generic cartan invariants for frobenius kernels and chevalley groups chastkofsky 2 described the generic behavior of the composition factor multiplicities cartan invariants of indecomposable projective modules for finite groups of lie type in the defining characteristic more recently ye lo provided a different approach along with interesting refinements in each case the reason for the existence of generic cartan invariants is somewhat hidden behind the complicated character manipulations involved here we offer a more conceptual treatment emphasizing the much simpler picture for the group schemes g t associated with the frobenius kernels g in the ambient algebraic group this is closer in spirit to lo than to 2 but avoids the messy explicit calculations found in both papers

cohomological degrees and hilbert functions of graded modules making use of the recent construction of cohomological degrees functions we give several estimates on the relationship between number of generators and degrees of ideals and modules with applications to hilbert functions they extend results heretofore known from generalized cohen macaulay local rings to nearly arbitrary local rings the rules of computation these functions satisfy enables comparison with castelnuovo mumford s regularity in the graded case as application we derive sharp improvements on predicting the outcome of effecting noether normalizations in tangent cones

a new approach to linear interval differential equations as a first step toward solving fuzzy differential abstract this study uses a concept of interval differentiability which was recently introduced to formulate interval initial value problems involving linear interval differential equations differently from the approaches that use the gh differentiability this study does not make use of a criterion of choice for switching points in order to obtain solutions for such problems the method herein presented provides solutions in a simple straightforward and computationally tractable way moreover these solutions are intuitive because they coincide with the solutions given by a differential inclusion method the efficiency and practicality of our approach are illustrated through some examples that have appeared in other articles

extensions of simple modules for finite chevalley groups let g be an almost simple algebraic group over an algebraically closed field k of positive characteristic p we assume that g is defined and split over the prime field 5 for it 3 1 we denote by g n the finite group consisting of the points of g over the field with p elements our first result states that for a certain class of simple modules for g n the only extensions that can occur are those which are restrictions of g extensions see theorem 2 8 the full story about g extensions is not known yet except for a few rather small groups however much more is known about extensions for the algebraic group than about those for the finite group for instance the linkage principle l gives an upper bound for which g extensions may occur moreover a proof of the lusztig conjecture on the irreducible characters for g also implies strong statements about g extensions see 3 proposition 2 81 it is an open question whether all extensions between simple g n modules can be found via the extension theory for g we prove that in generic cases this is in fact so see theorem 3 2 moreover for the groups x 3 p and sp 4 p we are able to determine completely all extensions between simple modules from the corresponding known results for the algebraic groups x 3 k and sp 4 k our results extend previous results by cline parshall and scott s 91 who consider extensions of the trivial module by a minimal module and smith ls who considers extensions between certain minimal modules the method we use is a generalization of the one used by humphreys 131 who proves the existence of certain selfextensions for sp 4 p it should be pointed out that our methods do not work for p small on the other hand everything we do generalizes easily to the twisted chevalley groups see theorem 3 4 we demonstrate this by working out all extensions for su 3 p see section 4 3

interconnection between set and fuzzy differential equations abstract the original formulation of fuzzy differential equations suffers from the disadvantage since the solutions increases as time increases in this paper employing the results of set differential equations obtained in v laksmikantham s leela a s vatsala setvalued hybrid differential equations and stability in terms of two measures j hybrid systems 2 2 2002 169 188 a new approach is suggested in order to capture the vagueness and the rich properties of solutions without fuzziness

castelnuovo mumford regularity and extended degree our main result shows that the castelnuovo mumford regularity of the tangent cone of a local ring a is effectively bounded by the dimension and any extended degree of a from this it follows that there are only a finite number of hilbert samuel functions of local rings with given dimension and extended degree

on set valued stochastic integrals and fuzzy stochastic equations in this paper we extend the notion of set valued and fuzzy stochastic integrals to semimartingale integrators we present their main properties and finally we establish the existence of solutions of a fuzzy integral stochastic equation driven by a brownian motion the approach is based on the existence of solutions of an appropriately formulated martingale problem for a system of stochastic inclusions and the theorem of negoita and ralescu

almost periodicity of set valued functions and set dynamic equations on time scales in this paper we first propose the concept of the almost periodicity of set valued functions on almost periodic time scales then based on the exponential dichotomy we discuss the existence and uniqueness of almost periodic solutions of set dynamic equations on time scales as an application several examples are presented to exhibit that one can obtain the existence of almost periodic solutions to interval valued dynamic equations or fuzzy dynamic equations and their discrete counterparts in a united means

a generalization of pillen s theorem for principal series modules ii abstract let g be a connected semisimple and simply connected algebraic group and g q the corresponding finite chevalley group over the finite field of order q p r in a recent paper the author determined a direct sum decomposition of the k g q submodule generated by a highest weight vector of a certain weyl module when q is not too small which is a generalization of pillen s result in 1997 in this article we claim that the result does not need the assumption on q

comparing castelnuovo mumford regularity and extended degree the borderline cases castelnuovo mumford regularity and any extended degree function can be thought of as complexity measures for the structure of finitely generated graded modules a recent result of doering gunston and vasconcelos shows that both can be compared in the case of a graded algebra we extend this result to modules and analyze when the estimate is in fact an equality a complete classification is obtained if we choose as extended degree the homological or the smallest extended degree the corresponding algebras are characterized in three ways by relations among the algebra generators by using generic initial ideals and by their hilbert series

existence theorems for solutions to random fuzzy differential equations abstract we examine a random fuzzy initial value problem with two kinds of fuzzy derivatives for both cases we establish the results of existence and uniqueness of local solutions to random fuzzy differential equations the existence of global solutions is also obtained

some results on sheaf solutions of sheaf set control problems abstract in this paper we study the existence and some properties of solutions of the so called set control differential equations scde and sheaf solutions of sheaf set control problems

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

extended degree functions and monomial modules the arithmetic degree the smallest extended degree and the homological degree are invariants that have been proposed as alternatives of the degree of a module if this module is not cohen macaulay we compare these degree functions and study their behavior when passing to the generic initial or the lexicographic submodule this leads to various bounds and to counterexamples to a conjecture of gunston and vasconcelos respectively particular attention is given to the class of sequentially cohen macaulay modules the results in this case lead to an algorithm that computes the smallest extended degree

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

on random fuzzy differential equations we consider a cauchy problem in a random fuzzy setting under the condition of lipschitzean right hand side the existence and uniqueness of the solution is proven also the continuous dependence on the right hand side and initial condition is shown some kind of boundedness of the solution is established

bezout s theorem and cohen macaulay modules we define very proper intersections of modules and projective subschemes it turns out that equidimensional locally cohen macaulay modules intersect very properly if and only if they intersect properly we prove a bezout theorem for modules which meet very properly furthermore we show for equidimensional subschemes x and y if they intersect properly in an arithmetically cohen macaulay subscheme of positive dimension then x and y are arithmetically cohen macaulay the module version of this result implies splitting criteria for reflexive sheaves

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

stability criteria for set dynamic equations on time scales very recently a new theory known as set dynamic equations on time scales has been built in this paper notions of stability for the solutions of set dynamic equations on time scales using lyapunov like functions are considered criteria for the equistability equiasymptotic stability uniform and uniform asymptotic stability are developed

structure of cohomology of line bundles on gb for semisimple groups let g be a connected and simply connected semisimple algebraic group over an algebraically closed field k of characteristic p 0 t a maximal torus of g and b a borel subgroup containing t each weight in x t determines a line bundle on the flag varietyg b it turns out that cohomology of the line bundle is isomorphic to the derived functor of the induction functors from the category of b modules to the category of g modules for each 1 dimensional b module defined by a weight in x t further indgb h g b turns out to be the dual of a weyl module one of the main problems is to calculate the characters of the irreducible g modules for p 0 the character of the irreducible g module of highest weight is given by weyl s character formula and the g module structures of the cohomology of line bundles are well understood however for p 0 the story is quite different and many of the results remain conjectural since characters of weyl modules are given by weyl s character formula understanding the structure ofh g b turns out to be the main problem it is also interesting to understand the structure of the higher cohomology which might help us to understand h h andersen has a series of papers toward the understanding of the higher cohomology such as the simple socle of h and filtrations of h i 4 in 5 using the representations of infinitesimal subgroup schemes of g he proved generically that h g b w has simple socle and simple head and their highest weights can be calculated in 6 he proved that the socle series of h comes generically from the

stability and periodicity in fuzzy differential equations formulations of fuzzy differential equations des in terms of the hukuhara derivative do not reproduce the rich and varied behavior of crisp des another interpretation in terms of differential inclusions expressed level setwise overcomes much of this deficiency and opens up for profitable investigation such properties as stability attraction periodicity and the like this is especially important for investigating continuous systems which are uncertain or incompletely specified this paper studies studies lyapunov stability of fuzzy des and the periodicity of the fuzzy solution set for both the time dependent and autonomous cases

castelnuovo mumford regularity of canonical and deficiency modules abstract we give two kinds of bounds for the castelnuovo mumford regularity of the canonical module and the deficiency modules of a ring respectively in terms of the homological degree and the castelnuovo mumford regularity of the original ring

the permutation module on flag varieties in cross characteristic let mathbf g be a connected reductive group defined over mathbb f q the finite field with q elements where q is a power of the prime number p with the standard frobenius map f let mathbf b be an f stable borel subgroup let bbbk be a field may not be algebraically closed of characteristic 0 le r ne p in this paper we completely determine the composition factors of the permutation module module mathbb m text tr bbbk mathbf g otimes bbbk mathbf b text tr here bbbk mathbf h is the group algebra of the group mathbf h and text tr is the trivial mathbf b module in particular we find a large family of infinite dimensional absolutely irreducible abstract representations of mathbf g over bbbk

castelnuovo mumford regularity of ext modules and homological degree bounds for the castelnuovo mumford regularity of ext modules over a polynomial ring over a field are given in terms of the initial degrees castelnuovo mumford regularities and the number of generators of the two graded modules involved these general bounds are refined in the case where the second module is the ring other estimates for instance on the size of graded pieces of these modules are given we also derive a bound on the homological degree in terms of the castelnuovo mumford regularity this answers positively a question raised by vasconcelos

phase spaces and periodic solutions of set functional dynamic equations with infinite delay abstract very recently a new theory known as set dynamic equations on time scales has been built in this paper a phase space is built for set functional dynamic equations with infinite delay on time scales and sufficient criteria are established for the existence of periodic solutions of such equations which generalize and incorporate as special cases some known results for set differential equations and for set difference equations when the time scale is the real number set or the integer set respectively moreover for differential inclusions and difference inclusions if the variable under consideration is a single valued mapping our results show that one can unify the study of some continuous or discrete problems in the sense of set dynamic equations on general time scales

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

the homological degree of a module a homological degree of a graded module m is an extension of the usual notion of multiplicity tailored to provide a numerical signature for the module even when m is not cohen macaulay we construct a degree hdeg m that behaves well under hyperplane sections and the modding out of elements of finite support when carried out in a local algebra this degree gives a simulacrum of complexity a la castelnuovo mumford s regularity several applications for estimating reduction numbers of ideals and predictions on the outcome of noether normalizations are given

generalized hukuhara differentiability of interval valued functions and interval differential equations abstract in the present paper we use a recently proposed generalization of the hukuhara difference for compact convex sets to introduce and study a generalization of the hukuhara differentiability to the case of interval valued functions we also consider several possible definitions for the derivative of an interval valued function recently proposed in the literature and we study connections between them and their properties using these concepts we study interval differential equations local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ode systems several examples are also shown

on connections between stochastic differential inclusions and set valued stochastic differential equations driven by semimartingales abstract in the paper we study properties of solutions to stochastic differential inclusions and set valued stochastic differential equations with respect to semimartingale integrators we present new connections between their solutions in particular we show that attainable sets of solutions to stochastic inclusions are subsets of values of multivalued solutions of certain set valued stochastic equations we also show that every solution to stochastic inclusion is a continuous selection of a multivalued solution of an associated set valued stochastic equation the results obtained in the paper generalize results dealing with this topic known both in deterministic and stochastic cases

set valued and fuzzy stochastic differential equations driven by semimartingales abstract in the paper we present set valued and fuzzy stochastic integrals with respect to semimartingale integrators as well as their main properties then we study the existence of solutions to set valued and fuzzy set valued stochastic differential equations driven by semimartingales the stability of solutions is also established

generic patterns for extensions of simple modules for finite chevalley groups abstract this paper describes generic patterns for the extensions between simple modules of a finite chevalley group a one to one correspondence between these extensions and the extensions between certain simple modules of the ambient algebraic group are established it is shown that an extension appears in a principal series representation of the finite chevalley group if and only if the corresponding extension for the algebraic group appears in a weyl module