l2 analytic torsion abstract in this paper we introduce a new differential invariant called l2 analytic torsion for closed manifolds with positive decay and whose universal covers have trivial l2 cohomology l2 analytic torsion can be thought of as a suitable generalization of the ray singer analytic torsion we establish various functorial properties of l2 analytic torsion and also compute it for odd dimensional closed hyperbolic manifolds with the help of results from fried

semi invariants and their use for stability analysis of planar systems semi invariants and relative characteristic functions extend to nonlinear systems the concept of eigenvector eigenvalue pair for linear systems and are therefore very useful to depict the behaviour of the system in this article semi invariants are used to construct explicitly lyapunov functions useful for studying the stability of the origin for continuous time systems moreover using well known tools from differential geometry such as orbital symmetries it is shown how semi invariants can be found for several classes of systems important connections with centre manifold theory are pointed out by using the proposed general techniques a new proof of a result claimed to bendixson bendixson i 1901 sur les courbes definies par des equations differentielles acta mathematics 24 1 88 is given

quadratic model for reservoir management application to the central valley project a quadratic optimization model is applied to a large scale reservoir system to obtain operation schedules the model has the minimum possible dimensionality treats spillage and penstock releases as decision variables and takes advantage of system dependent features to reduce the size of the decision space an efficient and stable quadratic programming active set algorithm is used to solve for the optimal release policies the stability and convergence of the solution algorithm are ensured by the factorization of the reduced hessian matrix and the accurate computation of the lagrange multipliers the quadratic model is compared with a simplified linear model and it is found that optimal release schedules are robust to the choice of model both yielding an increase of nearly 27 in the total annual energy production with respect to conventional operation procedures although the quadratic model is more flexible and of general applicability the adequate fulfillment of other system functions such as flood control and water supply is guaranteed via constraints on storage and spillage variables

the hyperelliptic limit cycles of the lienard systems abstract for lienard systems x y y f m x y g n x with f m and g n real polynomials of degree m and n respectively in h zoladek algebraic invariant curves for the lienard equation trans amer math soc 350 1998 1681 1701 the author showed that if m 3 and m 1 n 2 m there always exist lienard systems which have a hyperelliptic limit cycle llibre and zhang j llibre xiang zhang on the algebraic limit cycles of lienard systems nonlinearity 21 2008 2011 2022 proved that the lienard systems with m 3 and n 5 have no hyperelliptic limit cycles and that there exist lienard systems with m 4 and 5 n 8 which do have hyperelliptic limit cycles so it is still an open problem to characterize the lienard systems which have an algebraic limit cycle in cases m 4 and m 1 n 2 m in this paper we will prove that there exist lienard systems with m 5 and m 1 n 2 m which have hyperelliptic limit cycles

an approach to parameter estimation and stochastic control in water resources with an application to reservoir operation this paper presents an algorithm for the estimation of parameters in state space models that represent hydrologic processes in which the state variables are observed with error the algorithm is based on the maximization of the conditional expectation of the likelihood function of the state equation the estimation algorithm is numerically stable and guarantees local convergence under mild conditions it is also shown that the estimation algorithm can be coupled with an optimal control method to yield a combined control estimation technique that can be easily implemented an application of the theory and methods developed herein is given for flood routing via reservoir operation

optimal multireservoir network control by the discrete maximum principle an optimal control algorithm based on the discrete maximum principle is applied to multireservoir network control variable metric techniques are used for direct solution of the resulting two point boundary value problem state variable constraints are considered by use of penalty functions computer time and storage requirements are moderate and increase roughly linearly with the problem dimension application to a 10 reservoir network demonstrates efficacy of the proposed algorithm

torsion homology growth and cycle complexity of arithmetic manifolds let m be an arithmetic hyperbolic 3 manifold such as a bianchi manifold we conjecture that there is a basis for the second homology of m where each basis element is represented by a surface of low genus and give evidence for this we explain the relationship between this conjecture and the study of torsion homology growth

number of limit cycles of the lienard equation in this paper we study a lienard system of the form x 5y2f x y52x where f x is an odd polyno mial we introduce a method that gives a sequence of algebraic approximations to the equation of each limit cycle of the system this sequence seems to converge to the exact equation of each limit cycle we obtain also a sequence of polynomials rn x whose roots of odd multiplicity are related to the number and location of the limit cycles of the system s1063 651x 97 02809 2

improved dynamic programming methods for optimal control of lumped parameter stochastic systems new dynamic programming methods are developed to solve stochastic control problems with a larger number of state variables than previously possible these methods apply accurate interpolation to numerical approximation of continuous cost to go functions greatly reducing the number of discrete states that must be evaluated by efficiently incorporating information on first and second derivatives the approximation reduces computational effort by several orders of magnitude over traditional methods consequently it is practical to apply dynamic programming to complex stochastic problems with a larger number of state variables than traditionally possible results are presented for hypothetical reservoir control problems with up to seven state variables and two random inputs

a note on relaxation oscillators with exact limit cycles abstract in this note we give a family of planar polynomial differential systems with a prescribed hyperbolic limit cycle this family constitutes a corrected and wider version of an example given in the work m a abdelkader relaxation oscillators with exact limit cycles j math anal appl 218 1998 308 312 the result given in this note may be used to construct models of lienard differential equations exhibiting a desired limit cycle

on the growth of torsion in the cohomology of arithmetic groups in this paper we consider certain families of arithmetic subgroups of mathrm so 0 p q and mathrm sl 3 mathbb r respectively we study the cohomology of such arithmetic groups with coefficients in arithmetically defined modules we show that for natural sequences of such modules the torsion in the cohomology grows exponentially

darboux integrability and the inverse integrating factor abstract we mainly study polynomial differential systems of the form dx dt p x y dy dt q x y where p and q are complex polynomials in the dependent complex variables x and y and the independent variable t is either real or complex we assume that the polynomials p and q are relatively prime and that the differential system has a darboux first integral of the form h f 1 1 f p p exp h 1 g 1 n 1 1 exp h q g q n q q where the polynomials f i and g j are irreducible the polynomials g j and h j are coprime and the i and j are complex numbers when i 1 p and j 1 q prelle and singer proved that these systems have a rational integrating factor we improve this result as follows assume that h is a rational function which is not polynomial following to poincare we define the critical remarkable values of h then we prove that the system has a polynomial inverse integrating factor if and only if h has at most two critical remarkable values under some assumptions over the darboux first integral h we show first that the system has a polynomial inverse integrating factor and secondly that if the degree of the system is m the homogeneous part of highest degree of h is a multi valued function and the functions exp h j g j are exponential factors for j 1 q then the system has a polynomial inverse integrating factor of degree m 1 we also present versions of these results for real polynomial differential systems finally we apply these results to real polynomial differential systems having a darboux first integral and limit cycles or foci

constrained state formulation for the stochastic control of multireservoir systems a new formulation is presented for the analysis of multireservoir systems based on the development of first and second moment expressions for the stochastic storage state variables these expressions give explicit consideration to the maximum and minimum storage bounds in the reservoir system a feature not incorporated in some existing formulations based on traditional control theory using this analysis expected values of the storage states variances of storage release policies reliability levels and failure probabilities useful information in the context of reservoir operations and design can be obtained from a nonlinear programming solution the approach does not involve any discretization of the system variables the results for the means and standard deviations of the storage states for a multireservoir system compare favorably with those obtained from simulation

an inexact probabilistic possibilistic optimization framework for flood management in a hybrid uncertain environment flooding is one of the leading causes of loss due to natural catastrophes and at least one third of all losses due to natural forces can be attributed to flooding flood management systems involve a variety of complexities such as multiple uncertainties dynamic variations and policy implications this paper presents an inexact probabilistic possibilistic programming with fuzzy random coefficients ipp frc model for flood management in a hybrid uncertain environment ipp frc is capable not only of tackling multiple uncertainties in the form of intervals with fuzzy random boundaries but of addressing the dynamic complexity through capacity expansion planning within a multi region multi flood level and multi option context the possibility and necessity measures used in ipp frc are suitable for risk seeking and risk averse decision making respectively a case study is used to demonstrate the applicability of the proposed methodology for facilitating flood management the results indicate that the inexact degrees of possibility and necessity would decrease with increased probabilities of occurrence implying a potential tradeoff between fulfillment of objectives and associated risks a number of decision alternatives can be obtained under different policy scenarios they are helpful for decision makers to formulate the appropriate flood management policy according to practical situations the performance of ipp frc is analyzed and compared with a possibility based fractile model

on the period function of lienard systems abstract we study the period functiontof a centeroof a lienard system a sufficient condition for the monotonicity oft or for the isochronicity ofo is given such a condition is also necessary whenfandgare analytic andgis odd in this case a characterization of isochronous centers of lienard systems is given strict monotonicity and global monotonicity oftare also investigated

analytic torsion of complete hyperbolic manifolds of finite volume abstract in this paper we define the analytic torsion for a complete oriented hyperbolic manifold of finite volume it depends on a representation of the fundamental group for manifolds of odd dimension we study the asymptotic behavior of the analytic torsion with respect to certain sequences of representations obtained by restriction of irreducible representations of the group of isometries of the hyperbolic space to the fundamental group

a new method for the real time operation of reservoir systems this paper introduces a new method for the real time operation of reservoir systems the system is represented by a set of stochastic differential equations describing the reservoir and river dynamics in state space form the formulated reservoir operation problem calls for finding policies which maximize the expected benefits of one system s objective while satisfying the remaining objectives at prespecified reliability levels the solution is obtained by a new method named extended linear quadratic gaussian elqg controller elqg draws on and extends stochastic control theory results and it is well suited for the optimization of constrained dynamical systems it is a trajectory iteration algorithm theoretically expected to exhibit reliability and computational efficiency the new method is tested and evaluated in a real world case study

maximum amplitude of limit cycles in lienard systems we establish sufficient criteria for the existence of a limit cycle in the lienard system x over y f x y over x where f x is odd in their simplest form the criteria lead to the result that for all finite nonzero the amplitude of the limit cycle is less than and 0 a u where f a 0 and 0 u f x dx 0 we take the van der pol oscillator as a specific example and establish that for all finite nonzero the amplitude of its limit cycle is less than 2 0672 a value whose precision is limited by the capacity of our symbolic computation software package we show how the criterion for the upper bound can be extended to establish a bound on the amplitude of a limit cycle in systems where f x contains both odd and even components we also show how the criteria can be used to establish bounds for bifurcation sets

explicit non algebraic limit cycles for polynomial systems we consider a system of the form email protected p n x y xr m x y email protected q n x y yr m x y where p n x y q n x y and r m x y are homogeneous polynomials of degrees n n and m respectively with n

computing centre conditions for certain cubic systems abstract we present necessary and sufficient conditions for a critical point of certain two dimensional cubic differential systems to be a centre extensive use of the computer algebra system reduce is involved the search for necessary and sufficient conditions for a centre has long been of considerable interest in the theory of nonlinear differential equations it has proved to be a difficult problem and full conditions are known for very few classes of systems such conditions are also required in the investigation of hilbert s sixteenth problem concerning the number of limit cycles of polynomial systems

heat kernels on covering spaces and topological invariants it is well known that there are relationships between the heat flow acting on differential forms on a closed oriented manifold m and the topology of m from hodge theory one can recover the betti numbers of m from the heat flow furthermore ray and singer 42 defined an analytic torsion a smooth invariant of acyclic flat bundles on m and conjectured that it equals the classical reidemeister torsion this conjecture was proved to be true independently by cheeger 7 and muller 40 the analytic torsion is nonzero only on odd dimensional manifolds and behaves in some ways as an odd dimensional counterpart of the euler characteristic 20 if m is not simply connected then there is a covering space analog of the betti numbers using the heat flow on the universal cover m one can define the l2 betti numbers of m 1 by taking the trace of the heat kernel not in the ordinary sense but as an element of a certain type ii von neumann algebra more concretely this amounts to integrating the local trace of the heat kernel over a fundamental domain in m in 11 we summarize this theory we consider the covering space analog of the analytic torsion this i analytic torsion has the same relation to the i cohomology as the ordinary analytic torsion bears to de rham cohomology in 111 we define the i analytic torsion 9 m under a technical assumption which we discuss later and prove its basic properties we show that m is a smooth invariant of manifolds whose l2 betti numbers vanish the proof is similar to that of the analogous statement for the ordinary analytic torsion but requires some care because of the possible slow decay of the heat kernels for large time in order to know if there are interesting examples of m we compute it in the case where m admits a hyperbolic metric it is clear that is proportionate to the volume of the hyperbolic metric which is

irrigation control in the presence of salinity extended linear quadratic approach an intraseasonal irrigation scheduling problem is dealt with via extended linear quadratic elq control the elq control is well suited for constrained multidimensional problems and provides openloop feedback control rules over the control horizon a conceptual model is developed to describe the dynamics of water allocation and salt movement in the root zone of a crop moisture stress and osmotic stress are combined to obtain the integrated inhibitory effect of salinity on transpiration for the intraseasonal model to be effective against perennial salt accumulation in the root zone it should be able to yield control laws which will lead to favorable root zone conditions at the end of an irrigation season thus avoiding any significant leaching prior to the next growing season this long term aspect of salinity control is handled via probabilistic state constraints which impose desired salinity and moisture levels with desired confidence level the elq control is employed in a case study of expected net benefit maximization over an irrigation season of corn in fort morgan colorado the results in general correspond well with expected irrigation schedules under different conditions and provide valuable information on both short and long term aspects of irrigation control under saline conditions the elq control being an analytic iterative solution scheme with theoretically guaranteed fast convergence has a distinct computational advantage over state of the art procedures

stochastic control of reservoir systems using indicator functions new enhancements 1 in our previous works deterministic release policies were considered for the development of approximations of the two lower moments of the storage volume defined by the dynamic equation of the reservoir in discrete time but in continuous state space important innovation in that work was the incorporation of the lower and upper bounds of reservoir systems into the dynamic equation for the storage volume using indicator functions the current work which also does not use discretization looks at an extension of previous developments that incorporates standard operating policies and also a new randomized release policy both of which make the moments calculations exact under the assumptions that 1 the sum of current random inflow and the previous storage volume can be described by just the two lower moments and 2 only the means and variances of the inflows are known first and second moment expressions are derived for the stochastic storage state variable and include terms for the failure probabilities probabilities of spills or deficits expected values of the storage state variances of storage release policies and failure probabilities are obtained by solving the optimal reservoir operations problem using nonlinear programming the various statistics thus obtained from this optimization compare extremely well with those obtained from simulation for the single reservoir monthly operations problem studied the exact characterization of the mean and variance of the storage state variable is derived which is a difficulty in existing formulations based on linear quadratic gaussian methods for example the latter methods have been unable to estimate these moments reasonably accurately especially for long term operations whereas the traditional storage theory based on discretization brings on the curse of dimensionality the presentation herein is directed to both traditional reservoir storage theorists who are interested in the design of a reservoir where estimating the probabilities of spills and deficits is important and modern reservoir analysts who are interested in multiperiod optimal release decisions in the operations of reservoirs

real time forecasting and daily operation of a multireservoir system during floods by linear quadratic gaussian control the problem of short term daily operation of a multireservoir system during floods is examined the problem is how to use all available real time information to solve the tactical problem of regulating reservoirs to minimize the expected value of flood damage during a relatively short operating horizon and to do so consistently with the long term operating strategy a state space mathematical model is presented for short term forecasting of river flows and description of the dynamics of a multireservoir system the model includes reduced order state space unit hydrographs to forecast storm runoff from effective rainfall and state space representations of linear flood routing the optimization problem is solved in the framework of discrete time linear quadratic gaussian lqg control lqg control is a computationally efficient and flexible method of solving the large dimensional optimization problems associated with the solution of the tactical problem of short term operation of reservoirs for application of the lqg control methodology the system must be represented in state space form with quadratic cost functions subject to linear equality constraints given by the state transition equations the scheme was found to be suitable for operation under moderate flood conditions when capacity constraints are not likely to become binding the methodology has been applied and tested in a real world case

algorithmic derivation of centre conditions the conditions for a critical point of a polynomial differential system to be a centre are of particular significance because of the frequency with which they are required in applications we demonstrate how computer algebra can be effectively employed in the search for necessary and sufficient conditions for critical points of such systems to be centres we survey recent developments and illustrate our approach by means of examples

on the torsion in the cohomology of arithmetic hyperbolic 3 manifolds in this paper we consider the cohomology of a closed arithmetic hyperbolic 3 manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of sl 2 c the cohomology is defined over the integers and is a finite abelian group we show that the order of the 2nd cohomology grows exponentially as the local system grows we also consider the twisted ruelle zeta function of a closed arithmetic hyperbolic 3 manifold and we express the leading coefficient of its laurent expansion at the origin in terms of the orders of the torsion subgroups of the cohomology

variational approach to a class of nonlinear oscillators with several limit cycles we study limit cycles of nonlinear oscillators described by the equation x nuf x x 0 with f an odd function depending on the nonlinearity this equation may exhibit one or more limit cycles we show that limit cycles correspond to relative extrema of a certain functional analytical results in the limits nu 0 and nu infinity are in agreement with previously known criteria for intermediate nu numerical determination of the limit cycles can be obtained

risk based factorial probabilistic inference for optimization of flood control systems with correlated uncertainties abstract in this paper a risk based factorial probabilistic inference method is proposed to address the stochastic objective function and constraints as well as their interactions in a systematic manner to tackle random uncertainties decision makers risk preferences are taken into account in the decision process statistical significance for each of the linear nonlinear and interaction effects of risk parameters is uncovered through conducting a multi factorial analysis the proposed methodology is applied to a case study of flood control to demonstrate its validity and applicability a number of decision alternatives are obtained under various combinations of risk levels associated with the objective function and chance constraints facilitating an in depth analysis of trade offs between economic outcomes and associated risks dynamic complexities are addressed through a two stage decision process as well as through capacity expansion planning for flood diversion within a multi region multi flood level and multi option context findings from the factorial experiment reveal the multi level interactions between risk parameters and quantify their contributions to the variability of the total system cost the proposed method is compared against the fractile criterion optimization model and the chance constrained programming technique respectively

optimal short term hydro scheduling from the principle of progressive optimality this paper presents an algorithm based on the principle of progressive optimality for determining the optimal short term scheduling of multireservoir power systems the method takes into account water head variations spilling and time delays between upstream and downstream reservoirs the method is computationally efficient and has minimal storage requirements the convergence is monotonic and a global solution is reached contrary to dynamic programing the state variables do not have to be discretized with this method an example consisting of four hydroplants in series is solved and the results are presented

perturbation analysis of the limit cycle of the free van der pol equation a power series expansion in the damping parameter varepsilon of the limit cycle u t varepsilon of the free van der pol equation ddot u varepsilon u 2 1 dot u u 0 is constructed and analyzed coefficients in the expansion are computed up to o varepsilon 24 in exact rational arithmetic using the symbolic manipulation system macsyma and up to o varepsilon 163 using a fortran program the series is analyzed using pade approximants the convergence of the series for the maximum amplitude of the limit cycle is limited by two pairs of complex conjugate singularities in the complex varepsilon plane these singularities are the same as those which limit the convergence of the series expansion of the frequency of the limit cycle a new expansion parameter is introduced which maps these singularities to infinity and leads to a new expansion for the amplitude which converges for all real values of varepsilon amplitudes computed from this transformed series agree very we

relaxation oscillators with exact limit cycles as an inverse problem for relaxation oscillators modeled by the autonomous lienard differential equation we assume plausible exact expressions for the limit cycles in the phase plane which correspond to periodic solutions toward which all solutions converge or from which they recede as time tends to infinity we illustrate this by two examples one of which concerns an approximation to a van der pol oscillator the results could be utilized for the synthesis problem i e the construction of oscillator circuits yielding desired periodic behavior

error analysis of conventional discrete and gradient dynamic programming an asymptotic error analysis of the conventional discrete dynamic programming ddp method is presented and upper bounds of the error in the control policy i e the difference of the estimated and true optimal control at each operation period are computed this error is shown to be of the order of the state discretization interval as a result with significant implications in the optimization of multistate systems where the curse of dimensionality restricts the number of states to a relatively small number the error in the optimal cost varies with as 2 the analysis provides useful insights into the effects of state discretization on calculated control and cost functions the comparability of results from different discretizations and criteria about the required number of nodes in an effort to reduce the discretization error in the case of smooth cost functions a new discrete dynamic programming method termed gradient dynamic programming gdp is proposed gdp uses a piecewise hermite interpolation of the cost to go function at each stage which preserves the values of the cost to go function and of its first derivatives at the discretization nodes the error in the control policy is shown to be of the order of as 3 and the error in the cost to vary with as thus as as decreases gdp converges to the true optimum much more rapidly than ddp another major advantage of the new methodology is that it facilitates the use of newton type iterative methods in the solution of the nonlinear optimization problems at each stage the linear convergence of ddp and the superlinear convergence of gdp are illustrated in an example

the torsion in symmetric powers on congruence subgroups of bianchi groups in this paper we prove that for a fixed neat principal congruence subgroup of a bianchi group the order of the torsion part of its second cohomology group with coefficients in an integral lattice associated to the m th symmetric power of the standard representation of sl 2 c grows exponentially in m 2 we give upper and lower bounds for the growth rate our result extends a a result of w muller and s marshall who proved the corresponding statement for closed arithmetic 3 manifolds to the finite volume case we also prove a limit multiplicity formula for combinatorial reidemeister torsions on higher dimensional hyperbolic manifolds

extended linear quadratic gaussian control further extensions the extended linear quadratic gaussian elqg control method georgakakos and marks 1987 is a stochastic control algorithm for the optimal operation of multiobjective reservoirs mathematically this method optimizes a general functional of a stochastic system in state space form with upper and lower release constraints and probabilistic storage bounds elqg is a sequential algorithm which accounts for stochastic effects by preserving the first two statistical moments of the system s inputs and storages in this paper the method is first extended to handle nongaussian features which frequently characterize reservoir inputs second elqg s efficiency with respect to reliability storage constraints is discussed and a new barrier function method is researched these modifications are tested in case studies with the savannah river system

a new criterion for controlling the number of limit cycles of some generalized lienard equations abstract we consider a class of planar differential equations which include the lienard differential equations by applying the bendixson dulac criterion for l connected sets we reduce the study of the number of limit cycles for such equations to the condition that a certain function of just one variable does not change sign as an application this method is used to give a sharp upper bound for the number of limit cycles of some lienard differential equations in particular we present a polynomial lienard system with exactly three limit cycles

new control concepts for uncertain water resources systems 1 theory a major complicating factor in water resources systems management is handling unknown inputs stochastic optimization provides a sound mathematical framework but requires that enough data exist to develop statistical input representations in cases where data records are insufficient e g extreme events or atypical of future input realizations stochastic methods are inadequate this article presents a control approach where input variables are only expected to belong in certain sets the objective is to determine sets of admissible control actions guaranteeing that the system will remain within desirable bounds the solution is based on dynamic programming and derived for the case where all sets are convex polyhedra a companion paper yao and georgakakos this issue addresses specific applications and problems in relation to reservoir system management

the periodic solution of van der pol s equation we give the periodic solution of the van der pol equation ddot x x varepsilon 1 x 2 dot x in the form of a series converging for all values of the damping parameter varepsilon for varepsilon small the series solution reduces to a perturbation series in powers of varepsilon and is obtained from this series essentially by the analytical continuation method using a suitable transformation of the parameter the coefficients of the perturbation series and by virtue of the simplicity of the transformation proposed those of the transformed series as well are given in explicit form by recurrent analytical formulae so that both series can be developed in principle up to any order in varepsilon actually the coefficients in the expansions can be obtained by symbolic computation in exact rational number form up to a very high order the only limitation being the available computing resources

exponential growth of torsion in the cohomology of arithmetic hyperbolic manifolds for d 2n 1 a positive odd integer we consider sequences of arithmetic subgroups of text so 0 d 1 and yielding corresponding hyperbolic manifolds of finite volume and show that under appropriate and natural assumptions the torsion of the associated cohomology groups grows exponentially

analytic torsion and r torsion for unimodular representations let m be a closed c manifold of dimension n both r torsion and analytic torsion are smooth invariants of acyclic orthogonal or unitary representations p of the fundamental group 7i m the reidemeister franz torsion or r torsion tm p of p is defined in terms of the combinatorial structure of m given by its smooth triangulations the analytic torsion tm p was introduced by ray and singer rs as an analytic counterpart of r torsion in order to define the analytic torsion one has to choose a riemannian metric on m then tm p is a certain weighted alternating product of regularized determinants of the laplacians on differential qforms of m with values in the flat bundle e defined by p it was conjectured by ray and singer rs that tm p trm p for all acyclic orthogonal or unitary representations p this conjecture was proved independently by cheeger c and the author mu the restriction to orthogonal or unitary representations is certainly a limitation of the applicability of this result if 7i m is infinite because an infinite discrete group will have in general many nonorthogonal finite dimensional representations it is the purpose of the present paper to remove this limitation we call a representation p r m gl e on a finite dimensional real or complex vector space e unimodular if i det p y i 1 for all y e i m then we define r torsion and analytic torsion for unimodular representations and the main result is that for odd dimensional manifolds m the equality of the two torsions extends to all unimodular representations

the analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume abstract in this paper we study the regularized analytic torsion of finite volume hyperbolic manifolds we consider sequences of coverings x i of a fixed hyperbolic orbifold x 0 our main result is that for certain sequences of coverings and strongly acyclic flat bundles the analytic torsion divided by the index of the covering converges to the l 2 torsion our results apply to certain sequences of arithmetic groups in particular to sequences of principal congruence subgroups of so 0 d 1 z and to sequences of principal congruence subgroups or hecke subgroups of bianchi groups

realization problems for limit cycles of planar polynomial vector fields abstract we show that for any finite configuration of closed curves r 2 one can construct an explicit planar polynomial vector field that realizes up to homeomorphism as the set of its limit cycles with prescribed periods multiplicities and stabilities the only obstruction given on this data is the obvious compatibility relation between the stabilities and the parity of the multiplicities the constructed vector fields are darboux integrable and admit a polynomial inverse integrating factor

minimizing variance of reservoir systems operations benefits using soft computing tools soft computing based tools including fuzzy inference systems fis artificial neural networks ann and genetic algorithms ga are used here to tackle the minimization of variance of benefits from reservoir operation variance reduction is a very hard optimization problem and solvable only using implicit methods like simulation especially if the problem is nonlinear first a recently developed stochastic optimization method develops the optimal release policy which is simply the recommended release in each season of the system whose objective function maximizes the expected benefits the policy is then simulated for a long inflow series to provide the trajectories of optimal releases and storages of the reservoir these trajectories are then used as input output data to train an adaptive neuro fuzzy inference system anfis to obtain updated fuzzy operating rules the anfis based fuzzy rules are simulated and compared with policies developed using a multiple regression analysis a commonly used method in water resources optimization as the anfis performed better further a parameterized t norm operator is applied and its parameters numbering only two are optimized through ga but with the objective of variance reduction in the benefits achieved results compare the better performance of the anfis based policies with other methods such as stochastic dynamic programming and the original stochastic method to demonstrate the usefulness of ga optimized parameters of a t norm fuzzy operator for variance reduction

improving a method for the study of limit cycles of the lienard equation in recent papers we have introduced a method for the study of limit cycles of the li enard system mathrm x ifmmode dot else fi y ensuremath f x mathrm y ifmmode dot else fi ensuremath x where f x is an odd polynomial the method gives a sequence of polynomials r n x whose roots are related to the number and location of the limit cycles and a sequence of algebraic approximations to the bifurcation set of the system in this paper we present a variant of the method that gives very important qualitative and quantitative improvements

lienard systems limit cycles and melnikov theory lienard systems constitute a general class of two dimensional autonomous systems among which the van der pol equation is found recently giacomini and neukirch phys rev e 57 3809 1997 introduced a sequence of polynomials whose roots are related to the number and location of limit cycles of lienard systems we show that in the limit of these sequences the same information is given by a polynomial which melnikov theory associates with a given lienard system and discuss the relationship existing among them s1063 651x 98 00201 3

on a classification of polynomial differential operators with respect to the type of first integrals abstract this paper gives a classification of polynomial differential operators x x 1 x 1 x 2 1 x 2 x 1 x 2 2 i x i the classification is defined through an order derived from x let x x y be the associated differential polynomial the order is defined as the order of a differential ideal that is an essential extension of x the main result shows the order can only be four possible values 0 1 2 3 or furthermore when the order is finite the essential extension x a where a is a differential polynomial with coefficients obtained through a rational solution of a partial differential equation given explicitly by coefficients of x when the order is infinite the extension is identical with x in addition if and only if the order is 0 1 or 2 the associated polynomial differential equation has liouvillian first integrals examples and connections with godbillon vey sequences are also discussed