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

algebraic design techniques for reliable stabilization in this paper we study two problems in feedback stabilization the first is the simultaneous stabilization problem which can be stated as follows given plants g 0 g 1 g l does there exist a single compensator c that stabilizes all of them the second is that of stabilization by a stable compensator or more generally a least unstable compensator given a plant g we would like to know whether or not there exists a stable compensator c that stabilizes g if not what is the smallest number of right half place poles counted according to their mcmillan degree that any stabilizing compensator must have we show that the two problems are equivalent in the following sense the problem of simultaneously stabilizing l 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator which in turn can be stated as the following purely algebraic problem given 2l matrices a 1 a l b 1 b l where a i b i are right coprime for all i does there exist a matrix m such that a i mb i is unimodular for all i conversely the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l 1 plants the problem of determining whether or not there exists an m such that a bm is unimodular given a right coprime pair a b turns out to be a special case of a question concerning a matrix division algorithm in a proper euclidean domain we give an answer to this question and we believe this result might be of some independent interest we show that given two n times m plants g 0 and g 1 we can generically stabilize them simultaneously provided either n or m is greater than one in contrast simultaneous stabilizability of two single input single output plants g 0 and g 1 is not generic

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

simultaneous stabilization of multiplicatively perturbed plants necessary and sufficient conditions are obtained for simultaneous stabilizability of a given linear time invariant multi input multi output nominal plant and a multiplicatively perturbed plant where the multiplicative perturbation is stable these conditions are derived from the general parity interlacing property applicable to any two arbitrary plants and are expressed explicitly in terms of the real axis poles and zeros of the nominal plant and the perturbation the class of perturbations for which simultaneous stabilization is achievable are characterized using these conditions a special class of unknown diagonal perturbations is also considered and simultaneously stabilizing controllers are designed for the nominal plant under any such unknown perturbation

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

quadratic exponential vectors we give a necessary and sufficient condition for the existence of a quadratic exponential vector with test function in l2 rd l rd we prove the linear independence and totality in the quadratic fock space of these vectors using a technique different from the one used by accardi et al quantum probability and infinite dimensional analysis vol 25 p 262 2009 we also extend to a more general class of test functions the explicit form of the scalar product between two such vectors

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

the quadratic fock functor we construct the quadratic analog of the boson fock functor while in the first order linear case all contractions on the 1 particle space can be second quantized the semigroup of contractions that admit a quadratic second quantization is much smaller due to the nonlinearity the encouraging fact is that it contains as proper subgroups i e the contractions all the gauge transformations of second kind and all the a e invertible maps of rd into itself leaving the lebesgue measure quasi invariant in particular all diffeomorphism of rd this allows quadratic two dimensional quantization of gauge theories of representations of the witt group in fact it continuous analog of the zamolodchikov hierarchy and much more within this semigroup we characterize the unitary and the isometric elements and we single out a class of natural contractions

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

on the one mode quadratic weyl operators abstract it has been known in 7 that one mode quadratic weyl operators are well defined as unitary operators acting on the quadratic fock space these operators were defined by their action on the finite particles space however their action on the domain of the quadratic exponential vectors is still unknown in this paper we provide a new reformulation w r t 1 of the quadratic exponential vectors and we compute the action of the one mode quadratic weyl operators on the set of these exponential vectors then we prove the independence of the one mode quadratic weyl operators parameterized by the principal domain associated with the quadratic heisenberg group obtained in 7 this significant contribution to the program of developing the quadratic white noise calculus constitutes a step toward the c representation of the renormalized square of white noise algebra

on the order of simultaneously stabilizing compensators in this paper simultaneous strong stabilization problem is considered and it is shown that there is no upper bound for the minimal order of a simultaneously strongly stabilizing compensator in terms of the plant orders a similar problem was also considered by smith et al 1986 where it was shown that such a bound does not exist for the strong stabilization problem of a single plant but the examples given in the article were forcing an approximate unstable pole zero cancellation or forcing the distance between two distinct unstable zeros to go to zero in this paper it is shown that 1 if approximate unstable pole zero cancellation does not occur and the distances between distinct unstable zeros are bounded below by a positive constant then it is possible to find an upper bound for the minimal order of a strongly stabilizing compensator and 2 for the simultaneous strong stabilization problem even for the two plant case such a bound cannot be found

c quadratic quantization in the first part of the paper we introduce a new parametrization for the manifold underlying quadratic analogue of the usual heisenberg group introduced in accardi et al infin dimens anal quantum probab relat top 13 551 587 2010 which makes the composition law much more transparent in the second part of the paper the new coordinates are used to construct an inductive system of algebras each of which is isomorphic to a finite tensor product of copies of the one mode quadratic weyl algebra we prove that the inductive limit algebra is factorizable and has a natural localization given by a family of sub algebras each of which is localized on a bounded borel subset of mathbb r moreover we prove that the family of quadratic analogues of the fock states defined on the inductive family of algebras is projective hence it defines a unique state on the limit algebra finally we complete this algebra under the minimal regular c norm thus obtaining a c algebra

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

self adjointness and boundedness in quadratic quantization we construct a counter example showing for the quadratic quantization the identity t t is not necessarily true we characterize all operators on the one particle algebra whose quadratic quantization are self adjoint operators on the quadratic fock space finally we discuss the boundedness of the quadratic quantization

simultaneous stabilization of three or more plants conditions on the positive real axis do not suffice the problem of the simultaneous stabilizability of a finite family of single input single output time invariant systems by a time invariant controller is studied the link between stabilization and avoidance is shown and is used to derive necessary conditions for the simultaneous stabilization of k plants these necessary conditions are proved to be in general not sufficient this result also disproves a long standing conjecture on the stabilizability condition of a single plant with a stable minimum phase controller the main result is to show that unlike the case of two plants the existence of a simultaneous stabilizing controller for more than two plants is not guaranteed by the existence of a controller such that the closed loops have no real unstable poles

simultaneous stabilizability of p and kp it is shown that there exists a common controller which simultaneously stabilizes any given nominal plant p and perturbed plant kp for any given positive real constant k in the standard linear time invariant multi input multi output unity feedback system the class of plants such that p and kp can be simultaneously stabilized for negative k is also determined

a sufficient condition for simultaneous stabilization the condition under which it is possible to find a single controller that stabilizes k single input single output linear time invariant systems p sub i s i 1 k is investigated the concept of avoidance in the complex plane is introduced and used to derive a sufficient condition for k systems to be simultaneously stabilizable a method for constructing a simultaneous stabilizing controller is also provided and is illustrated by an example

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

simultaneously stabilizing controllers for a class of linear plants abstract it is shown that a class of linear time invariant multi input multi output plants that all have poles at zero but do not have other unstable poles can be simultaneously stabilized a procedure is proposed to design a stable and strictly proper simultaneously stabilizing controller all simultaneously stabilizing controllers for this class are also characterized in terms of a parameter matrix that has to satisfy a unimodularity condition

renormalized squares of white noise and other non gaussian noises as levy processes on real lie algebras it is shown how the relations of the renormalized squared white noise defined by accardi lu and volovich alv99 can be realized as factorizable current representations or levy processes on the real lie algebra 2 this allows to obtain its ito table which turns out to be infinite dimensional the linear white noise without or with number operator is shown to be a levy process on the heisenberg weyl lie algebra or the oscillator lie algebra furthermore a joint realization of the linear and quadratic white noise relations is constructed but it is proved that no such realizations exist with a vacuum that is an eigenvector of the central element and the annihilator classical levy processes are shown to arise as components of levy processes on real lie algebras and their distributions are characterized in particular the square of white noise analogue of the quantum poisson process is shown to have a 2 probability density and the analogue of the field operators to have a density proportional to where is the usual function and m 0 a real parameter