cohomology for frobenius kernels of sl2 abstract let sl 2 r be the r th frobenius kernels of the group scheme sl 2 defined over an algebraically closed field of characteristic p 0 in this paper we give for r 1 a complete description of the cohomology groups for sl 2 r we also prove that the reduced cohomology ring h sl 2 r k red is cohen macaulay geometrically we show for each r 1 that the maximal ideal spectrum of the cohomology ring for sl 2 r is homeomorphic to the fiber product g b u r finally we adapt our calculations to obtain analogous results for the cohomology of higher frobenius lusztig kernels of quantized enveloping algebras of type sl 2

permutable indistinguishability operators perfect vague groups and fuzzy subgroups permutability between t indistinguishability operators is a very interesting property that is related to the compatibility of the operators with algebraic structures it will be shown that the sup t product email protected f of two t indistinguishability operators is also a t indistinguishability operator if and only if e and f are permutable t indistinguishability operators i e email protected email protected e this property will be related to the study of fuzzy subgroups fuzzy normal subgroups and vague groups the aggregation of fuzzy subgroups will also be analyzed

cohomology of infinitesimal and discrete groups an infinitesimal algebraic group over a field k is a connected algebraic group scheme whose coordinate ring is a finite dimension al k algebra if one begins with a reductive algebraic group g defined over fp then one has a close relationship between the representation theory of g that of its infinitesimal subgroups g defined as the kernels of the frobenius endomorphisms rr g g and that of its finite chevalley subgroups g fq of fq rational points q pd a basic theorem of cline et al i 9 theorem 6 6 relates the cohomology of g in a rational representation to that of the finite groups g fq related work of cline et al 6 and of the authors 1 16 relates the cohomology of g to that of its infinitesimal subgroups a determination of the cohomology of the first infinitesimal subgroup g 1 of g with trivial coefficients has enabled the authors in 14 to make certain explicit cohomology calculations for the finite groups g fq with certain nontrivial coefficients in this paper we continue our study of infinitesimal subgroups of simple algebraic groups by proving various qualitative properties of their cohomology and by extending our computational knowledge in particular we provide a computation which applies in a range of degrees to chevalley groups over the prime field we also investigate the cohomology of infinitesimal subgroups of nilpotent groups enabling us for example to determine the cohomology of their lie algebras in certain cases we apply this lie algebra cohomology computation to determine the integral cohomology of various torsion free nilpotent groups modulo small prime torsion thereby extending a theorem of dwyer 12 in more detail the paper begins in sect 1 with a spectral sequence converging to the cohomology of the first infinitesimal subgroup g which was previously studied in 14 from this spectral sequence it easily follows that h g1 m is a noetherian h g1 k module for each finite dimensional rational gl module m we then apply the work of andersen and jantzen 1 on the gl cohomology of induced modules to obtain the analogous result for the second infinitesimal subgroup gz our analysis also permits a determination of the generic

fuzzy equivalence relations and their equivalence classes in this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice we give certain characterizations of fuzzy semi partitions and fuzzy partitions over a complete residuated lattice as well as over a linearly ordered complete heyting algebra in the latter case for a fuzzy equivalence relation over a linearly ordered complete heyting algebra we construct an algorithm for calculation of a minimal family of its equivalence classes which generates it most of the presented results are new but some of them are generalizations of known results given in a way which simplifies and clarifies them

modelling a linguistic variable as a hierarchical family of partitions induced by an indistinguishability operator this work shows a method to obtain a hierarchy of partitions on the universe 0 1 in such a way that each of them is compatible with a refinement of lukasiewicz indistinguishability operator the classes of the partition at a given level present a relation of antonymy between them moreover the partition at a certain level can be seen as the refinement of a previous level by means of a class of linguistic modifiers due to this fact they seem appropriate for modeling linguistic labels of a linguistic variable the associated indistinguishability operators show the increasing granularity when the number of classes rises up

aggregation of fuzzy subgroups abstract given an aggregation function and two fuzzy subgroups of a group this study investigates whether the aggregation of them is also a fuzzy subgroup it is proven that if the cardinal of the group is a prime power then it is always a fuzzy subgroup for a general group it is proven that the existence of a binary relation between the fuzzy subgroups determines if their aggregation is a fuzzy subgroup moreover given two fuzzy subgroups if the aggregation of them is a fuzzy subgroup for some strictly monotone aggregation function then it is a fuzzy subgroup for any aggregation function the present study also examines conditions on the aggregation function that characterise the case where the aggregation of two arbitrary fuzzy subgroups is also a fuzzy subgroup

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

second cohomology groups for frobenius kernels and related structures abstract let g be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p let b be a borel subgroup of g and u be its unipotent radical in this paper we determine the second cohomology groups of b and its frobenius kernels for all simple b modules we also consider the standard induced modules obtained by inducing a simple b module to g and compute all second cohomology groups of the frobenius kernels of g for these induced modules also included is a calculation of the second ordinary lie algebra cohomology group of lie u with coefficients in k

uniform fuzzy relations and fuzzy functions in this paper we introduce and study the concepts of a uniform fuzzy relation and a partially uniform f function we give various characterizations and constructions of uniform fuzzy relations and uniform f functions we show that the usual composition of fuzzy relations is not convenient for f functions so we introduce another kind of composition and we establish a mutual correspondence between uniform f functions and fuzzy equivalences we also give some applications of uniform fuzzy relations in approximate reasoning especially in fuzzy control and we show that uniform fuzzy relations are closely related to the defuzzification problem

broue s conjecture holds for principal 3 blocks with elementary abelian defect group of order 9 abstract in representation theory of finite groups there is a well known and important conjecture due to m broue he conjectures that for any prime p if a finite group g has an abelian sylow p subgroup p then the derived categories of the principal p blocks of g and of the normalizer ng p of p in g are equivalent we prove in this paper that broue inage kus conjecture holds for the principal 3 block of an arbitrary finite group g with an elementary abelian sylow 3 subgroup p of order 9 by using initiated works for the case where g is simple which are due to puig okuyama waki miyachi and the authors the result depends on the classification of finite simple groups

extensions of modules for algebraic groups let g be a connected semi simple and simply connected algebraic group over an algebraically closed field k of prime characteristic p in this paper we show how one can obtain the extensions of any two simple g modules if one knows the socles of all the g modules extg l l where g1 denotes the kernel of the frobenius endomorphism on g and l as well as l are simple g1 modules alternatively we prove how all such g extensions can be found once one knows the extensions of any two simple g modules belonging to a specified finite set of simple modules with small highest weight such information is available in certain low rank cases it should also be pointed out that it is possible from lusztig s conjecture on irreducible characters 11 to derive the dimensions of ext l1 l2 when l1 and l2 are simple with such small highest weights to obtain our results on exth we need to know that there exists no g1 extension of a simple module by itself this turns out to be somewhat

self equivalences of stable module categories let p be an abelian p group e a cyclic p group acting freely on p and k an algebraically closed field of characteristic p 0 in this work we prove that every self equivalence of the stable module category of k p rtimes e comes from a self equivalence of the derived category of k p rtimes e work of puig and rickard allows us to deduce that if a block b with defect group p and inertial quotient e is rickard equivalent to k p rtimes e then they are splendidly rickard equivalent that is broue s original conjecture implies rickard s refinement of the conjecture in this case all of this follows from a general result concerning the self equivalences of the thick subcategory generated by the trivial module

exponential stability of functional differential systems with impulsive effect on random moments abstract the model of nonlinear functional differential systems with impulsive effect on random moments is brought forward by authors in this paper then by means of liapunov s direct method coupled with razumikhin technique and comparison principle sufficient conditions for p moment exponential stability of the trivial solution to the systems are present where the parameter t in dv t x t dt are not required to be negative finally an example is discussed to show the applications of the obtained results

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

numerical meshfree path integration method for non linear dynamic systems abstract we present a new numerical meshfree path integration mpi method for non linear dynamic systems the obtained mpi method can be performed in the irregular computational domain and the probability density values of the random nodes in the domain can be calculated via the mpi method and the ordinary differential equations for the first and second order moments on the basis of gaussian closure method the piecewise linear interpolation based on adaptive least squares is utilized as a post processor to approximate the probability density values on arbitrary positions the good performance of the resulting method is finally shown in the numerical examples by using three specific non linear dynamic systems duffing oscillator subjected to both harmonic and stochastic excitations duffing rayleigh oscillator subjected to both harmonic and stochastic excitations and chen system driven by three different gaussian white noises

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

algebras of fuzzy sets in this paper we investigate two kinds of algebras of fuzzy sets which are obtained by using zadeh s extension principle we give conditions under which a homomorphism between two algebras induces a homomorphism between corresponding algebras of fuzzy sets we prove that if the structure of truth values is a complete residuated lattice the induced algebra of a subalgebra of an algebra a can be embedded into the induced algebra of fuzzy sets of a for direct products we give conditions under which the direct product of algebras of fuzzy sets could be embedded into the algebra of fuzzy sets of the direct product in the case of homomorphisms and direct products the two kinds of algebras of fuzzy sets behave in different ways

extensions for finite chevalley groups i let g be a connected semisimple algebraic group defined and split over the field fp with p elements and k be the algebraic closure of fp assume further that g is almost simple and simply connected and let g fq be the finite chevalley group consisting of fq rational points of g where q pr for a non negative integer r in this paper formulas are found relating extensions between simple kg fq modules and extensions over g considered as an algebraic group over k one of these formulas which only holds for primes p 3 h 1 where h is the coxeter number of g is then used to show the vanishing of self extensions between simple kg fq modules except for certain simple modules when r 1 and the underlying root system is of type a1 or cn

ergodicity of dissipative differential equations subject to random impulses differential equations subject to random impulses are studied randomness is introduced both through the time between impulses which is distributed exponentially and through the sign of the impulses which are fixed in amplitude and orientation such models are particular instances of piecewise deterministic markov processes and they arise naturally in the study of a number of physical phenomena particularly impacting systems the underlying deterministic semigroup is assumed to be dissipative and a general theorem which establishes the existence of invariant measures for the randomly forced problem is proved further structure is then added to the deterministic semigroup which enables the proof of ergodic theorems characteristic functions are used for the case when the deterministic component forms a damped linear problem and irreducibility measures are employed for the study of a randomly forced damped double well nonlinear oscillator with a gradient structure

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

fuzzy groups fuzzy functions and fuzzy equivalence relations to each fuzzy subgroup a fuzzy equivalence relation is associated and it is proved that a fuzzy subgroup is normal if and only if the operation of the group is compatible with its associated fuzzy equivalence relation in the definition of fuzzy subgroup only the subset is fuzzy whilst the group operation remains crisp in an opposite direction another structure can be constructed maintaining crisp the basic set x of the group and fuzzifying the group operation when in addition it is imposed to the fuzzy operation to be compatible with a given fuzzy equivalence relation this structure is called a vague group the results of the paper allow us to associate a vague group to every fuzzy subgroup in such a way that it can be interpreted as the fuzzy quotient group x special attention is paid to normal fuzzy groups and vague groups of the additive group r of the real numbers two examples related to triangular numbers illustrating the results of the paper are provided

extensions for frobenius kernels 1 1 let g be a simple algebraic group over an algebraically closed field k of characteristic p 0 for r 1 let gr be the rth frobenius kernel of g it is well known that the representations for g1 are equivalent to the restricted representations for lie g historically the cohomology for frobenius kernels has been best understood for large primes friedlander and parshall fp first computed the cohomology ring h g1 k for p 3 h 1 where h is the coxeter number of the underlying root system they proved that the cohomology ring can be identified with the coordinate algebra of the nullcone andersen and jantzen aj later verified this fact for p h furthermore they generalized this calculation by looking at h g1 h0 where h0 indb for p h their results had some restrictions on the type of root system involved kumar lauritzen and thomsen klt removed the restrictions on the root systems through the use of frobenius splittings the cohomology ring h g1 k modulo nilpotents can be identified in general with the coordinate algebra of the restricted nullcone n1 x lie g x p 0 for good primes nakano parshall and vella npv proved that this variety is irreducible and can be identified with the closure of some richardson orbit recently carlson lin nakano and parshall clnp have given an explicit description of n1 these recent results provide some indication that one can systematically study extensions of frobenius kernels for small primes by using general formulas which exhibit generic behavior for large primes

the euler scheme for random impulsive differential equations random impulsive differential equations rides are a kind of mathematical models with extensive applications in this paper the euler scheme for rides is first brought forward one of whose important applications is to generate the whole approximate trajectories of rides thus the proposed euler scheme allows us to approximate moments functionals and the distribution for the underlying process and perform monte carlo type analysis the obtained results show that the euler scheme is at least 1 order of step h when the right terms of equation satisfy lipschitz conditions and the waiting times of random impulses follow the mutually independent exponential distribution with the same parameter thus it is an efficient method for numerical simulation

frobenius splitting of cotangent bundles of flag varieties let g be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic p 0 let u be the unipotent radical of a borel subgroup b g and u the lie algebra of u springer 16 has shown for good primes that there is a b equivariant isomorphism u u where b acts through conjugation on u and through the adjoint action on u for g sln one has the well known equivariant isomorphism a 7 a i between unipotent and nilpotent upper triangular matrices let p be a good prime for g then there is an isomorphism of homogeneous bundles x g b u g b u where the latter can be identified with the cotangent bundle t g b of g b motivated in part by 12 we establish a link between the g invariant form on the steinberg module st h0 g b p 1 cf 1 8 and frobenius splittings 15 of the cotangent bundle of g b the representation h0 g b 2 p 1 is a quotient of the space of functions h x ox on x here h0 g b m denotes the g module induced from the b module m and half the sum of the roots r opposite to the roots

extensions for finite chevalley groups ii let g be a semisimple simply connected algebraic group defined and split over the field fp with p elements let g fq be the finite chevalley group consisting of the fq rational points of g where q p r and let g r be the rth frobenius kernel the purpose of this paper is to relate extensions between modules in mod g f q and mod g r with extensions between modules in mod g among the results obtained are the following for r 2 and p 3 h 1 the g f q extensions between two simple g f q modules are isomorphic to the g extensions between two simple p r restricted g modules with suitably twisted highest weights for p 3 h 1 we provide a complete characterization of h 1 g f q h0 where h 0 ind g b a and a is p r restricted furthermore for p 3 h 1 necessary and sufficient bounds on the size of the highest weight of a g module v are given to insure that the restriction map h 1 g v h 1 g f q v is an isomorphism finally it is shown that the extensions between two simple p r restricted g modules coincide in all three categories provided the highest weights are close together

second cohomology groups for algebraic groups and their frobenius kernels let g be a simple simply connected algebraic group scheme defined over an algebraically closed field of characteristic p 0 let t be a maximal split torus in g b t be a borel subgroup of g and u its unipotent radical let f g g be the frobenius morphism for r 1 define the frobenius kernel gr to be the kernel of f iterated with itself r times define ur respectively br to be the kernel of the frobenius map restricted to u respectively b let x t be the integral weight lattice and x t be the dominant integral weights the computations of particular importance are h2 u1 k h2 br for x t h2 gr h0 for x t and h2 b for x t the above cohomology groups for the case when the field has characteristic 2 are computed in this paper these computations complete the picture started by bendel nakano and pillen 2007 5 for p 3 furthermore the computations show h2 gr h0 has a good filtration

computation of the greatest right and left invariant fuzzy quasi orders and fuzzy equivalences abstract right invariant fuzzy quasi orders for fuzzy automata are broadly studied in the recent literature as they arise as solutions to particular systems of fuzzy relation equations and inequalities some of their applications include determinization and state reduction procedures as well as simulations and bisimulations for fuzzy automata in this paper we provide a procedure for computing the greatest right invariant fuzzy quasi order for a given fuzzy automaton over a complete residuated lattice the proposed procedure terminates in a finite number of steps whenever the underlying structure of the fuzzy automaton is locally finite when the previous condition is not satisfied we show that the greatest right invariant fuzzy quasi order can be obtained by taking the limit value of the convergent array of fuzzy quasi orders for fuzzy automata over bl algebras on the real unit interval 0 1 analogous procedures for computing the greatest left invariant fuzzy quasi order as well as the greatest right and left invariant fuzzy equivalences for a fuzzy automaton are also presented in addition the faster algorithm for computing the greatest right invariant equivalence on a nondeterministic automaton is also presented

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

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

on the relationship between fuzzy subgroups and indistinguishability operators abstract fuzzy subgroups are revisited considering their close relationship with indistinguishability operators fuzzy equivalences invariant under translations different ways to obtain new fuzzy subgroups from a given one are provided and different ways to characterize normal fuzzy subgroups are obtained the idea of double coset of two crisp subgroups allows us to relate them via their equivalence classes this is generalized to the fuzzy framework the conditions in which a fuzzy relation r on a group g can be considered a fuzzy subgroup of g g are obtained

fuzzy submonoids fuzzy preorders and quasi metrics let l be a residuated lattice then given a monoid m we define a galois connection from the lattice of the compatible l preorders in m and the lattice of l submonoids of m given a set s we define a galois connection between the lattice of the l preorders in s and the lattice of l submonoids of the monoid ss i a link with the notion of quasi metric is also established

the gln q module structure of the symmetric algebra around the steinberg module abstract we determine the graded composition multiplicity in the symmetric algebra s v of the natural gl n q module v or equivalently in the coinvariant algebra of v for a large class of irreducible modules around the steinberg module this was built on a computation via connections to algebraic groups of the steinberg module multiplicity in a tensor product of s v with other tensor spaces of fundamental weight modules

local averaged path integration method approach for nonlinear dynamic systems abstract traditional path integration method always has a limited time step length whose minimum is based on the subdivision degree of integral space in order to numerically analyze a smaller time step probability evolution without significant increase in computing time a new local averaged path integration lapi method is investigated the basic idea of this method is giving a local averaged value of transition probability density function tpdf then the numerical integration of chapman kolmogorov ck equation to the local averaged probability density function pdf will be calculated in lower space subdivision degree second order partial derivative of pdf can be used to correct the deviation between local averaged pdf and the real one it is demonstrated that the proposed method can give a more efficient calculation compared with traditional gauss legendre path integration glpi in the case of our method smaller time step value is permitted with the same space subdivision degree and smaller space subdivision degree is needed to keep the accuracy of stationary pdf with the same time step value this merits also exist when analyzing transient pdf which leads to a better capability of capturing the evolution of the pdf

probabalistic solution of nonlinear oscillators under external and parametric poisson impulses this paper deals with the stationary probability density function solution of the dynamic response for nonlinear oscillators excited by poisson impulses the probability density function solution is governed by the generalized fokker planck kolmogorov equation the exponential polynomial closure method is used to obtain an approximate solution to the generalized fokker planck kolmogorov equation to evaluate the effectiveness of the exponential polynomial closure method in the case of poisson excitations duffing oscillators van der pol oscillators and an additional nonlinear oscillator excited by external and parametric poisson impulses with different levels of nonlinearity in stiffness and damping are studied the numerical results show good agreement with the probability distribution obtained with monte carlo simulations including the tail regions which is of significance for reliability analyses

convergence and stability of euler method for impulsive stochastic delay differential equations this paper deals with the mean square convergence and mean square exponential stability of an euler scheme for a linear impulsive stochastic delay differential equation isdde first a method is presented to take the grid points of the numerical scheme based on this method a fixed stepsize numerical scheme is provided based on the method of fixed stepsize grid points an euler method is given the convergence of the euler method is considered and it is shown the euler scheme is of mean square convergence with order 1 2 then the mean square exponential stability is studied using lyapunov like techniques the sufficient conditions to guarantee the mean square exponential stability are obtained the result shows that the mean square exponential stability may be reproduced by the euler scheme for linear isddes under the restriction on the stepsize at last examples are given to illustrate our results

stability of the zero solution of impulsive differential equations by the lyapunov second method abstract the paper is concerned with the stability of the zero solution of the impulsive system dx dt f t x t i x x t i x j i x i n 1 2 where x t x x x limt x t the lyapunov second method is used as a tool in obtaining the criteria for stability asymptotic stability and instability of the trivial solution

a characterization of fuzzy subgroups abstract an ordinary subgroup of a group g is 1 a subset of g 2 closed under the group operation in a fuzzy subgroup it is precisely these two notions that lose their deterministic character a fuzzy subgroup of a group g associates with each group element a number the larger the number the more certainly that element belongs to the fuzzy subgroup the closure property is captured by the inequality x y t x y in a rosenfeld s original definition t was the function minimum however any t norm t provides a meaningful generalization of the closure property two classes of fuzzy subgroups are investigated the fuzzy subgroups in one class are subgroup generated those in the other are function generated each fuzzy subgroup in these classes satisfies the above inequality with t given by t a b max a b 1 0 while the two classes look different each fuzzy subgroup in either is isomorphic to one in the other it is shown that a fuzzy subgroup satisfies the above inequality with t minimum if and only if it is subgroup generated of a very special type finally these notions are applied to some abstract pattern recognition problems

topological and lattice structures of l fuzzy rough sets determined by lower and upper sets this paper builds the topological and lattice structures of l fuzzy rough sets by introducing lower and upper sets in particular it is shown that when the l relation is reflexive the upper resp lower set is equivalent to the lower resp upper l fuzzy approximation set then by the upper resp lower set it is indicated that an l preorder is the equivalence condition under which the set of all the lower resp upper l fuzzy approximation sets and the alexandrov l topology are identical however associating with an l preorder the equivalence condition that l interior resp closure operator accords with the lower resp upper l fuzzy approximation operator is investigated at last it is proven that the set of all the lower resp upper l fuzzy approximation sets forms a complete lattice when the l relation is reflexive

t fuzzy mapping abstract we introduce the concepts of t fuzzy mapping surjective t fuzzy mapping injective t fuzzy mapping t fuzzy monomorphism t fuzzy epimorphism and t fuzzy partion and we prove some of their properties

fuzzy functions and their fundamental properties in this paper it is noticed that the definition of fuzzy equality of sasaki 1993 is not concordant with that of es and coker 1995 and sostak 1985 1988 furthermore the construction of fuzzy equality and fuzzy function is far from the way of natural fuzzification for this purpose the fuzzy equality of two points of a crisp set which is more convenient than that in es and coker 1995 and sostak 1985 1988 and the fuzzy function having weaker conditions than that is given in sasaki 1993 are redefined in much more natural way then various types of fuzzy functions are introduced and their fundamental properties are established

preserving fuzzy subgroups and indistinguishability operators abstract in this paper we study the class of all fuzzy subgroups defined with respect to a given t norm on a fixed group we find necessary and sufficient conditions over two t norms to guarantee that the class of all fuzzy subgroups induced by the first t norm is contained in the class of all fuzzy subgroups with respect to by the second one we characterize the functions that transform fuzzy subgroups into fuzzy subgroups the close relationship between indistinguishability operators and fuzzy subgroups allows us to obtain similar results for some classes of indistinguishability operators

vague groups and generalized vague subgroups on the basis of 0 1 le in this paper various elementary properties of vague groups and some properties of vague binary operations related with their associativity aspects are obtained furthermore the concept of vague isomorphism is defined and some basic properties of this concept are studied the concept of external direct product of vague groups is established later the definition of generalized vague subgroup which is a generalization of the vague subgroup defined by demirci is introduced and the validity of some classical results in this setting is investigated on the basis of the particular integral commutative complete quasi monoidal lattice 0 1

fuzzy t transitive relations eigenvectors and generators abstract after some preliminaries the set of generators of a generalized equality relation t indistinguishability operator is studied this set is identified with the set of the eigenvectors of the relation the relation between the fuzzy and metric topologies derived from these equalities is stablished the concept of basis is introduced and the construction of a procedure in order to calculate explicitly a basis of a t indistinguishability operator for t archimedean is proposed

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

population dynamics control in regard to minimizing the time necessary for the regeneration of a biomass taken away from the population the dynamics of a population whose mathematical model is the equation of verhulst is considered from the population by discrete external effects in the form of impulses a certain quantity of biomass is taken away or added this process is described by means of differential equations with impulses at fixed moments the moments and the magnitudes of the impulses are determined so that the time of regeneration of the quantity of biomass taken away from the population be minimal

broue s conjecture for the nonprincipal block of sl 2 q with full defect abstract m broue gives an important conjecture which is called broue s abelian defect group conjecture this conjecture says that a p block where p is a prime number of a finite group with an abelian defect group is derived equivalent to its brauer correspondent in the normalizer of the defect group in this paper we prove that this conjecture is true for the nonprincipal block of sl 2 p n for a positive integer n

splendid equivalences derived categories and permutation modules we introduce the notion of a splendid equivalence between blocks of finite groups which is a strengthening of the notion of an equivalence of derived categories such an equivalence is always induced by a two sided tilting complex whose terms are summands of permutation modules a feature that is compatible with the belief that these complexes should be related to the adic cohomology of deligne lusztig varities in the case of reductive groups over finite fields we show that a splendid equivalence has good consequences over and above those that follow from a derived equivalence in particular it gives for principal blocks a structural explanation of brou s concept of an isotypy a compatible family of perfect isometries we give several examples including examples of morita equivalences that are splendid in an unexpectedly non trivial way

oscillation stability and boundedness of second order differential systems with random impulses the model of second order linear differential systems with random impulses is brought forward in this paper then necessary and sufficient conditions for oscillation in mean p moment stability and p moment boundedness are obtained by several theorems that compare solutions of this system with the corresponding nonimpulsive differential system at last an example is presented to show the application of obtained results

the euler scheme and its convergence for impulsive delay differential equations this paper presents a fixed stepsize euler scheme for linear impulsive delay differential equations and considers its convergence we propose a method to take the partition nodes for the euler scheme employing the induction and the technique of inequality we obtain the order of convergence for euler scheme an example is given to illustrate the efficiency of our result

extensions for finite chevalley groups iii rational and generic cohomology abstract let g be a connected reductive algebraic group and b be a borel subgroup defined over an algebraically closed field of characteristic p 0 in this paper the authors study the existence of generic g cohomology and its stability with rational g cohomology groups via the use of methods from the authors earlier work new results on the vanishing of g and b cohomology groups are presented furthermore vanishing ranges for the associated finite group cohomology of g f q are established which generalize earlier work of hiller in addition to stability ranges for generic cohomology which improve on seminal work of cline parshall scott and van der kallen

second cohomology for finite groups of lie type let g be a simple simply connected algebraic group defined over mathbb f p given a power q p r of p let g mathbb f q subset g be the subgroup of mathbb f q rational points let l lambda be the simple rational g module of highest weight lambda in this paper we establish sufficient criteria for the restriction map in second cohomology h 2 g l lambda rightarrow h 2 g mathbb f q l lambda to be an isomorphism in particular the restriction map is an isomorphism under very mild conditions on p and q provided lambda is less than or equal to a fundamental dominant weight even when the restriction map is not an isomorphism we are often able to describe h 2 g mathbb f q l lambda in terms of rational cohomology for g we apply our techniques to compute h 2 g mathbb f q l lambda in a wide range of cases and obtain new examples of nonzero second cohomology for finite groups of lie type

stochastic stationary response of a variable mass system with mass disturbance described by poisson white noise abstract variable mass systems have received widespread attention and show prominent significance with the explosive development of micro and nanotechnologies so there is a growing need to study the influences of mass disturbances on systems this paper is devoted to investigating the stochastic response of a variable mass system subject to weakly random excitation in which the mass disturbance is modeled as a poisson white noise firstly the original system is approximately replaced by the associated conservative system with small disturbance based on the taylor expansion technique then the stationary response of the approximate system is obtained by applying the stochastic averaging method at last a representative variable mass oscillator is worked out to illustrate the effectiveness of the analytical solution by comparing with monte carlo simulation the relative change of mean square displacement is used to measure the influences of mass disturbance on system responses results reveal that the stochastic responses are more sensitive to mass disturbance for some system parameters it is also found that the influences of poisson white noise as the mass disturbance on system responses are significantly different from that of gaussian white noise of the same intensity

permutable fuzzy consequence and interior operators and their connection with fuzzy relations fuzzy operators are an essential tool in many fields and the operation of composition is often needed in general composition is not a commutative operation however it is very useful to have operators for which the order of composition does not affect the result in this paper we analyze when permutability appears that is when the order of application of the operators does not change the outcome we characterize permutability in the case of the composition of fuzzy consequence operators and the dual case of fuzzy interior operators we prove that for these cases permutability is completely connected to the preservation of the operator type we also study the particular case of fuzzy operators induced by fuzzy relations through zadeh s compositional rule and the inf composition for this cases we connect permutability of the fuzzy relations using the sup composition with permutability of the induced operators special attention is paid to the cases of operators induced by fuzzy preorders and similarities finally we use these results to relate the operator induced by the transitive closure of the composition of two reflexive fuzzy relations with the closure of the operator this composition induces

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

myhill nerode type theory for fuzzy languages and automata the myhill nerode theory is a branch of the algebraic theory of languages and automata in which formal languages and deterministic automata are studied through right congruences and congruences on a free monoid in this paper we develop a general myhill nerode type theory for fuzzy languages with membership values in an arbitrary set with two distinguished elements 0 and 1 which are needed to take crisp languages in consideration we establish connections between extensionality of fuzzy languages w r t right congruences and congruences on a free monoid and recognition of fuzzy languages by deterministic automata and monoids and we prove the myhill nerode type theorem for fuzzy languages we also prove that each fuzzy language possess a minimal deterministic automaton recognizing it we give a construction of this automaton using the concept of a derivative automaton of a fuzzy language and we give a method for minimization of deterministic fuzzy recognizers in the second part of the paper we introduce and study nerode s and myhill s automata assigned to a fuzzy automaton with membership values in a complete residuated lattice the obtained results establish nice relationships between fuzzy languages fuzzy automata and deterministic automata

on stability of perturbed impulsive differential systems the notion of lipschitz stability of impulsive systems of differential equations des was introduced in this paper we will extend the notion of eventual stability to impulsive systems of des and extend the notion of lipschitz stability of impulsive systems of des to a new type of stability called eventual lipschitz stability some criteria and results are given our technique depends on liapunov s direct method and comparison principle

probability density function modeling of evaporating droplets dispersed in isotropic turbulence a statistical closure scheme is used to obtain an approximate equation for probability density function to predict the statistical properties of interest of collisionless evaporating droplets suspended in isothermal isotropic turbulent flows the resulting fokker planck equation has nonlinear time dependent drift and diffusion coefficients that depend on the statistical properties of the droplet slip velocity approximate analytical expressions for these properties are derived and the equation is solved numerically after implementing the path integral approach the time evolution of various statistical properties related to the droplet diameter are then calculated

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

some properties of fuzzy topological groups abstract it is well known that if s is a subgroup of a topological group g so is s if s is a normal subgroup of g so is s and if u is the system of all neighborhoods of the identity e in a topological group x then b bu ub for any subset b of x we extend these to the corresponding theorems in fuzzy topological groups that is based on the q neighborhood system of fuzzy points and the structure of a fuzzy topological group defined by ma and yu fuzzy sets and systems 12 1984 289 we characterize some basic properties of fuzzy topological groups show that if h is a fuzzy subgroup of a fuzzy topological group x and h h h h so is h and if h is a fuzzy normal subgroup of x and h h h h so is h and show that for any fuzzy subset a of a fuzzy topological group if x p aw wa and 0 5 then x p a and if x p a and p q then aw x wa x 0 where w is the system of all q neighborhoods of the identity e q

equivalence operators in nilpotent systems a consistent connective system generated by nilpotent operators is not necessarily isomorphic to the ukasiewicz system using more than one generator function consistent nilpotent connective systems so called bounded systems can be obtained with the advantage of three naturally derived negation operators and thresholds in this paper equivalences in bounded systems are examined here three different types of operators are studied and a paradox of the equivalence i e there is no equivalence relation in a non boolean setting which fulfils x e x x 1 and e x n x 0 is resolved by aggregating the implication based equivalence and its dual operator we will also show that the aggregated equivalence has nice properties like associativity threshold transitivity and t transitiviy

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