orlicz lorentz hardy martingale spaces abstract martingale hardy spaces are widely studied in the field of mathematical physics and probability in this paper we develop the theory of orlicz lorentz hardy martingale spaces which are much more wider than the classical lorentz hardy martingale spaces more precisely we first investigate several basic properties of orlicz lorentz spaces and then construct the atomic decomposition theorems of these martingale function spaces also we establish the dual theorem of orlicz lorentz hardy spaces for martingales furthermore we study the boundedness of generalized fractional integral operators i in this new framework where is a non negative concave function the results partially extend the very recent results jiao et al 2017 21

on the inductive construction of galois coverings of algebras let a be a finite dimensional algebra over an algebraically closed field k in order to study the category moda of finitely generated left a modules we may assume that a is basic and connected by 8 there is a finite quiver i e oriented graph qa and a surjective homomorphism of algebras kqa a such that i ker kqa 2 where kqa denotes the path algebra associated to qa and kq a denotes the ideal of kqa generated by all arrows in qa for each pair qa i called a presentation of a we can define the fundamental group 1 qa i see 9 13 or section 1 2 below then there is surjective homomorphism of algebras f a a defined by the action of 1 qa i on a called the universal galois covering of a with respect to as in 8 we shall consider algebras as locally finite k categories galois coverings have proved to be a powerful tool for the study of the module category moda indeed a is representation finite if and only if a is locally support finite and locally representation finite in this case if chark 2 then a kq where q is a quiver without oriented cycles 4 9 14 if a is tame then a is tame 6 but the converse does not hold 11 certain a modules may be described via the push down functor f mod a moda see 5 in this paper we shall consider only triangular algebras that is algebras a kqa i such that qa has no oriented cycles in this case a vertex x in qa is said to be separating if for the indecomposable decomposition radpx m1 ms there is a decomposition into connected components q a q1 qt with t s of the induced full subquiver q a of qa with vertices those y which are not predecessors

martingale hardy spaces for 0 p 1 the relation between the usually considered martingale hardy spaces is examined it will be shown that if the stochastic basis is regular then all these hardy spaces are equivalent an atomic description of the spacesh andp p will be given for 0 p 1 and with the help of this we shall consider the duals of these spaces moreover the dual space ofvmo p generated by the dual space ofh p will be determined in the case when all algebras are generated by countably many atoms furthermore we shall show in this case that the dual ofvmo 1 ish 1

convergence and stability of the compensated split step theta method for stochastic differential equations with piecewise continuous arguments driven by poisson random measure abstract this paper deals with the numerical solutions of stochastic differential equations with piecewise continuous arguments sdepcas driven by poisson random measure in which the coefficients are highly nonlinear it is shown that the compensated split step theta csst method with 0 1 is strongly convergent in p th p 2 moment under some polynomially lipschitz continuous conditions it is also obtained that the convergence order is close to 1 p in terms of the stability it is proved that the csst method with 1 2 1 reproduces the exponential mean square stability of the underlying system under the monotone condition and some restrictions on the step size without any restriction on the step size there exists 1 2 1 such that the csst method with 1 is exponentially stable in mean square moreover if the drift and jump coefficients satisfy the linear growth condition the csst method with 0 1 2 also preserves the exponential mean square stability some numerical simulations are presented to verify the conclusions

the tits and euler forms of a tame algebra since the introduction by gabriel in 1972 of the tits quadratic form associated to finite dimensional hereditary algebras quadratic forms have played an important role in the representation theory of algebras and related categories there have been quadratic forms associated to finite dimensional algebras partially ordered sets and other objects in order to determine their representation type and the classes of indecomposable representations in the corresponding grothendieck group see for example 3 9 14 21 the class of finite dimensional algebras associative with in identity over an algebraically closed field k is divided into two disjoint classes one class consists of tame algebras for which the indecomposable modules occur in each dimension in a finite number of discrete and one parameter families the second class consist of wild algebras whose representation theory is as complicated as the classification of pairs of matrices up to simultaneous conjugation in this paper we establish properties of the tits and euler forms associated to tame simply connected algebras let a be a basic connected finite dimensional algebra over k then a has a presentation a kqa i where qa q0 q1 is the ordinary quiver of a with set of vertices q0 and set of arrows q1 and i is an admissible ideal of the path algebra kqa see 9 we shall assume that qa has no oriented cycles and hence g dim a then the euler form a zq0 z is defined by

atomic decomposition and interpolation for hardy spaces of noncommutative martingales we prove that atomic decomposition for the hardy spaces h1 and h1 is valid for noncommutative martingales we also establish that the conditioned hardy spaces of noncommutative martingales hp and bmo form interpolation scales with respect to both complex and real interpolations

martingale weak orlicz karamata hardy spaces associated with concave functions abstract in this paper we introduce the weak orlicz karamata hardy spaces associated with concave functions and establish the atomic decompositions as applications we prove several martingale inequalities and obtain a duality theorem

tame algebras and tits quadratic forms abstract we solve a long standing problem concerning the connection between the tameness of simply connected algebras and the weak nonnegativity of the associated tits integral quadratic forms and derive some consequences

global stability and positive recurrence of a stochastic sis model with levy noise perturbation abstract focusing on epidemic model in random environments this paper uses white noise and levy noise to model the dynamics of the sis epidemic model subject to the random changes of the external environment we show that the jump encourages the extinction of the disease in the population we first give a rigorous proof of the global stability of the disease free equilibrium state we also establish sufficient conditions for the persistence of the disease the presented results are demonstrated by numerical simulations

global flows for stochastic differential equations without global lipschitz conditions we consider stochastic differential equations driven by wiener processes the vector fields are supposed to satisfy only local lipschitz conditions the lipschitz constants of the drift vector field valid on balls of radius r are supposed to grow not faster than log r while those of the diffusion vector fields are supposed to grow not faster than sqrt log r we regularize the stochastic differential equations by associating with them approximating ordinary differential equations obtained by discretization of the increments of the wiener process on small intervals by showing that the flow associated with a regularized equation converges uniformly to the solution of the stochastic differential equation we simultaneously establish the existence of a global flow for the stochastic equation under local lipschitz conditions

theta schemes for sddes with non globally lipschitz continuous coefficients this paper establishes the boundedness convergence and stability of the two classes of theta schemes namely split step theta sst scheme and stochastic linear theta slt scheme for stochastic differential delay equations sddes with non globally lipschitz continuous coefficients when the drift f x y satisfies one sided lipschitz condition with respect to the present state x and the diffusion g x y obeys the global lipschitz condition with respect to the present term x but the delay terms y in the drift and diffusion may be highly nonlinear this paper first examines the strong convergence rates of the theta schemes for sddes it is also proved that the two classes of theta schemes for 1 2 1 converge strongly to the exact solution with the order 1 2 but for 0 1 2 the linear growth condition on drift f x y in x is needed for the strong convergence rates the exponential mean square stability of the theta schemes with 1 2 1 is also investigated for highly nonlinear sddes

the truncated em method for stochastic differential equations with poisson jumps abstract in this paper we use the truncated euler maruyama em method to study the finite time strong convergence for sdes with poisson jumps under the khasminskii type condition we establish the finite time l r r 2 convergence order when the drift and diffusion coefficients satisfy the super linear growth condition and the jump coefficient satisfies the linear growth condition the result shows that the optimal l r convergence order is close to 1 this is significantly different from the result on sdes without jumps when all the three coefficients of sdes are allowing to grow super linearly the l r 0 r 2 convergence results are also investigated and the optimal l r convergence order is shown to be not greater than 1 4 moreover we prove that the truncated em method preserves nicely the mean square exponential stability and asymptotic boundedness of the underlying sdes with poisson jumps several examples are given to illustrate our results

strongly simply connected schurian algebras and multiplicative bases abstract in this paper we define concepts of crowns and quasi crowns valid in an arbitrary schurian algebra and which generalise the corresponding concepts in an incidence algebra we show first that a triangular schurian algebra is strongly simply connected if and only if it is simply connected and contains no quasi crown we then prove that the absence of quasi crowns in a triangular schurian algebra implies the existence of a multiplicative basis

algorithms for weakly nonnegative quadratic forms abstract let q z n z with q v n i 1 v t 2 i j a ij v i v j be a unit form we present an algorithm that allows one to check if q is weakly nonnegative i e q v 0 for any vector v n n the algorithm also calculates the set of critical vectors of q we sketch the relation of this problem to the representation theory of finite dimensional algebras

a note on almost sure asymptotic stability of neutral stochastic delay differential equations with markovian switching in this paper we consider neutral stochastic delay differential equations with markovian switching our key aim is to establish lasalle type stability theorems for the underlying equations the key techniques used in this paper are the method of lyapunov functions and the convergence theorem of nonnegative semi martingales the key advantage of our new results lies in the fact that our results can be applied to more general non autonomous equations

some new φ moment martingale inequalities associated with concave functions abstract the paper is devoted to study the moment inequalities for martingales where is a concave function we mainly employ the method of atomic decomposition involving moment martingale inequalities

the classifying space of a bound quiver abstract we associate to a bound quiver q i a cw complex which we denote by b q i and call the classifying space of q i we show that the fundamental group of b q i is isomorphic to the fundamental group of q i moreover we show that this construction behaves well with respect to coverings on the other hand we study the co homology groups of b q i and compare them with the simplicial and the hochschild co homology groups of the algebra a kq i more precisely we give sufficient conditions for these groups to be isomorphic this generalizes a theorem due to gerstenhaber and schack j pure appl algebra 30 1983 143 156

new martingale inequalities and applications to fourier analysis abstract let f p be a probability space and 0 0 be a musielak orlicz function in this article the authors prove that the doob maximal operator is bounded on the musielak orlicz space l using this and extrapolation method the authors then establish a fefferman stein vector valued doob maximal inequality on l as applications the authors obtain the dual version of the doob maximal inequality and the stein inequality for l which are new even in weighted orlicz spaces the authors then establish the atomic characterizations of martingale musielak orlicz hardy spaces h s p q h s and h m from these atomic characterizations the authors further deduce some martingale inequalities between different martingale musielak orlicz hardy spaces which essentially improve the corresponding results in orlicz space case and are also new even in weighted orlicz spaces by establishing the davis decomposition on h s and h m the authors obtain the burkholder davis gundy inequality associated with musielak orlicz functions finally using the previous martingale inequalities the authors prove that the maximal fejer operator is bounded from h 0 1 to l 0 1 which further implies some convergence results of the fejer means these results are new even for the weighted hardy spaces

convergence rate of numerical solutions to sfdes with jumps abstract in this paper we are interested in numerical solutions of stochastic functional differential equations with jumps under a global lipschitz condition we show that the p th moment convergence of euler maruyama numerical solutions to stochastic functional differential equations with jumps has order 1 p for any p 2 this is significantly different from the case of stochastic functional differential equations without jumps where the order is 1 2 for any p 2 it is therefore best to use the mean square convergence for stochastic functional differential equations with jumps moreover under a local lipschitz condition we reveal that the order of mean square convergence is close to 1 2 provided that local lipschitz constants valid on balls of radius j do not grow faster than log j

numerical methods for nonlinear stochastic differential equations with jumps we present and analyse two implicit methods for ito stochastic differential equations sdes with poisson driven jumps the first method ssbe is a split step extension of the backward euler method the second method cssbe arises from the introduction of a compensated martingale form of the poisson process we show that both methods are amenable to rigorous analysis when a one sided lipschitz condition rather than a more restrictive global lipschitz condition holds for the drift our analysis covers strong convergence and nonlinear stability we prove that both methods give strong convergence when the drift coefficient is one sided lipschitz and the diffusion and jump coefficients are globally lipschitz on the way to proving these results we show that a compensated form of the euler maruyama method converges strongly when the sde coefficients satisfy a local lipschitz condition and the pth moment of the exact and numerical solution are bounded for some p 2 under our assumptions both ssbe and cssbe give well defined unique solutions for sufficiently small stepsizes and ssbe has the advantage that the restriction is independent of the jump intensity we also study the ability of the methods to reproduce exponential mean square stability in the case where the drift has a negative one sided lipschitz constant this work extends the deterministic nonlinear stability theory in numerical analysis we find that ssbe preserves stability under a stepsize constraint that is independent of the initial data cssbe satisfies an even stronger condition and gives a generalization of b stability finally we specialize to a linear test problem and show that cssbe has a natural extension of deterministic a stability the difference in stability properties of the ssbe and cssbe methods emphasizes that the addition of a jump term has a significant effect that cannot be deduced directly from the non jump literature

almost sure convergence rate of θ em scheme for neutral sddes abstract this paper is concerned with almost sure convergence of em scheme for neutral stochastic differential delay equations under local lipschitz conditions with 1 2 1 it is revealed that the strong convergence rate under local lipschitz condition is 1 2 while almost sure convergence rate is less than 1 2 provided the local lipschitz constant valid on balls of radius r grows not faster than log r

finite dimensional algebras with vanishing hochschild cohomology let a be a finite dimensional algebra associative with unit over an algebraically closed field k for a finitely generated bimodule x the hochschild cohomology a a i w x i groups h a x i g 0 were introduced in 10 we shall denote h a i i s h a a the vanishing of the low dimensional groups h a i f 2 has classical interpretations and plays an important role in the representaw x tion theory of algebras gernstenhaber 7 has shown some connections 2 w x 1 between h a and the deformation theory of a see also 5 h a is

hardy lorentz spaces for b valued martingales abstract in this paper we study the hardy lorentz spaces for banach space valued martingales the dual spaces are characterized and several martingale inequalities are established the proofs mainly depend on the classical tool of atomic decompositions as usual these conclusions are closely related to the geometrical properties of the underlying banach spaces

strong convergence rates for backward euler on a class of nonlinear jump diffusion problems we generalise the current theory of optimal strong convergence rates for implicit euler based methods by allowing for poisson driven jumps in a stochastic differential equation sde more precisely we show that under one sided lipschitz and polynomial growth conditions on the drift coefficient and global lipschitz conditions on the diffusion and jump coefficients three variants of backward euler converge with strong order of one half the analysis exploits a relation between the backward and explicit euler methods