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

sierpinski and carmichael numbers abstract we establish several related results on carmichael sierpinski and riesel numbers first we prove that almost all odd natural numbers k have the property that 2 exp n k 1 is not a carmichael number for any n epilson n this implies the existence of a set k of positive lower density such that for any k epsilon k the number 2 exp n k 1 is neither prime nor carmichael for every k epilson n next using a recent result of matomaki we show that there are x1 5 carmichael numbers up to x that are also sierpinski and riesel finally we show that if 2 exp n k 1 is lehmer then n 150 omega k 2 log k where omega k is the number of distinct primes dividing k

mathematics of granular materials this is a short and somewhat informal review on the most mathematical parts of the kinetic theory of granular media intended for physicists and for mathematicians outside the field

two sided bounds on the rate of convergence to a stable law we derive necessary and sufficient conditions for several characterizations of the rate of convergence of a sum of independent variables to a stable law the technique used is to obtain upper and lower bounds on the rate in terms of functions depending in a very simple way on the common underlying distribution this permits a general approach to the problem of rates of convergence

self similar solutions of the boltzmann equation and their applications we consider a class of solutions of the boltzmann equation with infinite energy using the fourier transformed boltzmann equation we prove the existence of a wide class of solutions of this kind they fall into subclasses labelled by a parameter a and are shown to be asymptotic in a very precise sense to the self similar one with the same value of a and the same mass specializing to the case of a maxwell isotropic cross section we give evidence to the effect that the only self similar closed form solutions are the bkw mode and the two solutions recently found by the authors all the self similar solutions discussed in this paper are eternal i e they exist for t which shows that a recent conjecture cannot be extended to solutions with infinite energy eternal solutions with finite moments of all orders and different from a maxwellian are also studied it is shown that these solutions cannot be positive moreover all such solutions partly negative must be asymptotically for large negative times close to the exact eternal solution of bkw type

improving convergence and solution quality of hopfield type neural networks with augmented lagrange multipliers hopfield type networks convert a combinatorial optimization to a constrained real optimization and solve the latter using the penalty method there is a dilemma with such networks when tuned to produce good quality solutions they can fail to converge to valid solutions and when tuned to converge they tend to give low quality solutions this paper proposes a new method called the augmented lagrange hopfield alh method to improve hopfield type neural networks in both the convergence and the solution quality in solving combinatorial optimization it uses the augmented lagrange method which combines both the lagrange and the penalty methods to effectively solve the dilemma experimental results on the travelling salesman problem tsp show superiority of the alh method over the existing hopfield type neural networks in the convergence and solution quality for the ten city tsps alh finds the known optimal tour with 100 success rate as the result of 1000 runs with different random initializations for larger size problems it also finds remarkably better solutions than the compared methods while always converging to valid tours

primal dual solution for the linear programming problems using neural networks in this paper we represent two new methods for the solution of canonical form linear programming problems in order to solve this linear programming problem we must minimize energy function of the corresponding neural network here energy function is considered as a liapunov function and we use treated hopfield neural network first new method finds optimal solution for primal problem using neural network while second new method composes primal and dual problem and therefore finds optimal solution for both problems numerical results compared with simplex solution and find that the convergence of two new methods to the correct solution is too fast even faster than neguyen s method the new methods are fully stable

discrete time zhang neural network for online time varying nonlinear optimization with application to manipulator motion generation in this brief a discrete time zhang neural network dtznn model is first proposed developed and investigated for online time varying nonlinear optimization otvno then newton iteration is shown to be derived from the proposed dtznn model in addition to eliminate the explicit matrix inversion operation the quasi newton broyden fletcher goldfarb shanno bfgs method is introduced which can effectively approximate the inverse of hessian matrix a dtznn bfgs model is thus proposed and investigated for otvno which is the combination of the dtznn model and the quasi newton bfgs method in addition theoretical analyses show that with step size h 1 and or with zero initial error the maximal residual error of the dtznn model has an o tau 2 pattern whereas the maximal residual error of the newton iteration has an o tau pattern with tau denoting the sampling gap besides when h neq 1 and h in 0 2 the maximal steady state residual error of the dtznn model has an o tau 2 pattern finally an illustrative numerical experiment and an application example to manipulator motion generation are provided and analyzed to substantiate the efficacy of the proposed dtznn and dtznn bfgs models for otvno

proof of a mckean conjecture on the rate of convergence of boltzmann equation solutions the present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous boltzmann equation for maxwellian molecules under really mild conditions on the initial datum and a weak physically consistent angular cutoff hypothesis our main result theorem 1 contains the first precise statement that the total variation distance between the solution and the limiting maxwellian distribution admits an upper bound of the form c e varlambda b t varlambda b being the least negative eigenvalue of the linearized collision operator and c a constant depending only on the initial datum the validity of this quantification was conjectured about fifty years ago by henry p mckean as to the proof of our results we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory highlighted by mckean

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

on the distribution of pseudoprimes let v x denote the pseudoprime counting function with l x exp log x log log log x log log x we prove vp x x l x 2 for large x an improvement on the 1956 work of erdos we conjecture that vp x x l x i l

probabilistic view of explosion in an inelastic kac model let mu cdot t t ge 0 be the family of probability measures corresponding to the solution of the inelastic kac model introduced in pulvirenti and toscani j stat phys 114 1453 1480 2004 it has been proved by gabetta and regazzini j stat phys 147 1007 1019 2012 that the solution converges weakly to equilibrium if and only if a suitable symmetrized form of the initial data belongs to the standard domain of attraction of a specific stable law in the present paper it is shown that for initial data which are heavier tailed than the aforementioned ones the limiting distribution is improper in the sense that it has probability 1 2 adherent to infty and probability 1 2 adherent to infty it is explained in which sense this phenomenon is amenable to a sort of explosion and the main result consists in an explicit expression of the rate of such an explosion the presentation of these statements is preceded by a discussion about the necessity of the assumption under which their validity is proved this gives the chance to make an adjustment to a portion of a proof contained in the above mentioned paper by gabetta and regazzini

a discrete time neural network for solving nonlinear convex problems with hybrid constraints this paper investigates a discrete time neural network model for solving nonlinear convex programming problems with hybrid constraints the neural network finds the solution of both primal and dual problems and converges to the corresponding exact solution globally we prove here that the proposed neural network is globally exponentially stable furthermore we extend the proposed neural network for solving a class of monotone variational inequality problems with hybrid constraints compared with other existing neural networks for solving such problems the proposed neural network has a low complexity for implementation without a penalty parameter and converge an exact solution to convex problem with hybrid constraints some numerical simulations for justifying the theoretical analysis are also given the numerical simulations are shown that in the new model note only the cost of the hardware implementation is not relatively expensive but also accuracy of the solution is greatly good

entropy inequalities for stable densities and strengthened central limit theorems we consider the central limit theorem for stable laws in the case of the standardized sum of independent and identically distributed random variables with regular probability density function by showing decay of different entropy functionals along the sequence we prove convergence with explicit rate in various norms to a levy centered density of parameter lambda 1 this introduces a new information theoretic approach to the central limit theorem for stable laws in which the main argument is shown to be the relative fractional fisher information recently introduced in toscani ricerche mat 65 1 71 91 2016 in particular it is proven that with respect to the relative fractional fisher information the levy density satisfies an analogous of the logarithmic sobolev inequality which allows to pass from the monotonicity and decay to zero of the relative fractional fisher information in the standardized sum to the decay to zero in relative entropy with an explicit decay rate

discrete time neural network for fast solving large linear l_ 1 estimation problems and its application to image restoration there is growing interest in solving linear l1 estimation problems for sparsity of the solution and robustness against non gaussian noise this paper proposes a discrete time neural network which can calculate large linear l1 estimation problems fast the proposed neural network has a fixed computational step length and is proved to be globally convergent to an optimal solution then the proposed neural network is efficiently applied to image restoration numerical results show that the proposed neural network is not only efficient in solving degenerate problems resulting from the nonunique solutions of the linear l1 estimation problems but also needs much less computational time than the related algorithms in solving both linear l1 estimation and image restoration problems

some new results for mckean s graphs with applications to kac s equation the main goal of the present paper is to sharpen some results about the error made when the wild sums used to represent the solution of the kac analog of boltzmann s equation are truncated at the n th stage more precisely in carlen carvalho and gabetta j funct anal 220 362 387 2005 one finds a bound for the above mentioned error which depends on an e on the one hand it is shown that the least negative eigenvalue of the linearized collision operator is the best possible exponent on the other hand e is an extra strictly positive number and a a positive coefficient which depends on e too thus it is interesting to check whether e can be removed from the above bound according to the aforesaid reference this problem is studied here by means of the probability distribution of the depth of a leaf in a mckean random tree in fact an accurate study of the probability generating function of such a depth leads to conclude that the above bound can be replaced with a n

new discrete time recurrent neural network proposal for quadratic optimization with general linear constraints in this brief the quadratic problem with general linear constraints is reformulated using the wolfe dual theory and a very simple discrete time recurrent neural network is proved to be able to solve it conditions that guarantee global convergence of this network to the constrained minimum are developed the computational complexity of the method is analyzed and experimental work is presented that shows its high efficiency

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

probabilistic study of the speed of approach to equilibrium for an inelastic kac model this paper deals with a one dimensional model for granular materials which boils down to an inelastic version of the kac kinetic equation with inelasticity parameter p 0 in particular the paper provides bounds for certain distances such as specific weighted distances and the kolmogorov distance between the solution of that equation and the limit it is assumed that the even part of the initial datum which determines the asymptotic properties of the solution belongs to the domain of normal attraction of a symmetric stable distribution with characteristic exponent 2 1 p with such initial data it turns out that the limit exists and is just the aforementioned stable distribution a necessary condition for the relaxation to equilibrium is also proved some bounds are obtained without introducing any extra condition sharper bounds of an exponential type are exhibited in the presence of additional assumptions concerning either the behaviour close to the origin of the initial characteristic function or the behaviour at infinity of the initial probability distribution function

a novel neural network for generally constrained variational inequalities this paper presents a novel neural network for solving generally constrained variational inequality problems by constructing a system of double projection equations by defining proper convex energy functions the proposed neural network is proved to be stable in the sense of lyapunov and converges to an exact solution of the original problem for any starting point under the weaker cocoercivity condition or the monotonicity condition of the gradient mapping on the linear equation set furthermore two sufficient conditions are provided to ensure the stability of the proposed neural network for a special case the proposed model overcomes some shortcomings of existing continuous time neural networks for constrained variational inequality and its stability only requires some monotonicity conditions of the underlying mapping and the concavity of nonlinear inequality constraints on the equation set the validity and transient behavior of the proposed neural network are demonstrated by some simulation results

the distribution of integers with a divisor in a given interval we determine the order of magnitude of h x y z the number of integers n x having a divisor in y z for all x y and z we also study hr x y z the number of integers n x having exactly r divisors in y z whenr 1 we establish the order of magnitude ofh1 x y z for allx y z satisfying z x 1 2 for every r 2 c 1 and 0 we determine the order of magnitude of hr x y z uniformly for y large and y y logy log 4 1 z min y c x 1 2 as a consequence of these bounds we settle a 1960 conjecture of erd os and some conjectures of tenenbaum one key element of the proofs is a new result on the distribution of uniform order statistics

classical and modular approaches to exponential diophantine equations i fibonacci and lucas perfect powers this is the first in a series of papers whereby we combine the classical approach to exponential diophantine equations linear forms in logarithms thue equations etc with a modular approach based on some of the ideas of the proof of fermat s last theorem in this paper we give new improved bounds for linear forms in three logarithms we also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the fibonacci sequence are 0 1 8 and 144 and the only perfect powers in the lucas sequence are 1 and 4

a recurrent neural network for solving bilevel linear programming problem in this brief based on the method of penalty functions a recurrent neural network nn modeled by means of a differential inclusion is proposed for solving the bilevel linear programming problem blpp compared with the existing nns for blpp the model has the least number of state variables and simple structure using nonsmooth analysis the theory of differential inclusions and lyapunov like method the equilibrium point sequence of the proposed nns can approximately converge to an optimal solution of blpp under certain conditions finally the numerical simulations of a supply chain distribution model have shown excellent performance of the proposed recurrent nns

neural network for quadratic optimization with bound constraints a recurrent neural network is presented which performs quadratic optimization subject to bound constraints on each of the optimization variables the network is shown to be globally convergent and conditions on the quadratic problem and the network parameters are established under which exponential asymptotic stability is achieved through suitable choice of the network parameters the system of differential equations governing the network activations is preconditioned in order to reduce its sensitivity to noise and to roundoff errors the optimization method employed by the neural network is shown to fall into the general class of gradient methods for constrained nonlinear optimization and in contrast with penalty function methods is guaranteed to yield only feasible solutions

central limit theorem for a class of one dimensional kinetic equations we introduce a class of kinetic type equations on the real line which constitute extensions of the classical kac caricature the collisional gain operators are defined by smoothing transformations with rather general properties by establishing a connection to the central limit problem we are able to prove long time convergence of the equation s solutions toward a limit distribution for example we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law then the limit is a scale mixture of under some additional assumptions explicit exponential rates for the convergence to equilibrium in wasserstein metrics are calculated and strong convergence of the probability densities is shown

discrete time neural inverse optimal control for nonlinear systems via passivation this paper presents a discrete time inverse optimal neural controller which is constituted by combination of two techniques 1 inverse optimal control to avoid solving the hamilton jacobi bellman equation associated with nonlinear system optimal control and 2 on line neural identification using a recurrent neural network trained with an extended kalman filter in order to build a model of the assumed unknown nonlinear system the inverse optimal controller is based on passivity theory the applicability of the proposed approach is illustrated via simulations for an unstable nonlinear system and a planar robot

characterization of weak convergence of probability valued solutions of general one dimensional kinetic equations for a general inelastic kac like equation recently proposed this paper studies the long time behaviour of its probability valued solution in particular the paper provides necessary and sufficient conditions for the initial datum in order that the corresponding solution converges to equilibrium the proofs rest on the general clt for independent summands applied to a suitable skorokhod representation of the original solution evaluated at an increasing and divergent sequence of times it turns out that roughly speaking the initial datum must belong to the standard domain of attraction of a stable law while the equilibrium is presentable as a mixture of stable laws

convergence analysis of a discrete time recurrent neural network to perform quadratic real optimization with bound constraints presents a model of a discrete time recurrent neural network designed to perform quadratic real optimization with bound constraints the network iteratively improves the estimate of the solution always maintaining it inside of the feasible region several neuron updating rules which assure global convergence of the net to the desired minimum have been obtained some of them also assure exponential convergence and maximize a lower bound for the convergence degree simulation results are presented to show the net performance

a lagrangian network for kinematic control of redundant robot manipulators a recurrent neural network called the lagrangian network is presented for the kinematic control of redundant robot manipulators the optimal redundancy resolution is determined by the lagrangian network through real time solution to the inverse kinematics problem formulated as a quadratic optimization problem while the signal for a desired velocity of the end effector is fed into the inputs of the lagrangian network it generates the joint velocity vector of the manipulator in its outputs along with the associated lagrange multipliers the proposed lagrangian network is shown to be capable of asymptotic tracking for the motion control of kinematically redundant manipulators

stepsize range and optimal value for taylor zhang discretization formula applied to zeroing neurodynamics illustrated via future equality constrained quadratic programming in this brief future equality constrained quadratic programming fecqp is studied via a zeroing neurodynamics method a continuous time zeroing neurodynamics ctzn model is presented by using taylor zhang discretization formula to discretize the ctzn model a taylor zhang discrete time zeroing neurodynamics tz dtzn model is presented to perform fecqp furthermore we focus on the critical parameter of the tz dtzn model i e stepsize by theoretical analyses we obtain an effective range of the stepsize which guarantees the stability of the tz dtzn model in addition we further discuss the optimal value of the stepsize which makes the tz dtzn model possess the optimal stability i e the best stability with the fastest convergence finally numerical experiments and application experiments for motion generation of a robot manipulator are conducted to verify the high precision of the tz dtzn model and the effective range and optimal value of the stepsize for fecqp

central limit theorems for solutions of the kac equation speed of approach to equilibrium in weak metrics this paper is part of our efforts to show how direct application of probabilistic methods pertaining to central limit general theory can enlighten us about the convergence to equilibrium of the solutions of the kac equation here we consider convergence with respect to the following metrics kolmogorov s uniform metric 1 and 2 gini s dissimilarity indices widely known as 1 and 2 wasserstein metrics weighted metrics our main results provide new bounds or improvements on already well known ones for the corresponding distances between the solution of the kac equation and the limiting gaussian maxwellian distribution the study is conducted both under the necessary assumption that initial data have finite energy without assuming existence of moments of order greater than 2 and under the condition that the 2 moment of the initial distribution is finite for some 0

a recurrent neural network for solving nonlinear convex programs subject to linear constraints in this paper we propose a recurrent neural network for solving nonlinear convex programming problems with linear constraints the proposed neural network has a simpler structure and a lower complexity for implementation than the existing neural networks for solving such problems it is shown here that the proposed neural network is stable in the sense of lyapunov and globally convergent to an optimal solution within a finite time under the condition that the objective function is strictly convex compared with the existing convergence results the present results do not require lipschitz continuity condition on the objective function finally examples are provided to show the applicability of the proposed neural network

taylor o h 3 discretization of znn models for dynamic equality constrained quadratic programming with application to manipulators in this paper a new taylor type numerical differentiation formula is first presented to discretize the continuous time zhang neural network znn and obtain higher computational accuracy based on the taylor type formula two taylor type discrete time znn models termed taylor type discrete time znnk and taylor type discrete time znnu models are then proposed and discussed to perform online dynamic equality constrained quadratic programming for comparison euler type discrete time znn models called euler type discrete time znnk and euler type discrete time znnu models and newton iteration with interesting links being found are also presented it is proved herein that the steady state residual errors of the proposed taylor type discrete time znn models euler type discrete time znn models and newton iteration have the patterns of o h 3 o h 2 and o h respectively with h denoting the sampling gap numerical experiments including the application examples are carried out of which the results further substantiate the theoretical findings and the efficacy of taylor type discrete time znn models finally the comparisons with taylor type discrete time derivative model and other lagrange type discrete time znn models for dynamic equality constrained quadratic programming substantiate the superiority of the proposed taylor type discrete time znn models once again

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

tanaka theorem for inelastic maxwell models we show that the euclidean wasserstein distance is contractive for inelastic homogeneous boltzmann kinetic equations in the maxwellian approximation and its associated kac like caricature this property is as a generalization of the tanaka theorem to inelastic interactions even in the elastic classical boltzmann equation we give a simpler proof of the tanaka theorem than the ones in 29 31 consequences are drawn on the asymptotic behavior of solutions in terms only of the euclidean wasserstein distance

metrics for probability distributions and the trend to equilibrium for solutions of the boltzmann equation this paper deals with the trend to equilibrium of solutions to the spacehomogeneous boltzmann equation for maxwellian molecules with angular cutoff as well as with infinite range forces the solutions are considered as densities of probability distributions the tanaka functional is a metric for the space of probability distributions which has previously been used in connection with the boltzmann equation our main result is that if the initial distribution possesses moments of order 2 e then the convergence to equilibrium in his metric is exponential in time in the proof we study the relation between several metrics for spaces of probability distributions and relate this to the boltzmann equation by proving that the fourier transformed solutions are at least as regular as the fourier transform of the initial data this is also used to prove that even if the initial data only possess a second moment then v r f v t v 2 dv 0 asr and this convergence is uniform in time

a discrete time neural network for optimization problems with hybrid constraints recurrent neural networks have become a prominent tool for optimizations including linear or nonlinear variational inequalities and programming due to its regular mathematical properties and well defined parallel structure this brief presents a general discrete time recurrent network for linear variational inequalities and related optimization problems with hybrid constraints in contrary to the existing discrete time networks this general model can operate not only on bound constraints but also on hybrid constraints comprised of inequality equality and bound constraints the model has dynamical properties of global convergence asymptotical and exponential convergences under some weaker conditions numerical examples demonstrate its efficacy and performance

asymptotic properties of the inelastic kac model we discuss the asymptotic behavior of certain models of dissipative systems obtained from a suitable modification of kac caricature of a maxwellian gas it is shown that global equilibria different from concentration are possible if the energy is not finite these equilibria are distributed like stable laws and attract initial densities which belong to the normal domain of attraction if the initial density is assumed of finite energy with higher moments bounded it is shown that the solution converges for large time to a profile with power law tails these tails are heavily dependent of the collision rule

solving complementarity and variational inequalities problems using neural networks in this paper we propose a recurrent neural network model for solving a class of monotone variational inequalities problem with linear constraints the neural network is stable in the sense of lyapunov and globally convergent to an optimal solution compared with the existing convergence results the present proof do not require lipschitz continuity condition on the objective function this neural network model has no adjustable parameter thus its structure is very simple variational inequalities problem with general set of constraints plus a general form of the complementarity problems are solved using the proposed neural networks some examples demonstrated to show the applicability of the proposed neural networks to solve various nonlinear optimization problems numerically

dynamics of discrete time continuous state hopfield networks the dynamics of discrete time continuous state hopfield networks is driven by an energy function in this paper we use this tool to prove under mild hypotheses that any trajectory converges to a fixed point for the sequential iteration and to a cycle of length 2 or a fixed point for the parallel iteration perhaps surprisingly it seems that no rigorous proof of these results was published before