a simple homotopy method for determining all isolated solutions to polynomial systems a new homotopy method for solving systems of polynomial equations is presented the homotopy equation is extremely simple it is linear with respect to the homotopy parameter and only one auxiliary parameter is needed to regularize the problem within some limits an arbitrary starting problem can be chosen as long as its solution set is known no restrictions on the polynomial systems are made a few numerical tests are reported which show the influence of the auxiliary parameter resp the starting problem upon the computa tional cost of the method

multiple objective programming with polynomial objectives and constraints abstract an approach to solving multiple objective programming problems with polynomial objectives and polynomial constraints is developed the notions of pre efficient solution and local efficient solution are investigated relative to the single objective problem of benson illustrative examples are presented in which the solutions are obtained by means of homotopy continuation

full factors and co amenable inclusions we show that if m is a full factor and n subset m is a co amenable subfactor with expectation then n is also full this answers a question of popa from 1986 we also generalize a theorem of tomatsu by showing that if m is a full factor and sigma g curvearrowright m is an outer action of a compact group g then sigma is automatically minimal and m g is a full factor which has w spectral gap in m finally in the appendix we give a proof of the fact that several natural notions of co amenability for an inclusion n subset m of von neumann algebras are equivalent thus closing the cycle of implications given in anantharaman delaroche s paper in 1995

a valued semicircular systems abstract to a von neurnann algebra a and a set of linear maps ij a a i j i such that a ij ij i a a b l2 i is normal and completely positive we associate a von neumann algebra a this von neumann algebra is generated by a and an a valued semicircular system xi i i associated to in many cases there is a faithful conditional expectation e a a if a is tracial then under certain assumptions on a also has a trace one can think of the construction a as an analogue of a crossed product construction we show that most known algebras arising in free probability theory can be obtained from the complex field by iterating the construction of a particular interest are free krieger algebras which by analogy with crossed products and ordinary krieger factors are defined to be algebras of the form l 0 1 the cores of free araki woods factors are free krieger algebras we study the free krieger algebras and as a result obtain several non isomorphism results for free araki woods factors as another source of classification results for free araki woods factors we compute the invariant of connes for free products of von neumann algebras this computation generalizes earlier work on computation of t s and sd invariants for free product algebras

symmetric homotopy construction abstract homotopy continuation methods can be applied to compute all finite solutions to a given polynomial system computations will be performed more efficiently if the symmetric structure of the system can be exploited this paper presents the construction of a symmetric homotopy using this homotopy only the paths according to the generating solutions have to be traced during continuation

l 2 homology for von neumann algebras the aim of this paper is to introduce a notion of l homology in the context of von neumann algebras finding a suitable co homology theory for von neumann algebras has been a dream for several generations see kr71a kr71b jkr72 ss95 and references therein one s hope is to have a powerful invariant to distinguish von neumann algebras unfortunately little positive is known about the kadison ringrose cohomology h b m m except that it vanishes in many cases furthermore there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von neumann algebra our interest in developing an l cohomology theory was revived by the introduction of lcohomology invariants in the field of ergodic equivalence relations in the paper of gaboriau gab02 his results in particular imply that l betti numbers 2 i of a discrete group are the same for measure equivalent groups i e for groups that can generate isomorphic ergodic measure preserving equivalence relations parallels between the worlds of von neumann algebras and measurable equivalence relations have been noted for a long time starting with the parallel between the work of murray and von neumann mvn and that of h dye dy thus there is hope that an invariant of a group that survives measure equivalence will survive also von neumann algebra equivalence i e will be an invariant of the von neumann algebra of the group the original motivation for our construction comes from the well understood analogy between the theory of ii1 factors and that of discrete groups based on the theory of correspondences con con94 this analogy has been remarkably efficient to transpose analytic properties such as amenability or property t from the group context to the factor context con80 cj and more recently in the breakthrough work of popa popa con03 we use the theory of correspondences together with the algebraic description of l betti numbers given by luck his definition involves the computation of the algebraic group homology with coefficients in the group von neumann algebra following the guiding idea that the category of bimodules over a von neumann algebra is the analogue of the category of modules over a group we are led to the following algebraic definition of l homology of a von neumann algebra m

bezout number calculations for multi homogeneous polynomial systems the bezout number of a system of multi homogeneous polynomial equations is the largest number of nonsingular solutions such a system can have and it is also the number of solution paths used to compute all geometrically isolated solutions of the system using multi homogeneous polynomial continuation any polynomial system can be homogenized in a variety of ways typically yielding different bezout numbers this paper presents an efficient algorithm for computing bezout numbers and describes a procedure for finding the minimal bezout number over all possible homogenizations

a hybrid polynomial system solving method for mixed trigonometric polynomial systems mixed trigonometric polynomial systems arise in many fields of science and engineering commonly this class of systems is transformed into polynomial systems by variable substituting and adding some quadratic equations and then solved by some polynomial system solving method in this paper by exploiting the special structure of the additional quadratic equations an efficient hybrid method for solving polynomial systems coming from mixed trigonometric polynomial systems is presented it combines the homotopy method in which the homotopy is a combination of coefficient parameter homotopy and the random product homotopy with decomposition variable substitution and reduction techniques numerical tests are given to show its effectiveness and it is applied to solve a practical problem

numerical linear algebra aspects of globally convergent homotopy methods probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one these methods are theoretically powerful and if constructed and implemented properly are robust numerically stable accurate and practical the concomitant numerical linear algebra problems deal with rectangular matrices and good algorithms require a delicate balance not always achieved of accuracy robustness and efficiency in both space and time the author s experience with globally convergent homotopy algorithms is surveyed here and some of the linear algebra difficulties for dense and sparse problems are discussed

compressions of free products of von neumann algebras a reduction formula for compressions of von neumann algebra ii 1 factors arising as free products is proved this shows that the fundamental group is bf r for some such algebras additionally by taking a sort of free product with an unbounded semicircular element continuous one parameter groups of trace scaling automorphisms on ii infty factors are constructed this produces type iii 1 factors with core mathcal m otimes b mathcal h where mathcal m can be a full ii 1 factor without the haagerup approximation property

a semi finite algebra associated to a subfactor planar algebra we canonically associate to any planar algebra two type ii factors m the subfactors constructed previously by the authors in guionnet et al 2010 6 are isomorphic to compressions of m to finite projections we show that each m is isomorphic to an amalgamated free product of type i von neumann algebras with amalgamation over a fixed discrete type i von neumann subalgebra in the finite depth case existing results in the literature imply that m m is the amplification a free group factor on a finite number of generators as an application we show that the factors mj constructed in guionnet et al in press 6 are isomorphic to interpolated free group factors l f rj rj 1 2 2j 1 i where 2 is the index of the planar algebra and i is its global index other applications include computations of laws of jones wenzl projections

approximation properties and absence of cartan subalgebra for free araki woods factors abstract we show that all the free araki woods factors h r u t have the complete metric approximation property using ozawa popa s techniques we then prove that every nonamenable subfactor n h r u t which is the range of a normal conditional expectation has no cartan subalgebra we finally deduce that the type iii1 factors constructed by connes in the 70s can never be isomorphic to any free araki woods factor which answers a question of shlyakhtenko and vaes

a new start system for solving deficient polynomial systems using continuation we are interested in all finite solutions of polynomial systems the start system normally used in the homotopy causes divergent continuation paths when applied to a deficient system our aim is to avoid this wasted computation by constructing a new start system the start system presented in this paper respects the m homogeneous structure of the system we want to solve

free products of measured equivalence relations abstract for given countable standard equivalence relations over the same unit space we construct their free product which is a measured groupoid related topics such as measures and 2 cocyles on the free product needed to deal with groupoid von neumann algebras are also discussed

an efficient start system for multi homogeneous polynomial continuation when solving a system of polynomial equations using homotopy continuation the number of solution paths can often be significantly reduced by casting the equations in multi homogeneous form this requires a multi homogeneous start system having a full set of nonsingular solutions that can be easily computed we give a general form of such a start system that works for any multi homogeneous structure and that can be used efficiently in continuation

some minimization problems for the free analogue of the fisher information abstract we consider the free non commutative analogue introduced by d voiculescu of the concept of fisher information for random variables we determine the minimal possible value of a a if a is a non commutative random variable subject to the constraint that the distribution of a a is prescribed more generally we obtain the minimal possible value of a ij a ij 1 i j d if a ij 1 i j d is a family of non commutative random variables such that the distribution of a a is prescribed where a is the matrix a ij d i j 1 the d d generalization is obtained from the case d 1 via a result of independent interest concerning the minimal value of a ij a ij 1 i j d when the matrix a a ij d i j 1 and its adjoint have a given joint distribution a version of this result describes the minimal value of b ij 1 i j d when the matrix b b ij d i j 1 is selfadjoint and has a given distribution we then show how the minimization results obtained for lead to maximization results concerning the free entropy also defined by voiculescu

the norm of polynomials in large random and deterministic matrices let bf x n x 1 n ldots x p n be a family of n n independent normalized random matrices from the gaussian unitary ensemble we state sufficient conditions on matrices bf y n y 1 n ldots y q n possibly random but independent of xn for which the operator norm of p bf x n bf y n bf y n converges almost surely for all polynomials p limits are described by operator norms of objects from free probability theory taking advantage of the choice of the matrices yn and of the polynomials p we get for a large class of matrices the no eigenvalues outside a neighborhood of the limiting spectrum phenomena we give examples of diagonal matrices yn for which the convergence holds convergence of the operator norm is shown to hold for block matrices even with rectangular gaussian blocks a situation including non white wishart matrices and some matrices encountered in mimo systems

finding all solutions to a system of polynomial equations given a polynomial equation of degree d over the complex domain the fundamen tal theorem of algebra tells us that there are d solutions assuming that the solutions are counted by multiplicity these solutions can be approximated by deforming a standard n th degree equation into the given equation and following the solutions through the deformation this is called the homotopy method the fundamental theorem of algebra can be proved by the same technique in this paper we extend these results and methods to a system of n polynomial equations in n complex variables we show that the number of solutions to such a system is the product of the degrees of the equations assuming that infinite solutions are included and solutions are counted by multiplicity the proof is based on a homotopy or deformation from a standard system of equations with the same degrees and known solutions this homotopy provides a computational method of approximating all solutions computational results demonstrating the feasibility of this method are also presented

combinatorial optimization in nash games abstract games in which at least one player must solve a combinatorial optimization problem are studied after a suitable transformation some cases may be solved by consideration of a polynomial system of equations by finding all the solutions of this system one compuutes a all the nash equiblibria and the pre equilibrium set which may be of independent interest the question of a lack of equilibrium is also resolved for these games

on the ground states of the bernasconi model the ground states of the bernasconi model are binary sequences of length n with low autocorrelations we introduce the notion of perfect sequences binary sequences with one valued off peak correlations of minimum amount if they exist they are ground states using results from the mathematical theory of cyclic difference sets we specify all values of n for which perfect sequences exist and how to construct them for other values of n we investigate almost perfect sequences i e sequences with two valued off peak correlations of minimum amount numerical and analytical results support the conjecture that almost perfect sequences exist for all values of n but that they are not always ground states we present a construction for low energy configurations that works if n is the product of two odd primes

a new construction of factors of type iii1 abstract we give in this paper a new construction of factors of type iii 1 under certain assumptions we can thanks to a result by popa give a complete classification for this family of factors although these factors are never full we can nevertheless in many cases compute connes invariant we obtain a new example of an uncountable family of pairwise non isomorphic factors of type iii 1 with the same invariant

asymptotic structure of free araki woods factors the purpose of this paper is to investigate the structure of shlyakhtenko s free araki woods factors using the framework of ultraproduct von neumann algebras we first prove that all the free araki woods factors gamma h mathbb r u t prime prime are omega solid in the following sense for every von neumann subalgebra q subset gamma h mathbb r u t prime prime that is the range of a faithful normal conditional expectation and such that the relative commutant q cap m omega is diffuse we have that q is amenable next we prove that the continuous cores of the free araki woods factors gamma h mathbb r u t prime prime associated with mixing orthogonal representations u mathbb r rightarrow mathcal o h mathbb r are omega solid type mathrm ii infty factors finally when the orthogonal representation u mathbb r rightarrow mathcal o h mathbb r is weakly mixing we prove a dichotomy result for all the von neumann subalgebras q subset gamma h mathbb r u t prime prime that are globally invariant under the modular automorphism group sigma t varphi u of the free quasi free state varphi u

a homotopy for solving polynomial systems this paper presents a continuation method for finding all solutions to polynomial systems it features a simpler homotopy than has been previously published

on the classification of full factors of type iii we introduce a new invariant s m for type iii factors m with no almost periodic weights we compute this invariant for certain free araki woods factors we show that connes invariant r cannot distinguish all isomorphism classes of free araki woods factors we show that there exists a continuum of mutually non isomorphic free araki woods factors each without almost periodic weights

a globally convergent parallel algorithm for zeros of polynomial systems certain classes of nonlinear systems of equations such as polynomial systems have properties that make them particularly amenable to solution on distributed computing systems some algorithms considered unfavorably on a single processor serial computer may be excellent on a distributed system this paper considers the solution of polynomial systems of equations via a globally convergent homotopy algorithm on a hypercube some computational results are reported

strong solidity of free araki woods factors we show that shlyakhtenko s free araki woods factors are strongly solid meaning that for any diffuse amenable von neumann subalgebra that is the range of a normal conditional expectation the normalizer remains amenable this provides the first class of nonamenable strongly solid type iii factors

errata computing all solutions to polynomial systems using homotopy continuation in a previous paper we described a new method for defining homotopies for finding all solutions to polynomial systems a major feature of this new approach is that the start system for the homotopy need not be a random or generic system also homotopy paths are strictly increasing in the homotopy parameter in this paper we establish some principles of implementation and report on the performance of programs that use the new homotopies a feature of our implementation is that we eliminate divergent paths entirely we include performance statistics for homotopies derived from more traditional approaches for comparison generally the new approach is faster and more reliable