a necessary condition for feedback stabilization abstract we give a new necessary condition for the existence of a static or dynamic continuous asymptotically stabilizing feedback control for the system x dot f x u we apply our condition to give an example of a system in r 3 which is small time locally controllable satisfies brockett s second and third condition and cannot be locally asymptotically stabilized by employing a static or dynamic continuous feedback law

low density series expansions for directed percolation on square and triangular lattices greatly extended series have been derived for moments of the pair connectedness for bond and site percolation on the directed square and triangular lattices the length of the various series has been at least doubled to more than 110 100 terms for the square lattice bond site problem and more than 55 terms for the bond and site problems on the triangular lattice analysis of the series leads to very accurate estimates for the critical parameters and generally seems to rule out simple rational values for the critical exponents the values of the critical exponents for the average cluster size parallel and perpendicular connectedness lengths are estimated by d 2 277 69 4 k d 1 733 825 25 and d 1 096 844 14 respectively an improved estimate for the percolation probability exponent is obtained from the scaling relation d k c 2 d 0 276 49 4 in all cases the leading correction to scaling term is analytic

nilpotent and high order approximations of vector field systems this paper is concerned with the study of structurally stable properties associated with systems described by one or several real analytic vector fields via approximating systems which retain inf

moment ratios for absorbing state phase transitions we determine the first through fourth moments of the order parameter and various ratios for several one and two dimensional models with absorbing state phase transitions we perform a detailed analysis of the system size dependence of these ratios and confirm that they are indeed universal for three models the contact process the a model and the pair contact process belonging to the directed percolation universality class our studies also yield a refined estimate for the critical point of the pair contact process

low density series expansions for directed percolation i a new efficient algorithm with applications to the square lattice a new algorithm for the derivation of low density series for percolation on directed lattices is introduced and applied to the square lattice bond and site problems numerical evidence shows that the computational complexity grows exponentially but with a growth factor which is much smaller than the growth factor of the previous best algorithm for bond site percolation on the directed square lattice the series has been extended to order 171 158 analysis of the series yields sharper estimates of the critical points and exponents

the mean field scaling function of the universality class of absorbing phase transitions with a conserved field we consider two mean field like models which belong to the universality class of absorbing phase transitions with a conserved field in both cases we analytically derive the order parameter as a function of the control parameter and of an external field conjugated to the order parameter this allows us to calculate the universal scaling function of the mean field behaviour the obtained universal function is in perfect agreement with recently obtained numerical data of the corresponding five and six dimensional models showing that four is the upper critical dimension of this particular universality class

how to construct random functions a constructive theory of randomness for functions based on computational complexity is developed and a pseudorandom function generator is presented this generator is a deterministic polynomial time algorithm that transforms pairs g r where g is any one way function and r is a random k bit string to polynomial time computable functions r 1 2 k 1 2 k these r s cannot be distinguished from random functions by any probabilistic polynomial time algorithm that asks and receives the value of a function at arguments of its choice the result has applications in cryptography random constructions and complexity theory

equation of state for directed percolation using field theoretic renormalization group methods we calculate the equation of state for non equilibrium systems belonging to the universality class of directed percolation to second order in by introducing a parametric representation the result can be written to this order in a very simple form we use our result to obtain a universal amplitude ratio to second order in

asymptotically stabilizing feedback controls abstract the problem considered is the existence and construction of an asymptotically stabilizing feedback control for a rest solution of an n dimensional single input affine control system the method is to construct high order homogeneous approximations of the vector fields defining the control system which then describe an approximating control system such that an asymptotically stabilizing control of the latter system is a local asymptotically stabilizing control of the original system unlike linear approximations there are many possible nonlinear homogeneous approximations these are constructed from lie algebra filtrations obtained by assigning various weights to the defining vector fields one seeks a weighting hence an approximating system which satisfies the various known necessary conditions for the existence of an asymptotically stabilizing feedback control and for which such a control can be constructed as the solution to a homogeneous hamilton jacobi bellman equation associated with an optimization regulator problem

local asymptotic stabilization for a class of uncertain upper triangular systems abstract this paper considers local asymptotic stabilization of a class of uncertain upper triangular systems it shows that by appropriately increasing the powers of the states in a linear controller an uncertain upper triangular system can be locally asymptotically stabilized a nested nonlinear controller is designed by introducing the notion of homogeneity with strictly decreasing degrees for the stability analysis a common lyapunov chetaev function is constructed and a necessary and sufficient condition for the local asymptotic stability is established

scaling behavior of the order parameter and its conjugated field in an absorbing phase transition around the upper critical dimension we analyze numerically the critical behavior of an absorbing phase transition in a conserved lattice gas in an external field the external field is realized as a spontaneous creation of active particles that drives the system away from criticality nevertheless the order parameter obeys certain scaling laws for sufficiently small external fields these scaling laws are investigated and the corresponding exponents are determined in various dimensions d 2 3 4 5 at the so called upper critical dimension d mathrm c 4 one has to modify the usual scaling laws by logarithmic corrections

universal scaling behaviour of directed percolation and the pair contact process in an external field we consider the scaling behaviour of directed percolation and of the pair contact process with a conjugated field in particular we determine numerically the equation of state and show that both models are characterized by the same universal scaling function furthermore we derive the equation of state for the pair contact process within a mean field approach which again agrees with the mean field equation of state of the directed percolation universality class

universal scaling behavior of directed percolation around the upper critical dimension in this work we consider the steady state scaling behavior of directed percolation around the upper critical dimension in particular we determine numerically the order parameter its fluctuations as well as the susceptibility as a function of the control parameter and the conjugated field additionally to the universal scaling functions several universal amplitude combinations are considered we compare our results with those of a renormalization group approach

homogeneous lyapunov function for homogeneous continuous vector field the goal of this article is to provide a construction of a homogeneous lyapunov function p associated with a system of differential equations j f x x r n 1 under the hypotheses 1 f c r n vanishes at x 0 and is homogeneous 2 the zero solution of this system is locally asymptotically stable moreover the lyapunov function v x tends to infinity with 1 x i and belongs to c r 0 r n cp with p e as large as wanted as application to the theory of homogeneous systems we present two well known results of robustness in a slightly extended form and with simpler proofs

controller design for nonlinear affine systems by control lyapunov functions abstract this paper addresses the stabilization problems for nonlinear affine systems first of all the explicit feedback controller is developed for a nonlinear multiple input affine system by assuming that there exists a control lyapunov function next based upon the homogeneous property sufficient conditions for the continuity of the derived controller are developed and then the developed control design methodology is applied to stabilize a class of nonlinear affine cascaded systems it is shown that under some homogeneous assumptions on control lyapunov functions and the interconnection term the cascaded system can be globally stabilized finally some interesting results of finite time stabilization for nonlinear affine systems are also obtained

asymptotic stabilization of a generic class of three dimensional homogeneous quadratic systems abstract we give necessary and sufficient conditions for the asymptotic stabilizability of a generic class of single input three dimensional homogeneous quadratic systems

finite size scaling of directed percolation above the upper critical dimension critical phenomena such as second order phase transitions are characterized by singularities causing a discontinuous behavior of various quantities at the transition point e g the speciflc heat the susceptibility and the correlation length these singularities are described by power laws deflning the well known critical exponents studying the phase transition of a given system the aim of investigation is to identify the set of critical exponents which characterizes together with certain universal scaling functions the so called universality class since most systems are not analytically tractable their critical behavior is often investigated via numerical methods for example monte carlo simulations or transfer matrix calculations in these cases the obtained data are limited to flnite system sizes therefore flnite size scaling fss is widely used to extrapolate to the behavior of the inflnite systems in particular fss is an e cient method

asymptotically stabilizing feedback controls and the nonlinear regulator problem continuous asymptotically stabilizing feedback controls are constructed for two dimensional and certain three dimensional small time locally controllable affine systems the lie products which de

how to construct pseudorandom permutations from pseudorandom functions we show how to efficiently construct a pseudorandom invertible permutation generator from a pseudorandom function generator goldreich goldwasser and micali how to construct random functions p

how to generate cryptographically strong sequences of pseudo random bits we give a set of conditions that allow one to generate 50 50 unpredictable bits based on those conditions we present a general algorithmic scheme for constructing polynomial time deterministic algorithms that stretch a short secret random input into a long sequence of unpredictable pseudo random bits we give an implementation of our scheme and exhibit a pseudo random bit generator for which any efficient strategy for predicting the next output bit with better than 50 50 chance is easily transformable to an equally efficient algorithm for solving the discrete logarithm problem in particular if the discrete logarithm problem cannot be solved in probabilistic polynomial time no probabilistic polynomial time algorithm can guess the next output bit better than by flipping a coin if head guess 0 if tail guess 1

on the implementation of huge random objects we initiate a general study of the feasibility of implementing huge random objects and demonstrate its applicability to a number of areas in which random objects occur naturally we highlight two types of measures of the quality of the implementation with respect to the desired specification the first type corresponds to various standard notions of indistinguishability applied to function ensembles whereas the second type is a novel notion that we call truthfulness intuitively a truthful implementation of a random object of type t must always be an object of type t and not merely be indistinguishable from a random object of type t our formalism allows for the consideration of random objects that satisfy some fixed property or have some fixed structure as well as the consideration of objects supporting complex queries for example we consider the truthful implementation of random hamiltonian graphs as well as supporting complex queries regarding such graphs e g providing the next vertex along a fixed hamiltonian path in such a graph

average case complete problems many interesting combinatorial problems were found to be np complete since there is little hope to solve them fast in the worst case researchers look for algorithms which are fast just on average this matter is sensitive to the choice of a particular np complete problem and a probability distribution of its instances some of these tasks are easy and some not but one needs a way to distinguish the difficult on average problems such negative results could not only save positive efforts but may also be used in areas like cryptography where hardness of some problems is a frequent assumption it is shown in 1 that the tiling problem with uniform distribution of instances has no polynominal on average algorithm unless every np problem with every simple probability distribution has it

a pseudorandom generator from any one way function pseudorandom generators are fundamental to many theoretical and applied aspects of computing we show how to construct a pseudorandom generator from any one way function since it is easy to construct a one way function from a pseudorandom generator this result shows that there is a pseudorandom generator if and only if there is a one way function

tricritical directed percolation we consider a modification of the contact process incorporating higher order reaction terms the original contact process exhibits a non equilibrium phase transition belonging to the universality class of directed percolation the incorporated higher order reaction terms lead to a non trivial phase diagram in particular a line of continuous phase transitions is separated by a tricritical point from a line of discontinuous phase transitions the corresponding tricritical scaling behavior is analyzed in detail i e we determine the critical exponents various universal scaling functions as well as universal amplitude combinations

asymptotic stabilizability of homogeneous polynomial systems of odd degree we give some tools for the construction of the homogeneous feedback witch stabilizes a generic class of single input two dimensional homogeneous systems

homogeneous feedback controls for homogeneous systems let x x j 2n aj x xj be a real analytic n dimensional vector field with x 0 0 which is homogeneous of degree m 0 with respect to a dilation erx er1x1 ernxn and y xj we show that control system x x x uy admits a continuous asymptotically stabilizing feedback control u x then a one dimensional dynamic extension of the system admits a homogeneous asymptotically stabilizing dynamic feedback control u x such that the vector field w x 1 m u x 1 j 2aj x xj describing the extended controlled system is homogeneous of degree m with respect to the extended dilation er x e er1x1 ernxn

finite size scaling of directed percolation in the steady state recently considerable progress has been made in understanding finite size scaling in equilibrium systems here we study finite size scaling in non equilibrium systems at the instance of directed percolation dp which has become the paradigm of non equilibrium phase transitions into absorbing states above at and below the upper critical dimension we investigate the finite size scaling behavior of dp analytically and numerically by considering its steady state generated by a homogeneous constant external source on a d dimensional hypercube of finite edge length l with periodic boundary conditions near the bulk critical point in particular we study the order parameter and its higher moments using renormalized field theory we derive finite size scaling forms of the moments in a one loop calculation moreover we introduce and calculate a ratio of the order parameter moments that plays a similar role in the analysis of finite size scaling in absorbing nonequilibrium processes as the famous binder cumulant in equilibrium systems and that in particular provides a new signature of the dp universality class to complement our analytical work we perform monte carlo simulations which confirm our analytical results