l shape 2 dimensional arrival angle estimation with propagator method it is known that computational loads of the propagator method pm can be significantly smaller e g one or two order than those of music and esprit because the pm does not require any eigenvalue decomposition evd of the cross correlation matrix and singular value decomposition svd of the received data however the pm of the parallel shape array has nonnegligible drawbacks such as 1 requirement of pair matching between the 2 d azimuth and elevation angle estimation which is an exhaustive search and 2 estimation failure problems when elevation angles are between 70 spl deg and 90 spl deg the purpose of this paper is to show a way to remove these problems in the pm without additional computational loads this paper will employ one or two l shape arrays because the parallel shape used in the pm may cause the aforesaid problems simulation results verify that the pm with one or two l shape configurations can remove these problems and improve the performance of the pm significantly e g almost 5 db in signal to noise ratio for the parameters used in this paper

reply to comments on l shape two dimensional arrival angle estimation with propagator method in this reply to the paper by shu liu and lu see ibid p 1502 3 2008 a novel method with low complexity but showing neither estimation failure nor the pair matching problems is described for the two dimensional doa estimation the authors in the previous papers kikuchi et al 2006 gu et al 2008 also proposed two methods different from that in van der veen et a 1992 to achieve the automatic pair matching by employing the l shaped array configuration the method in this reply is also different from the ones in kikuchi et al 2006 gu et al 2008 in this reply an independently developed algorithm is presented by using the unitary root music and unitary music and changing the l shape to a cross shaped array configuration

a weighted configuration model and inhomogeneous epidemics a random graph model with prescribed degree distribution and degree dependent edge weights is introduced each vertex is independently equipped with a random number of half edges and each half edge is assigned an integer valued weight according to a distribution that is allowed to depend on the degree of its vertex half edges with the same weight are then paired randomly to create edges an expression for the threshold for the appearance of a giant component in the resulting graph is derived using results on multi type branching processes the same technique also gives an expression for the basic reproduction number for an epidemic on the graph where the probability that a certain edge is used for transmission is a function of the edge weight reflecting how closely connected the corresponding vertices are it is demonstrated that if vertices with large degree tend to have large small weights on their edges and if the transmission probability increases with the edge weight then it is easier harder for the epidemic to take off compared to a randomized epidemic with the same degree and weight distribution a recipe for calculating the probability of a large outbreak in the epidemic and the size of such an outbreak is also given finally the model is fitted to three empirical weighted networks of importance for the spread of contagious diseases and it is shown that r0 can be substantially over or underestimated if the correlation between degree and weight is not taken into account

complex geometry and the asymptotics of harish chandra modules for real reductive lie groups i let g be a connected semisimple real matrix group it is now apparent that the representation theory of g is intimately connected with the complex geometry of the flag variety 9 by studying appropriate orbit structures on 9 we are naturally led to representation theory in the category of harish chandra modules x or the representation theory of category 9 the jacquet functor j f 9 has proved a useful tool in converting yfw problems into 9 problems which are often more tractable in this paper we advance the philosophy that the complex geometry of 9 associated to yf and 9 interacts in a natural way with the functor j leading to deep new information on the structure of jacquet modules this in turn gives new insight into the structure of certain nilpotent cohomology groups associated to harish chandra modules our techniques are based upon many of the ideas present in the proof of the kazhdan lusztig conjectures and bernstein s proof of the jantzen conjecture

a class of algebras similar to the enveloping algebra of sl 2 fix f e c x define r c a b h subject to the relations ha ah a hb bh b ab ba f h we study these algebras for different f and in particular show how they are similar to and different from u sl 2 the enveloping algebra of sl 2 c there is a notion of highest weight modules and a category a for such r for each n 0 if f x x 1 n l xn 1 then r has precisely n simple modules in each finite dimension and every finite dimensional r module is semisimple

on the ext groups between weyl modules for gln abstract this paper studies extension groups between certain weyl modules for the algebraic group gl n over the integers main results include 1 a complete determination of ext groups between weyl modules whose highest weights differ by a single root and 2 determination of ext 1 between an exterior power of the defining representation and any weyl module the significance of these results for modular representation theory of gl n is discussed in several remarks notably the first result leads to a calculation of ext groups between neighboring weyl modules for gl n and also recovers the gl n case of a recent result of andersen some generalities about ext groups between weyl modules and a brief overview of known results about these groups are also included

fragmentation of random trees we study fragmentation of a random recursive tree into a forest by repeated removal of nodes the initial tree consists of n nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly selected existing node as nodes are removed from the tree one at a time the tree dissolves into an ensemble of separate trees namely a forest we study statistical properties of trees and nodes in this heterogeneous forest and find that the fraction of remaining nodes m characterizes the system in the limit we obtain analytically the size density of trees of size s the size density has power law tail with exponent therefore the tail becomes steeper as further nodes are removed and the fragmentation process is unusual in that exponent increases continuously with time we also extend our analysis to the case where nodes are added as well as removed and obtain the asymptotic size density for growing trees

extensions of thickened verma modules of the virasoro algebra abstract let g denote the virasoro lie algebra h its cartan subalgebra and s h the symmetric algebra on h in this paper we consider thickened verma modules m which are u g s h bimodules satisfying m s h c m where m is the usual verma module with highest weight h we determine ext 1 m m to be s h s h where is up to a c algebra automorphism of s h a product of irreducible factors of the determinant of the shapovalov matrix this result provides a conceptual explanation of the factorization of the shapovalov determinant and implies that the inverse of the shapovalov matrix has only simple poles

hochschild and cyclic homology of quantum groups for an arbitrary complex linear semisimple lie groupg we consider hopf algebras of the deformations of the formal and algebraic functions ong the hochschild and cyclic homology of these hopf algebras are computed when the value of the deformation parameter is generic

jacquet modules and dirac cohomology abstract we show that dirac cohomology of the jacquet module of a harish chandra module is a harish chandra module for the corresponding levi subgroup we obtain an explicit formula of dirac cohomology of the jacquet module for most of the principal series based on our determination of dirac cohomology of irreducible generalized verma modules with regular infinitesimal characters

scale free models for the structure of business firm networks we study firm collaborations in the life sciences and the information and communication technology sectors we propose an approach to characterize industrial leadership using k shell decomposition with top ranking firms in terms of market value in higher k shell layers we find that the life sciences industry network consists of three distinct components a nucleus which is a small well connected subgraph tendrils which are small subgraphs consisting of small degree nodes connected exclusively to the nucleus and a bulk body which consists of the majority of nodes industrial leaders i e the largest companies in terms of market value are in the highest k shells of both networks the nucleus of the life sciences sector is very stable once a firm enters the nucleus it is likely to stay there for a long time at the same time we do not observe the above three components in the information and communication technology sector we also conduct a systematic study of these three components in random scale free networks our results suggest that the sizes of the nucleus and the tendrils in scale free networks decrease as the exponent of the power law degree distribution lambda increases and disappear for lambda or 3 we compare the k shell structure of random scale free model networks with two real world business firm networks in the life sciences and in the information and communication technology sectors we argue that the observed behavior of the k shell structure in the two industries is consistent with the coexistence of both preferential and random agreements in the evolution of industrial networks

graph partitioning induced phase transitions we study the percolation properties of graph partitioning on random regular graphs with n vertices of degree k optimal graph partitioning is directly related to optimal attack and immunization of complex networks we find that for any partitioning process even if nonoptimal that partitions the graph into essentially equal sized connected components clusters the system undergoes a percolation phase

graded characters of induced representations ii classification of principal series modules for complex groups let g be a connected complex reductive algebraic group which we view here as a real lie group by taking its complex points and then ignoring its complex structure the corresponding real lie group obtained in this way will be denoted g the purpose of this paper is the classification of principal series modules for g with integral regular infinitesimal character up to isomorphism of harish chandra modules using translation functors this parametrizes all lie algebra modules that occur as the space of k finite sections of an integral regular line bundle on the flag manifold of g see 2 14 here the term integral regular means that we consider only line bundles that are gr equivariant and whose space of c sections have an integral regular infinitesimal character the same as that of a finite dimensional g module our main result is the proof in this complex case of a conjecture that was stated in 4 see os we also give some estimates on ext groups 2 16 at the end of section 2

quasi hereditary algebras via generator cogenerators of local self injective algebras and transfer of ringel duality the dominant dimension of algebras in the class mathbf a of 1 quasi hereditary algebras introduced in pucinskait j lond math soc 2 87 2 478 496 2013 is at least two by the morita tachikawa theorem this implies that mathbf a is related to a certain class mathbf b of pairs consisting of an another algebra and a module satisfying the double centralizer condition in this paper we determine the class mathbf b and show the connection between the partial order of a 1 quasi hereditary algebra and the structure of the related pair in mathbf b see theorem a if the first component of a pair in mathbf b is a commutative algebra then the corresponding algebra in mathbf a receives additional features described in theorem b finally we introduce the ringel dual for objects in mathbf b of particular interest are those pairs in mathbf b which occur as ringel dual see theorem c

comments on l shape 2 dimensional arrival angle estimation with propagator method for original paper see n tayem et al ibid vol 53 no 5 p 1622 30 2005 we indicate that the two dimensional 2 d direction of arrival doa estimation method proposed by n tayem has an error in the pair matching between the azimuth and elevation angle estimates a feasible solution for pair matching is proposed

bgg algebras and the bgg reciprocity principle the bgg reciprocity principle is a famous result in the representation theary of complex semisimple lie algebras proved by bernstein gelfand and gelfand in bgg 21 it may be viewed as an analogue for the category 6 of the brauer reciprocity principle in the modular representation theory of finite groups and of humphreys reciprocity principle in the representation theory of restricted lie algebras indeed the result of humphreys served as motivation for the bgg result similar results have since been proved in a number of other settings by various approaches for all these results one works in a suitable category of modules in which each simple module l has a projective cover p in addition there is a family of intermediate modules mi such that each p has a filtration with m s as quotients then the appropriate reciprocity principle states that m occurs as a filtration quotient in p as often as l occurs as a composition factor in mi in the case of the category 0 the simple modules are the simple highest weight modules the category is introduced as a setting for projective covers to exist and the intermediate modules are the verma modules the proof of bgg reciprocity in bgg 21 is fairly straightforward once the basic facts about the category 0 are developed the crucial point is the construction of enough projective modules in 0 once these are constructed the existence of projective covers the existence of suitable filtrations on these and the reciprocity principle are ail proved with reference to the construction however another proof of reciprocity exists which is more natural in the sense that it makes no further reference to the construction of projective modules this proof can be used in some of the other settings as well although in any given setting no drastic simplifications result i believe it is still worthwhile to draw attention to this more conceptual approach the properties of the category 0 which are needed for the proof may be

on a common generalization of koszul duality and tilting equivalence abstract we propose a new definition of koszulity for graded algebras where the degree zero part has finite global dimension but is not necessarily semi simple the standard koszul duality theorems hold in this setting

verma modules over p adic arens michael envelopes of reductive lie algebras abstract let k be a locally compact p adic field g a split reductive lie algebra over k and u g its universal enveloping algebra we investigate the category c g of coadmissible modules over the p adic arens michael envelope u g of u g let p g be a parabolic subalgebra the main result gives a canonical equivalence between the classical parabolic bgg category of g relative to p and a certain explicitly given highest weight subcategory of c g this completely clarifies the verma module theory over u g

highest weight categories for dedekind domains abstract this paper examines the concept of a stratified exact category in the context of dedekind domains and corresponding finite galois groups bgg reciprocity and duality are proven for these categories making them highest weight categories the strong connections between the structure of the category and ramification in the ring are explored

n cohomology of simple highest weight modules on walls and purity letg p b be respectively a complex connected linear algebraic semisimple group a parabolic subgroup and a borel subgroup the first main result is the following theorem let be a pure complex ong b smooth with respect to bruhat cells then its restriction to anyp orbit is pure as well of the same weight as a consequence we are able to compute then cohomology of simple highest weight modules on walls

quasi hereditary algebras and generalized koszul duality abstract we present an easily applicable sufficient condition for standard koszul algebras to be koszul with respect to if a quasi hereditary algebra is koszul with respect to then and the yoneda extension algebra of are koszul dual in a sense explained below implying in particular that their bounded derived categories of finitely generated graded modules are equivalent we also prove that the extension algebra of is koszul in the classical sense

skew weyl modules for gln and degree reduction for schur algebras abstract we prove a strong characteristic free analogue of the classical adjoint formula s s f s f in the ring of symmetric functions this is done by showing that the representative of a suitably chosen functor involving a tensor product is the skew weyl module by strong we mean that this representative preserves not only hom groups but higher ext groups also a fact which can be used to compute some homological invariants of weyl modules for gln via recursion on degree we use the following main tools existence of weyl filtrations in tensor products of weyl modules the akin buchsbaum weyman constructions of weyl modules and certain vanishing properties of ext groups

quasi hereditary structure of twisted split category algebras revisited abstract let k be a field of characteristic 0 let c be a finite split category let be a 2 cocycle of c with values in the multiplicative group of k and consider the resulting twisted category algebra a k c several interesting algebras arise that way for instance the brauer algebra moreover the category of biset functors over k is equivalent to a module category over a condensed algebra eae for an idempotent e of a in 2 the authors proved that a is quasi hereditary with respect to an explicit partial order on the set of irreducible modules and standard modules were given explicitly here we improve the partial order by introducing a coarser order leading to the same results on a but which allows to pass the quasi heredity result to the condensed algebra eae describing biset functors thereby giving a different proof of a quasi heredity result of webb see 21 the new partial order has not been considered before even in the special cases and we evaluate it explicitly for the case of biset functors and the brauer algebra it also puts further restrictions on the possible composition factors of standard modules

brauer type reciprocity for a class of graded associative algebras classical brauer reciprocity can be stated roughly as follows let g be a finite group and let k be an algebraically closed field of characteristic p 0 if s is a simple kg module and p s is its projective cover then the multiplicity of a simple module l as a composition factor of a characteristic zero lift of p s is the same as the multiplicity of s as a composition factor of a modulo p reduction of l all modules in this paper are assumed to be finite dimensional one thinks of the module l as playing an intermediate role between p s and s reciprocities similar to brauer reciprocity here called brauer type reciprocities have subsequently been found to occur in many other settings for instance if g is a classical modular lie algebra definition given in section 2 then there exists a set 9 of g modules with the following property given a simple g module s the projective cover p s of s has a filtration with each successive quotient isomorphic to a module in 5 and for each such filtration the number of times ze 9 occurs is the same as the multiplicity of s as a composition factor of z this was proved for g of type a i by pollack lo and for arbitrary g by humphreys 3 the reciprocity in this setting is often called humphreys reciprocity inspired by humphreys result bernstein gelfand and gelfand sought and found a brauer type reciprocity bgg reciprocity in a certain truncated category their category 8 of modules for a complex semisimple lie algebra see 11 later this was generalized by mirollo and

percolation theory applied to measures of fragmentation in social networks we apply percolation theory to a recently proposed measure of fragmentation f for social networks the measure f is defined as the ratio between the number of pairs of nodes that are not connected in the fragmented network after removing a fraction q of nodes and the total number of pairs in the original fully connected network we compare f with the traditional measure used in percolation theory p infinity the fraction of nodes in the largest cluster relative to the total number of nodes using both analytical and numerical methods from percolation we study erdos renyi and scale free networks under various types of node removal strategies the removal strategies are random removal high degree removal and high betweenness centrality removal we find that for a network obtained after removal all strategies of a fraction q of nodes above percolation threshold p infinity approximately 1 f 1 2 for fixed p infinity and close to percolation threshold q qc we show that 1 f better reflects the actual fragmentation close to qc for a given p infinity 1 f has a broad distribution and it is thus possible to improve the fragmentation of the network we also study and compare the fragmentation measure f and the percolation measure p infinity for a real social network of workplaces linked by the households of the employees and find similar results

analysis of percolation behaviors of clustered networks with partial support dependence relations we carry out a study of percolation behaviors of clustered networks with partial support dependence relations by adopting two different attacking strategies attacking only one network and both networks which help to further understand real coupled networks for two different attacking strategies we find that the system changes from a second order phase transition to a first order phase transition as coupling strengthq increases we also notice that the first order region becomes smaller and the secondorder region becomes larger as average degree or clustering coefficient increases and as the average supported degree approaches infinity coupled clustered networks become independent and only the second order transition is observed which is similar toq 0 furthermore we find that clustering coefficient has a significant impact on robustness of the system for strong coupling strength but for weak coupling strength it has little influence especially for attacking both networks the study implies that we can obtain a more robust network by reducing clustering coefficient and increasing average degree for strong coupling strength however for weak coupling strength a more robust network is obtained only by increasing average degree for the same support average degree

category o for the schrodinger algebra we study category o for the centrally extended schrodinger algebra we determine the quivers for all blocks and relations for blocks of nonzero central charge we also describe the quiver and rel