hyperbolic gradient operator and hyperbolic back propagation learning algorithms in this paper we first extend the wirtinger derivative which is defined for complex functions to hyperbolic functions and derive the hyperbolic gradient operator yielding the steepest descent direction by using it next we derive the hyperbolic backpropagation learning algorithms for some multilayered hyperbolic neural networks nns using the hyperbolic gradient operator it is shown that the use of the wirtinger derivative reduces the effort necessary for the derivation of the learning algorithms by half simplifies the representation of the learning algorithms and makes their computer programs easier to code in addition we discuss the differences between the derived hyperbolic bp rules and the complex valued backpropagation learning rule complex bp finally we make some experiments with the derived learning algorithms as a result we find that the convergence rates of the hyperbolic bp learning algorithms are high even if the fully activation functions are used and discover that the hyperbolic bp learning algorithm for the hyperbolic nn with the split type hyperbolic activation function has an ability to learn hyperbolic rotation as its inherent property

on nilpotent multipliers of some verbal products of groups the paper is devoted to finding a homomorphic image for the c nilpotent multiplier of the verbal product of a family of groups with respect to a variety v when v nc or nc v also a structure of the c nilpotent multiplier of a special case of the verbal product the nilpotent product of cyclic groups is given in fact we present an explicit formula for the c nilpotent multiplier of the nth nilpotent product of the group g z n nz nzr1 n nzrt where ri 1 divides ri for all i 1 i t 1 and p r1 1 for any prime p less than or equal to n c for all positive integers n c

on the relation between upper central quotients and lower central series of a group let h be a group with a normal subgroup n contained in the upper central subgroup zch in this article we study the influence of the quotient group g h n on the lower central subgroup yc lh in particular for any finite group g we give bounds on the order and exponent of yc l h for g equal to a dihedral group or quaternion group or extra special group we list all possible groups that can arise as yc lh our proofs involve i the baer invariants of g ii the schur multiplier m l g of g relative to a normal subgroup l and iii the nonabelian tensor product of groups some results on the nonabelian tensor product may be of independent interest

the solution of singularly perturbed semilinear elliptic problems by iterative domain decomposition algorithms abstract parallel domain decomposition algorithms for the solution of singularly perturbed semilinear elliptic problems are considered convergence properties and numerical stability of these iterative algorithms are established numerical experiments are provided

iterative domain decomposition algorithms for the solution of 2 d eddy current problem abstract iterative domain decomposition algorithms are applied to the solution of two dimensional eddy current problem the system of differential equations describing this problem is considered as singularly perturbed problem convergence of these algorithms is established several numerical experiments are presented

iterative algorithms of domain decomposition for the solution of a quasilinear elliptic problem this paper deals with iterative algorithms for domain decomposition applied to the solution of a quasilinear elliptic problem two iterative algorithms are examined the first one is the schwarz alternating procedure and the second algorithm is suitable for parallel computing convergence results are established in the two domain and multidomain decomposition cases some issues of parallel implementation of these algorithms are discussed

on pro p groups with potent filtrations abstract in this note we prove that if g is pf group of finite exponent then the exponent of the second homology group h 2 g m divides the exponent of g for every profinite trivial z g module m we introduce the notion of the exponential rank of a pro p group and find a bound for the exponential rank of a pf group

schur multipliers and power endomorphisms of groups abstract we obtain new bounds for the exponent of the schur multiplier of a given p group we prove that the exponent of the schur multiplier can be bounded by a function depending only on the exponent of a given group as a consequence we show that the exponent of the schur multiplier of any group of exponent four divides eight and that this bound is best possible the notion of the exponential rank of a p group is introduced we show that powerful p groups have exponential rank either zero or one

the unitary cover of a finite group and the exponent of the schur multiplier abstract for a finite group we introduce a particular central extension the unitary cover having minimal exponent among those satisfying the projective lifting property we obtain new bounds for the exponent of the schur multiplier relating to subnormal series and we discover new families for which the bound is the exponent of the group finally we show that unitary covers are controlled by the zelmanov solution of the restricted burnside problem for 2 generator groups

some computations of non abelian tensor products of groups on considere le produit tensoriel generalise g h des groupes g h on presente le calcul de g g

overlapping schwarz methods for a singularly perturbed semilinear elliptic problem and their parallel implementation domain decomposition algorithms based on the schwarz alternating method are applied to the numerical solution of a singularly perturbed semilinear elliptic problem the emphasis is given to the description of the algorithms multidomain schwarz alternating method and two level schwarz method as well as their computer realization on a distributed memory multiprocessor computer numerical experiments related to a model problem illustrate the performance of the algorithms on parallel environment as well as their behavior with respect to the critical parameters such as the perturbation parameter the amount of overlapping and sizes of subdomains application of the two level schwarz method to a model semiconductor problem is also discussed

a second order parameter uniform overlapping schwarz method for reaction diffusion problems with boundary layers the problem of constructing a parameter uniform numerical method for a singularly perturbed self adjoint ordinary differential equation is considered it is shown that a suitably designed discrete schwarz method based on a standard finite difference operator with a uniform mesh on each subdomain gives numerical approximations which converge in the maximum norm to the exact solution uniformly with respect to the singular perturbation parameter this parameter uniform convergence is shown to be essentially second order that this new discrete schwarz method is efficient in practice is demonstrated by numerical experiments

adaptive identifier for uncertain complex nonlinear systems based on continuous neural networks this paper presents the design of a complex valued differential neural network identifier for uncertain nonlinear systems defined in the complex domain this design includes the construction of an adaptive algorithm to adjust the parameters included in the identifier the algorithm is obtained based on a special class of controlled lyapunov functions the quality of the identification process is characterized using the practical stability framework indeed the region where the identification error converges is derived by the same lyapunov method this zone is defined by the power of uncertainties and perturbations affecting the complex valued uncertain dynamics moreover this convergence zone is reduced to its lowest possible value using ideas related to the so called ellipsoid methodology two simple but informative numerical examples are developed to show how the identifier proposed in this paper can be used to approximate uncertain nonlinear systems valued in the complex domain

hyperbolic hopfield neural networks in recent years several neural networks using clifford algebra have been studied clifford algebra is also called geometric algebra complex valued hopfield neural networks chnns are the most popular neural networks using clifford algebra the aim of this brief is to construct hyperbolic hnns hhnns as an analog of chnns hyperbolic algebra is a clifford algebra based on lorentzian geometry in this brief a hyperbolic neuron is defined in a manner analogous to a phasor neuron which is a typical complex valued neuron model hhnns share common concepts with chnns such as the angle and energy however hhnns and chnns are different in several aspects the states of hyperbolic neurons do not form a circle and therefore the start and end states are not identical in the quantized version unlike complex valued neurons hyperbolic neurons have an infinite number of states

a robust overlapping schwarz domain decomposition algorithm for time dependent singularly perturbed reaction diffusion problems in this work we consider a singularly perturbed parabolic problem of reaction diffusion type to solve this problem numerically we develop an overlapping schwarz domain decomposition method where we use the asymptotic behaviour of the exact solution for domain partitioning we prove that the method gives uniform numerical approximations of first order in time and almost second order in space furthermore we address the much faster convergence of the algorithm for small perturbation parameter e to be more specific we prove that when e is small just one iteration is required to achieve the desired accuracy we then extend the method to a system of singularly perturbed parabolic problems of reaction diffusion type numerical experiments support the theoretical results and demonstrate the effectiveness of the method

some notes on the baer invariant of a nilpotent product of groups abstract w haebich 1972 bull austral math soc 7 279 296 presented a formula for the schur multiplier of a regular product of groups in this paper first it is shown that the baer invariant of a nilpotent product of groups with respect to the variety of nilpotent groups has a homomorphic image and in finite case a subgroup of haebich s type second a formula will be presented for the baer invariant of a nilpotent product of cyclic groups with respect to the variety of nilpotent groups

autocommutator subgroups of finite groups abstract we present some new results about the absolute center and autocommutator subgroup of a group concepts which were introduced in an earlier paper

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

two level parallel schwarz methods for singularly perturbed semilinear elliptic problems abstract two level domain decomposition methods based on the schwarz alternating procedure are proposed for the numerical solution of singularly perturbed semilinear elliptic problems convergence properties of the two level schwarz methods are established numerical experiments for a test problem illustrate the behaviour of the methods with respect to the critical parameters such as the perturbation parameter the amount of overlapping and sizes of subdomains application of one of the two level schwarz methods to a model semiconductor problem is also discussed

some baer invariants of free nilpotent groups abstract we present an explicit structure for the baer invariant of a free nth nilpotent group the nth nilpotent product of infinite cyclic groups z n z n n z with respect to the variety v with the set of words v c 1 1 c 2 1 for all c 1 c 2 and 2 c 2 c 1 2 n 2 also an explicit formula for the polynilpotent multiplier of a free nth nilpotent group is given for any class row c 1 c 2 c t where c 1 n

iterative domain decomposition algorithms for the solution of singularly perturbed parabolic problems abstract a combination of a time discretization and domain decomposition methods for the solution of singularly perturbed semilinear parabolic problems is considered on this basis iterative algorithms suitable for parallelization are constructed convergence properties of these algorithms are established the implementation of the algorithms on a shared memory multiprocessor is examined numerical results for a test singularly perturbed problem are presented

computational method for a singular perturbation problem via domain decomposition and its parallel implementation abstract in this paper a computation method for the nonlinear problem with a small perturbation parameter via domain decomposition is presented we apply a combination of the two approaches to solving the problem iterative algorithms for domain decomposition and a grid refinement technique on subdomains two iterative algorithms are examined the first one is the schwarz alternating procedure and the second algorithm is highly suitable for parallel computing numerical examples are provided