two dimensional velocity distribution in open channels using renyi entropy in this work the concept of entropy based on probability is applied in modeling the vertical distribution of velocity in open channel turbulent flow using the principle of maximum entropy one dimensional velocity distribution is derived by maximizing the renyi entropy subject to some constraints by assuming dimensionless velocity as a random variable the renyi entropy based equation is capable of modeling the velocity distribution from the channel bed to the water surface the derived velocity distribution is tested with field and laboratory observations and is also compared with existing entropy based velocity distributions the present model has shown good agreement with observed data and its prediction accuracy is superior than the other existing models

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

quantum groups via hall algebras of complexes we describe quantum enveloping algebras of symmetric kac moody lie algebras via a nite eld hall algebra construction involving z2 graded complexes of quiver representations

the hall algebra approach to drinfeld s presentation of quantum loop algebras abstract the quantum loop algebra u v l g was defined as a generalization of the drinfeld s new realization of the quantum affine algebra to the loop algebra of any kac moody algebra g it has been shown by schiffmann that the hall algebra of the category of coherent sheaves on a weighted projective line is closely related to the quantum loop algebra u v l g for some g with a star shaped dynkin diagram in this paper we study drinfeld s presentation of u v l g in the double hall algebra setting based on schiffmann s work we explicitly find out a collection of generators of the double composition algebra dc coh x and verify that they satisfy all the drinfeld relations

a method based on the tsallis entropy for characterizing threshold channel bank profiles abstract as a stable channel forms a point is reached when particles shift to motion threshold threshold channel as such the most common challenges related to alluvial channels include the variation in the transverse bank slope s t and channel geometric profile formation the present study introduces a novel reliable method of estimating the transverse slope distribution of stable channel bank profiles in different hydraulic conditions by using the tsallis entropy concept accordingly an equation for calculating s t and subsequently the vertical boundary level of bank profiles is presented based on the principle of maximum entropy the threshold channel bank profile features are defined according to the probability density function and cumulative density function to adjust the general form of the tsallis entropy for this specific engineering problem an extensive detailed study of lagrange multipliers is carried out and a wide range of observational data are used to determine the tsallis entropy model accuracy in measuring the s t of stable channel banks the results indicate that the tsallis entropy model is capable of detecting the ascending trend governing the profiles of stable channel banks in different hydraulic conditions this model exhibits excellent predictive performance with a mean absolute relative error mare of 0 265 and a coefficient of determination r 2 of 0 975 the proposed tsallis entropy based equation can be used to predict river morphology changes for river management plans

on the stack of semistable g bundles over an elliptic curve in a recent paper ben zvi and nadler proved that the induction map from b bundles of degree 0 to semistable g bundles of degree 0 over an elliptic curve is a small map with galois group isomorphic to the weyl group of g we generalize their result to all connected components of text bun g for an arbitrary reductive group g we prove that for every degree i e topological type there exists a unique parabolic subgroup such that any semistable g bundle of this degree has a reduction to it and moreover the induction map is small with galois group the relative weyl group of the levi this provides new examples of simple automorphic sheaves which are constituents of eisenstein sheaves for the trivial local system

the hopf algebra of skew shapes torsion sheaves on a f1n and ideals in hall algebras of monoid representations abstract we study ideals in hall algebras of monoid representations on pointed sets corresponding to certain conditions on the representations these conditions include the property that the monoid act via partial permutations that the representation possess a compatible grading and conditions on the support of the module quotients by these ideals lead to combinatorial hopf algebras which can be interpreted as hall algebras of certain sub categories of modules in the case of the free commutative monoid on n generators we obtain a co commutative hopf algebra structure on n dimensional skew shapes whose underlying associative product amounts to a stacking operation on the skew shapes the primitive elements of this hopf algebra correspond to connected skew shapes and form a graded lie algebra by anti symmetrizing the associative product we interpret this hopf algebra as the hall algebra of a certain category of coherent torsion sheaves on a f 1 n supported at the origin where f 1 denotes the field of one element this hopf algebra may be viewed as an n dimensional generalization of the hopf algebra of symmetric functions which corresponds to the case n 1

the hall algebra of the category of coherent sheaves on the projective line to an abelian category a of homological dimension one satisfying certain finiteness conditions one can associate an algebra called the hall algebra kapranov studied this algebra when a is the category of coherent sheaves over a smooth projective curve defined over a finite field and observed analogies with quantum affine algebras we recover here in an elementary way his results in the case when the curve is the projective line

derivation of rouse equation for sediment concentration using shannon entropy abstract sediment concentration is fundamental for determining sediment transport in open channels the rouse equation one of several methods for computing sediment concentration has been derived using deterministic hydraulic principles this study derives the rouse equation using the shannon entropy theory the derivation requires a hypothesis on the cumulative probability distribution function of sediment concentration in terms of flow depth which is formulated in a general form and can specialize in several specific forms reported in the literature the advantage of using the entropy theory is that it permits quantification of uncertainty associated with concentration and determination of parameters in terms of specified information such as mean concentration

branch and bound crossed with ga to solve hybrid flowshops this article deals with an optimal methods for solving a k stage hybrid flowshop scheduling problem this problem is known to be np hard in 1991 brah and hunsucker proposed a branch and bound algorithm to solve this problem however for some medium size problems the computation time is not acceptable the aim of this article is to present an improvement of this algorithm as a matter of fact we prove that the value of their lower bound lb may decrease along a path of the search tree first of all we present an improvement of their lb then we introduce several heuristics at the beginning of the search in order to compute an initial upper bound and genetic algorithm ga to improve during the search the value of the upper bound more precisely our ga takes into account the set of partial decisions made by the branch and bound and builds a series of populations of complete solutions with the aim of improving the upper bound the best found criterion value corresponding to a complete solution experimentation show that the optimality of branch and bound is more often reached and the value of criterion is improved when our improvements are taken into account

on the hall algebra of an elliptic curve i these rings admit numerous algebraic and geometric realizations but one of the historically first constructions dating to the work of steinitz in 1900 completed later by hall was given in terms of what is now called the classical hall algebra h see ma chapter ii this algebra has a basis consisting of isomorphism classes of abelian q groups where q is a fixed prime power and the structure constants are defined by counting extensions between such abelian groups in fact these structure constants are polynomials in q and we can therefore consider h as a c q 1 algebra a theorem of steinitz and hall provides an isomorphism h q c q x1 x2 under this isomorphism the natural basis of h resp the natural scalar product is mapped to the basis of hall littlewood polynomials resp the hall littlewood scalar product in addition zelevinsky z endowed q with a structure of a cocommutative hopf algebra and the whole algebra q c q 1

torus knots and the rational daha author s gorsky e oblomkov a rasmussen j shende v abstract 2014 we conjecturally extract the triply graded khovanov rozansky homology of the m n torus knot from the unique finite dimensional simple representation of the rational daha of type a rank n 1 and central character m n the conjectural differentials of gukov dunfield and the third author receive an explicit algebraic expression in this picture yielding a prescription for the doubly graded khovanov rozansky homologies we match our conjecture to previous conjectures of the first author relating knot homology to q t catalan numbers and to previous conjectures of the last three authors relating knot homology to hilbert schemes on singular curves

topology of the compactified jacobians of singular curves we compute the euler number of the compactified jacobian of a curve whose minimal unibranched normalization has only plane irreducible singularities with characteristic puiseux exponents p q 4 2q s 6 8 s or 6 10 s further we derive a combinatorial method to compute the betti numbers of the compactified jacobian of an unibranched rational curve with singularities like above some of the betti numbers can be stated explicitly

branching rules for quantum toroidal gln abstract we construct an analog of the subalgebra u gl n u gl m u gl m n in the setting of quantum toroidal algebras and study the restrictions of various representations to this subalgebra

root categories and simple lie algebras abstract by using the t 2 orbit category of the derived category of a hereditary algebra which is proved to be a triangulated category too we give a complete realization of a simple lie algebra this is a global version of ringel s work for the positive parts of simple lie algebras in this realization the grothendieck group provides the cartan subalgebra the indecomposable objects correspond to the chevalley basis and the structure constants are given by the evaluations of hall polynomials

quantum toroidal mathfrak g mathfrak l _ 1 and bethe ansatz we establish the method of bethe ansatz for the xxz type model obtained from the r matrix associated to quantum toroidal we do this by using shuffle realizations of the modules and by showing that the hamiltonian of the model is obtained from a simple multiplication operator by taking an appropriate quotient we expect this approach to be applicable to a wide variety of models

the q agt w relations via shuffle algebras we construct the action of the q deformed w algebra on its level r representation geometrically using the moduli space of u r instantons on the plane and the double shuffle algebra we give an explicit ldu decomposition for the action of w algebra currents in the fixed point basis of the level r representation and prove a relation between the carlsson okounkov ext operator and intertwiners for the deformed w algebra we interpret this result as a q deformed version of the agt w relations

derivation of hunt equation for suspension distribution using shannon entropy theory abstract in this study the hunt equation for computing suspension concentration in sediment laden flows is derived using shannon entropy theory considering the inverse of the void ratio as a random variable and using principle of maximum entropy probability density function and cumulative distribution function of suspension concentration is derived a new and more general cumulative distribution function for the flow domain is proposed which includes several specific other models of cdf reported in literature this general form of cumulative distribution function also helps to derive the rouse equation the entropy based approach helps to estimate model parameters using suspension data of sediment concentration which shows the advantage of using entropy theory finally model parameters in the entropy based model are also expressed as functions of the rouse number to establish a link between the parameters of the deterministic and probabilistic approaches

a review of exact solution methods for the non preemptive multiprocessor flowshop problem abstract the hybrid or flexible flowshop problem is a generalization of the flowshop in such a way that every job can be processed by one among several machines on each machine stage in recent years a number of effective exact methods have been developed a major reason for this progress is the development of new job and machine based lower bounds as well as the rapidly increasing importance of constraint programming in this paper we provide the first comprehensive and uniform overview on exact solution methods for flexible flowshops with branching bounding and propagation of constraints under two different objective functions minimizing the makespan of a schedule and the mean flow time

drinfeld double and ringel green theory of hall algebras abstract as a continuation of the work of ringel and green on hall algebras the hopf algebra structure of a extended twisted hall algebra is presented here by introducing the ringel pairing of hall algebras the drinfeld double of a hall algebra is constructed the reduced form of this drinfeld double is applicable to the generic composition algebra which gives a complete realization of quantum group this ringel green theory makes it possible to give the formulae for the computation of commutators in quantum groups in terms of hall algebras

modules over plane curve singularities in any ranks and daha abstract we generalize the construction of geometric superpolynomials for unibranch plane curve singularities from our prior paper from rank one to any ranks explicit formulas are obtained for torus knots the new feature is the definition of counterparts of jacobian factors directly related to compactified jacobians for higher ranks which is parallel to the classical passage from invertible sheaves to vector bundles over algebraic curves this is an entirely local theory connected with affine springer fibers for non reduced germs of spectral curves we conjecture and justify numerically the connection of our geometric polynomials in arbitrary ranks with the corresponding daha superpolynomials for any algebraic knots colored by columns

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

indecomposable vector bundles and stable higgs bundles over smooth projective curves we prove that the number of geometrically indecomposable vector bundles of xed rank r and degree d over a smooth projective curve x dened over a nite eld is given by a polynomial depending only on r d and the genus g of x in the weil numbers of x we provide a closed formula expressed in terms of generating series for this polynomial we also show that the same polynomial computes the number of points of the moduli space of stable higgs bundles of rank r and degree d over x this entails a closed formula for the poincar e polynomial of the moduli spaces of stable higgs bundles over a compact riemann surface and hence also for the poincar e polynomials of the twisted character varieties for the groups gl r

a multi stage parallel processor flowshop problem with minimum flowtime abstract most algorithms on flowshop scheduling assume that every stage has a single processor or machine only with a view to achieve the practical objective of minimizing total flowtime a branch and bound algorithm for obraining a permutation schedule for the multi stage flowshop with parallel processorsis presented computational experience with the proposed algorithm and the existing single processor algorithms revelas that most of the parallel processor problems are solved faster and easier the performance of the existing procedures is also evaluated in the parallel processor environment since the flowshop scheduling problem is np complete a heuristic algorithm for scheduling in a two stage parallel processor flowshop is proposed the heuristic solutions when compared with the optimal solutions given by the proposed branch and bound method are found to be near optimal

a new family of power transformations to improve normality or symmetry summary we introduce a new power transformation family which is well defined on the whole real line and which is appropriate for reducing skewness and to approximate normality it has properties similar to those of the box cox transformation for positive variables the large sample properties of the transformation are investigated in the contect of a single random sample

the homflypt skein algebra of the torus and the elliptic hall algebra we give a generators and relations presentation of the homflypt skein algebra h of the torus t2 and we give an explicit description of the module corresponding to the solid torus using this presentation we show that h is isomorphic to the 1 specialization of the elliptic hall algebra of burban and schiffmann as an application for an iterated cable k of the unknot we use the elliptic hall algebra to construct a 3 variable polynomial that specializes to the colored homflypt polynomial of k we show that this polynomial also specializes to one constructed by cherednik and danilenko using the gln double affine hecke algebra this proves one of the connection conjectures in their recent work

estimation of two dimensional velocity distribution profile using general index entropy in open channels abstract estimation of velocity distribution profile is a challenging subject of open channel hydraulics in this study an entropy based method is used to derive two dimensional velocity distribution profile the general index entropy gie can be considered as the generalized form of shannon entropy which is suitable to combine with the different form of cumulative distribution function cdf using the principle of maximum entropy pome the velocity distribution is defined by maximizing the gie by treating the velocity as a random variable the combination of gie and a cdf proposed by marini et al 2011 was utilized to introduce an efficient entropy model whose results are comparable with several well known experimental and field data consequently in spite of less sensitivity of the related parameters of the model to flow conditions and less complexity in application of the model compared with other entropy based methods more accuracy is obtained in estimating velocity distribution profile either near the boundaries or the free surface of the flow

branch and bound algorithm for the flow shop with multiple processors abstract the sequencing of a flow shop with multiple processors at each stage is a general case of the flow shop problem it involves sequencing n jobs in a flow shop where for at least one stage the facility has one or more identical machines the purpose of this paper is to present a branch and bound algorithm to solve scheduling problems of such facilities for optimizing the maximum completion time the lower bounds and elimination rules developed in this research are based upon the generalization of the flow shop problem furthermore a computational algorithm along with results is also presented the branch and bound algorithm can also be used to optimize other measures of performance in a flow shop with multiple processors

hall algebras hereditary algebras and quantum groups let r be an associative hereditary algebra over a finite field k and let r fin be the full subcategory of r mod whose objects are those left r modules x which are finite as sets ix oc assume also that r isfinitary in c ringel s sense i e that iext l s s c for all simple s s in r fin this condition is met for example if r is finitely generated as k algebra 4 pp 435 436 let be the set of all isomorphism classes in r fin if 2 e then u will denote an r module in class 2 the class of all zero left r modules is denoted 0 let i c be the set of all isomorphism classes of simple modules in r fin thus u i e i is a complete set of simple finite left r modules we identify the grothendieck group k0 r fin with the free abelian group e1 v i v e 7z having i as free basis so that if x e r fin then the corresponding element of k0 r fin is the dimension vector dimx y i vii where for each i e vi is the multiplicity of the simple module u in any composition series of x clearly d imx lies in the subsemigroup

the affine yangian of gl1 revisited abstract the affine yangian of gl 1 has recently appeared simultaneously in the work of maulik okounkov 11 and schiffmann vasserot 20 in connection with the alday gaiotto tachikawa conjecture while the presentation from 11 is purely geometric the algebraic presentation in 20 is quite involved in this article we provide a simple loop realization of this algebra which can be viewed as an additivization of the quantum toroidal algebra of gl 1 in the same way as the yangian y h g is an additivization of the quantum loop algebra u q l g for a simple lie algebra g we also explain the similarity between the representation theories of the affine yangian and the quantum toroidal algebras of gl 1 by generalizing the main result of 10 to the current settings

representations of rational cherednik algebras with minimal support and torus knots abstract in this paper we obtain several results about representations of rational cherednik algebras and discuss their applications our first result is the cohen macaulayness property as modules over the polynomial ring of cherednik algebra modules with minimal support our second result is an explicit formula for the character of an irreducible minimal support module in type a n 1 for c m n and an expression of its quasispherical part i e the isotypic part of hooks in terms of the homfly polynomial of a torus knot colored by a young diagram we use this formula and the work of calaque enriquez and etingof to give explicit formulas for the characters of the irreducible equivariant d modules on the nilpotent cone for sl m our third result is the construction of the koszul bgg complex for the rational cherednik algebra which generalizes the construction of the koszul bgg resolution from 3 and 21 and the calculation of its homology in type a we also show in type a that the differentials in the koszul bgg complex are uniquely determined by the condition that they are nonzero homomorphisms of modules over the cherednik algebra finally our fourth result is the symmetry theorem which identifies the quasispherical components in the representations with minimal support over the rational cherednik algebras h m n s n and h n m s m in fact we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras this symmetry was essentially established in 8 in the spherical case and in 24 in the case gcd m n 1 and it has a natural interpretation in terms of invariants of torus knots

uncertainty analysis of shear stress estimation in circular channels by tsallis entropy abstract accurate prediction of the shear stress distribution is essential for the successful design of stable erodible bed channels and for the sediment transport studies considerable attention in recent years has been given to the estimation of velocity distribution using entropy concept in open channels despite the importance of knowledge about shear stress distribution there are very few studies on the application of the entropy methods for prediction of the shear stress distribution in open channels the tsallis entropy has been employed in this study for estimating the shear stress in open channels in this approach a pair of mean and maximum shear stresses are used to evaluate the shear stress distribution on the entire channel cross section we then calculated the prediction uncertainty of the shear stress obtained from the tsallis entropy in a circular open channel moreover the distribution of prediction error for the tsallis approach is examined in two cases both before and after data normalization the quantitative results from this uncertainty analysis showed satisfactory results for the tsallis entropy model for estimating shear stress in the entire section the 95 confidence bounds cb are obtained for the shear stress distribution predicted by the model closely match the observed values

double hall algebras and derived equivalences abstract we show that the reduced drinfeld double of the ringel hall algebra of a hereditary category is invariant under derived equivalences by associating an explicit isomorphism to a given derived equivalence we also extend the results of burban and schiffmann 5 6 sevenhant and van den bergh 1999 25 and xiao and yang 2001 31

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

derived categories tilted algebras and drinfel d doubles abstract given a finite dimensional hereditary algebra over a finite field k in the derived category db we obtain some formulae on hall numbers associated to triangles using them we prove that for any tilted algebra of there is an embedding of the twisted ringel hall algebra of into the drinfel d double d of the twisted ringel hall algebra of furthermore if is also hereditary we show that this embedding can be extended as an isomorphism between the two corresponding drinfel d doubles these formulae are also used to construct an automorphism of d directly associated to auslander reiten translation and this automorphism follows by b sevenhant m van den bergh s construction in the case of quivers

stable pairs and the homfly polynomial given a planar curve singularity we prove a conjecture of oblomkov shende relating the geometry of its hilbert scheme of points to the homfly polynomial of the associated algebraic link more generally we prove an extension of this conjecture due to diaconescu hua soibelman relating stable pair invariants on the conifold to the colored homfly polynomial of the algebraic link our proof uses wall crossing techniques to prove a blowup identity on the algebro geometric side we prove a matching identity for the colored homfly polynomials of a link using skein theoretic techniques

the alexander polynomial of a plane curve singularity and the ring of functions on it we prove two formulae that express the alexander polynomial delta sp c of several variables of a plane curve singularity c in terms of the ring mathscr o sb c of germs of analytic functions on the curve one of them expresses delta sp c in terms of dimensions of some factors corresponding to a multi indexed filtration on the ring mathscr o sb c the other one gives the coefficients of the alexander polynomial delta sp c as euler characteristics of some explicitly described spaces complements to arrangements of projective hyperplanes

affine springer fibers of type a and combinatorics of diagonal coinvariants abstract we calculate the borel moore homology of affine springer fibers of type a associated with some regular semisimple nil elliptic elements as a result we obtain bigraded s n modules whose bigraded frobenius series are a generalization of the symmetric functions introduced by haglund haiman loehr remmel and ulyanov

the harder narasimhan system in quantum groups and cohomology of quiver moduli methods of harder and narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations using the hall algebra approach to quantum groups an analog of the harder narasimhan recursion is constructed inside the quantized enveloping algebra of a kac moody algebra this leads to a canonical orthogonal system the hn system in this algebra using a resolution of the recursion an explicit formula for the hn system is given as an application explicit formulas for betti numbers of the cohomology of quiver moduli are derived generalizing several results on the cohomology of quotients in linear algebra type situations

counting rational curves on text k3 surfaces the aim of these notes is to explain the remarkable formula found by yau and zaslow y z to express the number of rational curves on a k3 surface projective k3 surfaces fall into countably many families fg g 1 a surface in fg admits a gdimensional linear system of curves of genus g a naive count of constants suggests that such a system will contain a positive number say n g of rational highly singular curves the formula is

noncommutative projective curves and quantum loop algebras we show that the hall algebra of the category of coherent sheaves on a weighted projective line over a finite field provides a realization of the quantized enveloping algebra of a certain nilpotent subalgebra of the affinization of the corresponding kac moody algebra in particular this yields a geometric realization of the quantized enveloping algebra of elliptic or 2 toroidal algebras of types d 4 1 1 e 1 1 6 e 1 1 7 and e 8 1 1 in terms of coherent sheaves on weighted projective lines of genus one or equivalently in terms of equivariant coherent sheaves on elliptic curves

formulating the shear stress distribution in circular open channels based on the renyi entropy the principle of maximum entropy is employed to derive the shear stress distribution by maximizing the renyi entropy subject to some constraints and by assuming that dimensionless shear stress is a random variable a renyi entropy based equation can be used to model the shear stress distribution along the entire wetted perimeter of circular channels and circular channels with flat beds and deposited sediments a wide range of experimental results for 12 hydraulic conditions with different froude numbers 0 375 to 1 71 and flow depths 20 3 to 201 5 mm were used to validate the derived shear stress distribution for circular channels model performance enhanced with increasing flow depth mean relative error re of 0 0414 and only deteriorated slightly at the greatest flow depth re of 0 0573 for circular channels with flat beds the renyi entropy model predicted the shear stress distribution well at lower sediment depth the renyi entropy model results were also compared with shannon entropy model results both models performed well for circular channels but for circular channels with flat beds the renyi entropy model displayed superior performance in estimating the shear stress distribution the renyi entropy model was highly precise and predicted the shear stress distribution in a circular channel with re of 0 0480 and in a circular channel with a flat bed with re of 0 0488

hall algebras associated to triangulated categories by counting triangles and applying the octahedral axiom we find a direct way to prove the formula of toen in t section 5 1 for a triangulated category satisfying the left homological finiteness condition

derived hall algebras the purpose of this work is to define a derived hall algebra mathcal dh t associated to any dg category t under some finiteness conditions our main theorem states that mathcal dh t is associative and unital it is shown that mathcal dh t contains the usual hall algebra mathcal h t when t is an abelian category we will also prove an explicit formula for the derived hall numbers purely in terms of invariants of the triangulated category associated to t as an example we describe the derived hall algebra of an hereditary abelian category