the weighted sum of linear and quadratic cp distances which gives us the composite metric while the linear terms pursue achievement the quadratic ones pursue balanced non corner solutions because some decision makers fear imbalance while others prefer large achievements even to the detriment of balance the paper defines an aversion to imbalance ratio so that the composite linear quadratic metric should conform to this ratio depending on the decision maker s preferences and attitudes as the problem of selecting an appropriate metric is an ongoing issue in cp the paper is a contribution to theory and practice for the sole purpose of suggesting industrial applications an example is developed

motion of level sets by mean curvature ii we give a new proof of short time existence for the classical motion by mean curvature of a smooth hypersurface our method consists in studying a fully nonlinear uniformly parabolic equation satisfied by the signed distance function to the surface

twisted endoscopy and reducibility of induced representations for p adic groups 1 introduction this is the first in a series of papers in which we study the reducibility of representations induced from discrete series representations of the levi factors ofmaximal parabolic subgroups of p adic groups on the unitary axis the problem is equivalent to determining certain local langlands l functions which are of arithmetic significance by themselves one of the main aims of this paper is to interpret our results in the direction of the parametrization problem by means of the theory of twisted endoscopy for which our method seems to be very suitable when the representation is supercuspidal we also study the reducibility of the induced representations off the unitary axis let f be a p adic field of characteristic zero fix a positive integer n 1 and let g be either of the groups sp2n s02n or s02n l in all three cases there is a conjugacy class of maximal parabolic subgroups which has gl as its levi factor let p mn be the standard parabolic subgroup ofg in this conjugacy class then m gl let tr be a discrete series representation of m gln f and given s c let i s tr be the representation of g g f induced from tr r idet ls let i tr i 0 tr

le lemme fondamental pondere ii enonces cohomologiques dans cet article on etudie la cohomologie de la fibration de hitchin tronquee introduite dans un article precedent on etend les principaux theoremes de ngo sur la cohomologie de la partie elliptique de la fibration de hitchin comme consequence on obtient une demonstration du lemme fondamental pondere d arthur in this paper we study the cohomology of the truncated hitchin fibration which was introduced in a previous paper we extend ngo s main theorems on the cohomology of the elliptic part of the hitchin fibration as a consequence we get a proof of arthur s weighted fundamental lemma

deformations of singularities and variation of git quotients we study the deformations of the minimally elliptic surface sin gularity n16 a standard argument reduces the study of the deformations of n16 to the study of the moduli space of pairs c l consisting of a plane quintic curve and a line we construct this moduli space in two ways via the periods of k3 surfaces and by using geometric invariant theory git the git construction depends on the choice of the linearization in particular for one choice of linearization we recover the space constructed via k3 surfaces and for another we obtain the full deformation space of n16 the two spaces are related by a series of explicit flips in conclusion by using the flexibil ity given by git and the standard tools of hodge theory we obtain a good understanding of the deformations of n16