a method for computing lowest eigenvalues of symmetric polynomial differential operators by semidefinite programming abstract a method for computing global minima of real multivariate polynomials based on semidefinite programming was developed by n z shor j b lasserre and p a parrilo the aim of this article is to extend a variant of their method to noncommutative symmetric polynomials in variables x and y satisfying y x x y 1 and x x y y global minima of such polynomials are defined and showed to be equal to minima of the spectra of the corresponding differential operators we also discuss how to exploit sparsity and symmetry several numerical experiments are included the last section explains how our theory fits into the framework of noncommutative real algebraic geometry

lower dimensional defect transitions in self avoiding walks and percolation the author studies self avoiding walks and percolation on d dimensional lattices focusing on the existence of transitions associated with a defect of dimensionality d or 2 in contrast to percolation for which they are absent if d 1 an apparent contradiction with scaling arguments is resolved a simple renormalisation group approximation yields phase diagrams consistent with these results and provides numerical estimates for crossover exponents and the splitting between special and homogeneous critical points

adsorption and collapse of self avoiding walks in three dimensions a monte carlo study we consider self avoiding walks on the simple cubic lattice with a unit degree vertex at the origin and confined to the half space in addition the walks interact with the plane z 0 and have a vertex vertex interaction within the walk so that the walk can adsorb at the surface z 0 and also undergo collapse we investigate the interaction between collapse and adsorption using monte carlo methods and explore the form of the phase diagram

exact evaluation of the collapse phase boundary for two dimensional directed polymers the phase boundary of the collapsed phase for a directed polymer on a square lattice in the presence of an attractive wall and monomer monomer interactions is calculated exactly using transfer matrix techniques the position of the multicritical point is also identified

non commutative moment problems we are concerned with two problems called the generalized moment problem gmp and the quantum moment problem qmp both problems can be considered as non commutative versions of the classical moment problem et g be a connected lie group and 7b e g the universal enveloping algebra of the lie algebra of g suppose f is a given linear functional on 7b e g and is a fixed unitary representation of g we then ask gmp when do there exist a unitary representation u of g and a cyclic c vector for u such that f x x 7b e g qmp when is f of the form f x trdv x x 7b e g where is an operator such that dv x is of trace class for every x 7b e g

strong majorization in a free algebra we study in the spirit of modern real algebra the interplay between left ideals of the free algebra mathbb f with n generators and their suitably defined zero sets and similarly between quadratic submodules of mathbb f and their positivity sets

composite models of polygons a composite polygon is composed of a lattice polygon in the square lattice which contains in its interior an internal structure which may also be a lattice polygon or a lattice tree or a lattice animal or a lattice disc or a collection of these the properties of composite polygons are considered in this manuscript in particular i shall consider the growth constants and generating functions of these models as well as the statistical mechanics of interacting models of composite polygons it is shown that there is an adsorption transition of the internal structure on the containing polygon and a transition which corresponds to the inflation of the containing polygon by the internal structure

critical exponents hyperscaling and universal amplitude ratios for two and three dimensional self avoiding walks we make a high precision monte carlo study of two and three dimensional self avoiding walks saws of length up to 80 000 steps using the pivot algorithm and the karp luby algorithm we study the critical exponents v and 2 triangle sub 4 gamma as well as several universal amplitude ratios in particular we make an extremely sensitive test of the hyperscaling relation dv 2 triangle sub 4 gamma in two dimensions we confirm the predicted exponent v 3 4 and the hyperscaling relation we estimate the universal ratios 0 14026 0 00007 0 43961 0 00034 and psi 0 66296 0 00043 68 confidence limits in three dimensions we estimate v 0 5877 0 0006 with a correction to scaling exponent triangle sub 1 0 56 0 03 subjective 68 confidence limits this value for v agrees excellently with the field theoretic renormalization group prediction but there is some discrepancy for triangle 1 earlier monte carlo estimates of v which were approximately 0 592 are now seen to be biased by corrections to scaling we estimate the universal ratios 0 1599 0 0002 and psi 0 2471 0 0003 since psi 0 hyperscaling holds the approach to psi is from above contrary to the prediction of the two parameter renormalization group theory we critically reexamine this theory and explain where the error lies in an appendix we prove rigorously modulo some standard scalingmore assumptions the hyperscaling relation dv 2 triangle sub 4 gamma for two dimensional saws less

chain polymers near an adsorbing surface we present improved estimates of critical exponents for 3d polymers near a plane adsorbing surface the improvements were possible by simulating rather long chains up to n 2000 by means of a new algorithm apart from minor but significant adjustments in the location of the tricritical point and in the various gamma exponents our main result is that the crossover exponent for critical adsorption is compatible with phi 1 2 this is much lower than previous estimates and suggests that this exponent might be superuniversal

surface critical exponents for models of polymer collapse and adsorption the universality of the theta and theta points the surface critical exponents are estimated both in ordinary and in special regimes from exact series expansions of up to 28 terms for a model of the theta point they are found to be in agreement with those derived for the theta point confirming the conjecture that the theta and theta points lie in the same universality class some essential features of the phase diagram for a self interacting self avoiding polymer in solution are also calculated to a higher degree of accuracy than previously obtained

multiple markov chain monte carlo study of adsorbing self avoiding walks in two and in three dimensions a self avoiding walk adsorbing on a line in the square lattice and on a plane in the cubic lattice is studied numerically as a model of an adsorbing polymer in dilute solution the walk is simulated by a multiple markov chain monte carlo implementation of the pivot algorithm for self avoiding walks vertices in the walk that are visits in the adsorbing line or plane are weighted by e t he critical value of where the walk adsorbs on the adsorbing line or adsorbing plane is determined by considering energy ratios and approximations to the free energy we determine that the critical values of are c 0 565 0 010 in the square lattice 0 288 0 020 in the cubic lattice in addition the value of the crossover exponent is determined 0 501 0 015 in the square lattice 0 5005 0 0036 in the cubic lattice metric quantities including the mean square radius of gyration are also considered as well as rescaling of the specific heat and free energy as the critical point is approached

self avoiding walks interacting with a surface discusses two models of polymer adsorption in one model a self avoiding walk on a d dimensional lattice interacts with a d 1 dimensional hyperplane in the other model this walk must also lie in or on one side of this hyperplane both models exhibit non analytic behaviour corresponding to a phase transition but these phase transitions do not occur at the same point the authors give numerical estimates of the locations of these transitions

interacting self avoiding walks and polygons in three dimensions self interacting walks and polygons on the simple cubic lattice undergo a collapse transition at the point we consider self avoiding walks and polygons with an additional interaction between pairs of vertices which are unit distance apart but not joined by an edge of the walk or polygon we prove that these walks and polygons have the same limiting free energy if the interactions between nearest neighbour vertices are repulsive the attractive interaction regime is investigated using monte carlo methods and we find evidence that the limiting free energies are also equal here in particular this means that these models have the same point in the asymptotic limit the dimensions and shapes of walks and polygons are also examined as a function of the interaction strength

a real nullstellensatz for free modules abstract let a be the algebra of all n n matrices with entries from r x 1 x d and let g 1 g m f a we will show that f a v 0 for every a r d and v r n such that g i a v 0 for all i if and only if f belongs to the smallest real left ideal of a which contains g 1 g m here a left ideal j of a is real if for every h 1 h k a such that h 1 t h 1 h k t h k j j t we have that h 1 h k j we call this result the one sided real nullstellensatz for matrix polynomials we first prove by induction on n that it holds when g 1 g m f have zeros everywhere except in the first row this auxiliary result can be formulated as a real nullstellensatz for the free module r x 1 x d n

q positive definiteness and related operators abstract we are in progress of extending the family of q deformed operators considered in the previous papers by joining to them q subnormal as well as q formally subnormal ones it turns out that q positive definiteness a notion generalizing halmos standard positive definiteness of bounded subnormal operators is likewise central for our new scheme

minimum width double strip and parallelogram annulus abstract in this paper we study the problem of computing a minimum width double strip or parallelogram annulus that encloses a given set of n points in the plane a double strip is a closed region in the plane whose boundary consists of four parallel lines and a parallelogram annulus is a closed region between two edge parallel parallelograms we present several first algorithms for these problems among them are o n 2 and o n 3 log n time algorithms that compute a minimum width double strip and parallelogram annulus respectively when their orientations can be freely chosen

a strict positivstellensatz for the weyl algebra let c be an element of the weyl algebra open image in new window which is given by a strictly positive operator in the schrodinger representation it is shown that under some conditions there exist certain elements b1 bd from open image in new window such that dj 1bjcb j is a finite sum of squares

a strict positivstellensatz for enveloping algebras let g be a connected and simply connected real lie group with lie algebra open image in new window semialgebraic subsets of the unitary dual of g are defined and a strict positivstellensatz for positive elements of the universal enveloping algebra open image in new window of open image in new window is proved

approximating extent measures of points we present a general technique for approximating various descriptors of the extent of a set p of n points in rd when the dimension d is an arbitrary fixed constant for a given extent measure and a parameter e 0 it computes in time o n 1 eo 1 a subset q p of size 1 eo 1 with the property that 1 e p q p the specific applications of our technique include e approximation algorithms for i computing diameter width and smallest bounding box ball and cylinder of p ii maintaining all the previous measures for a set of moving points and iii fitting spheres and cylinders through a point set p our algorithms are considerably simpler and faster in many cases than previously known algorithms

adsorption and collapse of self avoiding walks at a defect plane we consider self avoiding walks and polygons on the simple cubic lattice with a vertex vertex interaction i e a contact interaction which also interact with a defect plane z 0 the walks or polygons can undergo a collapse transition and can also adsorb at the defect plane for the polygon case we prove that the limiting free energy exists and show that there is an adsorption transition for each finite value of the vertex vertex interaction we use monte carlo methods to investigate the form of the phase diagram for the walk problem

adsorbing and collapsing trees self interacting branched polymers interacting with a wall can be modelled by lattice trees in a half space with a nearest neighbour contact fugacity and a visit fugacity conjugate to the number of visits the tree makes to the wall which is the boundary of the half space we show that there is a limiting free energy in this model and that it is a non analytic function of the visit fugacity for every finite and fixed value of the contact fugacity this implies the existence of an adsorption transition in this model and there is a critical curve in the phase diagram which separates the phase of desorbed trees from a phase of adsorbed trees moreover we show that for all finite values of the contact fugacity and the adsorption transition occurs at a strictly attractive value of the interaction between the tree and the wall

new monte carlo algorithms for interacting self avoiding walks in this paper we present two new monte carlo algorithms for simulating interacting self avoiding walks these algorithms are built on the berretti sokal and reptation algorithms for simulating non interacting self avoiding walks respectively by adding a metropolis step we enhance each algorithm by introducing special local moves in addition the reptation algorithm is further enhanced by running the simulations as a multiple markov chain each of these algorithms are studied here through an analysis of their autocorrelation properties in the case of the berretti sokal algorithm an increase in efficiency by a factor of around 40 is achieved for non interacting self avoiding walks

computing a minimum width square annulus in arbitrary orientation in this paper we address the problem of computing a minimum width square annulus in arbitrary orientation that encloses a given set of n points in the plane a square annulus is the region between two concentric squares we present an o n 3 log n time algorithm that finds such a square annulus over all orientations

adsorption and collapse of self avoiding walks and polygons in three dimensions we consider self avoiding walks and polygons on the simple cubic lattice confined to the half space and interacting with the plane z 0 in addition there is a short range vertex vertex interaction in the walk or polygon which can lead to a collapse transition we explore the interaction between collapse and adsorption in these systems and discuss the form of the phase diagram key results include a proof of the existence of an adsorption transition for polygons for every value of the vertex vertex interaction a corresponding proof for walks when the vertex vertex interaction term is repulsive and a proof that if polygons exhibit a collapse transition then the phase boundary between the expanded and desorbed phase and the collapsed and desorbed phase must be a straight line

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

positive noncommutative polynomials are sums of squares hilbert s 17th problem concerns expression of polynomials on r n as a sum of squares it is well known that many positive polynomials are not sums of squares see re d a for excellent surveys in this paper we consider symmetric noncommutative polynomials and call one matrix positive if whenever matrices of any size are substituted for the variables in the polynomial the matrix value which the polynomial takes is positive semidefinite the result in this paper is a polynomial is matrix positive if and only if it is a sum of squares

minimum width rectangular annulus in this paper we identify a minimum width rectangular annulus that encloses a given set of n points in a plane we propose an o n 2logn time and o n space algorithm for this problem to the best of our knowledge this is the first sub cubic algorithm for a rectangular annulus for arbitrary orientation

algorithms for a minimum volume enclosing simplex in three dimensions we develop a combinatorial algorithm for determining a minimum volume simplex enclosing a set of points in cal r 3 if the convex hull of the points has n vertices then our algorithm takes theta n 4 time combining our exact but slow algorithm with a simple but crude approximation technique we also develop an varepsilon approximation algorithm the algorithm computes in o n 1 varepsilon 6 time a simplex whose volume is within 1 varepsilon factor of the optimal for any varepsilon 0