a variational formulation of the fpk equations with application to the first passage problem in random vibration by using the technique of eigenfunction expansions a variational formulation of the fokker planck kolmogorov fpk equations is developed in which the boundary conditions are incorporated within the functional this allows an approximate solution to the equations to be sought by using the rayleight ritz procedure with trial functions which do not satisfy the often complex boundary conditions application of the method to the first passage problem in random vibration shows that the common assumption of independent outcrossings from the safe region is equivalent to a one term rayleigh ritz solution of the fpk equations in which the exact solutions to the stationary vibration problem are used as trial functions the method is applied to the first passage probability of a linear oscillator and duffing s oscillator and good agreement with published results is found

acceleration of limit periodic continued fractions by the t m transformation we consider limit periodic continued fractions b 0 a 1 1 a n 1 where lim n a n a is finite and not on the ray 1 4 of the negative real axis using related properties of continued fractions we obtain the t m process considered by gray and clark and the process considered by w j thron and h waadeland it follows the possibility of choosing the best process according to the convergence behaviour of the sequences a n

dynamic response of non linear systems to poisson distributed random impulses abstract the dynamic response of non linear systems to external excitations in form of a poisson distributed train of random impulses is considered the state vector of the dynamical system under consideration is a poisson driven markov vector process the equations governing its joint statistical moments are obtained by making use of a generalized ito s differential rule the resulting infinite hierarchy of equations is truncated with the help of a cumulant neglect closure of different orders for a duffing oscillator the time histories of the mean value the variance and the skewness coefficient of the response are evaluated analytically and compared with those obtained from monte carlo simulations it is found that if the departure of the poisson driven excitation process from gaussianity ranges from small to moderately large i e the expected arrival rate of impulses is not very low the application of the fourth order cumulant neglect closure technique i e neglecting the cumulants above the fourth order yields very good estimates of the response mean values and variances

response of nonlinear systems in probability domain using stochastic averaging determination of probability density function pdf of the response for strongly nonlinear single degree of freedom system subjected to both multiplicative and additive random excitations using stochastic averaging technique is faced with few difficulties firstly the size of excitations should be small such that the response of the system converges weakly to a markov process secondly the excitations should preferably be broad banded so that the analytical results are reasonably accurate and finally the nonlinear functions should be integrable if closed formed expressions for the result are desired the above issues are examined in the paper with a view to show a limiting values of the size parameter of excitations for which stochastic averaging technique can be applied to obtain reasonable estimates of pdf and mean square value of response b the effect of the nature of the frequency contents of excitation on the response and c the use of the stochastic averaging method employing generalized harmonic functions for nonintegrable functions representing the dynamic system it is shown that the stochastic averaging procedure provides results which compare very well with those obtained from simulation analysis not only for wide band excitations but also for narrow band excitations for a wide range of the size parameter of excitation further the procedure can be easily implemented for highly irregular functions by employing a numerical scheme with fft

higher adeles and non abelian riemann roch abstract we show a riemann roch theorem for group ring bundles over an arithmetic surface this is expressed using the higher adeles of beilinson parshin and the tame symbol via a theory of adelic equivariant chow groups and chern classes the theorem is obtained by combining a group ring coefficient version of the local riemann roch formula as in kapranov vasserot with results on k groups of group rings and an explicit description of group ring bundles over p 1 our set up provides an extension of several aspects of the classical frohlich theory of the galois module structure of rings of integers of number fields to arithmetic surfaces

on the congruence subgroup problem for anisotropic groups of inner type a_n an let g be an absolutely almost simple simply connected algebraic group of inner form of type a n defined and anisotropic over a number field k let v denote a complete set of inequivalent valuations of k and v f resp infty the subset of non archimedean resp archimedean valuations in v this work deals with the congruence subgroup problem for groups g of inner form of type a n with respect to a particular class of valuations s subset v which we term as d amenable fix a realization g subset mathrm gl n as a k subgroup and set lambda s g k cap gl n mathcal o s where mathcal o s is the ring of s integers in k then we show that if s is d amenable and gamma is any subgroup of finite index in lambda s there exists a finite set s gamma and positive integers r v mid v in s gamma such that gamma contains gamma in lambda s mid gamma in 1 mathfrak p v r v text for v in s gamma the d amenability property holds for any s with v smallsetminus s finite we also show that the set of primes in v in an arithmetic progression is d amenable for any d so that we recover the result of prasad and rapinchuk proc steklov inst math 292 1 216 246 2016

generators of large subgroups of units of integral group rings of nilpotent groups let g be a finite nilpotent group so that all simple components d n n n 2 of qg satisfy the congruence subgroup theorem suppose that for all odd primes p dividing g the hamiltonian quaternions h split over the pth cyclotomic field q p then new units b3 are introduced so that b1 b2 b 2 b3 is of finite index in u zg

statistics of nonlinear stochastic dynamical systems under levy noises by a convolution quadrature approach this paper describes a novel numerical approach to find the statistics of the non stationary response of scalar nonlinear systems excited by levy white noises the proposed numerical procedure relies on the introduction of an integral transform of the wiener hopf type into the equation governing the characteristic function once this equation is rewritten as partial integro differential equation it is then solved by applying the method of convolution quadrature originally proposed by lubich here extended to deal with this particular integral transform the proposed approach is relevant for two reasons 1 statistics of systems with several different drift terms can be handled in an efficient way independently from the kind of white noise 2 the particular form of wiener hopf integral transform and its numerical evaluation both introduced in this study are generalizations of fractional integro differential operators of potential type and grunwald letnikov fractional derivatives respectively

statistical linearization of the duffing oscillator under non gaussian external excitation the duffing oscillator under external non gaussian excitations is investigated by means of statistical linearization the input process is modelled as a polynomial of a gaussian process or as a renewal driven impulse process four criteria of statistical linearization are considered the interarrival times of the renewal process are distributed according to a pearson type iii law predictions of the stationary variance are compared with monte carlo simulations

response of stochastic dynamical systems driven by additive gaussian and poisson white noise solution of a forward generalized kolmogorov equation by a spectral finite difference method abstract a numerical method is given for the solution of the probability density function of the response process of memoryless one and two state dynamical systems having polynomial restoring forces and which are subjected to a combination of gaussian and poisson white noises the method employs the fourier transformed forward generalized kolmogorov equation to arrive at an initial boundary value problem for the characteristic function which is solved using a high order finite difference procedure the probability density function is recovered by numerical inverse fourier transformation several examples are given the results of which are compared with analytical solutions where available and with simulation otherwise

probabilistic solution of some multi degree of freedom nonlinear systems under external independent poisson white noises this paper studies the stationary probability density function pdf of the response of multi degree of freedom nonlinear systems under external independent poisson white noises the pdf is governed by the high dimensional generalized fokker planck kolmogorov fpk equation the state space split 3s method is adopted to reduce the high dimensional generalized fpk equation to a low dimensional equation subsequently the exponential polynomial closure epc method is further used to solve the reduced fpk equation for the pdf solution two illustrative examples are presented to examine the accuracy of the 3s epc solution procedure one example involves a two degree of freedom coupled nonlinear system the other example is concerned with a ten degree of freedom system with cubic terms in displacement a monte carlo simulation is also performed for simulating the pdf solution of the response the comparison with the simulated result shows that the 3s epc solution procedure can provide satisfactory pdf solut

stochastic response of vibro impact duffing oscillators under external and parametric gaussian white noises abstract this study presents a solution procedure for the stationary probability density function pdf of the response of vibro impact duffing oscillators under external and parametric gaussian white noises first the zhuravlev non smooth coordinate transformation is adopted to convert a vibro impact oscillator into an oscillator without barriers the stationary pdf of the converted oscillator is governed by the fokker planck fp equation the fp equation is solved by the exponential polynomial closure epc method illustrative examples are presented with vibro impact duffing oscillators under external and parametric gaussian white noises to show the effectiveness of the solution procedure the parametric excitation is acting in displacement and the constraint is a unilateral zero offset barrier the restitution coefficient of impacts is taken as 0 90 comparison with the simulated results shows that the proposed solution procedure can provide good approximate pdfs for displacement and velocity although a little difference exists in the tail of these pdfs this difference may be due to the weak approximation on the response of the vibro impact oscillators using a continuous markov process when the restitution coefficient is not very close to unity

k1 of a p adic group ring i the determinantal image abstract we study the k group k 1 of the group ring for a finite group over a coefficient ring which is p adically complete and admits a lift of frobenius in this paper we consider the image of k 1 under the determinant map the central tool is the group logarithm which we can define using the frobenius lift using this we prove a fixed point theorem for the determinantal image of k 1

exact stationary probability density for second order non linear systems under external white noise excitation abstract in this paper exact stationary solutions are constructed for non linear dynamical systems subjected to stochastic excitation the solution of the equations 6 7 in this paper is demonstrated to be unique and then the results of references 10 11 are shown to be generalized therefore new exact stationary solutions that satisfies the fokker plank kolmogorov fpk equation is obtainable

subgroup growth and congruence subgroups letk be a global field o its ring of integers g an almost simple simply connected connected algebraic subgroups ofgl m defined overk and g o which is assumed to be infinite let n resp n be the number of all resp congruence subgroups of index at mostn in we show a if char k 0 then i c 1 frac log 2 n log log n leqq log gamma n gamma leqq c 2 frac log 2 n log log n for suitable constantsc1 andc2 under some mild assumptions we also have ii has the congruence subgroup property if and only if log n o log2n iii if is boundedly generated the has the congruence subgroup property this confirms a conjecture of rapinchuk r1 which was also proved by platonov and rapinchuk pr3 b if char k 0 and under somewhat stronger conditions ong then for suitable constantsc3 andc4 c3 log2n log n c4log3n

probabilistic solution of nonlinear oscillators excited by combined gaussian and poisson white noises abstract the stationary probability density function pdf solution of the stochastic response of nonlinear oscillators is investigated in this paper the external excitation is assumed to be a combination of gaussian and poisson white noises the pdf solution is governed by the generalized kolmogorov equation which is solved by the exponential polynomial closure epc method in order to evaluate the effectiveness of the epc method different nonlinear oscillators are considered in numerical analysis nonlinearity exists either in displacement or in velocity for these nonlinear oscillators the impulse arrival rate mono modal pdf and bi modal pdf are also considered in this study compared to the pdf given by monte carlo simulation the epc method presents good agreement with the simulated result which can also be observed in the tail region of the pdf solution

group algebras of kleinian type and groups of units the algebras of kleinian type are finite dimensional semisimple rational algebras a such that the group of units of an order in a is commensurable with a direct product of kleinian groups we classify the schur algebras of kleinian type and the group algebras of kleinian type as an application we characterize the group rings rg with r an order in a number field and g a finite group such that rgis virtually a direct product of free by free groups the study of kleinian groups goes back to the works of poincar poi and bianchi bia and it has been an active field of research ever since in the last decades it is strongly related to the geometrization program of thurston for the classification of 3 manifolds egm mr mas thu the use of the methods of kleinian groups to the study of the groups of units of group rings was started in prr and led to the notion of algebras of kleinian type and finite groups of kleinian type let k be a number field a a central simple k algebra and r a z order in a let r 1 denote the group of units of r of reduced norm 1 every embedding k c induces an embedding a md c where d is the degree of a furthermore r 1 sld c we say that a is of kleinian type if either a k or a is a quaternion algebra over k and r 1 is a discrete subgroup of sl2 c for some embeddingof k in c more generally an algebra of kleinian type prr is by definition a direct sum of simple algebras of kleinian type a finite group g is of kleinian type if and only if the rational group algebra qg is of kleinian type the finite groups of kleinian type have been classified in jprrz where it has been also proved that a finite group g is of kleinian type if and only if the group of units zg of its integral group ring zg is commensurable with a direct product of free by free groups recently alan reid asked us in a private communication about the consequences of replacing the ring of rational integers by another ring of integers this leads to the following two problems problem 1 classify the group algebras of kleinian type of finite groups over number fields problem 2 given a group algebra of kleinian type kg describe the structure of the group of units of the group ring rg for r an order in k

generators relations and coverings of algebraic groups ii theorem 1 1 let g be the group of rational points of a simply connected simple afine algebraic group defined and split over a fmite j eld of q elements then the schur multiplier m of g is trivial with the following exceptions a zf g is a 4 a 2 a 3 c 2 f 2 or g 4 then h4 is z cyclic of order 2 if g is a 9 b 3 or g 3 it is iz b zf g is a 4 resp d 2 then m is the direct product of z resp z with itself

response of dynamic systems to poisson white noise abstract a method is developed for finding response statistics and reliability for a non linear system subjected to poisson white noise the poisson white noise can be viewed as a sequence of independent indentically distributed pulses arriving in time according to a poisson counting process the method is based on the observation that the system experiences free vibrations between consecutive pulses of the input two algorithms are used for analysis the first uses the definition of the poisson white noise and the second is based on an approximate representation of the poisson white noise referred to as binomial white noise the binomial white noise is an independent time series with a fixed time step that can be zero with a finite probability the time step of the first and second algorithms are random and deterministic respectively the algorithms are used to calculate transition probabilities and estimate response statistics numerical examples are presented to demonstrate the proposed method of response and reliability analysis

applications of a computer implementation of poincare s theorem on fundamental polyhedra poincarcs theorem asserts that a group f of isometries of hyperbolic space h is discrete if its generators act suitably on the boundary of some polyhedron in h and when this happens a presentation of f can be derived from this action we explain methods for deducing the precise hypotheses of the theorem from calculation in f when f is algorithmi cally defined and we describe a file of fortran programs that use these methods for groups f acting on the upper half space model of hyperbolic 3 space h we exhibit one modest example of the application of these programs and we summarize computations of repesenta tions of groups psl 2 0 where 6 is an order in a complex quadratic number field in the early 1880 s h poincare discovered a general theorem allowing one to deduce the discreteness of and a presentation for a group g of isometries of hyperbolic space from its action on a hyperbolic polyhedron under certain condi tions theorems of this sort are part of the foundations of his theories of fuchsian and kleinian groups that have become very popular again and h seifert has recently given us a modern proof of a fairly general version of poincare s theorem in 12 see also 7 this theorem has been little used over the past century perhaps partly because its hypotheses have seemed very difficult to verify for a given group g except in very special circumstances one reason for doubting that poincare s theorem is unreasonably difficult to apply to fairly general discrete groups is that no general alternative method for accomplishing its tasks has been proposed the present paper is devoted to demonstrating that poincare s theorem can indeed be applied to given groups in apparently difficult cases and that much of the work can be done by a computer our experience suggests that the theorem is really very helpful in guiding the user to an understanding of the details of the action of g starting from a state of near ignorance an outline of this paper is as follows in section 1 we begin by stating seifert s version of poincare s theorem and explaining how we would apply it to an algorithmically defined group g of isometries of hyperbolic space h o in section 2 we specialize to the situation 00 for which we wrote our file poincare of fortran

numerical solution of stochastic differential equations with poisson and levy white noise a fixed time step method is developed for integrating stochastic differential equations sde s with poisson white noise pwn and l evy white noise lwn the method for integrating sde s with pwn has the same structure as that proposed by kim et al phys rev e 76 011109 2007 but is established by using different arguments the integration of sde s with lwn is based on a representation of l evy processes by sums of scaled brownian motions and compound poisson processes it is shown that the numerical solutions of sde s with pwn and lwn converge weakly to the exact solutions of these equations so that they can be used to estimate not only marginal properties but also distributions of functionals of the exact solutions numerical examples are used to demonstrate the applications and the accuracy of the proposed integration algorithms

construction of units in group rings of monomial and symmetric groups let rg be the group ring of a finite group g over an integral domain rand u rg its group of units in 11 for nilpotent groups g with a few exceptions if the order igi is even generators of a subgroup of finite index in the unit group u u 7lg of the integral group ring tlgare given these generators consist of the bass cyclic units which are of the form 1 m u l a ai i m d fi a 1 d l a a a d ord a

on fundamental domains of arithmetic fuchsian groups let k be a totally real algebraic number field and o an order in a quaternion algebra a over k assume that the group o 1 of units in o with reduced norm equal to 1 is embedded into psl 2 r as an arithmetic fuchsian group it is shown how ford s algorithm can be effectively applied in order to determine a fundamental domain of o 1 as well as a complete system of generators of o 1

successive modifications of limit periodic continued fractions it is well known that limit periodic continued fractions can be accelerated by modifications using converging factors in this paper the repeated use of modifications is studied in the case of a constant converging factor and in the case of aitken s d 2 process based on these modifications a method for controlling the error is proposed

k2of cyclic group rings over λ rings the theory developed in previous papers of the author is used to compute the algebraick2of group rings of cyclicp groups with coefficients in an arbitrary ring

maximum entropy principle and nonlinear stochastic oscillators abstract the principle of maximum entropy introduced first in physics and successfully applied in many other fields e g statistics reliability estimation simulation has recently been developed to analyze the systems governed by stochastic differential equations in this paper we extend the maximum entropy method to the general class of stochastic nonlinear systems and apply it to stochastic nonlinear oscillators to indicate the effectiveness of the method a comparison of the results with predictions of the known procedures is shown graphically the analysis presented in this paper indicates that the maximum entropy principle can serve as a general method for determining the probability distributions of a wide class of stochastic systems when we have only a partial information in the form of moments or equations for moments

stochastic response of a ϕ6 oscillator subjected to combined harmonic and poisson white noise excitations abstract the transient and stationary probability density functions pdfs of stochastic response of the 6 duffing oscillator under combined harmonic and external and parametric poisson white noises excitations are investigated by the generalized cell mapping method in this paper based on the digraph analysis method the global qualitative properties are obtained such as attractors basins of attraction basin boundaries saddles and invariant manifolds the evolutionary process of transient and stationary pdfs are shown based on the matrix analysis method it is observed that there is a close relationship between evolutionary direction of pdf and the unstable manifold monte carlo mc simulation is used to verify the accuracy of the matrix analysis method

free groups in quaternion algebras abstract in juriaans et al 2009 9 we constructed pairs of units u v in z orders of a quaternion algebra over q d d a positive and square free integer with d 7 mod 8 such that u n v n is free for some n n here we extend this result to any imaginary quadratic extension of q thus including matrix algebras more precisely we show that u n v n is a free group for all n 1 and d 2 and for d 2 and all n 2 the units we use arise from pell s and gauss equations

numerical analysis for stochastic age dependent population equations a numerical method proposed to approximate the solution of a class of stochastic age dependent populations we show that under certain conditions weaker than linear growth and global lips chits the euler scheme applied to stochastic age dependent population converges to the analytic solution and provide information on the order of approximation

on the acceleration of limit periodic continued fractions a continued fraction c f k a n 1 is called limit periodic if mathop lim limits n to infty a n a fora anda 1 4 a 0 thron waadeland 1980 examined a modification of a limit periodic c f for accelerating the convergence this acceleration remains modest if thea n converge only logarithmically thus it is proposed to apply an euler summability method to the series equivalent to the c f properties of the equivalent function are derived these properties are used for choosing appropriate parameters for the summability method such that a considerable acceleration can be expected even if thea n converge logarithmically

on the congruence subgroup problem determination of the metaplectic kernel let f be a global field i e either a number field or a function field in one variable over a finite field and let oo be the set of its archimedean places let g be a finite non empty set of places of f containing oo let o o denote the ring of g integers of f let a denote the ring of g adeles i e the restricted direct product of the completions f v for r eg let f be an absolutely simple simply connected subgroup of sl defined over f recall that a subgroup f of c f is an g arithmetic subgroup if fnsl n o has finite index in f as well as in o fy f c sl n o an g arithmetic subgroup f is a congruence subgroup if there exists a non zero ideal a in o o g such that

first eigenvalue of laplace operator on locally symmetric space this paper is concerned with the lower bound for the first positive eigenvalue lambda 1 gamma of the laplace operator bigtriangleup on the space of square integrable functions on some locally symmetric space gamma backslash g k where g so 2n 1 mathbb c k is a maximal compact subgroup of g and gamma is a lattice in g which arises from some imaginary quadratic field it is well known that the casimir operator fancyscript c acts by a scalar lambda pi on every irreducible admissible representation pi of g by diximir s schur lemma in this paper we prove that when n ge 5 there exists a spectral gap between lambda 1 gamma and the infimum lambda 1 g of lambda pi when pi passes through all non trivial irreducible spherical unitary representations of g

a partial differential equation for the characteristic function of the response of non linear systems to additive poisson white noise abstract a method has been developed for finding the characteristic function u t e exp i u x t of the solution x t of a non linear stochastic differential equation driven by poisson white noise poisson white noise can be viewed as a sequence of independent identically distributed pulses arriving at poisson times if the drift is a polynomial of x t and the diffusion is independent of this process u t satisfies a partial differential equation that can be solved numerically two approaches are used to establish the partial differential equation of the characteristic function of x t the first approach is based on a generalized version of the ito differentiation formula for stochastic differential equations with poisson white noise the second approach uses elementary arguments to determine the rate of change of u t in time linear and non linear systems subjected to poisson white noise are used to demonstrate and evaluate the proposed method the solution x t of a linear differential equation with poisson white noise is a filtered poisson process and the characteristic function for this type of process is available in closed form it is shown that the characteristic function of the filtered poisson process x t satisfies the partial differential equation of u t because the characteristic function of the state x t of a non linear system subjected to poisson white noise is not known numerical solutions of the partial differential equation of u t are compared with estimates of this characteristic function calculated from independent realization of x t obtained by monte carlo simulation these comparisons show that the proposed method for calculating the characteristic function of x t as the solution of a partial differential equation is accurate

groups of units of integral group rings commensurable with direct products of free by free groups research partially supported by onderzoeksraad of vrije universiteit brussel fonds voor wetenschappelijk onderzoek belgium d g i of spain fundacion seneca of murcia cnpq and fapdf ams classification index primary 16u60 secondary 11r27 16a26

dynamics of offshore structures part ii stochastic averaging analysis this is the second of two companion papers concerned with an investigation into the non linear dynamics of drag dominated offshore structures excited by waves the equations of motion are approximated by a linear system with a time varying damping term the excitation however is a non linear function of the wave particle kinematics to study the effect of the time varying damping the resonant response is here examined by means of a narrow band idealization a stochastic averaging of the governing equations is applied and the fokker planck equation for the response amplitude probability densities is thereby obtained from the stationary solution of this equation second order statistics are derived together with an expression for the hydrodynamic damping which is equivalent to the time varying term numerical results are given to illustrate the magnitude of this term for a typical framed structure and to provide comparisons with results obtained by the conventional technique of linearization

an improved path integration method for nonlinear systems under poisson white noise excitation abstract in order to overcome some unsatisfactory trends and limitations of the traditional path integration pi method for poisson white noise a novel pi method is proposed in this paper which includes two improved schemes the first one is a new transition probability density function tpdf approximation which considers the randomness of the impulse happening time during each time intervals the second one is a transformation of chapman kolmogorov ck equation by a variable substitution instead of directly using it whose numerical calculation is based on the back stepping runge kutta scheme and the triangulation based interpolation monte carlo simulations mcs are utilized to measure the accuracy of the improved algorithm with three illustrative nonlinear systems the results show that compared with the traditional pi method the improved pi method can give a more accurate description of the tpdf values and provide more precise stationary probability density function pdf results whenever the mean arrival rate is large or small the improved algorithm has a wider range of choices in time interval values to maintain the accuracy of stationary pdf results besides it is discovered that cubic interpolation deserves to be applied in the improved pi method more than linear and natural interpolations

reliability of non linear oscillators subject to poisson driven impulses abstract the reliability problem of a sdof non linear and non hysteretic oscillator subject to a stationary poisson driven train of impulses is considered in the first passage time problem of the oscillator time invariant double barriers with a deterministic start in the safe domain are assumed initially the kolmogorov feller backwards equation for the reliability function along with associated initial and boundary conditions is specified next a petrov galerkin approach with shape and weighting functions matching all differentiability and continuity requirements of the weak formulation is applied to solve the consequent partial differential equation as an example a duffing oscillator with linear damping is considered

sk 1 of finite abelian groups i g which can be used both for specific computations and to prove general theo rems for certain classes of groups d st2 in collaboration with oliver we have greatly extended the power of our computational methods the results of all computations as well as the ad ditional methods leading to them will appear in the sequel a d o s to this paper in a future issue of inventiones there we derive a very efficient algo rithm for computing

presentations of the unit group of an order in a non split quaternion algebra abstract we give an algorithm to determine a finite set of generators of the unit group of an order in a non split classical quaternion algebra h k over an imaginary quadratic extension k of the rationals we then apply this method to obtain a presentation for the unit group of h z 1 7 2 as a consequence a presentation is discovered for the orthogonal group so 3 z 1 7 2 these results provide the first examples of a characterization of the unit group of some group rings that have an epimorphic image that is an order in a non commutative division algebra that is not a totally definite quaternion algebra

tame component of the schur multipliers of finite groups of lie type let f be a finite field of characteristic p let be an almost f simple simply connected algebraic group defined over f for any f subgroup a of y we shall denote by r f the finite group of f rational points of z and shall consider q z as an r f module with trivial x f action let g f f let j be the p primary component of q z and j be the sum of all primeto p primary components of q z then q z jo j and the schur multiplier h2 g q z is the direct sum of h g j and h g j the subgroup h2 g j is the p primary component of the schur multiplier of g we shall call it the wild component of the schur multiplier and h2 g j which is the sum of prime to p primary components of the schur multiplier the tame component it is a classical result of schur 41 that the tame component of the schur multiplier of sl 2 5 is trivial for any finite field 5 griess has proved 3 1 that the schur multiplier of su 3 5 is trivial we shall use these two results and the tits building of y f to give a new proof of the following theorem of steinberg 6 71 and griess 31

numerical solution of high dimensional stationary fokker planck equations via tensor decomposition and chebyshev spectral differentiation abstract this paper focuses on the curse of dimensionality in the numerical solution of the stationary fokker planck equation for systems with state independent excitation a tensor decomposition approach is combined with chebyshev spectral differentiation to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases following the enforcement of the normality condition via a penalty method the discretized system is solved using alternating least squares algorithm numerical results for a variety of systems including separable non separable systems linear nonlinear systems and systems with without closed form stationary solutions up to ten dimensional state space are presented to illustrate the effectiveness of the proposed method

stochastic averaging of quasi partially integrable hamiltonian systems under combined gaussian and poisson white noise excitations abstract a stochastic averaging method for predicting the response of quasi partially integrable and non resonant hamiltonian systems to combined gaussian and poisson white noise excitations is proposed for the case with r 1 r n independent first integrals which are in involution an r dimensional averaged generalized fokker planck kolmogorov gfpk equation for the transition probability density of r independent first integrals is derived from the stochastic integro differential equations sides of the original quasi partially integrable and non resonant hamiltonian systems by using the stochastic jump diffusion chain rule and the stochastic averaging theorem an example is given to illustrate the applications of the proposed stochastic averaging method and a combination of the finite difference method and the successive over relaxation method is used to solve the reduced gfpk equation to obtain the stationary probability density of the system the results are well verified by a monte carlo simulation

extreme value prediction for non linear stochastic oscillators via numerical solutions of the stationary fpk equation abstract the accuracy of two well established numerical methods is demonstrated and the importance of bandwidth examined for computationally efficient markov based extreme value predictions associated with finite duration stationary sample paths of a non linear oscillator driven by gaussian white noise by making the poisson assumption of independent upcrossings extreme exceedance probabilities are predicted via the mean threshold crossing rate using numerical solutions of the stationary fokker planck fpk equation with bandwidth initially ignored predicted exceedances using the weighted residual methods of bhandari and sherrer soize and kunert and the finite element method of langley are compared with nominally exact predictions for a heavily damped duffing type model obtained by using an explicit fpk solution the fe method being established as superior predictions via fe solutions are then compared with very long monte carlo simulations in which bandwidth effects are included two lightly damped non linear oscillator models are examined both with cubic stiffness but different damping mechanisms one model again being of simple duffing type with linear damping the other being appropriate to single co ordinate random vibration of a clamped clamped beam with wholly non linear damping the realistic damping parameter values assigned to the beam model are statistically equivalent to the linear damping level chosen for the simple model at just above 1 at this overall damping level results clearly demonstrate that for the probability levels and durations considered bandwidth is only important for the linearly damped model for the beam model with non linear damping bandwidth can be ignored allowing accurate extreme exceedance predictions by using only the stationary fpk equation the paper also demonstrates that the limiting decay rate of the first passage probability advocated by crandall roberts and others as a criterion for deciding the barrier height above which first passage times can be assumed independent proves grossly over conservative as a corresponding criterion for deciding the independence level for use in stationary extreme value prediction

a finite element method for the statistics of non linear random vibration the transitional probability density function for the random response of a certain class of non linear system satisfies the fokker planck kolmogorov equation this paper concerns the numerical solution of the stationary form of this equation yielding the stationary probability density function of response the weighted residual statement for the problem is integrated by parts to yield the weak form of the equations which are then solved by the finite element method the method is applied to a duffing oscillator and good agreement is found with the exact result and the method is compared favourably with a galerkin solution method given by bhandari and sherrer 1 also the method is applied to the ship rolling problem and good agreement is found with an approximate analytical result due to roberts 2

beyond the markovian assumption response excitation probabilistic solution to random nonlinear differential equations in the long time uncertainty quantification for dynamical systems under non white excitation is a difficult problem encountered across many scientific and engineering disciplines difficulties originate from the la