the number of small amplitude limit cycles in arbitrary polynomial systems abstract in this paper we study the number of small amplitude limit cycles in arbitrary polynomial systems it is found that almost all the results for the number of small amplitude limit cycles are obtained by calculating lyapunov constants and determining the order of the corresponding hopf bifurcation it is well known that the difficulty in calculating the lyapunov constants increases with the increasing of the degree of polynomial systems so it is necessary and valuable for us to achieve some general results about the number of small amplitude limit cycles in arbitrary polynomial systems with degree m which is denoted by m m in this paper by applying the method developed by c christopher and n lloyd in 1995 and m han and j li in 2012 we first obtain the lower bounds for m 6 m 14 and then prove that m m m 2 if m 23 finally we obtain that m m grows as least as rapidly as 18 25 1 2 ln 2 m 2 2 ln m 2 for all large m it is proved by m han j li lower bounds for the hilbert number of polynomial systems j differential equations 252 2012 3278 3304 that the number of all limit cycles in arbitrary polynomial systems with degree m grows as least as rapidly as 1 2 ln 2 m 2 2 ln m 2

control oriented system identification a worst case deterministic approach in h sub infinity the authors formulate and solve two related control oriented system identification problems for stable linear shift invariant distributed parameter plants in each of these problems the assumed a priori information is minimal consisting only of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant and an upper bound on the noise level the first of these problems involves identification of a point sample of the plant frequency response from a noisy finite output time series obtained in response to an applied sinusoidal input with frequency corresponding to the frequency point of interest this problem leads naturally to the second problem which involves identification of the plant transfer function in h sub infinity from a finite number of noisy point samples of the plant frequency response concrete plans for identification algorithms are provided for each of these two problems

on fermionic novikov algebras novikov algebras were introduced in connection with the poisson brackets of hydrodynamic type and hamiltonian operators in formal variational calculus they are a class of left symmetric algebras with commutative right multiplication operators which can be viewed as bosonic fermionic novikov algebras are a class of left symmetric algebras with anti commutative right multiplication operators they correspond to a certain hamiltonian superoperator in a supervariable in this paper we commence a study on fermionic novikov algebras from the algebraic point of view we will show that any fermionic novikov algebra in dimension 3 must be bosonic moreover we give the classification of real fermionic novikov algebras on four dimensional nilpotent lie algebras and some examples in higher dimensions as a corollary we obtain kinds of four dimensional real fermionic novikov algebras which are not bosonic all of these examples will serve as a guide for further development including the application in physics

bifurcation of small limit cycles in z5 equivariant planar vector fields of order 5 abstract in this paper we consider bifurcation of small limit cycles from hopf type singular points in z 5 equivariant planar vector fields of order 5 we apply normal form theory and the technique of solving coupled multivariate polynomial equations to prove that the maximal number of small limit cycles that such vector fields can have is 25 in addition we show that no large limit cycles exist thus h 5 25 where h n denotes the hilbert number of the n th degree polynomial vector fields this improves the best result of h 5 24 existing in the current literature

on the time complexity of worst case system identification in this paper we treat a general worst case system identification problem this problem is worst case with respect to both noise and system modeling uncertainty we consider this problem under various a priori information structures we determine bounds on the minimum duration identification experiment that must be run to identify the plant to within a specified guaranteed worst case error bound our results are algorithm independent we show that this minimum duration is prohibitively long based on our results we suggest that worst case with respect to noise system identification requires unrealistic amounts of experimental data

computation of normal forms via a perturbation technique considering qualitative behavior of a non linear dynamical system often leads to first simplifying the differential equations or finding their normal forms a perturbation technique for computing normal forms is presented this technique associated with the method of multiple scales can be used to systematically find a unique form for a given set of differential equations the technique is discussed in detail through the analysis of hopf bifurcation it is shown that for hopf bifurcation the method only requires solving two dimensional matrix systems for any higher order normal forms of a generaln dimensional system with the aid of the symbolic language maple this approach is straightforward and is computationally efficient and fast furthermore a simple verification scheme is given for verifying the normal forms and associated non linear transformations obtained using any methodology examples are presented to demonstrate the applicability of the perturbation technique and the computation efficiency of the maple program

identification and application of bounded parameter models abstract the place of parameter bounding algorithms in identification methodology is discussed such algorithms provide a radical alternative to the computation of parameter point estimates and covariances instead of p d f or mean and covariance for the noise and prior parameter estimates they require bounds in an essentially deterministic model formulation the paper is a preliminary review of some standard tasks namely experiment design testing for outliers toleranced prediction and worst case control design in the context of parameter bounding identification computational requirements of these tasks are examined and the limitations of existing parameter bounding algorithms are noted

the real cohomology of virtually nilpotent groups abstract in this paper we present a method to compute the real cohomology of any finitely generated virtually nilpotent group the main ingredient in our setup consists of a polynomial crystallographic action of this group as any finitely generated virtually nilpotent group admits such an action which can be constructed quite easily the approach we present applies to all these groups our main result is an algorithmic way of computing these cohomology spaces as a first application we prove a kind of poincare duality also in the nontorsion free case and we derive explicit formulas in the virtually abelian case

quasi filiform lie algebras of maximum length revisited abstract we revisit the problem of classification of quasi filiform lie algebras of maximum length the lists found in the literature are either incomplete or non explicit en route we touch upon the classification of graded filiform lie algebras generated by two elements of degree one and three respectively and as a byproduct obtain a classification of n graded infinite dimensional lie algebras with one dimensional homogeneous components of coclass two

detection function method and its application to a class of quintic hamiltonian systems with quintic perturbations by using the bifurcation theory of planar dynamical systems and the method of detection functions to investigate the bifurcation of limit cycles of a perturbed quintic hamiltonian system it is shown that under two different sets of controlled parameters the given system has at least 23 limit cycles having two different configurations of the compound eyes respectively

worst case system identification in h sub spl infin validation of a priori information essentially optimal algorithms and error bounds resolves several important open issues pertaining to a worst case control oriented system identification problem known as identification in h sub spl infin first a method is presented for developing confidence that certain a priori information available for identification is not invalid this method requires the solution of a certain nondifferentiable convex program second an essentially optimal identification algorithm is constructed this algorithm is worst case strongly optimal to within a factor of two finally new upper and lower bounds on the optimal identification error are derived and used to estimate the identification error associated with the given algorithm interestingly the development of each of these results draws heavily upon the classical nevanlinna pick interpolation theory as such the authors results establish a clear link between the areas of system identification and optimal interpolation theory both the formulation and techniques in this paper are applicable to problems where the frequency data available for identification may essentially be arbitrarily distributed

optimal asymptotic identification under bounded disturbances the intrinsic limitation of worst case identification of linear time invariant systems using data corrupted by bounded disturbances when the unknown plant is known to belong to a given model set is studied this is done by analyzing the optimal worst case asymptotic error achievable by performing experiments using any bounded input and estimating the plant using any identification algorithm it is shown that under some topological conditions on the model set there is an identification algorithm which is asymptotically optimal for any input and the optimal asymptotic error is characterized as a function of the inputs these results which hold for any error metric and disturbance norm are applied to three specific identification problems identification of stable systems in the l sub 1 norm identification of stable rational systems in the h sub infinity norm and identification of unstable rational systems in the gap metric for each of these problems the general characterization of optimal asymptotic error is used to find near optimal inputs to minimize the error

bifurcations of limit cycles in a z4 equivariant planar polynomial vector field of degree 7 the present paper intends to explore the number and distribution of limit cycles for a class of perturbed z 4 equivariant planar hamiltonian system of degree 7 the bifurcation theory of planar dynamical systems and the method of detection functions are applied to study the given perturbed system by controlling the perturbation parameters an example of a special perturbed z 4 equivariant vector field having 41 limit cycles is found the distribution of the above 41 limit cycles is also given the results are significant to the study of weaken hilbert s 16th problem and to the acquaintance of the characteristics of higher order equivariant vector fields

on the worst case divergence of the least squares algorithm in this paper we provide a h norm lower bound on the worst case identification error of least squares estimation when using fir model structures this bound increases as a logarithmic function of model complexity and is valid for a wide class of inputs characterized as being quasi stationary with covariance function falling off sufficiently quickly

least squares type algorithms for identification in the presence of modeling uncertainty the celebrated least squares and lms least mean squares are system identification approaches that are easily implementable need minimal a priori assumptions and have very nice identification properties when the uncertainty in measurements is only due to noises and not due to unmodeled behavior of the system when there is uncertainty present due to an unmodeled part of the system as well however the performance of these algorithms can be poor here the authors propose a modified weighted least squares algorithm that is geared toward identification in the presence of both unmodeled dynamics and measurement disturbances the algorithm uses very little a priori information and is easily implementable in a recursive fashion through an example the authors demonstrate the improved performance of the proposed approach motivated by a certain worst case property of the lms algorithm an h sub spl infin estimation algorithm is also proposed for the same objective of identification in the presence of modeling uncertainty

bifurcations of limit cycles in equivariant quintic planar vector fields abstract in this paper we obtain 23 limit cycles for a z 3 equivariant near hamiltonian system of degree 5 which is the perturbation of a z 6 equivariant quintic hamiltonian system the configuration of these limit cycles is new and different from the configuration obtained by h s y chan k w chung and j li where the unperturbed system is a z 3 equivariant quintic hamiltonian system our unperturbed system is different from the unperturbed systems studied by y wu and m han the limit cycles are obtained by poincare pontryagin theorem and poincare bendixson theorem

limit cycles near a homoclinic loop by perturbing a class of integrable systems abstract in this paper the problem of limit cycle bifurcation is investigated by perturbing a class of integrable systems with a homoclinic loop under the assumption that the homoclinic loop passes through a degenerate singular point at the origin the asymptotic expansion of the melnikov function along the level curves of the first integral inside the homoclinic loop is studied near the loop meanwhile the formulas for the first coefficients in the expansion are given which can be used to study the number of limit cycles near the homoclinic loop finally an example is provided to demonstrate the obtained results

a transfer function approach to the problems of discrete time systems h sub infinity optimal linear control and filtering a solution is derived to the problems of h sub infinity optimal linear state regulation and filtering the solution method for both problems is based on a transfer function approach which applies standard spectral factorization return difference relations are given which are extensions of the relations associated with the linear quadratic problem

left symmetric algebras for mathfrak gl n we study the classification problem for left symmetric algebras with commutation lie algebra gr n in characteristic 0 the problem is equivalent to the classification of etale affine representations of g n algebraic invariant theory is used to characterize those modules for the algebraic group sl n which belong to affine etale representations of gr n from the classification of these modules we obtain the solution of the classification problem for gl n as another application of our approach we exhibit left symmetric algebra structures on certain reductive lie algebras with a one dimensional center and a non simple semisimple ideal

on the number of limit cycles of a z4 equivariant quintic polynomial system abstract in this paper we study the number of limit cycles of a near hamiltonian system under z 4 equivariant quintic perturbations using the methods of hopf and heteroclinic bifurcation theory we found that the perturbed system can have 13 limit cycles