Let G be a group and H an abelian group. Let J ∗ (G, H) be the set of solutions f : G → H of the Jensen functional equation f(xy) + f(xy−1 ) = 2f(x) satisfying the condition f(xyz) − f(xzy) = f(yz) − f(zy) for all x, y,z ∈ G. Let Q ∗ (G, H) be the set of solutions f : G → H of the quadratic equation f(xy) + f(xy−1 ) = 2f(x) + 2f(y) satisfying the Kannappan condition f(xyz) = f(xzy) for all x,y, z ∈ G. In this paper we determine solutions of the Whitehead equation on groups. We show that every solution f : G → H of the Whitehead equation is of the form 4f = 2ϕ + 2ψ, where 2ϕ ∈ J ∗ (G, H) and 2ψ ∈ Q ∗ (G, H). Moreover, if H has the additional property that 2h = 0 implies h = 0 for all h ∈ H, then every solution f : G → H of the Whitehead equation is of the form 2f = ϕ+ψ, where ϕ ∈ J ∗ (G, H) and 2ψ(x) = B(x,x) for some symmetric bihomomorphism B : G × G → H.
This paper studies a class of discrete-time discounted semi-Markov control model on Borel spaces. We assume possibly unbounded costs and a non-stationary exponential form in the discount factor which depends of on a rate, called the discount rate. Given an initial discount rate the evolution in next steps depends on both the previous discount rate and the sojourn time of the system at the current state. The new results provided here are the existence and the approximation of optimal policies for this class of discounted Markov control model with non-stationary rates and the horizon is finite or infinite. Under regularity condition on sojourn time distributions and measurable selector conditions, we show the validity of the dynamic programming algorithm for the finite horizon case. By the convergence in finite steps to the value functions, we guarantee the existence of non-stationary optimal policies for the infinite horizon case and we approximate them using non-stationary ϵ−optimal policies. We illustrated our results a discounted semi-Markov linear-quadratic model, when the evolution of the discount rate follows an appropriate type of stochastic differential equation.
We establish new efficient conditions sufficient for the unique solvability of the initial value problem for two-dimensional systems of linear functional differential equations with monotone operators.
This paper deals with the generalized nonlinear third-order left focal problem at resonance (p(t)u ′′(t))′ − q(t)u(t) = f(t, u(t), u′ (t), u′′(t)), t ∈ ]t0, T[, m(u(t0), u′′(t0)) = 0, n(u(T), u′ (T)) = 0, l(u(ξ), u′ (ξ), u′′(ξ)) = 0, where the nonlinear term is a Carathéodory function and contains explicitly the first and second-order derivatives of the unknown function. The boundary conditions that we study are quite general, involve a linearity and include, as particular cases, Sturm-Liouville boundary conditions. Under certain growth conditions on the nonlinearity, we establish the existence of the nontrivial solutions by using the topological degree technique as well as some recent generalizations of this technique. Our results are generalizations and extensions of the results of several authors. An application is included to illustrate the results obtained.
The problem of solvability of a system of fuzzy relation equations with
respect to unknown fuzzy relation is considered. A number of new criteria of the so called Mamdani relation to be a solntion to the system is suggested. At the same time those criteria are sufficient coiiditions of a solvability of the system in general. A new, easy-to-check criterion of a solvability of the system with special fuzzy pararneters is found.
A class of q-nonlinear parabolic systems with a nondiagonal principal matrix and strong nonlinearities in the gradient is considered.We discuss the global in time solvability results of the classical initial boundary value problems in the case of two spatial variables. The systems with nonlinearities q ∈ (1, 2), q = 2, q > 2, are analyzed.
This paper presents a numerical approach to solve the Hamilton-Jacobi-Bellman (HJB) problem which appears in feedback solution of the optimal control problems. In this method, first, by using Chebyshev pseudospectral spatial discretization, the HJB problem is converted to a system of ordinary differential equations with terminal conditions. Second, the time-marching Runge-Kutta method is used to solve the corresponding system of differential equations. Then, an approximate solution for the HJB problem is computed. In addition, to get more efficient and accurate method, the domain decomposition strategy is proposed with the pseudospectral spatial discretization. Five numerical examples are presented to demonstrate the efficiency and accuracy of the proposed hybrid method.
n this paper, based on a generalized Karush-Kuhn-Tucker (KKT) method a modified recurrent neural network model for a class of non-convex quadratic programming problems involving a so-called Z-matrix is proposed. The basic idea is to express the optimality condition as a mixed nonlinear complementarity problem. Then one may specify conditions for guaranteeing the global solutions of the original problem by using results from the S-lemma. This process is proved by building up a dynamic system from the optimality condition whose equilibrium point is exactly the solution of the mixed nonlinear complementarity problem. By the study of the resulting dynamic system it is shown that under given assumptions, steady states of the dynamic system are stable. Numerical simulations and comparisons with the other methods are presented to illustrate the efficiency of the practical technique that is proposed in this paper.
It is generally accepted that most benchmark problems known today
can be solved by artificial neural networks with one single hidden layer. Networks with more than one hidden layer normally slow down learning dramatically. Furthermore, generalisation to new input patterns is generally better in small networks [1], [2], However, most benchmark problems only involve a small training data set which is normally discrete (such as binary values 0 and 1) in nature. The ability of single hidden layer supervised networks to solve problems with large and continuous type of data (e.g. most engineering problems) is virtually unknown. A fast learning method for solving continuous type problems has been proposed by Evans et al. [3]. However, the method is based on the Kohonen competitive, and ART unsupervised network models. In addition, almost every benchmark problem has the training set containing all possible input patterns, so there is no study of the generalisation behaviour of the network [4]. This study attempts to show that Single hidden layer supervised networks can be used to solve large and continuous type problems within measurable algorithmic complexities.
A large sample of high velocity HI detections have been collected from several surveys of HCV´s. All detections of HI that could be associated with a known object were subtracted. The main asymmetries in the velocity-position diagrams, shown by the very high velocity component (VHVC´s) were apparently reproduced with a kinematical model which, in addition, could explain other pecullarities of the HCV´s distiribution. The parameters of this model indicate that the VHCV´s might be at distances of the order if the size of the Local Group of Galaxies.