nondegenerate 2d complex euclidean superintegrable systems and algebraic varieties a classical or quantum superintegrable system is an integrable n dimensional hamiltonian system with potential that admits 2n 1 functionally independent constants of the motion polynomial in the momenta the maximum possible if the constants are all quadratic the system is second order superintegrable the kepler coulomb system is the best known example such systems have remarkable properties multi integrability and multi separability an algebra of higher order symmetries whose representation theory yields spectral information about the schrodinger operator deep connections with special functions and with qes systems for complex riemannian spaces with n 2 the structure and classification of second order superintegrable systems is complete here however we present a new and conceptually simpler approach to the classification for complex euclidean 2 space in which the possible superintegrable systems with nondegenerate potentials correspond to points on an algebraic variety specifically we determine a variety in six variables subject to two cubic and one quartic polynomial constraints each point on the variety corresponds to a superintegrable system the euclidean group e 2 acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation

superintegrability in a two dimensional space of nonconstant curvature a hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion this property has been extensively studied in the case of two dimensional spaces of constant possibly zero curvature when all the independent integrals are either quadratic or linear in the canonical momenta in this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions this is done by examining in detail one of the spaces of revolution found by g koenigs we determine that there are essentially three distinct potentials which when added to the free hamiltonian of this space have this type of superintegrability separation of variables for the associated hamilton jacobi and schrodinger equations is discussed the classical and quantum quadratic algebras associated with each of these potentials are determined

the complexity of manipulative attacks in nearly single peaked electorates many electoral control and manipulation problems which we will refer to in general as manipulative actions problems are np hard in the general case it has recently been noted that many of these problems fall into polynomial time if the electorate is single peaked i e is polarized along some axis issue however real world electorates are not truly single peaked there are usually some mavericks and so real world electorates tend merely to be nearly single peaked this paper studies the complexity of manipulative action algorithms for elections over nearly single peaked electorates we do this for many notions of nearness and for a broad range of election systems we provide instances where even one maverick jumps the manipulative action complexity up to np hardness but we also provide many instances where some number of mavericks can be tolerated without increasing the manipulative action complexity

tropicalization of the moduli space of stable maps let x be an algebraic variety and let s be a tropical variety associated to x we study the tropicalization map from the moduli space of stable maps into x to the moduli space of tropical curves in s we prove that it is a continuous map and that its image is compact and polyhedral loosely speaking when we deform algebraic curves in x the associated tropical curves in s deform continuously moreover the locus of realizable tropical curves inside the space of all tropical curves is compact and polyhedral our main tools are berkovich spaces formal models balancing conditions vanishing cycles and quantifier elimination for rigid subanalytic sets

a solution of a problem of sophus lie normal forms of two dimensional metrics admitting two projective vector fields we give a complete list of normal forms for the two dimensional metrics that admit a transitive lie pseudogroup of geodesic preserving transformations and we show that these normal forms are mutually non isometric this solves a problem posed by sophus lie

second order superintegrable systems in conformally flat spaces ii the classical two dimensional stackel transform this paper is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in conformally flat spaces here we study the stackel transform or coupling constant metamorphosis as an invertible mapping between classical superintegrable systems on different spaces through the use of this tool we derive and classify for the first time all two dimensional 2d superintegrable systems the underlying spaces are exactly those derived by koenigs in his remarkable paper giving all 2d manifolds with zero potential that admit at least three second order symmetries our derivation is very simple and quite distinct we also show that every superintegrable system is the stackel transform of a superintegrable system on a constant curvature space

on weak commutativity in groups abstract for a group g we study homological and homotopical properties of the group g g g g g 1 for g g in particular we show that the operator preserves the soluble of type f p property

completeness of superintegrability in two dimensional constant curvature spaces we classify the hamiltonians h px2 py2 v x y of all classical superintegrable systems in two dimensional complex euclidean space with two additional second order constants of the motion we similarly classify the superintegrable hamiltonians h j12 j22 j32 v x y z on the complex two sphere where x2 y2 z2 1 this is achieved in all generality using properties of the complex euclidean group and the complex orthogonal group

two dimensional metrics admitting precisely one projective vector field we give a complete list of 2 dimensional metrics that admit an essential projective vector field this solves a problem explicitly posed by sophus lie in 1882

tropical geometric interpretation of ultradiscrete singularity confinement using the interpretation of the ultradiscretization procedure as a non archimedean valuation we use results of tropical geometry to show how roots and poles manifest themselves in piece wise linear systems as points of non differentiability this will allow us to demonstrate a correspondence between singularity confinement for discrete integrable systems and ultradiscrete singularity confinement for ultradiscrete integrable systems

homogeneity and rigidity of the brookes groves invariant for the non commutative torus in 4 bieri and strebel defined a geometric invariant for finitely generated modules over the group algebras of finitely generated abelian groups they used this to define a criterion for when metabelian groups are finitely presented this invariant was further developed by bieri strebel and groves and has many interesting applications in 2 bieri and groves showed that when the group algebra is defined over a dedekind domain the complement of must be a closed rational spherical polyhedral cone in 6 7 brookes and groves defined a similar invariant for modules over the crossed product of a division ring by a free finitely generated abelian group such a crossed product is often known as the coordinate ring of the non commutative torus since in the special case where it is commutative it is the coordinate ring of an algebraic torus if in the commutative case we take the complement of and identify points that differ by a positive scalar multiple we obtain brookes and groves were unable to prove that their invariant must be a rational polyhedral cone although using the methods of 2 they do prove a weaker version of the result they show that for any finitely generated module m m must contain a rational polyhedral cone m of dimension equal to the gelfand kirillov dimension of m and moreover that the complement m m must be contained inside a rational polyhedral cone of strictly smaller dimension

pure dimension and projectivity of tropical polytopes abstract we study how geometric properties of tropical convex sets and polytopes which are of interest in many application areas manifest themselves in their algebraic structure as modules over the tropical semiring our main results establish a close connection between pure dimension of tropical convex sets and projectivity in the sense of ring theory these results lead to a geometric understanding of idempotency for tropical matrices as well as their direct interest our results suggest that there is substantial scope to apply ideas and techniques from abstract algebra in particular ring theory in tropical geometry

metrisability of two dimensional projective structures we carry out the programme of r liouville 19 to construct an explicit local obstruction to the existence of a levi civita connection within a given projective structure on a surface the obstruction is of order 5 in the components of a connection in a projective class it can be expressed as a point invariant for a second order ode whose integral curves are the geodesics of or as a weighted scalar projective invariant of the projective class if the obstruction vanishes we find the sufficient conditions for the existence of a metric in the real analytic case in the generic case they are expressed by the vanishing of two invariants of order 6 in the connection in degenerate cases the sufficient obstruction isof order at most 8

second order superintegrable systems in conformally flat spaces i two dimensional classical structure theory this paper is part of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in real or complex conformally flat spaces here we consider classical superintegrable systems with nondegenerate potentials in three dimensions we show that there exists a standard structure for such systems based on the algebra of 3 3 symmetric matrices and that the quadratic algebra always closes at order 6 we show that the spaces of truly second third fourth and sixth order constants of the motion are of dimension 6 4 21 and 56 respectively and we construct explicit bases for the fourth and sixth order constants in terms of products of the second order constants

addition theorems and the drach superintegrable systems we propose a new construction of the polynomial integrals of motion related to the addition theorems as an example we reconstruct drach systems and get some new two dimensional superintegrable stackel systems with third fifth and seventh order integrals of motion

degenerations of amoebae and berkovich spaces we prove a continuity result for the fibers of the berkovich analytification of a complex algebraic variety with respect to the maximum of the archimedean norm and the trivial norm as a consequence we obtain generalizations of a result of mikhalkin and rullgard about degenerations of amoebae onto tropical varieties

rank functions of tropical matrices abstract we consider rank functions important in tropical linear algebra the tropical rank equal to the topological dimension of the tropical linear span of the columns of a given matrix the factor rank equal to the smallest number of vectors containing these columns in their span the kapranov rank related to problems of tropical algebraic geometry and the determinantal and gondran minoux ranks which generalize the classical linear algebraic notion of rank we are interested in studying the arithmetic properties of the rank functions their mutual behavior and the related computational problems we discuss different open problems related to rank functions of tropical matrices and illustrate them by the number of examples

invariant classification of second order conformally flat superintegrable systems in this paper we continue the work of kalnins et al in classifying all second order conformally superintegrable laplace type systems over conformally flat spaces using tools from algebraic geometry and classical invariant theory the results obtained show through stackel equivalence that the list of known nondegenerate superintegrable systems over three dimensional conformally flat spaces is complete in particular a seven dimensional manifold is determined such that each point corresponds to a conformal class of superintegrable systems this manifold is foliated by the nonlinear action of the conformal group in three dimensions two systems lie in the same conformal class if and only if they lie in the same leaf of the foliation this foliation is explicitly described using algebraic varieties formed from representations of the conformal group the proof of these results rely heavily on grobner basis calculations using the computer algebra software packages maple and singular

a criterion for compatibility of conformal and projective structures in a space time m a conformal structure is defined by the distribution of light cones geodesics are traced by freely falling particles and the collection of all unparameterized geodesics determines the projective structure of m the article contains a formulation of the necessary and sufficient conditions for these structures to be compatible i e to come from a metric tensor which is then unique up to a constant factor the theorem applies to all dimensions and signatures

superintegrable systems in darboux spaces almost all research on superintegrable potentials concerns spaces of constant curvature in this paper we find by exhaustive calculation all superintegrable potentials in the four darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta in addition to the hamiltonian these are two dimensional spaces of nonconstant curvature it turns out that all of these potentials are equivalent to superintegrable potentials in complex euclidean 2 space or on the complex 2 sphere via coupling constant metamorphosis or equivalently via stackel multiplier transformations we present a table of the results

the σ2 conjecture for metabelian groups the general case abstract the bieri neumann strebel invariant m g of a group g is a certain subset of a sphere that contains information about finiteness properties of subgroups of g in case of a metabelian group g the set 1 g completely characterizes finite presentability and it is conjectured that it also contains complete information about the higher finiteness properties fpm conjecture the m conjecture states how the higher invariants are obtained from 1 g in this paper we prove the 2 conjecture

metrisability of projective surfaces and pseudo holomorphic curves we show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo holomorphic curves a projective structure mathfrak p and a volume form sigma on an oriented surface m equip the total space of a certain disk bundle z rightarrow m with a pair j mathfrak p mathfrak j mathfrak p sigma of almost complex structures a conformal structure on m corresponds to a section of z rightarrow m and mathfrak p is metrisable by the metric g if and only if g m rightarrow z is a pseudo holomorphic curve with respect to j mathfrak p and mathfrak j mathfrak p da g