low dimensional real lie algebras all real lie algebras of dimension up to 8 that admit a nontrivial levi decomposition are found

approximate symmetries and conservation laws for ito and stratonovich dynamical systems a new definition for the approximate symmetries of ito dynamical system is given determining systems of approximate symmetries for ito and stratonovich dynamical systems have been obtained it has been shown that approximate conservation laws can be found from the approximate symmetries of stochastic dynamical systems which do not arise from a hamiltonian the results have been applied to an example

a homological characterization of piecewise hereditary algebras let be a finite dimensional algebra over a field k we will show here that is piecewise hereditary if and only if its strong global dimension is finite

monomorphism categories cotilting theory and gorenstein projective modules abstract the monomorphism category s n x is introduced where x is a full subcategory of the module category a mod of an artin algebra a the key result is a reciprocity of the monomorphism operator s n and the left perpendicular operator for a cotilting a module t there is a canonical construction of a cotilting module m t over the upper triangular matrix algebra t n a such that s n t m t as applications s n x is a resolving contravariantly finite subcategory in t n a mod with s n x t n a mod if and only if x is a resolving contravariantly finite subcategory in a mod with x a mod for a gorenstein algebra a the category t n a g proj of gorenstein projective t n a modules can be explicitly determined as s n a also self injective algebras a can be characterized by the property t n a g proj s n a finally we obtain a characterization of those categories s n a which have finite representation type in terms of auslander s representation dimension

tame wild dichotomy of birkhoff type problems for nilpotent linear operators abstract inspired by recent results of ringel schmidmeier kussin lenzing meltzer xiao wu chen and pu zhang given a field k m 1 and a finite poset i i with a unique maximal element we study the category m o n i f m of mono representations of i over the frobenius k algebra f m k t t m of k dimension m viewed as k vector spaces u with an m nilpotent k linear operator t u u together with t invariant subspaces u i u j u for all i j in i the problem of when the krull schmidt k category m o n i f m is of wild resp tame representation type is called a wild resp tame birkhoff type problem for m nilpotent operators in case when k is algebraically closed we give a complete solution of the problem by describing all minimal pairs i m resp all pairs with m 1 such that category m o n i f m is of wild resp tame representation type we reduce the problem to a birkhoff type problem for the category fspr i f m m o n i f m of subprojective representations over f m of a larger poset i i the tame wild dichotomy for the category m o n i f m is also proved surprisingly in case when i i a b is the union of two incomparable chains i and i of length i a 1 1 and i b 1 1 with i i the problem is equivalent with the wildness resp tameness of the category coh x p of coherent sheaves over the weighted projective line x p for the weight triple p a b m with a b m 2 studied by kussin lenzing and meltzer 19 in relation with the hypersurface singularity f x 1 a x 2 b x 3 m

the strong global dimension of piecewise hereditary algebras let a be a finite dimensional piecewise hereditary algebra over an algebraically closed field this text investigates the strong global dimension of a this invariant is characterised in terms of the lengths of sequences of tilting mutations relating a to a hereditary abelian category in terms of the generating hereditary abelian subcategories of the derived category of a and in terms of the auslander reiten structure of that derived category

the acyclic closure of an exact category and its triangulation abstract for any exact category a with splitting idempotents a maximal exact category t a containing a as a biresolving subcategory is constructed important types of exact categories including n tilting torsion classes categories of cohen macaulay modules over a cohen macaulay order or categories of gorenstein projectives are shown to be of the form t a the quotient category t a a in the sense of grothendieck always exists and carries a triangulated structure more generally it is proved that any biresolving subcategory a of an exact category m gives a triangulated localization m a for example the unbounded derived category d a of an exact category a is obtained directly with no passage through a homotopy category of complexes as applications some recent developments related to gorenstein projectivity non commutative crepant resolutions singularity categories and cohen macaulay representations are extended and improved in the new framework for example the concept of non commutative resolution of a noetherian frobenius category is extended to arbitrary exact categories which leads to an overarching connection with representation dimension of exact categories and n tilting

lie point symmetries and stochastic differential equations we discuss lie point symmetries of stochastic ordinary differential equations and the interrelations between these and analogous symmetries of the associated fokker planck equation for the probability measure

invariant subspaces of nilpotent linear operators i let k be a field we consider triples v u t where v is a finite dimensional k space u a subspace of v and t v v a linear operator with t n 0 for some n and such that t u u thus t is a nilpotent operator on v and u is an invariant subspace with respect to t we will discuss the question whether it is possible to classify these triples these triples v u t are the objects of a category with the krullremak schmidt property thus it will be sufficient to deal with indecomposable triples obviously the classification problem depends on n and it will turn out that the decisive case is n 6 for n 6 we deal with what is called wild representation type so no complete classification can be expected for n 6 we will exhibit a complete description of all the indecomposable triples let k be a field and v a finite dimensional k vector space we consider a linear operator t v v which is nilpotent say satisfying t n 0 for some fixed natural number n and we are interested in the subspaces u of v which are t invariant this means that t u u to be more precise we are going to consider the corresponding triples v u t and ask for a classification of all the isomorphism classes in order to do so we consider these triples v u t as the objects of a category s n or sk n in case a reference to the base field

integration of stochastic ordinary differential equations from a symmetry standpoint we derive an algorithm for calculating lie point symmetries of systems of stochastic ordinary differential equations sodes of any order from this algorithm the following facts emerge symmetries of a sode do not in general form a lie algebra the determining equations for ito s equation are in general stochastic linear partial differential equations whereas for sodes of order n 2 the determining equations are linear deterministic partial differential equations that form an overdetemined system which is solvable by classical methods for scalar second order sodes we provide a complete classification of equations admitting finite dimensional symmetry lie algebras this classification is applied to the integration of scalar second order sodes in general a sode admitting a two dimensional symmetry algebra is not integrable by quadratures although it is reducible to a homogeneous ito equation in particular a scalar second order sode admitting a two dimensional symmetry algebra with connected operators is linearizable we also characterize integrable scalar second order sodes admitting three dimensional symmetry algebras finally we show that a sode can admit maximally a zero one two three or four dimensional lie algebra

wild canonical algebras canonical algebras p depending on a weight sequence p p1 pt of positive integers and a parameter sequence 3 t of pairwise distinct non zero elements from the base eld k were introduced and studied in 23 the nite dimensional representation theory mod is completely controlled by the category coh x of coherent sheaves on a non singular weighted projective line x x p since the derived categories d mod and d coh x are equivalent as triangulated categories 9 the curve x attached to has virtual genus gx 1 2 t 2 p t i 1 p pi where p l c m p1 pt the complexity of the classi cation problem for coh x and hence for mod is essentially determined by gx for gxi1 the algebra is concealed of extended dynkin type accordingly the classi cation problem for coh x and mod is equivalent to the classi cation of indecomposable modules over a tame hereditary algebra cf 18 9 or according to 10 to the classi cation problem for cohen macaulay modules over a simple surface singularity for gx 1 the algebra is of tubular type its representation theory is known from 23 while the classi cation problem for coh x 9 20 relates to atiyah s classi cation 1 of vector bundles over an elliptic curve very little was known on mod or coh x if gx 1 in this case 0 r is the one point extension of a wild hereditary algebra 0 accordingly the category mod 0 embeds into coh x as a full exact and extension closed subcategory implying that the classi cation problems for coh x and mod are wild from now on we assume gx 1 combining the sheaf theory 9 with the spectral investigation of wild hereditary algebras 6 we obtain a fairly complete picture on the structure of auslander reiten components for mod the asymptotic behaviour of the auslander reiten translations and

representation types of the category of subprojective representations of a finite poset over k t t m and a solution of a birkhoff type problem we determine the representation type wild tame polynomial growth of the category fspr i fm of filtered subprojective fm representations of a finite poset i in terms of m and i w herefm k t t m m 1 and k is an algebraically closed field criteria for tameness wildness and tameness of nonpolynomial growth of fspr i fm are given in theorems 1 1 and 1 2 as an application a solution of birkhoff s type problem g birkhoff subgroups of abelian groups proc london math soc 38 1934 385 401 for the category rep ft i fm of filtered i chains of fm modules is given in section 5 by determining the representation type repft i fm 2007 elsevier inc all rights reserved

mckay correspondence for the poincare series of kleinian and fuchsian singularities we give a uniform and to a large extent geometrical proof that the poincare series of the coordinate algebra of a kleinian singularity and of a fuchsian singularity of genus 0 is the quotient of the characteristic polynomials of two coxeter elements these coxeter elements are interpreted geometrically using 2 calabi yau triangulated categories and spherical twist functors

from submodule categories to the stable auslander algebra abstract we construct two functors from the submodule category of a representation finite self injective algebra to the module category of the stable auslander algebra of these functors factor through the module category of the auslander algebra of moreover they induce equivalences from the quotient categories of the submodule category modulo their respective kernels and said kernels have finitely many indecomposable objects up to isomorphism their construction uses a recollement of the module category of the auslander algebra induced by an idempotent and this recollement determines a characteristic tilting and cotilting module if is taken to be a nakayama algebra then said tilting and cotilting module is a characteristic tilting module of a quasi hereditary structure on the auslander algebra we prove that the self injective nakayama algebras are the only algebras with this property

triangle singularities ade chains and weighted projective lines abstract we study the singularity category of the triangle singularity f x a y b z c or s k x y z f attached to the weighted projective line x given by the weight triple a b c by investigating the stable category vect x of vector bundles on x obtained from the category of vector bundles by factoring out all the line bundles we show that vect x is fractional calabi yau whose cy dimension is a function of the euler characteristic of x and establish the existence of a tilting object having the shape of an a 1 b 1 c 1 cuboid the weight types 2 a b yield an explanation of happel seidel symmetry for a class of important nakayama algebras further the weight sequence 2 3 p corresponds to an ade chain the e n chain extrapolating the exceptional dynkin cases e 6 e 7 and e 8 to a whole sequence of triangulated categories

mutation of tilting bundles of tubular type abstract let x be a weighted projective line of tubular type and coh x the category of coherent sheaves on x the main purpose of this note is to show that the subgraph of the tilting graph consisting of all basic tilting bundles in coh x is connected this yields an alternative proof for the connectedness of the tilting graph of coh x our approach leads to the investigation of the change of slopes of a tilting sheaf in coh x under co apr mutations which may be of independent interest

triangulated categories of matrix factorizations for regular systems of weights with e 1 abstract we construct a full strongly exceptional collection in the triangulated category of graded matrix factorizations of a polynomial associated to a nondegenerate regular system of weights whose smallest exponents are equal to 1 in the associated grothendieck group the strongly exceptional collection defines a root basis of a generalized root system of sign l 0 2 and a coxeter element of finite order whose primitive eigenvector is a regular element in the expanded symmetric domain of type iv with respect to the weyl group

hereditary noetherian categories of positive euler characteristic we develop the fundamentals of hereditary noetherian categories with serre duality over an arbitrary field k where the category of coherent sheaves over a smooth projective curve over k serves as the prime example and others are coming from the representation theory of finite dimensional algebras the proper way to view such a category is to think of coherent sheaves on a possibly non commutative smooth projective curve we define for each such category notions like function field and euler characteristic determine its auslander reiten components and study stable and semistable bundles for an appropriate notion of degree we provide a complete classification of hereditary noetherian categories for the case of positive euler characteristic by relating these to finite dimensional representations of locally bounded hereditary k algebras whose underlying valued quiver admits a positive additive function

bimodule monomorphism categories and rss equivalences via cotilting modules abstract the monomorphism category m a m b induced by a bimodule m b a is the subcategory of mod consisting of x y such that m b y x is a monic a map where a m 0 b and a b are artin algebras in general m a m b is not the monomorphism category induced by quivers it could describe the gorenstein projective modules this monomorphism category is a resolving subcategory of mod if and only if m b is projective in this case it has enough injective objects and auslander reiten sequences and can be also described as the left perpendicular category of a unique basic cotilting module if m satisfies the condition ip see subsection 1 6 then the stable category of m a m b admits a recollement of additive categories which is in fact a recollement of singularity categories if m a m b is a frobenius category ringel schmidmeier simson equivalence between m a m b and its dual is introduced if m is an exchangeable bimodule then an rss equivalence is given by a bimodule which is a two sided cotilting module with a special property and the nakayama functor n gives an rss equivalence if and only if both a and b are frobenius algebras

on the definition of an admitted lie group for stochastic differential equations with multi brownian motion the definition of an admitted lie group of transformations for stochastic differential equations has been already presented for equations with one dimensional brownian motion the transformation of the dependent variables involves time as well and it has been proven that brownian motion is transformed to brownian motion in this paper we will discuss this concept for stochastic differential equations involving multi dimensional brownian motion and present applications to a variety of stochastic differential equations

extended canonical algebras and fuchsian singularities the authors introduce a new class of finite dimensional algebras called extended canonical and determine the shape of their derived categories extended canonical algebras arise from a canonical algebra by onepoint extension or coextension by an indecomposable projective module our main results concern the case of negative euler characteristic of the corresponding weighted projective line mathbb x more specifically we establish for a base field of arbitrary characteristic a link to the fuchsian singularity r of mathbb x which for the base field of complex numbers is isomorphic to an algebra of automorphic forms by means of a recent result of orlov we show that the triangulated category of the graded singularities of r in the sense of buchweitz and orlov admits a tilting object whose endomorphism ring is the corresponding extended canonical algebra of particular interest are those cases where the attached coxeter transformation has spectral radius one a k theoretic analysis then shows that this happens exactly for 38 cases including arnold s 14 exceptional unimodal singularities the paper is related to recent independent work by kajiura saito and takahashi

lie point symmetries of stratonovich stochastic differential equations this paper considers lie point symmetries of stochastic differential equations sdes in stratonovich form first we derive the determining equations of the random symmetries which correspond to lie group transformations acting in the space of the independent variable time the dependent variables and the brownian motion then the determining equations of the random symmetries are used to obtain the determining equations of the deterministic symmetries which correspond to lie group transformations acting in the space of the independent variable and dependent variables in the case of the deterministic symmetries the transformation of the brownian motion is induced by the random time change it is proved that the corresponding ito and stratonovich sdes admit the same random symmetries and the same deterministic symmetries

separated monic representations i gorenstein projective modules abstract for a finite acyclic quiver q an ideal i of a path algebra kq generated by monomial relations and a finite dimensional k algebra a we introduce the separated monic representations of a bound quiver q i over a they differ from the usual monic representations the category smon q i a of the separated monic representations of q i over a coincides with the category mon q i a of the usual monic representations if and only if i 0 and each vertex of q is the ending vertex of at most one arrow we give properties of the structural maps of separated monic representations and prove that smon q i a is a resolving subcategory of rep q i a we introduce the condition g let a k q i by the equivalence rep q i a mod of categories the main result claims that a module is gorenstein projective if and only if it is in smon q i a and has a local a gorenstein projective property g as consequences the separated monic modules are exactly the projective modules if and only if a is semi simple and they are exactly the gorenstein projective modules if and only if a is self injective and if and only if smon q i a is a frobenius category

symmetries of systems of stochastic differential equations with diffusion matrices of full rank lie point symmetries of a system of stochastic differential equations sdes with diffusion matrices of full rank are considered it is proved that the maximal dimension of a symmetry group admitted by a system of n sdes is n 2 in addition such systems cannot admit symmetry operators whose coefficients are proportional to a nonconstant coefficient of proportionality these results are applied to compute the lie group classification of a system of two sdes the classification is obtained with the help of non equivalent realizations of real lie algebras by fiber preserving vector fields in 1 2 variables possibilities of using symmetries for integration of sdes by quadratures are discussed

random lie symmetries of ito stochastic differential equations lie point symmetries of stochastic differential equations sdes which include transformations depending on the brownian motion are considered the corresponding lie group transformations acting in the space of the independent variable time the dependent variables and the brownian motion preserve the differential form of sdes and the properties of the brownian motion such symmetries are called random symmetries the focus of the paper is a definition of the random symmetries and their properties it is shown that the random symmetries form a lie algebra we also investigate relations between the random symmetries and the conserved quantities of sdes for several stochastic differential equations we find their random symmetries and compare them with the deterministic symmetries which correspond to lie group transformations in the space of the independent and dependent variables

homological properties of piecewise hereditary algebras let be a finite dimensional algebra over an algebraically closed field k we will investigate homological properties of piecewise hereditary algebras in particular we give lower and upper bounds of the strong global dimension show the behavior of the strong global dimension under one point extensions and tilting moreover we show that the pieces of mod have auslander reiten sequences

symmetry lie algebras of nth order ordinary differential equations abstract we show that an nth n 3 order linear ordinary differential equation has exactly one of n 1 n 2 or n 4 the maximum point symmetries the lie algebras corresponding to the respective numbers of point symmetries are obtained then it is shown that a necessary and sufficient conditon for an nth n 3 order equation to be linearizable via a point transformation is that it must admit the n dimensional abelian algebra na1 a1 a1 a1 we discuss in detail the symmetry realizations of n 1 a1 s a1 finally we prove that an nth n 3 order equation q n h t q qn 1 cannot admit exactly an n 3 dimensional algebra of point symmetries which is a subalgebra of na1 gl 2 r

training the random neural network using quasi newton methods abstract training in the random neural network rnn is generally specified as the minimization of an appropriate error function with respect to the parameters of the network weights corresponding to positive and negative connections we propose here a technique for error minimization that is based on the use of quasi newton optimization techniques such techniques offer more sophisticated exploitation of the gradient information compared to simple gradient descent methods but are computationally more expensive and difficult to implement in this work we specify the necessary details for the application of quasi newton methods to the training of the rnn and provide comparative experimental results from the use of these methods to some well known test problems which confirm the superiority of the approach

on the classification of higher auslander algebras for nakayama algebras abstract we give new improved bounds for the dominant dimension of nakayama algebras and use those bounds to give a classification of nakayama algebras with n simple modules that are higher auslander algebras with global dimension at least n the classification is then used to extend the results on the inequality for the global dimension of nakayama algebras obtained in 18

on the mahler measure of the coxeter polynomial of an algebra abstract let a be a finite dimensional algebra over an algebraically closed field k assume a is a basic connected and triangular algebra with n pairwise non isomorphic simple modules we consider the coxeter transformation a t as the automorphism of the grothendieck group k 0 a induced by the auslander reiten translation in the derived category der b mod a of the module category mod a of finite dimensional left a modules we say that a is of cyclotomic type if the characteristic polynomial a of a is a product of cyclotomic polynomials equivalently if the mahler measure m a 1 in 6 we have considered many examples of algebras of cyclotomic type in the representation theory literature in this paper we study the mahler measure of the coxeter polynomial of accessible algebras in 1933 d h lehmer found that the polynomial t 10 t 9 t 7 t 6 t 5 t 4 t 3 t 1 has mahler measure 0 1 176280 and he asked whether there exist any smaller values exceeding 1 in this paper we prove that for any accessible algebra a either m a 1 or m b 0 for some convex subcategory b of a we introduce interlaced tower of algebras a m a n with m n 2 satisfying a s 1 t 1 a s t a s 1 for m 1 s n 1 we prove that if spec a n s 1 r and a n is not of cyclotomic type then m a m m a n we construct several examples

on tensor products of path algebras of type a abstract we derive a formula for the coxeter polynomial of the s fold tensor product i 1 s f a n i 1 of path algebras of linearly oriented quivers of dynkin type a n i 1 in terms of the weights n 1 n s 2 and show that conversely the weights can be recovered from the coxeter polynomial of the tensor product our results have relevance in singularity theory in particular these algebras occur as endomorphism algebras of tilting objects in certain stable categories of vector bundles

matrix factorizations and representations of quivers ii type ade case abstract we study a triangulated category of graded matrix factorizations for a polynomial of type ade we show that it is equivalent to the derived category of finitely generated modules over the path algebra of the corresponding dynkin quiver also we discuss a special stability condition for the triangulated category in the sense of t bridgeland which is naturally defined by the grading