on groups of finite morley rank with a split bn pair of rank 1 abstract we study groups of finite morley rank with a split bn pair of tits rank 1 in the case where the normal complement to b n in b is infinite and abelian for such groups we give conditions ensuring that the standard action of g on the cosets of b is isomorphic to the natural action of psl 2 f on p 1 f for f an algebraically closed field in particular we show that sl 2 f and psl 2 f are the only infinite quasisimple l groups of finite morley rank possessing a split bn pair of tits rank 1 where the normal complement to b n in b is infinite abelian our approach is through the theory of moufang sets and is tied to work attempting to classify the abelian moufang sets of finite morley rank

a new neural network for solving linear and quadratic programming problems a new neural network for solving linear and quadratic programming problems is presented and is shown to be globally convergent the new neural network improves existing neural networks for solving these problems it avoids the parameter turning problem it is capable of achieving the exact solutions and it uses only simple hardware in which no analog multipliers for variables are required furthermore the network solves both the primal problems and their dual problems simultaneously

a stochastic burgers equation from a class of microscopic interactions we consider a class of nearest neighbor weakly asymmetric mass conservative particle systems evolving on mathbb z which includes zero range and types of exclusion processes starting from a perturbation of a stationary state when the weak asymmetry is of order o n gamma for 1 2 gamma leq1 we show that the scaling limit of the fluctuation field as seen across process characteristics is a generalized ornstein uhlenbeck process however at the critical weak asymmetry when gamma 1 2 we show that all limit points satisfy a martingale formulation which may be interpreted in terms of a stochastic burgers equation derived from taking the gradient of the kpz equation the proofs make use of a sharp boltzmann gibbs estimate which improves on earlier bounds

a pregenerator for burgers equation forced by conservative noise we consider the stationarity of a burgers equation with an external random force of gradient type in one space dimension the expected stationary measure is the white noise measure on the space of tempered distributions as a consequence the nonlinearity of the formal equation u t u u x u xx x is ill defined introducing a pregenerator we can formulate a generalized martingale problem leading to a meaningful version of the formal equation which was an open problem

a new nonlinear neural network for solving a class of constrained parametric optimization problems the paper deals with convex parametric programming problems in this paper convex parametric programming transform to a neural network model and then we solve neural network model with one of numerical methods finally simple numerical examples are provided for the sake of illustration

a review of hopfield neural networks for solving mathematical programming problems the hopfield neural network hnn is one major neural network nn for solving optimization or mathematical programming mp problems the major advantage of hnn is in its structure can be realized on an electronic circuit possibly on a vlsi very large scale integration circuit for an on line solver with a parallel distributed process the structure of hnn utilizes three common methods penalty functions lagrange multipliers and primal and dual methods to construct an energy function when the function reaches a steady state an approximate solution of the problem is obtained under the classes of these methods we further organize hnns by three types of mp problems linear non linear and mixed integer the essentials of each method are also discussed in details some remarks for utilizing hnn and difficulties are then addressed for the benefit of successive investigations finally conclusions are drawn and directions for future study are provided

an efficient simplified neural network for solving linear and quadratic programming problems we present a high performance and efficiently simplified new neural network which improves the existing neural networks for solving general linear and quadratic programming problems the network having no need for parameter setting results in a simple hardware requiring no analog multipliers is shown to be stable and converges globally to the exact solution moreover using this network we can solve both linear and quadratic programming problems and their duals simultaneously high accuracy of the obtained solutions and low cost of implementation are among the features of this network we prove the global convergence of the network analytically and verify the results numerically

a general methodology for designing globally convergent optimization neural networks we present a general methodology for designing optimization neural networks we prove that the neural networks constructed by using the proposed method are guaranteed to be globally convergent to solutions of problems with bounded or unbounded solution sets in contrast with the gradient methods whose convergence is not guaranteed we show that the proposed method contains both the gradient methods and nongradient methods employed in existing optimization neural networks as special cases based on the theoretical results of the proposed method we study the convergence and stability of general gradient models in the case of unisolated solutions using the proposed method we derive some new neural network models for a very large class of optimization problems in which the equilibrium points correspond to exact solutions and there is no variable parameter finally some numerical examples show the effectiveness of the method

constrained least absolute deviation neural networks it is well known that least absolute deviation lad criterion or norm used for estimation of parameters is characterized by robustness i e the estimated parameters are totally resistant insensitive to large changes in the sampled data this is an extremely useful feature especially when the sampled data are known to be contaminated by occasionally occurring outliers or by spiky noise in our previous works we have proposed the least absolute deviation neural network ladnn to solve unconstrained lad problems the theoretical proofs and numerical simulations have shown that the ladnn is lyapunov stable and it can globally converge to the exact solution to a given unconstrained lad problem we have also demonstrated its excellent application value in time delay estimation more generally a practical lad application problem may contain some linear constraints such as a set of equalities and or inequalities which is called constrained lad problem whereas the unconstrained lad can be considered as a special form of the constrained lad in this paper we present a new neural network called constrained least absolute deviation neural network cladnn to solve general constrained lad problems theoretical proofs and numerical simulations demonstrate that the proposed cladnn is lyapunov stable and globally converges to the exact solution to a given constrained lad problem independent of initial values the numerical simulations have also illustrated that the proposed cladnn can be used to robustly estimate parameters for nonlinear curve fitting which is extensively used in signal and image processing

numerical solution for linear and quadratic programming problems using a recurrent neural network a recurrent neural network is proposed in this paper for solving linear and quadratic programming problems the main advantage of this network is that here we are not in need of parameters setting moreover using this network we can solve primal programming problems and their duals simultaneously we prove the global convergence of the neural network and demonstrate the advanced performance of the proposed network by means of simulation of several numerical examples

weakly split spherical tits systems in quasi reductive groups we will prove that any weakly split spherical tits system b n in g bf g k bf g a quasi reductive k group such as a connected reductive k group satisfying some natural conditions is standard in particular if bf g is anisotropic over k then such a tits system is trivial i e b g n

application of projection neural network in solving convex programming problems in this paper we present that solution of convex programming problems is equivalent with solution of projection formulation then we introduce neural network models for solving projection formulation and analysis stability conditions and convergence simulation shows that the introduced neural network is effective in solving convex programming problems

second order boltzmann gibbs principle for polynomial functions and applications in this paper we give a new proof of the second order boltzmann gibbs principle introduced in goncalves and jara arch ration mech anal 212 2 597 644 2014 the proof does not impose the knowledge on the spectral gap inequality for the underlying model and it relies on a proper decomposition of the antisymmetric part of the current of the system in terms of polynomial functions in addition we fully derive the convergence of the equilibrium fluctuations towards 1 a trivial process in case of super diffusive systems 2 an ornstein uhlenbeck process or the unique energy solution of the stochastic burgers equation as defined in gubinelli and jara spdes anal comput 1 325 350 2013 and gubinelli and perkowski arxiv 1508 07764 2015 in case of weakly asymmetric diffusive systems examples and applications are presented for weakly and partial asymmetric exclusion processes weakly asymmetric speed change exclusion processes and hamiltonian systems with exponential interactions

proper moufang sets with abelian root groups are special moufang sets are split bn pairs of rank one or the moufang buildings of rank one as such they have been studied extensively being the basic building blocks of all split bn pairs a moufang set is proper if it is not sharply 2 transitive we prove that a proper moufang set whose root groups are abelian is special this resolves an important conjecture in the area of moufang sets it enables us to apply the theory of quadratic jordan division algebras to such moufang sets

solving the kpz equation we introduce a new concept of solution to the kpz equation which is shown to extend the classical cole hopf solution this notion provides a factorisation of the cole hopf solution map into a universal measurable map from the probability space into an explicitly described auxiliary metric space composed with a new solution map that has very good continuity properties the advantage of such a formulation is that it essentially provides a pathwise notion of a solution together with a very detailed approximation theory in particular our construction completely bypasses the cole hopf transform thus laying the groundwork for proving that the kpz equation describes the uctuations of systems in the kpz universality class as a corollary of our construction we obtain very detailed new regularity results about the solution as well as its derivative with respect to the initial condition other byproducts of the proof include an explicit approximation to the stationary solution of the kpz equation a well posedness result for the fokker planck equation associated to a particle diusing in a rough space time dependent potential and a new periodic homogenisation result for the heat equation with a space time periodic potential one ingredient in our construction is an example of a non gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge

identities in moufang sets moufang sets were introduced by jacques tits as an axiomatization of the buildings of rank one that arise from simple algebraic groups of relative rank one these fascinating objects have a simple definition and yet their structure is rich while it is rigid enough to allow for at least partial classification in this paper we obtain two identities that hold in any moufang set these identities are closely related to the axioms that define a quadratic jordan algebra we apply them in the case when the moufang set is so called special and has abelian root groups in addition we push forward the theory of special moufang sets

numerical solutions for constrained quadratic problems using high performance neural networks two new classes of neural networks for solving constrained quadratic programming problems are presented the main advantage of these networks is the requirement to use economic analog multipliers for variables the numerical simulations demonstrate that in the new neural networks not only the cost of the hardware implementation is not relatively expensive but also accuracy of the solution is greatly good the network dynamic behaviors are discussed the numerical simulations are shown that an optimal solution of the quadratic problems is an equilibrium point of the neural dynamics and vise versa we show that these networks find the solution of both primal and dual problems and converge to the corresponding exact solutions globally the proposed new neural networks models are fully stable

neural network models and its application for solving linear and quadratic programming problems in this paper we consider two recurrent neural network model for solving linear and quadratic programming problems the first model is derived from an unconstraint minimization reformulation of the program the second model directly is obtained of optimality condition for an optimization problem by applying the energy function and the duality gap we will compare the convergence these models we also explore the existence and the convergence of the trajectory and stability properties for the neural networks models finally in some numerical examples the effectiveness of the methods is shown

a new recurrent neural network for solving convex quadratic programming problems with an application to the k winners take all problem in this paper a new recurrent neural network is proposed for solving convex quadratic programming qp problems compared with existing neural networks the proposed one features global convergence property under weak conditions low structural complexity and no calculation of matrix inverse it serves as a competitive alternative in the neural network family for solving linear or quadratic programming problems in addition it is found that by some variable substitution the proposed network turns out to be an existing model for solving minimax problems in this sense it can be also viewed as a special case of the minimax neural network based on this scheme a k winners take all k wta network with o n complexity is designed which is characterized by simple structure global convergence and capability to deal with some ill cases numerical simulations are provided to validate the theoretical results obtained more importantly the network design method proposed in this paper has great potential to inspire other competitive inventions along the same line

algebraic structures related to universal covers of special rank one groups abstract in this note we give a presentation of special rank one groups with abelian unipotent subgroups abbreviated aus depending on some algebraic structure k and then show that each special rank one group with aus is a center factor group of a so presented group as shown in t de medts r weiss moufang sets and jordan division algebras math ann 335 2006 415 433 one example of such an algebraic structure is unital quadratic jordan division algebras in subsequent work my student sebastian weis studied these structures called r structures systematically and showed among other things that the problem of classifying such special rank one groups with aus is equivalent to the classification of these r structures up to isotopy

linear and quadratic programming neural network analysis neural networks for linear and quadratic programming are analyzed the network proposed by m p kennedy and l o chua ieee trans circuits syst vol 35 pp 554 562 may 1988 is justified from the viewpoint of optimization theory and the technique is extended to solve optimization problems such as the least squares problem for quadratic programming the network converges either to an equilibrium or to an exact solution depending on whether the problem has constraints or not the results also suggest an analytical approach to solve the linear system bx b without calculating the matrix inverse the results are directly applicable to optimization problems with c sup 2 convex objective functions and linear constraints the dynamics and applicability of the networks are demonstrated by simulation the distance between the equilibria of the networks and the problem solutions can be controlled by the appropriate choice of a network parameter

commuting u operators in jordan algebras for elements x y in a non degenerate non unital jordan algebra over a commutative ring the relation x y 0 is shown to imply that the u operators of x and y commute uxuy uyux the proof rests on the ze 0 lmanov mccrimmon classication 15 of strongly prime quadratic jordan algebras

a new nonlinear neural network for solving convex nonlinear programming problems this paper presents a new recurrent neural network for solving convex nonlinear programming problems the new model is simpler and more intuitive than existing models and converge very fast to exact solution of the original problem we show that this new model is asymptotically stable

trees contraction groups and moufang sets we study closed subgroups g of the automorphism group of a locally finite tree t acting doubly transitively on the boundary we show that if the stabilizer of some end is metabelian then there is a local field k such that psl 2 k g pgl 2 k we also show that the contraction group of some hyperbolic element is closed and torsion free if and only if g is virtually a rank 1 simple p adic analytic group for some prime p a key point is that if some contraction group is closed then g is boundary moufang meaning that the boundary t is a moufang set we collect basic results on moufang sets arising at infinity of locally finite trees and we provide a complete classification in case the root groups are torsion free

local moufang sets and local jordan pairs abstract in this paper we extend the theory of special local moufang sets we construct a local moufang set from every local jordan pair and we show that every local moufang set satisfying certain natural conditions gives rise to a local jordan pair we also explore the connections between these two constructions

the infinitesimal generator of the stochastic burgers equation we develop a martingale approach for a class of singular stochastic pdes of burgers type including fractional and multi component burgers equations by constructing a domain for their infinitesimal generators it was known that the domain must have trivial intersection with the usual cylinder test functions and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions using the new domain we are able to prove existence and uniqueness for the kolmogorov backward equation and the martingale problem we also extend the uniqueness result for energy solutions of the stochastic burgers equation of gubinelli and perkowski j am math soc 31 2 427 471 2018 to a wider class of equations as applications of our approach we prove that the stochastic burgers equation on the torus is exponentially l 2 ergodic and that the stochastic burgers equation on the real line is ergodic

moufang sets from groups of mixed type abstract we consider the existence of moufang sets related to certain groups of mixed type this way we obtain new examples of moufang sets and new constructions of known classes the most interesting class of new examples is related to the moufang quadrangles of type f 4 and to the ree tits octagon over a nonperfect field and the root groups of each member have nilpotency class three

a recurrent neural network for nonlinear continuously differentiable optimization over a compact convex subset we propose a general recurrent neural network rnn model for nonlinear optimization over a nonempty compact convex subset which includes the bound subset and spheroid subset as special cases it is shown that the compact convex subset is a positive invariant and attractive set of the rnn system and that all the network trajectories starting from the compact convex subset converge to the equilibrium set of the rnn system the above equilibrium set of the rnn system coincides with the optimum set of the minimization problem over the compact convex subset when the objective function is convex the analysis of these qualitative properties for the rnn model is conducted by employing the properties of the projection operator of euclidean space onto the general nonempty closed convex subset a numerical simulation example is also given to illustrate the qualitative properties of the proposed general rnn model for solving an optimization problem over various compact convex subsets

space time discrete kpz equation we study a general family of space time discretizations of the kpz equation and show that they converge to its solution the approach we follow makes use of basic elements of the theory of regularity structures hairer in invent math 198 2 269 504 2014 as well as its discrete counterpart hairer and matetski in discretizations of rough stochastic pdes 2015 arxiv 1511 06937 since the discretization is in both space and time and we allow non standard discretization for the product the methods mentioned above have to be suitably modified in order to accommodate the structure of the models under study

descent of affine buildings i large minimal angles in this two part paper we prove an existence result for affine buildings arising from exceptional algebraic reductive groups combined with earlier results on classical groups this gives a complete and positive answer to the conjecture concerning the existence of affine buildings arising from such groups defined over a skew field with a complete valuation as proposed by jacques tits this first part lays the foundations for our approach and deals with the large minimal angle case

a delayed neural network for solving linear projection equations and its analysis in this paper we present a delayed neural network approach to solve linear projection equations the lyapunov krasovskii theory for functional differential equations and the linear matrix inequality lmi approach are employed to analyze the global asymptotic stability and global exponential stability of the delayed neural network compared with the existing linear projection neural network theoretical results and illustrative examples show that the delayed neural network can effectively solve a class of linear projection equations and some quadratic programming problems

special moufang sets their root groups and their μ maps we prove timmesfeld s conjecture that special abstract rank one groups are quasisimple we give two characterizations of the root groups in special moufang sets a normal subgroup of the point stabilizer is a root group if it is either regular or nilpotent and transitive we prove that if a root group of a special moufang set contains an involution then it is of exponent 2 we also show that the root groups are abelian if and only if the so called maps are involutions

convergence in networks with counterclockwise neural dynamics the notion of counterclockwise ccw input output i o dynamics introduced by angeli 2006 to deal with questions of multistability in interconnected dynamical systems is applied and further developed in order to analyze convergence and stability of neural networks by pursuing a modular approach we interpret a cellular nonlinear network cnn as a positive feedback of a parallel block of single input single output siso dynamical systems the neurons and a static multiple input multiple output mimo system that couples them typically the so called interconnection matrix the analysis extends previously known results by enlarging the class of allowed neural dynamics to higher order neurons

a theory of regularity structures we introduce a new notion of regularity structure that provides an algebraic framework allowing to describe functions and or distributions via a kind of jet or local taylor expansion around each point the main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose built for the problem at hand in particular this allows to describe the local behaviour not only of functions but also of large classes of distributions we then build a calculus allowing to perform the various operations multiplication composition with smooth functions integration against singular kernels necessary to formulate fixed point equations for a very large class of semilinear pdes driven by some very singular typically random input this allows for the first time to give a mathematically rigorous meaning to many interesting stochastic pdes arising in physics the theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems possibly modified by the addition of diverging counterterms these counterterms arise naturally through the action of a renormalisation group which is defined canonically in terms of the regularity structure associated to the given class of pdes our theory also allows to easily recover many existing results on singular stochastic pdes kpz equation stochastic quantisation equations burgers type equations and to understand them as particular instances of a unified framework one surprising insight is that in all of these instances local solutions are actually smooth in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions distributions that play the role of polynomials in the theory as an example of a novel application we solve the long standing problem of building a natural markov process that is symmetric with respect to the finite volume measure describing the phi 4 3 euclidean quantum field theory it is natural to conjecture that the markov process built in this way describes the glauber dynamic of 3 dimensional ferromagnets near their critical temperature

a simple and high performance neural network for quadratic programming problems in this paper a neural network for quadratic programming problems is simplified the simplicity is necessary for the high accuracy of solutions and low cost of implementation the proposed network is proved to be an extension of newton s optimal descent flow about constraints problems and is globally convergent the network dynamic behaviors are also discussed and these can get the feasible solution more easily the simulations demonstrate the reasonability of the theory and advantages of the network

energy solutions of kpz are unique the kardar parisi zhang kpz equation is conjectured to universally describe the fluctuations of weakly asymmetric interface growth here we provide the first intrinsic well posedness result for the kpz equation on the real line by showing that its energy solutions as introduced by goncalves and jara in gj14 and refined in gj13 are unique together with the convergence results of gj14 and many follow up papers this establishes the weak kpz universality conjecture for a wide class of models our proof builds on an observation of funaki and quastel fq15 a remarkable consequence is that the energy solution to the kpz equation is not equal to the cole hopf solution but it involves an additional drift t 12

the one dimensional kpz equation and its universality class our understanding of the one dimensional kpz equation alias noisy burgers equation has advanced substantially over the past 5 years we provide a non technical review where we limit ourselves to the stochastic pde and lattice type models approximating it

probability distribution of the free energy of the continuum directed random polymer in 1 1 dimensions we consider the solution of the stochastic heat equation tz d 1 2 x zz p w with delta function initial condition z t d0 x dixd0 whose logarithm with appropriate normalization is the free energy of the con tinuum directed polymer or the hopf cole solution of the kardar parisi zhang equation with narrow wedge initial conditions we obtain explicit formulas for the one dimensional marginal distributions the crossover distributions which interpolate between a standard gaussian dis tribution small time and the gue tracy widom distribution large time the proof is via a rigorous steepest descent analysis of the tracy widom formula for the asymmetric simple exclusion process with antishock initial data which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit the limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes 2010 wiley periodicals inc

weakly asymmetric non simple exclusion process and the kardar parisi zhang equation we analyze a class of non simple exclusion processes and the corresponding growth models by generalizing the discrete cole hopf transformation of gartner stoch process appl 27 233 260 1987 we identify the main non linearity and eliminate it by imposing a gradient type condition for hopping range at most 3 using the generalized transformation we prove the convergence of the exclusion process toward the kardar parisi zhang kpz equation this is the first universality result under the weak asymmetry concerning interacting particle systems while this class of exclusion processes are not explicitly solvable under the weak asymmetry we obtain the exact one point limiting distribution for the step initial condition by using the previous result of amir et al commun pure appl math 64 4 466 537 2011 and our convergence result

moufang sets and jordan division algebras we give a simple criterion which determines when a permutation group u and one additional permutation give rise to a moufang set we apply this criterion to show that every jordan division algebra gives rise in a very natural way to a moufang set to provide sufficient conditions for a moufang set to arise from a jordan division algebra and to give a characterization of the projective moufang sets over a commutative field of characteristic different from 2

neural network for solving extended linear programming problems a neural network for solving extended linear programming problems is presented and is shown to be globally convergent to exact solutions the proposed neural network only uses simple hardware in which no analog multiplier for variables is required and has no parameter tuning problem finally an application of the neural network to the l sub 1 norm minimization problem is given

a central limit theorem for the kpz equation we consider the kpz equation in one space dimension driven by a stationary centred space time random field which is sufficiently integrable and mixing but not necessarily gaussian we show that in the weakly asymmetric regime the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the hopf cole solution to the kpz equation driven by space time gaussian white noise while the limiting process depends only on the integrated variance of the driving field the diverging constants appearing in the definition of the reference frame also depend on higher order moments

moufang sets of finite morley rank of odd type abstract we show that for a wide class of groups of finite morley rank the presence of a split bn pair of tits rank 1 forces the group to be of the form psl 2 and the bn pair to be standard our approach is via the theory of moufang sets specifically we investigate infinite and so called hereditarily proper moufang sets of finite morley rank in the case where the little projective group has no infinite elementary abelian 2 subgroups and show that all such moufang sets are standard and thus associated to psl 2 f for f an algebraically closed field of characteristic not 2 provided the hua subgroups are nilpotent further we prove that the same conclusion can be reached whenever the hua subgroups are l groups and the root groups are not simple

scaling limits of wick ordered kpz equation consider the kpz equation x d where w t x is a space time white noise this paper investigates the question of whether for some exponents and z k u k z t kx converges in some sense as k to infty and if so what are the values of these exponents the non linear term in the kpz equation is interpreted as a wick product and the equation is solved in a suitable space of stochastic distributions the main tools for establishing the scaling properties of the solution are those of white noise analysis in particular the wiener chaos expansion a notion of convergence in law in the sense of wiener chaos is formulated and convergence in this sense of k u k z t kx as k is established for various values of and z depending on the dimension d

nonlinear neural networks for solving the shortest path problem in this paper a new nonlinear neural network for solving the shortest path problem is presented the our nonlinear neural networks are able to generate optimal solutions to the shortest path problem the performance of the our neural networks are demonstrated by means of illustrative example the our neural networks are shown to be capable of generating the shortest path and suitable for electronic implementation

open asep in the weakly asymmetric regime we consider asep on a bounded interval and on a half line with sources and sinks on the full line bertini and giacomin in 1997 proved convergence under weakly asymmetric scaling of the height function to the solution of the kpz equation we prove here that under similar weakly asymmetric scaling of the sources and sinks as well the bounded interval asep height function converges to the kpz equation on the unit interval with neumann boundary conditions on both sides different parameter for each side and likewise for the half line asep to kpz on a half line this result can be interpreted as showing that the kpz equation arises at the triple critical point maximal current high density low density of the open asep 2018 wiley periodicals inc

kpz equation limit of higher spin exclusion processes we prove that under a particular weak scaling the 4 parameter interacting particle system introduced by corwin and petrov comm math phys 343 2016 651 700 converges to the kardar parisi zhang kpz equation this expands the relatively small number of systems for which weak universality of the kpz equation has been demonstrated

stochastic burgers and kpz equations from particle systems we consider two strictly related models a solid on solid interface growth model and the weakly asymmetric exclusion process both on the one dimensional lattice it has been proven that in the diffusive scaling limit the density field of the weakly asymmetric exclusion process evolves according to the burgers equation and the fluctuation field converges to a generalized ornstein uhlenbeck process we analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the non linear burgers equation with a random noise on the density current for the solid on solid model we prove that the fluctuation field of the interface profile if suitably rescaled converges to the kardar parisi zhang equation this provides a microscopic justification of the so called kinetic roughening i e the non gaussian fluctuations in some non equilibrium processes our main tool is the cole hopf transformation and its microscopic version we also develop a mathematical theory for the macroscopic equations

the kpz limit of asep with boundary it was recently proved in corwin and shen cpam cs16 that under weakly asymmetric scaling the height functions for asep with sources and sinks converges to the hopf cole solution of the kpz equation with inhomogeneous neumann boundary conditions in their assumptions cs16 chose positive values for the neumann boundary condition and they assumed initial data which is close to stationarity by developing more extensive heat kernel estimates we clarify and extend their results to negative values of the neumann boundary parameters and we also show how to generalize their results to empty initial data which is very far from stationarity combining our result with barraquand et al duke math j bbcw17 we obtain the laplace transform of the one point distribution for half line kpz and use this to confirm t1 3 scale goe tracy widom long time fluctuations at the origin

nonlinear fluctuations of weakly asymmetric interacting particle systems we introduce what we call the second order boltzmann gibbs principle which allows one to replace local functionals of a conservative one dimensional stochastic process by a possibly nonlinear function of the conserved quantity this replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states as an application of this second order boltzmann gibbs principle we introduce the notion of energy solutions of the kpz and stochastic burgers equations under minimal assumptions we prove that the density fluctuations of one dimensional stationary weakly asymmetric conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic burgers equation we also show that the fluctuations of the height function associated to these models are given by energy solutions of the kpz equation in this sense unfortunately we lack a uniqueness result for these energy solutions we conjecture these solutions to be unique and we show some regularity results for energy solutions of the kpz burgers equation supporting this conjecture

stochastic pde limit of the six vertex model we study the stochastic six vertex model and prove that under weak asymmetry scaling i e when the parameter delta rightarrow 1 so as to zoom into the ferroelectric disordered phase critical point its height function fluctuations converge to the solution to the kardar parisi zhang kpz equation we also prove that the one dimensional family of stochastic gibbs states for the symmetric six vertex model converge under the same scaling to the stationary solution to the stochastic burgers equation our proofs rely upon the markov self duality of our model the starting point is an exact microscopic hopf cole transform for the stochastic six vertex model which follows from the model s known one particle markov self duality given this transform the crucial step is to establish self averaging for specific quadratic function of the transformed height function we use the model s two particle self duality to produce explicit expressions as bethe ansatz contour integrals for conditional expectations from which we extract time decorrelation and hence self averaging in time the crux of our markov duality method is that the entire convergence result reduces to precise estimates on the one particle and two particle transition probabilities previous to our work markov dualities had only been used to prove convergence of particle systems to linear gaussian spdes e g the stochastic heat equation with additive noise

toward the abelian root groups conjecture for special moufang sets abstract we prove that if a root group of a special moufang set contains an element of order p 1 mod 4 then it is abelian

a new gradient based neural network for solving linear and quadratic programming problems a new gradient based neural network is constructed on the basis of the duality theory optimization theory convex analysis theory lyapunov stability theory and lasalle invariance principle to solve linear and quadratic programming problems in particular a new function f x y is introduced into the energy function e x y such that the function e x y is convex and differentiable and the resulting network is more efficient this network involves all the relevant necessary and sufficient optimality conditions for convex quadratic programming problems for linear programming and quadratic programming qp problems with unique and infinite number of solutions we have proven strictly that for any initial point every trajectory of the neural network converges to an optimal solution of the qp and its dual problem the proposed network is different from the existing networks which use the penalty method or lagrange method and the inequality constraints are properly handled the simulation results show that the proposed neural network is feasible and efficient

lower tail of the kpz equation we provide the first tight bounds on the lower tail probability of the one point distribution of the kardar parisi zhang kpz equation with narrow wedge initial data our bounds hold for all sufficiently large times t and demonstrates a crossover between superexponential decay with exponent 52 and leading prefactor 415 t1 3 for tail depth greater than t2 3 and exponent 3 with leading prefactor at least 112 for tail depth less than t2 3

malliavin calculus for regularity structures the case of gpam malliavin calculus is implemented in the context of hairer 2014 16 this involves some constructions of independent interest notably an extension of the structure which accommodates a robust and purely deterministic translation operator in l2l2 directions between models in the concrete context of the generalized parabolic anderson model in 2d one of the singular spdes discussed in the afore mentioned article we establish existence of a density at positive times

a high performance neural network for solving linear and quadratic programming problems two classes of high performance neural networks for solving linear and quadratic programming problems are given we prove that the new system converges globally to the solutions of the linear and quadratic programming problems in a neural network network parameters are usually not specified the proposed models can overcome numerical difficulty caused by neural networks with network parameters and obtain desired approximate solutions of the linear and quadratic programming problems

day to day dynamic network disequilibria and idealized traveler information systems in this paper we present tatonnement models for calculating static wardropian user equilibria on congested networks with fully general demand and cost structures we present both a qualitative analysis of stability and numerical studies which show that such an approach provides a reliable means for determining static user equilibria we also describe circumstances for which these models depict day to day adjustments from one realizable disequilibrium state to another and how these adjustment processes differ depending on the quality of the information being provided by abstract traveler information systems specifically we demonstrate that such dynamic adjustment processes settle down to equilibria both when information is complete and when it is incomplete

special moufang sets with abelian hua subgroup abstract in this paper we complete the classification of special moufang sets m u with abelian hua subgroup it is already known that if u has odd characteristic then m u is the projective line over a commutative field we present a similar result which also deals with special moufang sets of even characteristic

novel artificial neural network with simulation aspects for solving linear and quadratic programming problems in this paper linear and quadratic programming problems are solved using a novel recurrent artificial neural network the new model is simpler and converges very fast to the exact primal and dual solutions simultaneously the model is based on a nonlinear dynamical system using arbitrary initial conditions in order to construct an economy model here we avoid using analog multipliers the dynamical system is a time dependent system of equations with the gradient of specific lyapunov energy function in the right hand side block diagram of the proposed neural network model is given fourth order runge kutta method with controlled step size is used to solve the problem numerically global convergence of the new model is proved both theoretically and numerically numerical simulations show the fast convergence of the new model for the problems with a unique solution or infinitely many this model converges to the exact solution independent of the way that we may choose the starting points i e inside outside or on the boundaries of the feasible region

a feedback neural network for solving convex constraint optimization problems in this paper a feedback neural network model is presented by two steps firstly a convex sub optimization problem with bound constraints is established by introducing an energy function and the neural subnetwork for solving the sub optimization problem is constructed based on the projection method secondly a feedback neural network is proposed by using the subnetwork and can converge to an exact optimal solution of primal optimization problem the distinguishing features of the proposed feedback network are no lagrange multipliers no dual variables and no penalty parameters it has the least number of state variables simple structure and is suitable for hardware implementation two simulation examples are provided to show the feasibility and efficiency of proposed method in the paper