uncertainty measures decomposability and admissibility abstract uncertainty measures are defined in a general manner as morphism between certain posets which permits a unified treatment of different measures for classical and for fuzzy events admissible measures are discussed in a decision theoretical notation making use of quasilinear entropies connections and differences to decomposable measures are illustrated

homological mirror symmetry for calabi yau hypersurfaces in projective space we prove homological mirror symmetry for a smooth d dimensional calabi yau hypersurface in projective space for any d 3 for example d 3 is the quintic threefold the main techniques involved in the proof are the construction of an immersed lagrangian sphere in the d dimensional pair of pants the introduction of the relative fukaya category and an understanding of its grading structure a description of the behaviour of this category with respect to branched covers via an orbifold fukaya cate gory a morse bott model for the relative fukaya category that allows one to make explicit computations and the introduction of certain graded categories of matrix factorizations mirror to the relative fukaya category

on the erdős distinct distances problem in the plane in this paper we prove that a set of n points in r 2 has at least c n log n distinct distances thus obtaining the sharp exponent in a problem of erd os we follow the setup of elekes and sharir which in the spirit of the erlangen program allows us to study the problem in the group of rigid motions of the plane this converts the problem to one of point line incidences in space we introduce two new ideas in our proof in order to control points where many lines are incident we create a cell decomposition using the polynomial ham sandwich theorem this creates a dichotomy either most of the points are in the interiors of the cells in which case we immediately get sharp results or alternatively the points lie on the walls of the cells in which case they are in the zero set of a polynomial of suprisingly low degree and we may apply the algebraic method in order to control points incident to only two lines we use the ecnode polynomial of the rev george salmon to conclude that most of the lines lie on a ruled surface then we use the geometry of ruled surfaces to complete the proof

uncertainty measures problems concerning additivity additivity of an uncertainty measure on an mv algebra has a clear meaning if the divisibility is dropped we come up to a so called girard algebra there we discuss strong resp weak additivity based on so called divisible disjoint unions resp on additivity for all sub mv algebras we obtain a description of those extensions from additive measures on an mv algebra to the canonical girard algebra extension of pairs which are strongly additive and valuation measures finally we prove the non existence of strongly additive measure extensions if the underlying mv algebra is a finite chain with more than two non trivial elements

extensions of states on mv quantales abstract we consider the lattice of the ordered right continuous families of elements of a dual quantale with the index set 0 1 called its rc extension and the lattice of the ordered n 1 tuples with n 1 of elements of this dual quantale its n extension equipped with a dual quantale structure that generalizes the canonical extension proposed by hohle and weber of particular interest is the existence of states on n extensions of finite mv chains we generalize the recent results of weber by incorporating effectible girard quantales which are introduced by the author

measure free conditioning and extensions of additive measures on finite mv algebras using the well known representation of any finite mv algebra as a product of finite mv chains as factors we obtain a representation of its canonical extension as a girard algebra product of the canonical extensions of the mv chain factors based on this representation and using the results from our last paper we characterize the additive measures on any finite mv algebra resp the weakly and the strongly additive measures on its canonical girard algebra extension and that as convex combinations of the corresponding measures on the respective factors after that we apply the results to measure free defined conditional events which for this reason are considered as elements of the canonical extension of a finite mv algebra of events this definition for conditional events is naturally equivalent to the well known definition as lattice intervals of events here the weak additivity is considered as a reasonable notion for a global additivity property of a measure extension these extensions are essentially different from the measure extension which has a form analogous to the classical conditional probability but which is only levelwise additive moreover the measure free approach of this paper is completely different from the classical approach in which conditional measures of events given a fixed condition event are considered based on additive measures states of events without having defined conditional events

weighted restriction estimates using polynomial partitioning we use the polynomial partitioning method of guth j amer math soc 29 2016 371 413 to prove weighted fourier restriction estimates in r3 with exponents p that range between 3 and 3 25 depending on the weight as a corollary to our main theorem we obtain new non weighted local and global restriction estimates for compact c surfaces s r3 with strictly positive second fundamental form for example we establish the global restriction estimate ef lp r3 f lq s in the full conjectured range of exponents for p 3 25 up to the sharp line and the global restriction estimate ef lp f l2 s for p 3 and certain sets r3 of infinite lebesgue measure as a corollary to our main theorem we also obtain new results on the decay of spherical means of fourier transforms of positive compactly supported measures on r3 with finite dimensional energies

conditioning on mv algebras and additive measures i abstract we present a lattice ordered semigroup approach for the foundation of conditional events which covers the special situations where the underlying unconditional events are boolean or fuzzy respectively our proposal is quite different from other ring theoretical approaches the problem of extending additivity of uncertainty measures from unconditional to conditional events will be discussed

the weak type l1 convergence of eigenfunction expansions for pseudodifferential operators soit m une variete compacte connexe lisse sans bord de dimension n 2 soit p d un operateur pseudodifferentiel elliptique d ordre 1 sur m qui est auto adjoint par rapport a une densite positive lisse dx on demontre la convergence faible de type l 1 pour les developpements en fonctions propres

conditional probability on mv algebras an appropriate definition of a conditional probability on an mv algebra was an open problem mentioned in riecan and mundici handbook of measure theory north holland amsterdam 2002 we propose some concept of conditional probability state on a complete mv algebra with product its basic properties will be proven and it will also be demonstrated that the conditional probability in classical probability theory is a special case of this definition moreover the paper contains also a discussion of the interpretation of fuzzy sets and conditioning in fuzzy probability theory

extremizability of fourier restriction to the paraboloid abstract in this article we prove that nearly all valid scale invariant fourier restriction inequalities for the paraboloid in r 1 d have extremizers and that l p normalized extremizing sequences are precompact modulo symmetries this result had previously been established for the case q 2 in the range where the boundedness of the restriction operator is still an open question our result is conditional on improvements toward the restriction conjecture

syz mirror symmetry for toric calabi yau manifolds we investigate mirror symmetry for toric calabi yau manifolds from the perspective of the syz conjecture starting with a non toric special lagrangian torus fibration on a toric calabi yau manifold x we construct a complex manifold check x using t duality modified by quantum corrections these corrections are encoded by fourier transforms of generating functions of certain open gromov witten invariants we conjecture that this complex manifold check x which belongs to the hori iqbal vafa mirror family is inherently written in canonical flat coordinates in particular we obtain an enumerative meaning for the inverse mirror maps and this gives a geometric reason for why their taylor series expansions in terms of the kahler parameters of x have integral coefficients applying the results in a formula equating open and closed gromov witten invariants and its applications to mirror symmetry to appear in pacific j math and a relation for gromov witten invariants of local calabi yau threefolds to appear in math res lett we compute the open gromov witten invariants in terms of local bps invariants and give evidences of our conjecture for several 3 dimensional examples including k mathbb p 2 and k mathbb p 1

gross fibrations syz mirror symmetry and open gromov witten invariants for toric calabi yau orbifolds given a toric calabi yau orbifold x whose underlying toric variety is semi projective we construct and study a non toric lagrangian torus bration on x which we call the gross bration we apply the strominger yau zaslow recipe to the gross bration of a toric modication of x to construct its instanton corrected mirror where the instanton corrections come from genus 0 open orbifold gromov witten invariants which are virtual counts of holomorphic orbi disks inx bounded by bers of the gross bration we explicitly evaluate all these invariants by rst proving an open closed equality and then employing the toric mirror theorem for suitable toric parital compactications of x our calculations are then applied to 1 prove a conjecture of gross siebert on a relation between genus 0 open orbifold gromov witten invariants and mirror maps ofx this is called the open mirror theorem which leads to an enumerative meaning of mirror maps and 2 demonstrate how open orbifold gromov witten invariants for toric calabi yau orb ifolds change under toric crepant resolutions this is an open analogue of ruan s crepant resolution conjecture

a bilinear approach to the restriction and kakeya conjectures the purpose of this paper is to investigate bilinear variants of the restriction and kakeya conjectures to relate them to the standard formulations of these conjectures and to give applications of this bilinear approach to existing conjectures the methods used are based on several observations and results of bourgain see 2 6 together with some refinements by moyua vargas and vega 17 18 this paper is organized as follows in the first section we discuss bilinear restriction estimates and show how one can pass back and forth between these estimates and the standard restriction estimates we also generalize the 12 7 bilinear restriction estimate of 18 to higher dimensions in the second section we give analogues of the above results for the kakeya operator in particular we give a bilinear improvement to wolff s kakeya theorem in arbitrary dimension in the third section we give applications of these bilinear estimates in three dimensions for example we are able to improve the 42 11 exponent in wolff s restriction theorem to 34 9 we are also able to prove a sharp lp lq restriction theorem which improves on the classical l2 l4 tomas stein theorem and also give some concrete progress on a bilinear restriction conjecture of klainerman and machedon we also give a non bilinear approach to these estimates which gives weaker results but is more direct and probably has a wider range of application

a complete characterization of all weakly additive measures and of all valuations on the canonical extension of any finite mv chain we consider extensions of the unique additive measure on a finite mv chain to uncertainty measures on its canonical girard algebra extension if the underlying mv chain has more than two non trivial elements in a previous paper we have proved the non existence of strongly additive measure extensions where strong additivity is defined as additivity not for all disjoint unions but only restricted to the so called divisible disjoint unions this negative result motivates to look for weakly additive measure extensions which are defined to be additive only on all mv subalgebras of the canonical girard algebra extension we obtain a characterization of all such mv subalgebras which are in fact mv chains and using this result we come up to a characterization of all weakly additive measure extensions moreover we characterize also all the valuation extensions and we prove the incompatibility of these two types of measure extensions except for underlying mv chains with one or two or four non trivial elements

homological mirror symmetry for toric del pezzo surfaces we prove the homological mirror conjecture for toric del pezzo surfaces in this case the mirror object is a regular function on an algebraic torus open image in new window we show that the derived fukaya category of this mirror coincides with the derived category of coherent sheaves on the original manifold

categorical mirror symmetry on cohomology for a complex genus 2 curve abstract motivated by observations in physics mirror symmetry is the concept that certain manifolds come in pairs x and y such that the complex geometry on x mirrors the symplectic geometry on y it allows one to deduce symplectic information about y from known complex properties of x strominger yau zaslow 61 described how such pairs arise geometrically as torus fibrations with the same base and related fibers known as syz mirror symmetry kontsevich 43 conjectured that a complex invariant on x the bounded derived category of coherent sheaves should be equivalent to a symplectic invariant of y the fukaya category see 9 29 49 1 this is known as homological mirror symmetry in this project we first use the construction of generalized syz mirrors for hypersurfaces in toric varieties following abouzaid auroux katzarkov 6 in order to obtain x and y as manifolds the complex manifold is the genus 2 curve 2 so of general type c 1 0 as a hypersurface in its jacobian torus its generalized syz mirror is a landau ginzburg model y v 0 equipped with a holomorphic function v 0 y c which we put the structure of a symplectic fibration on we then describe an embedding of a full subcategory of d b c o h 2 into a cohomological fukaya seidel category of y as a symplectic fibration while our fibration is one of the first nonexact non lefschetz fibrations to be equipped with a fukaya category the main geometric idea in defining it is the same as in seidel s construction for fukaya categories of lefschetz fibrations in 55 and in abouzaid seidel 3

restriction estimates using polynomial partitioning ii we improve the estimates in the restriction problem in dimension n geqslant 4 to do so we establish a weak version of a k linear restriction estimate for any k the exponents in this weak k linear estimate are sharp for all k and n