complete intersection k theory and chern classes the purpose of this paper is to investigate the theory of complete intersection in noetherian commutative rings from the k theory point of view by complete intersection theory we mean questions like when whether an ideal is the image of a projective module of appropriate rank the paper has two parts in part one section 1 5 we deal with the relationship between complete intersection and k theory the part two section 6 8 is essentially devoted to construction projective modules with certain cycles as the total chern class here chern classes will take values in the associated graded ring of the grothedieck filtration and as well in the chow group in the smooth case in this paper all our rings are commutative and schemes are noetherian to avoid unnecessary complications we shall assume that all our schemes are connected for a noetherian schemex k0 x will denote the grothendieck group of locally free sheaves of finite rank over x whenever it make sense for a coherent sheaf m over x m will denote the class of m in k0 x we shall mostly be concerned with x speca where a is a noetherian commutative ring and in this case we shall also use the notation k0 a for k0 x

chow witt groups and grothendieck witt groups of regular schemes let a be a noetherian commutative z 1 2 algebra of krull dimension d and let p be a projective a module of rank d we use derived grothendieck witt groups and euler classes to detect some obstructions for p to split off a free factor of rank one if d 3 we show that the vanishing of its euler class in the corresponding grothendieck witt group is a necessary and sufficient condition for p to have a free factor of rank one if d is odd we also get some results in that direction if a is regular we show that the chow witt groups defined by morel and barge appear naturally as some homology groups of a gersten type complex in grothendieck witt theory from this we deduce that if d 3 then the vanishing of the euler class of p in the corresponding chow witt group is a necessary and sufficient condition for p to have a free factor of rank one crown copyright c 2008 published by elsevier inc all rights reserved

strong euler class of a stably free module of odd rank abstract let r be a commutative noetherian ring of dimension n 3 following a suggestion of fasel we establish a group homomorphism from van der kallen s group um n 1 r e n 1 r to the n th euler class group e n r so that 1 when n is even coincides with the homomorphism given by bhatwadekar and sridharan through euler classes 2 when n is odd is non trivial in general for an important class of rings 3 the sequence um n 1 r e n 1 r e n r e 0 n r 0 is exact where e 0 n r is the n th weak euler class group if x spec r is a smooth affine variety of dimension n over r so that the complex points of x are complete intersections and the canonical module k r is trivial then the sequence is proved to be exact on the left as well more generally let r be a commutative noetherian ring of dimension d and n be an integer such that n d 2 n 3 we also indicate how to extend our arguments to this setup to obtain a group homomorphism from um n 1 r e n 1 r to e n r

erratum to on the complete intersection conjecture of murthy with an example in complete intersections j algebra 458 2016 156 170 abstract this is an erratum to 4 and as well to 3 this article provides examples to indicate inconsistencies in 3 4

burst erasures and the mean square error for cyclic parseval frames this paper investigates the performance of frames for the linear redundant encoding of vectors when consecutive frame coefficients are lost due to the occurrence of random burst errors we assume that the distribution of bursts is invariant under cyclic shifts and that the burst length statistics are known in analogy with rate distortion theory we wish to find frames of a given size which minimize the mean square reconstruction error for the encoding of vectors in a complex finite dimensional hilbert space we obtain an upper bound for the mean square reconstruction error for a given parseval frame and in the case of cyclic parseval frames we find a family of frames which minimizes this upper bound under certain conditions these minimizers are identical to complex bose chaudhuri hocquenghem codes discussed in the literature the accuracy of our upper bounds for the mean square error is substantiated by complementary lower bounds all estimates are based on convexity arguments and a discrete rearrangement inequality

global stability of a delayed sir epidemic model with density dependent birth and death rates an sir epidemic model with density dependent birth and death rates is formulated in our model it is assumed that the total number of the population is governed by logistic equation the transmission of infection is assumed to be of the standard form namely proportional to i t h n t h where n t is the total variable population size i t is the size of the infective population and a time delay h is a fixed time during which the infectious agents develop in the vector we consider transmission dynamics for the model stability of an endemic equilibrium is investigated the stability result is stated in terms of a threshold parameter that is a basic reproduction number r 0

on the equations defining points let a be a commutative noetherian ring of krull dimension n let i be a local complete intersection ideal of height n suppose a i is torsion in k0 a it is proved that i is a set theoretic complete intersection if one of the following conditions holds 1 n is two 2 n is odd 3 n 4 even and a contains the field of rational numbers

epidemic spreading in correlated complex networks we study a dynamical model of epidemic spreading on complex networks in which there are explicit correlations among the node s connectivities for the case of markovian complex networks showing only correlations between pairs of nodes we find an epidemic threshold inversely proportional to the largest eigenvalue of the connectivity matrix that gives the average number of links which from a node with connectivity k go to nodes with connectivity k ensuremath numerical simulations on a correlated growing network model provide support for our conclusions

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

joint source channel coding using real bch codes for robust image transmission in this paper a new still image coding scheme is presented in contrast with standard tandem coding schemes where the redundancy is introduced after source coding it is introduced before source coding using real bch codes a joint channel model is first presented the model corresponds to a memoryless mixture of gaussian and bernoulli gaussian noise it may represent the source coder the channel coder the physical channel and their corresponding decoder decoding algorithms are derived from this channel model and compared to a state of art real bch decoding scheme a further comparison with two reference tandem coding schemes and the proposed joint coding scheme for the robust transmission of still images has been presented when the tandem scheme is not accurately tuned the joint coding scheme outperforms the tandem scheme in all situations compared to a tandem scheme well tuned for a given channel situation the joint coding scheme shows an increased robustness as the channel conditions worsen the soft performance degradation observed when the channel worsens gives an additional advantage to the joint source channel coding scheme for fading channels since a reconstruction with moderate quality may be still possible even if the channel is in a deep fade

on the number of generators of ideals in polynomial rings for an ideal i in a noetherian ring r let i be the minimal number of generators of i it is well known that there is a sequence of inequalities i i i i i 1 that are strict in general however murthy conjectured in 1975 that i i i for ideals in polynomial rings whose height equals i i the purpose of this article is to prove a stronger form of the conjecture in case the base field is infinite of characteristic different from 2 namely the equality i i i holds for any ideal i irrespective of its height

euler class group of a laurent polynomial ring local case abstract let r be a noetherian commutative ring of dimension n 2 and let a r t t 1 assume that the height of the jacobson radical of r is at least 2 let p be a projective a module of rank n dim a 1 with trivial determinant we define an abelian group called the euler class group of a denoted by e a let be an isomorphism from a to det p to the pair p we associate an element of e a called the euler class of p denoted by e p then we prove that a necessary and sufficient condition for p to have a unimodular element is the vanishing of e p in e a earlier bhatwadekar and raja sridharan have defined the euler class group of r denoted by e r and have proved similar results for projective r module of rank n later m k das defined the euler class group of the polynomial ring r t denoted by e r t and proved similar results for projective r t modules of rank n with trivial determinant

stability of a stochastic sir system abstract we propose a stochastic sir model with or without distributed time delay and we study the stability of disease free equilibrium the numerical simulation of the stochastic sir model shows that the introduction of noise modifies the threshold of system for an epidemic to occur and the threshold stochastic value is found

vanishing of euler class groups abstract in this paper we prove some theorems about vanishing of euler class groups for example suppose b a x f 1 where a x is a polynomial ring f a x is a non zero divisor and dim b dim a 1 3 then we prove that the euler class group e b l 0 for any rank one projective b module l

esprit estimation of signal parameters via rotational invariance techniques an approach to the general problem of signal parameter estimation is described the algorithm differs from its predecessor in that a total least squares rather than a standard least squares criterion is used although discussed in the context of direction of arrival estimation esprit can be applied to a wide variety of problems including accurate detection and estimation of sinusoids in noise it exploits an underlying rotational invariance among signal subspaces induced by an array of sensors with a translational invariance structure the technique when applicable manifests significant performance and computational advantages over previous algorithms such as mem capon s mlm and music

on a conjecture of murthy abstract this article concerns murthy s conjecture on complete intersections made in 1975 the sole breakthrough on this conjecture has still been the result proved by mohan kumar in 1978 the conjecture is open in general in this article we improve mohan kumar s bound when the base field is f p as an application we prove that any local complete intersection surface in the affine space a f p d is a set theoretic complete intersection generalizing a result of bloch murthy szpiro

local coefficients and euler class groups abstract in this paper we establish an isomorphism between the euler class group e r x l for a real smooth affine variety x spec a and the 0 th homology group h 0 m c g with local coefficients in a bundle g of groups constructed from the line bundle l over m corresponding to the orientation rank 1 projective module l where m c is the compact part of the manifold m of real points in x then by steenrod s poincare duality between homology and cohomology groups with local coefficients this isomorphism is identified with the whitney class homomorphism

complex field coding for ofdm over fading wireless channels orthogonal frequency division multiplexing ofdm converts a time dispersive channel into parallel subchannels and thus facilitates equalization and de coding but when the channel has nulls close to or on the fast fourier transform fft grid uncoded ofdm faces serious symbol recovery problems as an alternative to various error control coding techniques that have been proposed to ameliorate the problem we perform complex field coding cfc before the symbols are multiplexed we quantify the maximum achievable diversity order for independent and identically distributed i i d or correlated rayleigh fading channels and also provide design rules for achieving the maximum diversity order the maximum coding gain is given and the encoder enabling the maximum coding gain is also found simulated performance comparisons of cfc ofdm with existing block and convolutionally coded ofdm alternatives favor cfc ofdm for the code rates used in a hiperlan2 experiment

epidemic incidence in correlated complex networks a few years ago watts and strogatz 1 introduced a model able to produce networks with properties of both regu lar lattices and random graphs with small diameter their model soon led to a burst of activity in the field 2 3 further spurred by barabasi and co workers who found that many seemingly diverse systems share several topological proper ties such as a power law behavior in their connectivity dis tributions when represented as networks 4 these complex networks are formed by a set of many elements or nodes that are linked together through edges or links if they in teract directly empirical evidence supports that in notable networks such as metabolic or communication webs the probability p k that any node has k links to other nodes is distributed accordingly to a power law p k k 2g 5 7

good invariants for bad ideals abstract we generalize the notion of the segre class of ideals defined by murthy and obtain various analogous results using the theory of the euler class groups

zero cycles and projective modules let a be a reduced affine k algebra of dimension n over an algebraically closed field k let fnko a denote the sub group of ko a generated by the images of all the residue fields of all smooth maximal ideals of height n for a module of finite projective dimension m let m denote the image of m in ko a for a projective a module p of rank n we define the nth chern class of p to be cn p e l i aip where p is the dual of p suppose fnko a has no n 1 torsion then our main result th 3 8 is that p has a free direct summand of rank one if and only if cn p 0 when the characteristic of k is zero or a is normal and n 3 it is known that fnko a is torsion free le sr hence our theorem is applicable in these cases also when a is regular fnko a coincides with chn x the chow group of zero cycles of spec a and cn p coincides with the usual nth chern class as defined by grothendieck see fu when n 3 and a is regular this result was proved in mkm when n 3 and the characteristic of k is not equal to two this result in mkm was extended to the singular case by m levine in this paper we extend the results of mkm to all dimensions with the assumption stated above on torsion in fnko a we first show that cor 3 4 if i c a is a local complete intersection of height n then a i is zero in ko a if and only if i is a complete intersection corollary 3 4 together with a result of mohan kumar mk cor 1 9 here at once gives theorem 3 8 in section 1 we give some preliminaries and generalize results in mk2 these results are crucially used in the rest of the paper the basic ideas in section 1 are all taken from mk2 in section 2 we prove theorem 2 2 which strengthens a result of boratynski bo for local complete intersections with trivial co normal bundle theorem 2 2 is one of the crucial ingredients in the proofs of the results in section 3 as an amusing application of theorem 2 2 we give a new proof of a theorem of srinivas sr about torsion in zero cycles

on triviality of the euler class group of a deleted neighbourhood of a smooth local scheme let k be an infinite perfect field and let x specr where r is the localization of a k algebra b of finite type with respect to a regular prime ideal p specb letd be a divisor onx with a decompositiond i 1di into its irreducible components di such that each di is nonsingular and they intersect transversally consider y x d a deleted neighbourhood of the closed point x0 of x let e be a vector bundle on y in this set up e is trivial in the following cases i dim x 3 gabber ga1 ii t 1 bhatwadekar rao br iii ranke min dimr 2 t rao ra1 the above exposition is borrowed from a paper of nisnevich ni which contains some beautiful applications of these purity theorems in the same set up now let y y be a closed point and let my be the corresponding maximal ideal of oy y it is natural to ask whether my is a complete intersection observing that the conormal bundle my my is trivial we consider a more general situation let i be the defining ideal of a closed subscheme z of y such that the conormal bundle i i2 defined on z is trivial is i a complete intersection the questions we are addressing in this paper are a little stronger in nature can a given basis of i i2 be lifted to a minimal set of generators of i we remark that with the hypothesis on x each component di is defined by a single equation fi which vanishes at the closed point x0 of x let m be the maximal ideal at x0 the condition of transversality of intersection of di s is equivalent to saying that the images of f1 ft in the a m vector space m m2 are linearly independent keeping this discussion in mind we now give an excerpt of our main results in commutative algebraic terms see 4 2 4 3 4 4

murthy s conjecture on 0 cycles we show that the levine weibel chow group of 0 cycles mathrm ch d a of a reduced affine algebra a of dimension d ge 2 over an algebraically closed field is torsion free among several applications it implies an affirmative solution to an old conjecture of murthy in classical k theory

on algebras which are locally mathbb a 1 in codimension one let r be a noetherian normal domain call an r algebra a locally a1 in codimension one if rp r a is a polynomial ring in one variable over rp for every height one prime ideal p in r we shall describe a general structure for any faithfully flat r algebra a which is locally a1 in codimensionone and deduce results giving sufficient conditions for such an r algebra to be a locally polynomial algebra we also give a recipe for constructing r algebras which are locally a1 in codimension one when r is a normal affine spot i e a normal local domain obtained by a localisation of an affine domain we give criteria for a faithfully flat r algebra a which is locally a1 in codimensionone to be krull and a further condition for a to be noetherian the results are used to construct intricate examples of faithfully flat r algebras locally a1 in codimension one which are noetherian normal but not finitely generated

generalized and extended subspace algorithms for error correction with quantized dft codes discrete fourier transform dft codes have been used to provide robustness against errors and erasures in various applications this paper focuses on improving error localization of the bose chaudhuri hocquenghem bch dft codes first we analyze how the subspace based error localization outperforms the coding theoretic one then we propose an extension of the subspace based error localization based on additional syndrome that improves the existing one and is naturally suitable for rate adaptive distributed source coding dsc further we propose a new generic subspace based algorithm to decode bch dft codes the proposed approach generalizes the encoding and decoding of this important class of dft codes it introduces many different decoding matrices for a dft code this diversity is then used to diminish the effect of the quantization noise and thus to improve the decoding finally the extended and generalized approaches are combined to maximize the decoding gain simulation results demonstrate the capability of the proposed algorithms to perform significantly better than the existing subspace based error localization in the presence of quantization noise

on the complete intersection conjecture of murthy abstract suppose a k x 1 x 2 x n is a polynomial ring over a field k and i is an ideal in a m p murthy conjectured that i i i 2 where denotes the minimal number of generators recently fasel 3 settled this conjecture affirmatively when k is an infinite perfect field with 1 2 k always we are able to do the same when k is an infinite field in fact we prove similar results for ideals i in a polynomial ring a r x that contains a monic polynomial and r is essentially smooth algebra over an infinite field k or r is a regular ring over a perfect field k

on invariance of the euler class groups under a subintegral base change abstract let r be a commutative noetherian ring and r s be a subintegral extension it is proved that the top euler class group of r is isomorphic to the top euler class group of s an example is given to show that similar conclusion does not hold for integral but not subintegral extension of rings

sis model on homogeneous networks with threshold type delayed contact reduction we study the dynamics of a delayed sis epidemic model on homogeneous networks where it is assumed that individuals modify their contact patterns upon realizing the risk of infection this decision is made with some time delay and it is threshold type when the density of infected nodes reaches a critical value the number of links is reduced by a given factor such assumptions lead to a delay differential equation with discontinuous right hand side we show that if the basic reproduction number r 0 1 then the disease will be eradicated while it persists for r 0 1 in the latter case there is a globally asymptotically stable endemic equilibrium except for a crucial interval of reproduction numbers where the system shows oscillations we construct explicitly the unique slowly oscillatory periodic solution which has strong attractivity properties and show the existence of rapidly oscillatory periodic solutions with any frequency the amplitude of the oscillations is determined by the time delay our results indicate that with such information delays the link density of a network has an important effect on the qualitative dynamics of infectious diseases

subspace algorithms for error localization with quantized dft codes recently a class of real number bose chaudhuri hocquengem codes known as discrete fourier transform dft codes have been considered as joint source and channel codes for providing robustness to erasures and errors over wireless networks we propose three subspace algorithms for error localization with quantized dft codes the algorithms are similar to the music the minimum norm and the esprit algorithms used in array signal processing for direction of arrival estimation they provide different but related formulations of the error localizations by first partitioning a vector space into the channel error subspace and its orthogonal complement the noise subspace the locations of the errors are determined from either the error subspace eigenvectors or the noise subspace eigenvectors we also present a brief performance analysis of the localization error in terms of the perturbation of the error subspace due to quantization simulation results show that their localization performances are similar and they perform better than the coding theoretic approach over a broad range of channel error to quantization noise ratios

permanence of a delayed sir epidemic model with density dependent birth rate in this paper we consider the permanence of a modified delayed sir epidemic model with density dependent birth rate which is proposed in m song w ma asymptotic properties of a revised sir epidemic model with density dependent birth rate and time delay dynamic of continuous discrete and impulsive systems 13 2006 199 208 it is shown that global dynamic property of the modified delayed sir epidemic model is very similar as that of the model in w ma y takeuchi t hara e beretta permanence of an sir epidemic model with distributed time delays tohoku math j 54 2002 581 591 w ma m song y takeuchi global stability of an sir epidemic model with time delay appl math lett 17 2004 1141 1145

multiple epidemic waves in delayed susceptible infected recovered models on complex networks we consider a delayed susceptible infected recovered epidemic model on an uncorrelated complex network and address the effect of time lag on the shape and multiple waves of epidemic curves we show that when the transmission rate is above a threshold a large delay can cause multiple waves with larger amplitudes in the second and subsequent waves

efficient generation of zero dimensional ideals in polynomial rings let a be a commuta t ive noether ian ring and i an ideal in a it is a wel l known fact that it i 2 2 see 5 when n 1 and r is a regular local ring such as a power series r ing over a field or a regular spot over an infinite perfect field bha twadekar has shown that it ill 2 i for all maximal i d e a l s i n r t see 2 in this paper we generalise these results for the case when i is a zero d imens iona l ideal of the fol lowing kind i fo r any zero d imensional ideal i in a r t 1 t we have it i i2 it i if n 2 2 let i b e a zero d imensional ideal in a r t then it i i2 i in each o f the following cases a r is a power series ring over a field b r is a regular local r ing of an affine a lgebra over an infinite perfect field in sect 2 we state some theorems without proof which we require later in sect 3 we prove the analogues of dav i s ge rami t a theorems in sect 4 we prove the result for a power series r ing or for a regular local r ing of an affine algebra the results of this pape r go th rough for laurent po lynomia l rings

algebraic cycles and vector bundles over affine three folds on any smooth affine surface over an algebraically closed field it is easy to construct rank two projective modules with given determinant as the first chern class and an arbitrary rational equivalence class of a zero cycle as the second chern class we will briefly indicate this method in section 1 here we construct projective modules over smooth affine 3 folds with prescribed chern classes and of the right rank that is to say we construct rank two projective modules with prescribed first and second chem classes and rank three projective modules with prescribed first second and third chern classes theorem 2 1 as a corollary using roitman s theorem on torsion in zero cycles and suslin s cancellation theorem one gets that a rank three projective module over a smooth affine threefold with top chem class zero has a free direct summand of rank one cor 2 4 for example over a rational threefold any projective module splits into a free module and a rank two projective module thus we obtain necessary and sufficient conditions in terms of appropriate chern classes for modules to be efficiently generated for example on an affine rational 3 fold any line bundle is 3 generated and any rank two projective module is 4 generated we also prove that any maximal ideal over a smooth affine variety of dimension d is the zero of a section of a rank d projective module theorem 3 1 for an affine 3 fold if a3 x 0 we prove the validity of the eisenbud evans estimate for finitely generated modules we do not know the answers for most of these questions in higher dimensions sometimes we assume that the characteristic of the ground field is different from 2 3 and 5 due to technical reasons

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

infection dynamics on scale free networks we discuss properties of infection processes on scale free networks relating them to the node connectivity distribution that characterizes the network considering the epidemiologically important case of a disease that confers permanent immunity upon recovery we derive analytic expressions for the final size of an epidemic in an infinite closed population and for the dependence of infection probability on an individual s degree of connectivity within the population as in an earlier study r pastor satorras and a vesipignani phys rev lett 86 3200 2001 phys rev e 63 006117 2001 for an infection that did not confer immunity upon recovery the epidemic process in contrast with many traditional epidemiological models does not exhibit threshold behavior and we demonstrate that this is a consequence of the extreme heterogeneity in the connectivity distribution of a scale free network finally we discuss effects that arise from finite population sizes showing that networks of finite size do exhibit threshold effects infections cannot spread for arbitrarily low transmission probabilities

frame theoretic analysis of dft codes with erasures discrete fourier transform dft codes have been used as joint source channel codes in order to provide robustness against channel errors and erasures this paper presents a frame theoretic analysis of lowpass dft codes with erasures first it is shown that such codes make a special class of frames then the message reconstructions by the frame theoretic signal space projection and coding theoretic syndrome decoding of erasures are analyzed and their equivalence is proved the error performance of lowpass dft codes is analyzed using the frame theory the analysis helps in classifying the subframes of a dft frame into several classes this classification leads to the development of packetization criteria in order to guarantee minimum mean square reconstruction error and thus to minimize the effect of the quantization error simulation results substantiate the presented theory and the proposed packetization schemes

the euler class group of a polynomial algebra with coefficients in a line bundle let a be a commutative noetherian ring and p be a projective a module of rank text dim a 1 an intriguing open question is to find the precise obstruction for p to split as p simeq q oplus a for some a module q in this paper we settle this question when a r t for some ring r containing the field of rationals and p is a projective a module of rank text dim r

projective modules over smooth real affine varieties let x spec a be a smooth affine variety of dimension n 2 over a field k and p be a projective a module of rank n a result of murthy mu theorem 3 8 says that if k is an algebraically closed field and if the top chern class cn p in ch0 x is zero then p splits off a free summand of rank 1 i e p a q we note that over any base field the vanishing of the top chern class is a necessary condition for p to split off a free summand of rank 1 however as is shown by the example of the tangent bundle of an even dimensional real sphere this condition is not sufficient in view of these examples it is of interest to explore whether one can classify examples of projective modules over smooth real varieties which have the property that the top chern class of the projective module vanishes but the projective module does not split off a free summand of rank one more precisely we ask the following

systematic dft frames principle eigenvalues structure and applications motivated by a host of recent applications requiring some amount of redundancy frames are becoming a standard tool in the signal processing toolbox in this paper we study a specific class of frames known as discrete fourier transform dft codes and introduce the notion of systematic frames for this class this is encouraged by a new application of frames namely distributed source coding that uses dft codes for compression studying their extreme eigenvalues we show that unlike dft frames systematic dft frames are not necessarily tight then we come up with conditions for which these frames can be tight in either case the best and worst systematic frames are established in the minimum mean squared reconstruction error sense eigenvalues of dft frames and their subframes play a pivotal role in this work particularly we derive some bounds on the extreme eigenvalues dft subframes which are used to prove most of the results these bounds are valuable independently

correcting dct codes with laurent euclidean algorithm and syndrome extension real number block codes derived from the discrete cosine transform dct are corrupted by a few large errors along with low level noise checking syndromes are encapsulated in laurent polynomials and a special euclidean algorithm determines the locations of large errors these locations are adjusted properly to give an error modeling polynomial that is used to extend syndromes which are in turn transformed to codeword error values a probabilistic analysis describes the effects of the low level noise on corrected values after they pass through the syndrome extension process simulations yield probability of codeword errors mean squared decoding errors and sample means and variances of low level noise effects

bott periodicity and calculus of euler classes on spheres a variety of computations regarding the euler class group e an an and the grothendieck group k0 an of the algebraic sphere spec an is done the euler class of the algebraic tangent bundle on spec an is computed it is also investigated whether every element in the euler class group e an an is the euler class of a projective an module of rank n

euler class groups and the homology of elementary and special linear groups abstract we improve homology stability ranges for elementary and special linear groups over rings with many units our result implies stability for unstable quillen k groups and proves a conjecture of bass for commutative local rings with infinite residue fields we show that the obstruction to further stability is given by milnor witt k theory as an application we construct euler classes of projective modules with values in the cohomology of the milnor witt k theory sheaf for d dimensional commutative noetherian rings with infinite residue fields we show that the vanishing of the euler class is necessary and sufficient for an oriented projective module p of rank d to split off a rank 1 free direct summand along the way we obtain a new presentation of milnor witt k theory and of symplectic k 2 simplifying the classical matsumoto moore presentation

the monoid structure on homotopy obstructions abstract let a be a commutative noetherian ring containing a field k with 1 2 k dim a d and let p be a projective a module with r a n k p n let l o p denote the set of all pairs i where i is an ideal of a and p i i 2 is a surjective map the homotopy relations on l o p induced by l o p t leads to a set 0 l o p of equivalence classes in l o p there are two distinguished elements e 0 e 1 0 l o p respectively the images of 0 0 and a 0 define the obstruction class e p e 0 0 l o p to be called the nori homotopy class of p the following results are under suitable smoothness or regularity hypotheses we prove if 2 n d 2 then 0 l o p has a natural structure of a monoid which is a group if p q a when 2 n d 3 we prove p q a e p e 1 the additive zero further we give a definition of a euler class group e p under suitable smoothness hypotheses we prove if p q a and 2 n d 3 then there is natural isomorphism e p 0 l o p of groups

on maximal ideals in polynomial and laurent polynomial rings abstract when r is a complete local ring containing a field of characteristic zero we show that any maximal ideal in r t t 1 is projectively generated if r is a regular ring containing a field of characteristic zero we show that any maximal ideal m of r t such that ht m dim r is efficiently generated

global stability analysis of an sir epidemic model with demographics and time delay on networks abstract in this paper a susceptible infected recovery sir epidemic model is governed with demographics and time delay on networks firstly the basic reproduction number r 0 is derived dependent on birth rate death rate recovery rate and transmission rate the disease free equilibrium of the model is stable when r 0 1 and unstable when r 0 1 secondly based on a jacobian matrix calculated along with the disease free equilibrium we find that the system does not occur hopf branch under the disease free equilibrium thirdly the global asymptotic stability of a disease free equilibrium and a unique endemic equilibrium are proved by structuring two lyapunov functions finally numerical simulations are performed to illustrate the analysis results

improved coding theoretic and subspace based decoding algorithms for a wider class of dct and dst codes the decoding of a class of discrete cosine transform dct and discrete sine transform dst codes that are maximum distance separable codes mds is considered in this paper these class of codes are considered for error correction over real fields all the existing algebraic decoding algorithms are capable of decoding only a subclass of these codes which can be characterized into the bose chaudhuri hocquenghem bch form and fails to decode the remaining even though they are mds in this paper we propose a new generic algorithm along the lines of coding theoretic and subspace methods to decode the entire class of mds dct and dst codes the proposed subspace approaches are similar to popular esprit and music algorithms the proposed algorithms also perform significantly better than the existing algorithms on the bch like subclass a perturbation analysis is also presented to study the effect of various parameters on the error localization due to the quantization noise simulation results are presented to demonstrate the capability of proposed algorithms to decode the entire class and to perform significantly better on the bch like subclass than the existing algorithm under the influence of quantization noise

addition and subtraction of ideals abstract letabe a noetherian ring andibe comaximal ideals ofa andpbe a projectivea module addition refers to being given a surjectionp and producing a surjectionp i this is useful for example whenpis free to determine how many elements it takes to generate the ideal i subtraction refers to being givenp iand producing somep a major use of this is when a to show a projective module has a unimodular element here we extend certain addition and subtraction results of r sridharan 1995 j algebra176 947 958 and s mandal and r sridharan 1996 j math kyoto univ 36 no 3 453 470 in both addition and subtraction we weaken the hypotheses imposed on and in a suitable fashion remove the hypothesis thatpmust have trivial determinant with certain restrictions we also now allow dim a i 1 rather than ht i dim a whenais an affine algebra over a fieldf mandal and sridharan had subtraction results for wheniis the intersection of finitely many maximal ideals whose residue fields are quadratically closed when i has this form if char f 2 or wheniis the maximal ideal of anf rational point in our generalization we allowito be the intersection of finitely many maximal ideals all of whose residue fields are quadratically closed except possibly one which instead defines anf rational point when char f 2 we only need to require i to have this form in particular we can subtract an ideal whose radical is the maximal ideal of anf rational point this has been used by s m bhatwadekar and r sridharan 1998 invent math 133 no 1 161 192 to construct a certain local complete intersection ideal which is not a complete intersection ideal

on the number of generators of modules over polynomial affine rings preprint enviat per a la seva publicacio en una revista cientifica mathematische zeitschrift 1991 vol 208 p 11 21 https doi org 10 1007 bf02571506

mathbb p 1 p 1 gluing for local complete intersections we prove an analogue of the affine horrocks theorem for local complete intersection ideals of height n in r t where r is a regular domain of dimension d which is essentially of finite type over an infinite perfect field of characteristic unequal to 2 and 2n ge d 3

a riemann roch theorem in this paper we prove an analogue of this theorem for weak euler classes weak euler classes take values in weak euler class groups the definitions of the weak euler class group and weak euler classes were achieved by relaxing the definitions of the euler class group and euler classes the original definitions of the euler class group and the euler classes for smooth affine algebras over fields were given by m v nori for background literature we refer to 1 2 8 11 12 15 for noetherian commutative rings a with dima n 2 and a line bundle l on spec a the weak euler class group e0 a l has been defined in 2 as the quotient

stability analysis of a time delayed sir epidemic model with nonlinear incidence rate stability of sir models has been studied extensively within the framework of disease epidemiology we formulate a nonlinear mathematical model to study the role of nonlinear incidence rates and the effect of time delay in a nonlinear logistically growing time delayed sir model with variable population size the existence and stability of the possible equilibria are examined in terms of a certain threshold condition 0 the basic reproduction number for mathematical tractability the stability of the endemic equilibrium is shown using lyapunov functional approach with linear incidence rate numerical simulations are carried out to investigate the influence of the key parameters on the spread of the disease to support the analytical conclusion and illustrate possible behavioral scenarios of the model

projective modules over smooth affine varieties over real closed fields abstract let x spec a be a smooth affine variety of dimension n 2 over the field r of real numbers let p be a projective a module of rankn such that its nth chern class c n p ch 0 x is zero in this set up bhatwadekar das mandal showed amongst many other results that p a q in the case that either n is odd or the topological space x r of real points of x does not have a compact connected component in this paper we prove that similar results hold for smooth affine varieties over an arbitrary real closed field r the proof is algebraic and does not make use of tarski s principle nor of the earlier result for r

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved

the euler class group of a polynomial algebra in this paper we continue our study on the theory of the euler class group of a polynomial algebra a t where a is a commutative noetherian ring containing q of dimension n for such a ring a in 9 we defined the notion of the nth euler class group en a t of a t for simplicity let us call it e a t in 9 we also studied the relations between e a t and e a where e a is the nth euler class group of a for example there is canonical map e a e a t which is an injective group homomorphism and it is an isomorphism when a is a smooth affine domain 9 proposition 5 7 in general these two groups are not isomorphic see discussion preceding 9 proposition 5 7 in this context the following question is natural