deformation theory of objects in homotopy and derived categories iii abelian categories abstract this is the third paper in a series in part i we developed a deformation theory of objects in homotopy and derived categories of dg categories here we show how this theory can be used to study deformations of objects in homotopy and derived categories of abelian categories then we consider examples from noncommutative algebraic geometry in particular we study noncommutative grassmanians that are true noncommutative moduli spaces of structure sheaves of projective subspaces in projective spaces

semiprime graded algebras of dimension two semiprime noetherian connected graded k algebras r of quadratic growth are described in terms of geometric data a typical example of such a ring is obtained as follows let y be a projective variety of dimension at most one over the base field k and let e be an oy order in a finite dimensional semisimple algebra a over k k y then for any automorphism of a that restricts to an automorphism of y and any ample invertible e bimodule b van den bergh constructs a noetherian twisted homogeneous coordinate ring b h0 y b b n 1 we show that r is noetherian if and only if some veronese ring r m of r has the form k i where i is a left ideal of such a ring b and where i b at each point p y at which has finite order this allows one to give detailed information about the structure of r and its modules

ince s limits for confluent and double confluent heun equations we find pairs of solutions to a differential equation which is obtained as a special limit of a generalized spheroidal wave equation this is also known as confluent heun equation one solution in each pair is given by a series of hypergeometric functions and converges for any finite value of the independent variable z while the other is given by a series of modified bessel functions and converges for z z0 where z0 denotes a regular singularity for short the preceding limit is called ince s limit after ince who have used the same procedure to get the mathieu equations from the whittaker hill ones we find as well that when z0 tends to zero the ince limit of the generalized spheroidal wave equation turns out to be the ince limit of a double confluent heun equation for which solutions are provided finally we show that the schrodinger equation for inverse fourth and sixth power potentials reduces to peculiar cases of the double confluent heun equation and its ince s limit respectively

convergence and applications of some solutions of the confluent heun equation we study the convergence of a group of solutions in series of confluent hypergeometric functions for the confluent heun equation these solutions are expansions in two sided infinite series summation from minus to plus infinity which are interpreted as a modified version of expansions proposed by leaver 1986 we show that the two sided solutions yield two nonequivalent groups of one sided series solutions summation from zero to plus infinity in the second place we find that one sided solutions of one of these groups can be used to solve an equation which describes a time dependent two level system of quantum optics for this problem in addition to finite series solutions we obtain infinite series wavefunctions which are convergent and bounded for any value of the time t and vanish when t goes to infinity

the q division ring and its fixed rings abstract we prove that the fixed ring of the q division ring k q x y under any finite group of monomial automorphisms is isomorphic to k q x y for the same q in a similar manner we also show that this phenomenon extends to an automorphism that is defined only on k q x y and does not restrict to k q x 1 y 1 we then use these results to answer several questions posed by artamonov and cohn about the endomorphisms and automorphisms of k q x y

the grothendieck duality theorem via bousfield s techniques and brown representability grothendieck proved that if f x y is a proper morphism of nice schemes then rf has a right adjoint which is given as tensor product with the relative canonical bundle the original proof was by patching local data deligne proved the existence of the adjoint by a global argument and verdier showed that this global adjoint may be computed locally in this article we show that the existence of the adjoint is an immediate consequence of brown s representability theorem lit follows almost as imme diately by smashing arguments that the adjoint is given by tensor product with a dualising complex verdier s base change theorem is an easy conse quence department of mathematics university of virginia charlottesville virginia 22903 e mail address an3rqvirginia edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 06 26 19 utc all use subject to http about jstor org terms

noncommutative instantons and twistor transform recently n nekrasov and a schwarz proposed a modification of the adhm construction of instantons which produces instantons on a noncommutative deformation of 4 in this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces we exhibit one to one correspondences between three classes of objects framed bundles on a noncommutative 2 certain complexes of sheaves on a noncommutative 3 and the modified adhm data the modified adhm construction itself is interpreted in terms of a noncommutative version of the twistor transform we also prove that the moduli space of framed bundles on the noncommutative 2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative 2 the natural complex structures on the two moduli spaces do not coincide but are related by an so 3 rotation finally we propose a construction of instantons on a more general noncommutative 4 than the one considered by nekrasov and schwarz a q deformed 4

graded modules of gelfand kirillov dimension one over three dimensional artin schelter regular algebras abstract let a be a three dimensional artin schelter regular algebra we give a description of the category of finitely generated a modules of gelfand kirillov dimension one modulo those of finite dimension over the ground field the proof is based upon a result by gabriel which says that locally finite categories can be described by module categories over topological rings

intersection theory on non commutative surfaces consider a non commutative algebraic surface x and an effective divisor y on x as defined by van den bergh we show that the riemann roch theorem the genus formula and the self intersection formula from classical algebraic geometry generalize to this setting we also apply our theory to some special cases including the blow up of x in a point and show that the self intersection of the exceptional divisor is 1 this is used to give an example of a non commutative surface with a commutative p1 which cannot be blown down because its self intersection is 1 rather than 1 we also get some results on hilbert polynomials of modules on x

frequency domain equalization techniques for multi h continuous phase modulation two receiver architectures are presented for equalization of multi h continuous phase modulation cpm in the first approach the receiver front end is a matched filter bank which is derived from laurent decomposition of the signal a circulant matrix model derived for the samples of the matched filter is used to perform the equalization in frequency domain the second technique employs a receiver with a simple front end consisting of a low pass filter and a fractional sampler we also develop a circulant matrix model for the latter approach which relates the laurent pseudo symbols to each polyphase component at the output of the sampler a linear frequency domain equalizer fde is also proposed using this model in addition to providing the details of these approaches the paper proposes several simplifications applicable to each of the two cases and provides a detailed discussion on the performance complexity trade off bit error rate ber results are presented for the irig 106 aeronautical telemetry dual h cpm waveform

integral relations for solutions of the confluent heun equation firstly we construct kernels for integral relations among solutions of the confluent heun equation che additional kernels are systematically generated by applying substitutions of variables secondly we establish integral relations between known solutions of the che that are power series and solutions that are series of special functions thirdly by using one of the integral relations as an integral transformation we obtain a new series solution of the ordinary spheroidal wave equation a particular che from this solution we construct new series solutions of the general che and show that these are suitable for solving the radial part of the two center problem in quantum mechanics

the geometry of arithmetic noncommutative projective lines abstract let k be a perfect field and let k k be a finite extension of fields an arithmetic noncommutative projective line is a noncommutative space of the form proj s k v where v be a k central two sided vector space over k of rank two and s k v is the noncommutative symmetric algebra generated by v over k defined by m van den bergh 26 we study the geometry of these spaces more precisely we prove they are integral we classify vector bundles over them we classify them up to isomorphism and we classify isomorphisms between them using the classification of isomorphisms we compute the automorphism group of an arithmetic noncommutative projective line

exact and quasiexact solvability of second order superintegrable quantum systems i euclidean space preliminaries we show that second order superintegrable systems in two dimensional and three dimensional euclidean space generate both exactly solvable es and quasiexactly solvable qes problems in quantum mechanics via separation of variables and demonstrate the increased insight into the structure of such problems provided by superintegrability a principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable most past separation of variables treatments of qes problems via partial differential equations have only incorporated separability not multiseparability also we propose another definition of es and qes the quantum mechanical problem is called es if the solution of schrodinger equation can be expressed in terms of hypergeometric functions fnm and is qes if the schrodinger equation admits polynomial solutions with coefficients necessarily satisfying a three term or higher order of recurrence relations in three dimensions we give an example of a system tha

bandwidth constrained signature waveforms and walsh signal space receivers for synchronous cdma systems synchronous cdma systems whose transmission bandwidth is quantified through the fractional out of band energy fobe constraint are considered either a conventional matched filter mf receiver or a minimum mean square error mmse receiver is employed for users data detection the total squared correlation tsc and the total mean square error tmse are proposed as the performance parameters for the mf and mmse receivers respectively these parameters need to be minimized in order to maximize the signal to interference ratios sirs at the receivers outputs for a given fobe bandwidth constraint the sets of signature waveforms that minimize either tsc or tmse are obtained from the prolate spheroidal wave functions pswfs furthermore if the number of users is the size of a hadamard matrix then optimal signature waveforms can be obtained to maximize the individual sir for every user due to the complicated nature of the pswfs simplified mf and mmse receivers based on the walsh signal space are developed

a translation principle for the four dimensional sklyanin algebras abstract in this paper we prove a translation principle for the central quotients of four dimensional sklyanin algebras which is analogous to the translation principle for semi simple lie algebras in the course of the proof we construct an elliptic analog of the sheaf of differential operators on2 1 the translation principle may be used to construct the fat points of a four dimensional non pi sklyanin algebra

a quasi optimum receiver for continuous phase modulation a simple receiver structure for any continuous phase modulation cpm scheme is introduced its front end is just the ordinary linear receiver followed by a subinterval sampler eliminating the need for the standard analog matched filters its design is based on the decomposition of the cpm signal in the walsh signal space this brings the hardware requirement to a minimum and near optimum performance can be easily obtained

a decomposition approach to cpm it is shown that any continuous phase modulation cpm system can be decomposed into a continuous phase encoder and a memoryless modulator in such a way that the former is a linear modulo some integer p time invariant sequential circuit and the latter is also time invariant this decomposition is exploited to obtain alternative realizations of the continuous phase encoder and hence of cpm and also to obtain alternative forms of the optimum decoding algorithm when p is a prime p so that the encoder is linear over the finite field gf p it is shown that cascading it with an outside convolutional encoder is equivalent to a single convolutional encoder it is pointed out that the cascade of the modulator the waveform channel which it is assumed is characterized by additive white gaussian noise and the demodulator that operates over one symbol interval yield a discrete memoryless channel that can be studied without the distractions introduced by continuous phase encoding

minimal solutions of three term recurrence relations and orthogonal polynomials we observe that the well known recurrence relation pn x z z a pn z bnp l z for orthogonal polynomials admits a minimal solution if z is outside the spectrum of the mass distribution ds t with respect to which the polynomials are orthogonal and if the moment problem for this distribution is determined the minimal solution indeed is f z m ds t z t and can be computed accurately by means of the author s continued fraction algorithm an application is made to special gauss type quadrature formulas 1 introduction minimal solutions of three term recurrence relations and their computational implications are discussed systematically in 7 effective algorithms for computing minimal solutions have been developed in 7 16 17 and continue to be the subject of further study see e g 14 26 5 2 24 25 3 with these powerful algorithms at hand it seems desirable to delineate large classes of recurrence relations for which the presence of minimal solutions can be ascertained and the minimal solution itself identified even more desirable is an understanding of the deeper reasons for the occurrence of minimal solutions in terms of intrinsic features of the subject area in which they arise few attempts have been made along these lines the work of thacher 22 23 on power series solution of linear differential equations with polynomial coeffi cients may be considered a beginning inasmuch as the presence of minimal solutions of the recurrence relation satisfied by the expansion coefficients is conjectured to be related in a specified way to the type of singularities in the differential equation here we wish to consider the recurrence relation

regularity of algebras related to the sklyanin algebra this paper continues the research of ss by finding further examples of artin schelter regular rings of dimension four unlike the threedimensional case studied in atv1 2 these examples show that the fourdimensional regular rings are not uniquely determined by their associated geometric data indeed we find a one parameter family of regular algebras associated to this data

noetherian connected graded algebras of global dimension 3 abstract let a be a noetherian connected graded algebra of global dimension 3 we show that a is regular in the sense of artin and schelter if one of the following conditions holds 1 a is generated by two elements 2 the graded simple module has a standard resolution 3 the degrees of minimal relations are the same this includes the quadratic case some general properties of a are studied without assuming regularity

a class of noncommutative projective surfaces let a k a 1 a 2 be a connected graded noetherian k algebra that is generated in degree one over an algebraically closed field k suppose that the graded quotient ring q a has the form q a k y t t 1 sigma where sigma is an automorphism of the integral projective surface y then we prove that a can be written as a naive blowup algebra of a projective surface x birational to y this enables one to obtain a deep understanding of the structure of these algebras for example generically they are not strongly noetherian and their point modules are not parametrized by a projective scheme this is despite the fact that the simple objects in the quotient category qgr a will always be in 1 1 correspondence with the closed points of the scheme x

a noncommutative version of beilinson s theorem abstract we prove that the category of representations of the n kronecker quiver is derived equivalent to the category qgr k x 1 x n i 1 n x i 2 which is considered as the category of coherent sheaves on the noncommutative projective scheme associated to k x 1 x n i 1 n x i 2 this theorem is easily proved by applying orlov s theorem on the other hand in our proof a method of noncommutative projective geometry is used and the quadratic relation i 1 n x i 2 naturally arises from auslander reiten theory

algebras associated to elliptic curves this paper completes the classification of artin schelter regular algebras of global dimension three for algebras generated by elements of degree one this has been achieved by artin schelter tate and van den bergh we are therefore concerned with algebras which are not generated in degree one we show that there exist some exceptional algebras each of which has geometric data consisting of an elliptic curve together with an automorphism just as in the case where the algebras are assumed to be generated in degree one in particular we study the elliptic algebras a a and a a where a p2 which were first defined in an earlier paper we omit a set s p2 consisting of 11 specified points where the algebras a a become too degenerate to be regular theorem let a represent a a or a a where a p2 s then a is an artin schelter regular algebra of global dimension three moreover a is a noetherian domain with the same hilbert series as the appropriately graded commutative polynomial ring in three variables this combined with our earlier results completes the classification

embedding a quantum nonsingular quadric in a quantum p3 a definition of regularity has been given for non commutative graded algebras and results of artin schelter tate and van den bergh classify the regular algebras of global dimension three that are generated by degree one elements our purpose is to classify a certain class of quadratic regular algebras of global dimension four letsbe a twisted homogeneous coordinate ring of a nonsingular quadricq p3 our interest is in algebrasrsuch that there is an embedding proj s proj r in this paper we classify all the quadratic regular algebrasrof global dimension four which have the same hilbert series as that of the polynomial ring on four variables and which map ontosvia a graded degree zero homomorphism our approach makes use of the point modules ofrand their associated geometric data we classify the algebrasraccording to their point scheme pand corresponding automorphism aut p those algebrasrwhich are determined by p belong to at most a five parameter family but those which are not determined by p belong to at most a four parameter family in the first case pis either p3or consists ofqtogether with a linel while in the second casep q it is also proved that under certain sufficient conditions the zero locus of the defining relations of a quadratic regular algebra of global dimension four is the graph of an automorphism

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

theory and computation of spheroidal wavefunctions in this paper we report on a package written in the mathematica computer algebra system which has been developed to compute the spheroidal wavefunctions of meixner and schafke 1954 mathieusche funktionen und spharoidfunktionen and is available online physics uwa edu au falloon spheroidal spheroidal html this package represents a substantial contribution to the existing software since it computes the spheroidal wavefunctions to arbitrary precision for general complex parameters and argument z existing software can only handle integer and does not give arbitrary precision the package also incorporates various special cases and computes analytic power series and asymptotic expansions in the parameter the spheroidal wavefunctions of flammer 1957 spheroidal wave functions are included as a special case of meixner s more general functions this paper presents a concise review of the general theory of spheroidal wavefunctions and a description of the formulae and algorithms used in their computation and gives high precision numerical examples

rings with auslander dualizing complexes abstract a ring with an auslander dualizing complex is a generalization of an auslander gorenstein ring we show that many results which hold for auslander gorenstein rings also hold in the more general setting on the other hand we give criteria for existence of auslander dualizing complexes which show these occur quite frequently the most powerful tool we use is the local duality theorem for connected graded algebras over a field filtrations allow the transfer of results to nongraded algebras we also prove some results of a categorical nature most notably the functoriality of rigid dualizing complexes

generalized double affine hecke algebras of rank 1 and quantized del pezzo surfaces abstract let d be a simply laced dynkin diagram of rank r whose affinization has the shape of a star i e d 4 e 6 e 7 e 8 to such a diagram one can attach a group g whose generators correspond to the legs of the affinization have orders equal to the leg lengths plus 1 and the product of the generators is 1 the group g is then a 2 dimensional crystallographic group g z l z 2 where l is 2 3 4 and 6 respectively in this paper we define a flat deformation h t q of the group algebra c g of this group by replacing the relations saying that the generators have prescribed orders by their deformations saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots which are deformation parameters the algebra h t q for d 4 is the cherednik algebra of type c c 1 which was studied by noumi sahi and stokman and controls askey wilson polynomials we prove that h t q is the universal deformation of the twisted group algebra of g and that this deformation is compatible with certain filtrations on c g we also show that if q is a root of unity then for generic t the algebra h t q is an azumaya algebra and its center is the function algebra on an affine del pezzo surface for generic q the spherical subalgebra e h t q e provides a quantization of such surfaces we also discuss connections of h t q with preprojective algebras and painleve vi

quantum deformations of projective three space abstract we describe the possible noncommutative deformations of complex projective three space by exhibiting the calabi yau algebras that serve as their homogeneous coordinate rings we prove that the space parametrizing such deformations has exactly six irreducible components and we give explicit presentations for the generic members of each family in terms of generators and relations the proof uses deformation quantization to reduce the problem to a similar classification of unimodular quadratic poisson structures in four dimensions which we extract from cerveau and lins neto s classification of degree two foliations on projective space corresponding to the exceptional component in their classification is a quantization of the third symmetric power of the projective line that supports bimodule quantizations of the classical schwarzenberger bundles

geometric algebras on projective surfaces abstract let x be a projective surface let aut x and let l be a ample invertible sheaf on x we study the properties of a family of subrings parameterized by geometric data of the twisted homogeneous coordinate ring b x l in particular we find necessary and sufficient conditions for these subrings to be noetherian we also study their homological properties their associated noncommutative projective schemes and when they are maximal orders in the process we produce new examples of maximal orders these are graded and have the property that no veronese subring is generated in degree 1 our results are used in the companion paper s j sierra 2009 sie09 to give defining data for a large class of noncommutative projective surfaces

rational cherednik algebras and hilbert schemes let h c be the rational cherednik algebra of type a n 1 with spherical subalgebra u c e h c e then u c is filtered by order of differential operators with associated graded ring mbox gr u c mathbb c mathfrak h oplus mathfrak h where w is the n th symmetric group we construct a filtered mathbb z algebra b such that under mild conditions on c the category b qgr of graded noetherian b modules modulo torsion is equivalent to u c mod the associated graded mathbb z algebra mbox gr b has mbox gr b lqgr simeq coh hilb n this can be regarded as saying that u c simultaneously gives a non commutative deformation of mathfrak h oplus mathfrak h w and of its resolution of singularities hilb n rightarrow mathfrak h oplus mathfrak h as we show elsewhere this result is a powerful tool for studying the representation theory of h c and its relationship to hilb n

some noncommutative minimal surfaces abstract in the ongoing programme to classify noncommutative projective surfaces connected graded noetherian domains of gelfand kirillov dimension three a natural question is to determine the minimal models within any birational class in this paper we show that the generic noncommutative projective plane corresponding to the three dimensional sklyanin algebra as well as noncommutative analogues of p 1 p 1 and the more general van den bergh quadrics satisfy very strong minimality conditions translated into an algebraic question where one is interested in a maximality condition we prove the following result theorem a let r be a sklyanin algebra or a van den bergh quadric that is infinite dimensional over its centre and let a r be any connected graded noetherian maximal order with the same graded quotient ring as r then up to taking veronese rings a is isomorphic to r let t be an elliptic algebra that is the coordinate ring of a noncommutative surface containing an elliptic curve then under an appropriate homological condition we prove that every connected graded noetherian overring of t is obtained by blowing down finitely many lines line modules of self intersection 1

residue complex for sklyanin algebras of dimension three we describe the residue complex for three dimensional sklyanin algebras which are the interesting special cases of quantum polynomial rings in three variables in particular we show that the multiplicities of the point modules in the residue complex are all one just as in the classical case of commutative polynomial rings in three variables we explain why the residue complex in the quantum case has the same multiplicities of point modules as that in the commutative case even if there are fewer point modules in the quantum case by pointing out two quantum anomalies

noncommutative symmetric algebras of two sided vector spaces abstract we generalize the definition of the symmetric algebra of a vector space in order to define noncommutative symmetric algebras of two sided vector spaces not all two sided vector spaces have noncommutative symmetric algebras the ones that do are called admissible and conditions for admissibility are given further for some classes of admissible two sided vector spaces the skew fields of fractions of their noncommutative symmetric algebras are computed the degree 0 components of these skew fields correspond to function fields of certain noncommutative ruled surfaces and hence allow us to determine birational equivalence classes for such surfaces

some homological invariants of local pi algebras during the last several years artin jorgensen stafford yekutieli and others have generalized many homological results from the commutative case to the noncommutative connected graded case az sz1 jo1 jo2 ye in this paper we generalize some of those to the noncommutative local case our main interest is on noetherian local pi polynomial identity algebras though some results hold in a more general setting throughout a is an algebra over a base field k i is its jacobson radical and a0 a i if a0 is artinian a is called semilocal if a0 is a simple artinian algebra respectively a division algebra then a is called local respectively scalar local unless otherwise stated we are working with left modules and sometimes we use the term aop module for a right a module where aop is the opposite ring of a a finite a module means finitely generated over a we use noetherian respectively artinian for two sided noetherian respectively two sided artinian

naive noncommutative blowups at zero dimensional schemes abstract in an earlier paper d s keeler d rogalski j t stafford naive noncommutative blowing up duke math j 126 2005 491 546 mr 2120116 we defined and investigated the properties of the naive blowup of an integral projective scheme x at a single closed point in this paper we extend those results to the case when one naively blows up x at any suitably generic zero dimensional subscheme z the resulting algebra a has a number of curious properties for example it is noetherian but never strongly noetherian and the point modules are never parametrized by a projective scheme this is despite the fact that the category of torsion modules in qgr a is equivalent to the category of torsion coherent sheaves over x these results are used in the companion paper d rogalski j t stafford a class of noncommutative projective surfaces in press to prove that a large class of noncommutative surfaces can be written as naive blowups

homological properties of sklyanin algebras to a pair consisting of an elliptic curve and a point on it odeskii and feigin associate certain quadratic algebras sklyanin algebras having the hilbert series of a polynomial algebra in this paper we show that sklyanin algebras have good homological properties and we obtain some information about their so called linear modules we also show how the construction by odeskii and feigin may be generalized so as to yield other sklyanin type algebras

the grothendieck group of non commutative non noetherian analogues of p1 and regular algebras of global dimension two abstract let v be a finite dimensional positively graded vector space let b v v be a homogeneous element whose rank is dim v let a t v b the quotient of the tensor algebra tv modulo the 2 sided ideal generated by b let gr a be the category of finitely presented graded left a modules and fdim a its full subcategory of finite dimensional modules let qgr a be the quotient category gr a fdim a we compute the grothendieck group k 0 qgr a in particular if the reciprocal of the hilbert series of a which is a polynomial is irreducible then k 0 qgr a z r as ordered abelian groups where is the smallest positive real root of that polynomial when dim k v 2 qgr a is equivalent to the category of coherent sheaves on the projective line p 1 or a stacky p 1 if v is not concentrated in degree 1 if dim k v 3 results of piontkovski and minamoto suggest that qgr a behaves as if it is the category of coherent sheaves on a non commutative non noetherian analogue of p 1

irreducible representations of the 4 dimensional sklyanin algebra at points of infinite order abstract in 1982 sklyanin funct anal appl 16 1982 27 34 defined a family of graded algebras a e depending on an elliptic curve e and a point e which is not 4 torsion basic properties of these algebras were established in smith and stafford compositio math 83 1992 259 289 and a study of their representation theory was begun in levasseur and smith bull soc math france121 1993 35 90 the present paper classifies the finite dimensional simple a modules when is a point of infinite order sklyanin funct anal appl 17 1983 273 284 defines for each k n a representation of a in a certain k dimensional subspace of theta functions of order 2 k 1 we prove that these are irreducible representations and that any other simple module is obtained by twisting one of these by an automorphism of a the automorphism group of a is explicitly computed the method of proof relies on results in levasseur and smith in particular it is proved that every finite dimensional simple module is a quotient of a line module an important part of the analysis is a determination of the 1 critical a modules and the fact that such a module is equivalent to a quotient of a line module by a shifted line module

noncommutative graded domains with quadratic growth letk be an algebraically closed field and letr be a finitely generated connected gradedk algebra which is a domain of gelfand kirillov dimension two write the graded quotient ringq r ofr asd z z 1 for some automorphism of the division ringd we prove thatd is a finitely generated field extension ofk of transcendence degree one moreover we describer in terms of geometric data ifr is generated in degree one then up to a finite dimensional vector space r is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf over a projective curvey this implies in particular thatr is noetherian thatr is primitive when and thatr is a finite module over its centre when ifr is not generated in degree one thenr will still be noetherian and primitive if has infinite order butr need not be noetherian when has finite order

on some solutions to generalized spheroidal wave equations and applications expansions in series of coulomb and hypergeometric functions for the solutions of the generalized spheroidal wave equations gswes are analysed and written together in pairs each pair consists of a solution in series of hypergeometric functions and another in series of coulomb wavefunctions and has the same recurrence relations for the series coefficients but the solutions inside it present different radii of convergence expansions without a phase parameter are derived by truncating the series with a phase parameter for the whittaker hill equation solutions are found by treating that equation as a particular case of gswe while for the confluent gswe solutions with and without a phase parameter are given as pairs of series of coulomb wavefunctions amongst the applications there are equations for the time dependence of dirac test fields in some nonflat friedmann robertson walker spacetimes the radial schrodinger equation for an electron in the field of two coulombian centres and the schrodinger equation for the razavy type quasi exactly solvable potentials for these problems it is possible to find wavefunctions in terms of infinite series regular and convergent over the entire range of the independent variable by matching expansions belonging to one or more of the above pairs the infinite series solutions for the razavy type potentials in addition to the polynomial ones suggest that the whole energy spectra may be determined without appealing to perturbation theory or semi classical methods of approximation

a limit of the confluent heun equation and the schrodinger equation for an inverted potential and for an electric dipole we re examine and extend a group of solutions in series of bessel functions for a limiting case of the confluent heun equation and then apply such solutions to the one dimensional schrodinger equation with an inverted quasiexactly solvable potential as well as to the angular equation for an electron in the field of a point electric dipole for the first problem we find finite and infinite series solutions which are convergent and bounded for any value of the independent variable for the angular equation we also find expansions in series of jacobi polynomials

digitally phase modulated dpm signals some properties of digitally phase modulated dpm signals are presented phase modulation with overlapping pulses generated by a digital fir filter belong to this class which may be considered to be a practical approximation to continuous phase modulated cpm signals the power spectra of dpm signals are derived analytically the ability of these signals to operate through an additive white gaussian noise channel is assessed by calculating their minimum euclidean distance their noise and spectral properties are found to be similar to those of cpm signals assuming that a viterbi decoder is used to resolve symbol interference the out of band power tends to decrease as the pulse duration increases and the noise immunity is enhanced at the same time the receiver complexity grows exponentially hence noise immunity and spectrum compactness are achieved at the cost of higher received complexity modems for dpm signals are believed to be easier to implement than those for cpm signals this is because filter design is simple and a residual carrier component can be retained to facilitate carrier regeneration furthermore the accumulated carrier phase does not need to be continuously evaluated in order to perform matched filtering the analytic results derived are supported by measurements and simulations

solutions for confluent and double confluent heun equations this paper examines some solutions for confluent and double confluent heun equations in the first place we review two leaver s solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation leaver e w j math phys 27 1238 1986 and introduce an additional expansion in series of irregular confluent hypergeometric functions then we find the conditions under which one of these solutions can be written as a linear combination of the others in the second place by means of limiting procedures we generate solutions for the double confluent equation as well as for special limits of both the confluent and double confluent equations finally we present some problems which are ruled by one or the other of these four equations

naive noncommutative blowing up let b x mathscr l be the twisted homogeneous coordinate ring of an irreducible variety x over an algebraically closed field k with dim x 2 assume that c in x and in aut x are in sufficiently general position we show that if one follows the commutative prescription for blowing up x at c but in this noncommutative setting one obtains a noncommutative ring r r x c mathscr l with surprising properties 1 r is always noetherian but never strongly noetherian 2 if r is generated in degree one then the images of the r point modules in qgr r are naturally in one to one correspondence with the closed points of x however in both qgr r and gr r the r point modules are not parametrized by a projective scheme 3 while qgr r has finite cohomological dimension dim k h 1 mathscr o

orthogonalization by principal components applied to cpm the orthonormal basis for the space spanned by a given signal set can be chosen in many different ways however when the basis is truncated to fewer dimensions the quality of the approximated signals differs depending on the choice of the original basis we study two energy related quality measures and show that the optimal lower dimensional approximation is given by the principal components pc method which is also a simple and efficient alternative to gram schmidt techniques in addition we derive and bound the average decrease in squared euclidean distances over one symbol interval caused by the pc method this measure is relevant in serially concatenated continuous phase modulation where a manifold of signal pairs contributes to the bit error rate for low to medium signal to noise ratios by a numerical evaluation we find that for this measure the decrease is lower than that of a previous method by j huber and w liu see ieee j select areas commun vol 7 p 1437 49 1989 finally we compare the minimum squared euclidean distance for error events which is relevant for uncoded cpm systems here the loss with the pc method is generally larger than with huber and liu s method although examples of the opposite exist

homological properties of graded noetherian pi rings abstract let r be a connected graded noetherian pi ring if injdim r n r is auslander gorenstein and cohen macaulay with gelfand kirillov dimension equal to n if gldim r n r is a domain finitely generated as a module over its centre and a maximal order in its quotient division ring similar results hold if r is assumed to be local rather than connected graded alternatively suppose that r is a noetherian pi ring with gldim r r m 1 hd r m 2 for any two maximal ideals m i in the same clique then r is a direct sum of prime rings is integral over its centre and is auslander gorenstein if r is a prime ring then the centre z r of r is a krull domain and r equals its trace ring tr moreover hd r m height m for every maximal ideal m of r