on the factorization of consecutive integers a classical result of sylvester 21 see also 16 17 generalizing bertrand s postulate states that the greatest prime divisor of a product of k consecutive integers greater than k exceeds k more recent work in this vein well surveyed in 18 has focussed on sharpening sylvester s theorem or upon providing lower bounds for the number of prime divisors of such a product as noted in 18 a basic technique in these arguments is to make a careful distinction between small and large primes and then apply sophisticated results from multiplicative number theory along these lines if we write n k u v n 2k

linear and nonlinear estimators for one and two dimensional fourier transforms previously presented estimators for one dimensional fourier transforms are modified to provide two dimensional analogs which are computationally feasible for large systems of equations near optimal two dimensional estimators are obtained by an iterative approximation method which involves the inversion of block lower triangular matrices

the average amount of information lost in multiplication we show that if x and y are integers independently and uniformly distributed in the set 1 n then the information lost in forming their product which is given by the equivocation h x y xmiddoty is theta loglogn we also prove two extremal results regarding cases in which x and y are not necessarily independently or uniformly distributed first we note that the information lost in multiplication can of course be 0 we show that the condition h x y xmiddoty 0 implies 2log2n h x y omega loglogn furthermore if x and y are independent and uniformly distributed on disjoint sets of primes it is possible to have h x y xmiddoty 0 with log2n h x and log2n h y each o loglogn second we show that no matter how x and y are distributed h x y xmiddoty o logn loglogn furthermore there are distributions in which x and y are independent and uniformly distributed over sets of numbers having only small and distinct prime factors for which we have h x y xmiddoty omega logn loglogn

extrapolation algorithms for discrete signals with application in spectral estimation in this paper we present some new algorithms useful for extrapolation and spectral estimation of band limited sequences in one and two dimensions first we show that many of the existing extrapolation algorithms for noiseless observations are unified under the criterion of minimum norm least squares mnls extrapolation for example the iterative algorithms proposed in 2 and 8 10 are shown to be special cases of a one step gradient algorithm which has linear convergence convergence and other numerical properties are improved by going to a conjugate gradient algorithm for noisy observations these algorithms could be extended by considering a mean square extrapolation criterion which gives rise to a mean square extrapolation filter and also to a recursive extrapolation filter examples and application of these methods are given extension of these algorithms is made for problems where the signal is known to be periodic a new set of functions called the periodic discrete prolate spheroidal sequences p dpss analogous to dpss 21 22 are introduced and their properties are studied finally several of these algorithms are generalized to two dimensions and the relevant equations are given

modular an v theory abstract a series of associative algebras a n v for a vertex operator algebra v over an arbitrary algebraically closed field and nonnegative integers n are constructed such that there is a one to one correspondence between irreducible a n v modules which are not a n 1 v modules and irreducible v modules moreover it is proved that v is rational if and only if a n v are semisimple for all n in particular the homogeneous subspaces of any irreducible v module are finite dimensional for rational vertex operator algebra v

on zhu s algebra and c2 algebra for symplectic fermion vertex algebra sf d abstract in this paper we study the family of vertex operator algebras s f d known as symplectic fermions this family is of a particular interest because these voas are irrational and c 2 cofinite we determine zhu s algebra a s f d and show that the equality of dimensions of a s f d and the c 2 algebra p s f d holds for d 2 the case of d 1 was treated by t abe in 1 we use these results to prove a conjecture by y arike and k nagatomo 8 on the dimension of the space of one point functions on s f d

an approximation theoretic approach to maximum entropy spectral analysis it is shown that the theory of best approximation in weighted mean square can be used to provide a mathematical foundation for the analysis and extension of the maximum entropy method of spectral analysis

estimates of theta x k l for large values of x we extend a result of ramare and rumely 1996 about the chebyshev function in arithmetic progressions we find a map e x such that x k l x k 0 whereas e x is a constant now we are able to show that for x 1531 x 3 l x 2 x 2 ln x

short effective intervals containing primes in arithmetic progressions and the seven cubes problem for any e 0 and any non exceptional modulus q 3 we prove that for x large enough x e log 2 q the interval e x e 3x e contains a prime p in any of the arithmetic progressions modulo q we apply this result to establish that every integer n larger than exp 71000 is a sum of seven cubes

energy concentration in band limited extrapolation the authors present a novel bandlimited extrapolation method that incorporates qualitative prior knowledge of the distribution of signal energy in both the time domain spatial domain and the frequency domain the extrapolation is easily computed by simple noniterative algorithm the prior information is incorporated by means of a weighted energy measure and the resulting solution is derived by straightforward hilbert space optimization the algorithm accommodates complex valued sequences nonuniform sampling and arbitrary spectral support and is applicable in one or several dimensions many existing extrapolation algorithms are noted as special cases examples of signal extrapolation and spectrum estimation using the new algorithm are given in one and two dimensions algorithm modification for noisy data is investigated and error bounds are presented

associative algebras and universal enveloping algebras associated to voas abstract for any vertex operator algebra v and any finite automorphism g of v we prove that the associative algebra a g n v is isomorphic to some sub quotient of the universal enveloping algebra of v with respect to g

higher level twisted zhu algebras the study of twisted representations of graded vertex algebras is important for understanding orbifold models in conformal field theory in this paper we consider the general setup of a vertex algebra v graded by z for some subgroup of r containing z and with a hamiltonian operator h having real but not necessarily integer eigenvalues we construct the directed system of twisted level p zhu algebras zhup v and we prove the following theorems for each p there is a bijection between the irreducible zhup v modules and the irreducible twisted positive energy v modules and v is h rational if and only if all its zhu algebras zhup v are finite dimensional and semisimple the main novelty is the removal of the assumption of integer eigenvalues for h we provide an explicit description of the level p zhu algebras of a universal enveloping vertex algebra in particular of the virasoro vertex algebra virc and the universal affine kac moody vertex algebra vk g at non critical level we a

hopf actions on vertex operator algebras abstract the hopf actions on vertex operator algebras are investigated if the action is semisimple a schur weyl type decomposition is obtained when the hopf algebra is finite dimensional and the action is faithful the action is a group action moreover if the hopf algebra is finite dimensional and the action is semisimple and inner faithful the action is also a group action in this case inner faithfulness is equivalent to faithfulness

higher level zhu algebras are subquotients of universal enveloping algebras abstract we prove that higher level zhu algebras of a vertex operator algebra are isomorphic to subquotients of its universal enveloping algebra

optimality of certain iterative and non iterative data extrapolation procedures abstract a fundamental aspect of many modern spectral analysis and image restoration procedures is the extrapolation of the measured data to achieve improved resolution iterative extrapolation poses some mathematical questions about convergence and about the optimality of the resulting limit we consider these questions and discuss briefly the optimality of certain non iterative extrapolation methods

spectral extrapolation of spatially bounded images mri application a spectral extrapolation algorithm for spatially bounded images is presented an image is said to be spatially bounded when it is confined to a closed region and is surrounded by a background of zeros with prior knowledge of the spatial domain zeros the extrapolation algorithm extends the image s spectrum beyond a known interval of low frequency components the result which is referred to as the finite support solution has space variant resolution features near the edge of the support region are better resolved than those in the center the resolution of the finite support solution is discussed as a function of the number of known spatial zeros and known spectral components a regularized version of the finite support solution is included for handling the case where the known spectral components are noisy for both the noiseless and noisy cases the resolution of the finite support solution is measured in terms of its impulse response characteristics and compared to the resolution of the zerofilled and nyquist solutions the finite support solution is superior to the zerofilled solution for both the noisy and noiseless data cases when compared to the nyquist solution the finite support solution may be preferred in the noisy data case examples using medical image data are provided

modular categories and orbifold models in this paper we try to answer the following question given a modular tensor category with an action of a compact group g is it possible to describe in a suitable sense the quotient category g we give a full answer in the case when is the category of vector spaces in this case g turns out to be the category of representation of drinfeld s double d g this should be considered as the category theory analog of the topological identity pt g bg this implies a conjecture of dijkgraaf vafa e verlinde and h verlinde regarding so called orbifold conformal field theories if is a vertex operator algebra which has a unique irreducible module itself and g is a compact group of automorphisms of and some not too restrictive technical conditions are satisfied then g is finite and the category of representations of the algebra of invariants g is equivalent as a tensor category to the category of representations of drinfeld s double d g we also get some partial results in the non holomorphic case i e when has more than one simple module

symplectic fermions and a quasi hopf algebra structure on bar u _i sl 2 abstract we consider the finite dimensional restricted quantum group u q s l 2 at q i we show that u i s l 2 does not allow for a universal r matrix even though u v v u holds for all finite dimensional representations u v of u i s l 2 we then give an explicit coassociator and a universal r matrix r such that u i s l 2 becomes a quasi triangular quasi hopf algebra our construction is motivated by the two dimensional chiral conformal field theory of symplectic fermions with central charge c 2 there a braided monoidal category sf has been computed from the factorisation and monodromy properties of conformal blocks and we prove that rep u i s l 2 r is braided monoidally equivalent to sf

postsmoothing for the lms algorithm and a fixed point roundoff error analysis the effect of post algorithm smoothing on digital implementations of the least mean square lms algorithm is studied an expression is derived for the mean square error mse with post algorithm pa smoothing but without finite wordlength effects it is shown that the mse can be reduced from that of lms without pa smoothing pa smoothing results in reduced convergence speed the effect of fixed point finite precision on pa smoothing is studied an expression for the mse is derived monte carlo simulation results that support the result that digital lms with pa smoothing performs better than digital lms without pa smoothing are presented

on orbifold theory abstract let v be a simple vertex operator algebra and g a finite automorphism group of v such that v g is regular it is proved that every irreducible v g module occurs in an irreducible g twisted v module for some g g moreover the quantum dimensions of irreducible v g modules are determined and a global dimension formula for v in terms of twisted modules is obtained in particular the orbifold theory conjecture is completely solved if g is solvable

numerical computations concerning the erh this paper describes a computation which established the erh to height 10000 for all primitive dirichlet l series with conductor q 13 and to height 2500 for all q 72 all composite q 112 and other moduli the computations were based on euler maclaurin summation care was taken to obtain mathematically rigorous results the zeros were first located within 10 12 then rigorously separated using an interval arithmetic package a generalized turing criterion was used to show there were no zeros off the critical line statistics about the spacings between zeros were compiled to test the pair correlation conjecture and gue hypothesis

on the pair correlation of zeros of the riemann zeta function in 1972 montgomery 20 21 introduced a new method for studying the zeros of the riemann zeta function one of his main accomplishments was to determine partially the pair correlation of zeros and to apply his results to obtain new information on multiplicity of zeros and gaps between zeros perhaps more importantly he conjectured on number theoretic grounds an asymptotic formula for the pair correlation of zeros and found that the form of this correlation exactly agreed with the gaussian unitary ensemble gue model for random hermitian matrices which had been studied earlier by physicists he was therefore able to formulate a general n correlation conjecture for zeros during the 1980s odlyzko 23 24 performed extensive numerical calculations of the correlations for zeros in ranges up to the 10th zero and found excellent agreement with the gue model more recently hejhal 17 was able to prove the same partial result for triple correlation as montgomery proved for pair correlation and rudnick and sarnak 25 have done the same for n correlation rudnick and sarnak extended their results to a large class of l functions and also showed that the riemann hypothesis rh is not needed for smoother forms of the asymptotic result very recently bogomolny and keating 1 2 have used a prime twin type conjecture to derive heuristically the n correlation conjecture beyond the range where the results of rudnick and sarnak apply the conclusion of all this work is to rmly establish but not prove the gue distribution for zeros of many zeta functions there is a dual relationship between zeros of the riemann zeta function and prime numbers following montgomery s work it was realized that information on pair correlation of zeros could be used to obtain information on primes this connection was developed by gallagher and mueller 10 and heath brown 15 later goldston and montgomery 14 found an equivalence under the riemann hypothesis between the pair correlation of zeros and the variance for the number of primes in short intervals this equivalence arises out of the explicit formula via a parseval relation together with a tauberian theorem from this work one sees that to obtain new information on primes from zeros will require some new insight on the zeros and while the connections to statistical physics mentioned above are a possible source of this insight so far no progress has been made on this fundamental problem the motivation for this paper is the observation that there are additional tools

holomorphic symplectic fermions let v be the even part of the vertex operator super algebra of r pairs of symplectic fermions up to two conjectures we show that v admits a unique holomorphic extension if r is a multiple of 8 and no holomorphic extension otherwise this is implied by two results obtained in this paper 1 if r is a multiple of 8 one possible holomorphic extension is given by the lattice vertex operator algebra for the even self dual lattice d r with shifted stress tensor 2 we classify lagrangian algebras in mathcal s mathcal f mathfrak h a ribbon category associated to symplectic fermions the classification of holomorphic extensions of v follows from 1 and 2 if one assumes that mathcal s mathcal f mathfrak h is ribbon equivalent to mathrm rep v and that simple modules of extensions of v are in one to one relation with simple local modules of the corresponding commutative algebra in mathcal s mathcal f mathfrak h

minimum cross entropy spectral analysis the principle of minimum cross entropy minimum directed divergence minimum discrimination information minimum relative entropy is summarized discussed and applied to the classical problem of estimating power spectra given values of the autocorrelation function this new method differs from previous methods in its explicit inclusion of a prior estimate of the power spectrum and it reduces to maximum entropy spectral analysis as a special case the prior estimate can be viewed as a means of shaping the spectral estimator cross entropy minimization yields a family of shaped spectral estimators consistent with known autocorrelations results are derived in two equivalent ways once by minimizing the cross entropy of underlying probability densities and once by arguments concerning the cross entropy between the input and output of linear filters several example minimum cross entropy spectra are included

extrapolation and spectral estimation with iterative weighted norm modification an algorithm is developed to define from the data samples themselves a frequency weighted norm to use in minimum weighted norm extrapolation the iterative procedure developed consists of using a periodogram spectrum estimate obtained from some samples of the signal estimate extrapolation found at one iteration to define the weight that is used to estimate at the next iteration this algorithm usually converges in less than 10 iterations to an extrapolation which is characterized as a nonparametric frequency stationary extension of the data the frequency resolution and extrapolation length are controlled by the length of a time domain window used to obtain smooth spectral estimates between iterations examples are provided to illustrate the use of the algorithm for interpolation extrapolation the examples give comparable results to nonadaptive extrapolation methods without the need for a priori knowledge for a certain spectral estimation example the algorithm provides comparable resolution to the parametric methods with more accurate values of the relative strengths of the narrowband components

on the triplet vertex algebra w p abstract we study the triplet vertex operator algebra w p of central charge 1 6 p 1 2 p p 2 we show that w p is c 2 cofinite but irrational since it admits indecomposable and logarithmic modules furthermore we prove that w p is of finite representation type and we provide an explicit construction and classification of all irreducible w p modules and describe block decomposition of the category of ordinary w p modules all this is done through an extensive use of zhu s associative algebra together with explicit methods based on vertex operators and the theory of automorphic forms moreover we obtain an upper bound for dim a w p finally for p prime we completely describe the structure of a w p the methods of this paper are easily extendable to other w algebras and superalgebras

on associative algebras modules and twisted modules for vertex operator algebras abstract we give a new construction of functors from the category of modules for the associative algebras a n v and a g v associated with a vertex operator algebra v defined by dong li and mason to the category of admissible v modules and admissible twisted v modules respectively using the method developed in the joint work 5 with y z huang the functors were first constructed by dong li and mason but the importance of the new method as in 5 is that we can apply the method to study objects without the commutator formula in the representation theory of vertex operator algebras

a mathbb z _ 2 orbifold model of the symplectic fermionic vertex operator superalgebra we give an example of an irrational c 2 cofinite vertex operator algebra whose central charge is 2d for any positive integer d this vertex operator algebra is given as the even part of the vertex operator superalgebra generated by d pairs of symplectic fermions and it is just the realization of the c 2 triplet algebra given by kausch in the case d 1 we also classify irreducible modules for this vertex operator algebra and determine its automorphism group

the structure of zhu s algebras for certain w algebras abstract we introduce a new approach that allows us to determine the structure of zhu s algebra for certain vertex operator super algebras which admit horizontal z grading by using this method and an earlier description of zhu s algebra for the singlet w algebra we completely describe the structure of zhu s algebra for the triplet vertex algebra w p as a consequence we prove that zhu s algebra a w p and the related poisson algebra p w p have the same dimension we also completely describe zhu s algebras for the n 1 triplet vertex operator superalgebra sw m moreover we obtain similar results for the c 0 triplet vertex algebra w 2 3 important in logarithmic conformal field theory because our approach is internal we had to employ several constant term identities for purposes of getting right upper bounds on dim a v this work is in a way a continuation of the results published in adamovic and milas 2008 4

radford drinfeld and cardy boundary states in the 1 p logarithmic conformal field models we introduce p 1 pseudocharacters in the space of 1 p model vacuum torus amplitudes to complete the distinguished basis in the 2p dimensional fusion algebra to a basis in the whole 3p 1 dimensional space of torus amplitudes and the structure constants in this basis are not necessarily non negative integer numbers we obtain a generalized verlinde formula that gives these structure constants in the context of theories with boundaries we identify the space of vacuum torus amplitudes with the space of ishibashi states then we propose 3p 1 boundary states satisfying the cardy condition

on the zeros of the riemann zeta function in the critical strip iv etude de la localisation des zeros complexes de la fonction zeta de riemann verification numerique de l hypothese de riemann pour les 1 500 000 001 premiers zeros de la forme it dans la bande critique avec 0 t 545 439 823 215 tous ces zeros ayant pour partie reelle 1 2

universal two parameter even spin w algebra abstract we construct the unique two parameter vertex algebra which is freely generated of type w 2 4 6 and generated by the weights 2 and 4 fields subject to some mild constraints all vertex algebras of type w 2 4 2 n for some n can be obtained as quotients of this universal algebra this includes the b and c type principal w algebras the z 2 orbifolds of the d type principal w algebras and many others which arise as cosets of affine vertex algebras inside larger structures as an application we classify all coincidences among the simple quotients of the b and c type principal w algebras as well as the z 2 orbifolds of the d type principal w algebras finally we use our classification to give new examples of principal w algebras of b c and d types which are lisse and rational

a linear time varying system framework for noniterative discrete time band limited signal extrapolation a technique for the analysis and design of noniterative algorithms for discrete time band limited signal extrapolation is described the approach involves modeling the extrapolation process as a linear time varying ltv system or filter together with a previously developed fourier theory for ltv systems this model provides a frequency domain transfer function representation for the extrapolation system this representation serves as a powerful tool for characterizing and comparing the reconstruction properties of several well known least squares optimal algorithms for band limited extrapolation moreover the frequency domain setting provides a conceptually attractive means for understanding the process of extrapolation itself additionally a least squares approximation methodology for designing ltv filters for band limited extrapolation is developed the design technique is shown to unify a broad class of algorithms for extrapolating discrete time data and further to provide a means for designing new and improved extrapolation algorithms

bimodules and g rationality of vertex operator algebras this paper studies the twisted representations of vertex operator algebras let v be a vertex operator algebra and g an automorphism of v of finite order t for any m n 1 tz an a g n v a g m v bimodule a g n m v is constructed the collection of these bimodules determines any admissible g twisted v module completely a verma type admissible g twisted v module is constructed naturally from any a g m v module furthermore it is shown with the help of bimodule theory that a simple vertex operator algebra v is g rational if and only if its twisted associative algebra ag v is semisimple and each irreducible admissible g twisted v module is ordinary

the additive causal part of burg s maximum entropy estimator and data consistency the additive causal part of burg s maximum entropy me estimator spectrum is calculated in closed form an immediate corollary for data consistency is shown

vertex operator algebras generalized doubles and dual pairs let v be a simple vertex operator algebra and g a finite automorphism group then there is a natural right g action on the set of all inequivalent irreducible v modules let cal s be a finite set of inequivalent irreducible v modules which is closed under the action of g there is a finite dimensional semisimple associative algebra a alpha g cal s for a suitable 2 cocycle alpha naturally determined by the g action on cal s such that a alpha g cal s and the vertex operator algebra v g form a dual pair on the sum of v modules in cal s in the sense of howe in particular every irreducible v module is completely reducible v g module

explicit upper bounds for the remainder term in the divisor problem we first report on computations made using the gp pari package that show that the error term x in the divisor problem is mathscr m x 4 o 0 35 x 1 4 log x when x ranges 1 081 080 10 10 where mathscr m x 4 is a smooth approximation the remaining part and in fact most of the paper is devoted to showing that delta x le 0 397 x 1 2 when x ge 5 560 and that delta x le 0 764 x 1 3 log x when x ge 9 995 several other bounds are also proposed we use this results to get an improved upper bound for the class number of a quadractic imaginary field and to get a better remainder term for averages of multiplicative functions that are close to the divisor function we finally formulate a positivity conjecture concerning

which is the better entropy expression for speech processing s log s or log s in maximum entropy spectral analysis mesa one maximizes the integral of logs f where s f is a power spectrum the resulting spectral estimate which is equivalent to that obtained by linear prediction and other methods is popular in speech processing applications an alternative expression s f logs f is used in optical processing and elsewhere this paper considers whether the alternative expression leads to spectral estimates useful in speech processing we investigate the question both theoretically and empirically the theoretical investigation is based on generalizations of file two estimates the generalizations take into account prior estimates of the unknown power spectrum it is shown that both estimates result from applying a generalized version of the principle of maximum entropy but they differ concerning the quantities that are treated as random variables the empirical investigation is based on speech synthesized using the different spectral estimates although both estimates lead to intelligible speech speech based on the mesa estimate is qualitatively superior

an inverse problem in thermal imaging this paper examines uniqueness and stability results for an inverse problem in thermal imaging the goal is to identify an unknown boundary of an object by applying a heat flux and measuring the induced temperature on the boundary of the sample the problem is studied both in the case in which one has data at every point on the boundary of the region and the case in which only finitely many measurements are available an inversion procedure is developed and used to study the stability of the inverse problem for various experimental configurations

uniform product of ag n v for an orbifold model v and g twisted zhu algebra abstract let v be a vertex operator algebra and g a finite automorphism group of v for each g g and nonnegative rational number n z g an associative algebra ag n v plays an important role in the theory of vertex operator algebras but the given product in ag n v depends on the eigenspaces of g we show that if v has no negative weights then there is a uniform definition of products on v and we introduce a g twisted zhu algebra ag n v which covers all ag n v let v be a simple vertex operator algebra with no negative weights and let s be a finite set of inequivalent irreducible twisted v modules which is closed under the action of g there is a finite dimensional semisimple associative algebra a g s for a suitable 2 cocycle naturally determined by the g action on s we show that a duality theorem of schur weyl type holds for the actions of a g s and vg on the direct sum of twisted v modules in s as an application of the theory of ag n v it follows as a natural consequence of the result that for any g g every irreducible g twisted v module is a completely reducible vg module

logarithmic intertwining operators and vertex operators this is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of heisenberg and lattice vertex algebras in this paper we examine logarithmic intertwining operators associated with rank one heisenberg vertex operator algebra m 1 a of central charge 1 12a 2 we classify these operators in terms of depth and provide explicit constructions in all cases our intertwining operators resemble puncture operators appearing in quantum liouville field theory furthermore for a 0 we focus on the vertex operator subalgebra l 1 0 of m 1 0 and obtain logarithmic intertwining operators among indecomposable virasoro algebra modules in particular we construct explicitly a family of hidden logarithmic intertwining operators i e those that operate among two ordinary and one genuine logarithmic l 1 0 module

a braided monoidal category for free super bosons the chiral conformal field theory of free super bosons is generated by weight one currents whose mode algebra is the affinisation of an abelian lie super algebra h with non degenerate super symmetric pairing the mode algebras of a single free boson and of a single pair of symplectic fermions arise for even odd dimension 1 0 and 0 2 of h respectively in this paper the representations of the untwisted mode algebra of free super bosons are equipped with a tensor product a braiding and an associator in the symplectic fermion case i e if h is purely odd the braided monoidal structure is extended to representations of the z 2z twisted mode algebra the tensor product is obtained by computing spaces of vertex operators the braiding and associator are determined by explicit calculations from three and four point conformal blocks

sieve methods for odd perfect numbers using a new factor chain argument we show that 5 does not divide an odd perfect number indivisible by a sixth power applying sieve techniques we also find an upper bound on the smallest prime divisor putting this together we prove that an odd perfect number must be divisible by the sixth power of a prime or its smallest prime factor lies in the range 10 8 p 10 1000 these results are generalized to much broader situations

spectral estimators that extend the maximum entropy and maximum likelihood methods the theory of best linear approximation in weighted l 2 spaces is used to obtain a general procedure the pdft for linearly reconstructing the fourier transform from sampled data the pdft can be used either directly to reduce sidelobe structure and to extrapolate the data or indirectly to obtain high resolution spectral estimators the direct and indirect pdft include as special cases many of the commonly used spectral techniques including burg s maximum entropy method capon s maximum likelihood method the spectral estimators based on bandlimited extrapolation the eigenvalue eigenvector methods for detecting sinusoids in noise pisarenko method schmidt s music eigenvector power beamforming and the best linear unbiased estimator blue for regression coefficients by exploiting their relationship to the linear pdft these nonlinear techniques can be analyzed in terms of linear approximation theory in addition to providing a unifying formulation for many different spectral estimators the pdft

the hexagonal versus the square lattice schmutz schaller s conjecture regarding the lengths of the hexagonal versus the lengths of the square lattice is shown to be true the proof makes use of results from computational prime number theory using an identity due to selberg it is shown that in principle the conjecture can be resolved without using computational prime number theory by our approach however this would require a huge amount of computation

primes in arithmetic progressions strengthening work of rosser schoenfeld and mccurley we establish explicit chebyshev type estimates in the prime number theorem for arithmetic progressions for all moduli k 72 and other small moduli

vertex operator algebras and associative algebras letvbe a vertex operator algebra we construct a sequence of associative algebrasan v n 0 1 2 such thatan v is a quotient ofan 1 v and a pair of functors between the category ofan v modules which are notan 1 v modules and the category of admissiblev modules these functors exhibit a bijection between the simple modules in each category we also show thatvis rational if and only if allan v are finite dimensional semisimple algebras

on sums of seven cubes we show that every integer between 1290741 and 3 375 10 12 is a sum of 5 non negative cubes from which we deduce that every integer which is a cubic residue modulo 9 and an invertible cubic residue modulo 37 is a sum of 7 non negative cubes

bimodules associated to vertex operator algebras let v be a vertex operator algebra and m n 0 we construct an a n v a m v bimodule a n m v which determines the action of v from the level m subspace to level n subspace of an admissible v module we show how to use a n m v to construct naturally admissible v modules from a m v modules we also determine the structure of a n m v when v is rational

on the behavior of a capon type spectral density estimator in this paper we propose a capon type estimator for the spectrum of a stationary time series this estimator may be viewed as an alternative to classical periodogram based estimators its advantage is that it copes with the leakage effect by using implicity automatic adaptive windowing we show its asymptotic equivalence to a random variable which is a quadratic form in the observations thus obtaining the asymptotic normality of the capon estimator we also study its asymptotic bias and variance

hopf actions on vertex operator algebras ii smash product abstract the smash product v h of a hopf algebra h and an h module vertex operator algebra v are investigated a n v theory and contragredient module theory are founded for v h if h is the group algebra of a finite subgroup of aut v all the irreducible inequivalent admissible v h modules are classified we also prove the maschke theorem in vertex operator algebra version finally we prove weak duality theorems for actions and coactions in vertex operator algebras version

chebyshev s bias for composite numbers with restricted prime divisors let x d a denote the number of primes p x with p a modd chebyshev s bias is the phenomenon for which more often x d n x d r than the other way around where n is a quadratic nonresidue mod d and r is a quadratic residue mod d if x d n x d r for every x up to some large number then one expects that n x d n n x d r for every x here n x d a denotes the number of integers n x such that every prime divisor p of n satisfies p a mod d in this paper we develop some tools to deal with this type of problem and apply them to show that for example n x 4 3 n x 4 1 for every x in the process we express the so called second order landau ramanujan constant as an infinite series and show that the same type of formula holds for a much larger class of constants

nonsemisimple fusion algebras and the verlinde formula we find a nonsemisimple fusion algebra open image in new window associated with each 1 p virasoro model we present a nonsemisimple generalization of the verlinde formula which allows us to derive open image in new window from modular transformations of characters

a duality theorem for vertex operator algebras abstract in this paper we prove that v h h m n v as s local vertex operator algebras where v is a simple vertex operator algebra and h is the group algebra of a finite subgroup of aut v and n dim h

explicit estimates for the error term in the prime number theorem for arithmetic progressions we give explicit numerical estimates for the chebyshev functions 4 x k l and g x k l for certain nonexceptional moduli k for values of e and b a constant c is tabulated such that 14 x k l x 9p k l exp c log2 k and k 10b the methods are similar to those used by rosser and schoenfeld in the case k 1 but are based on explicit estimates of n t x and an explicit zero free region for dirichlet l functions

the relationship between skew group algebras and orbifold theory abstract let v be a simple vertex operator algebra and let g be a finite automorphism group of v in c dong g yamskulna math z to appear it was shown that any irreducible v module is a completely reducible v g module where v g is the g invariant vertex operator subalgebra of v in this paper we give an alternative proof of this fact using the theory of skew group algebras we also extend this result to any irreducible g twisted v module when g is in the center of g and v is a g rational vertex operator algebra

reconstruction from partial information with applications to tomography the problem of reconstruction from projections in hilbert space is treated an axiomatic basis is considered which leads to techniques which provide improved reconstructions by incorporating prior knowledge to tailor the hilbert space to the problem at hand when applied to reconstruction of the fourier transform of a function sampled at finitely many discrete points the procedures lead to previously derived optimal estimation techniques when applied to x ray tomography the procedures lead to new reconstruction techniques which are shown to include as a special case the minimum energy reconstruction of logan and shepp duke math j 42 1975 pp 645 659

on quantum galois theory the goals of the present paper are to initiate a program to systematically study and rigorously establish what a physicist might refer to as the operator content of orbifold models to explain what this might mean and to clarify the title of the paper we will assume that the reader is familiar with the algebraic formulation of 2 dimensional cft in the guise of vertex operator algebras voa see b flm and dm for more information on this point in the paper dvvv several ideas are proposed concerning the structure of a holomorphic orbifold in other words if v is a holomorphic voa and if g is a finite group of automorphisms of v then the sub voa v g of g invariants is itself a voa and the subject of dvvv is very much concerned with speculation on the nature of the v modules it turns out to be more useful at least for purpose of inductive proofs to take v to be a simple voa we will then see that v g is also simple whenever g is a finite group of automorphisms of v one consequence of our main results is the following

on the tensor structure of modules for compact orbifold vertex operator algebras suppose v g is the fixed point vertex operator subalgebra of a compact group g acting on a simple abelian intertwining algebra v we show that if all irreducible v g modules contained in v live in some braided tensor category of v g modules then they generate a tensor subcategory equivalent to the category mathrm rep g of finite dimensional representations of g with associativity and braiding isomorphisms modified by the abelian 3 cocycle defining the abelian intertwining algebra structure on v additionally we show that if the fusion rules for the irreducible v g modules contained in v agree with the dimensions of spaces of intertwiners among g modules then the irreducibles contained in v already generate a braided tensor category of v g modules these results do not require rigidity on any tensor category of v g modules and thus apply to many examples where braided tensor category structure is known to exist but rigidity is not known for example they apply when v g is c 2 cofinite but not necessarily rational when v g is both c 2 cofinite and rational and v is a vertex operator algebra we use the equivalence between mathrm rep g and the corresponding subcategory of v g modules to show that v is also rational as another application we show that a certain category of modules for the virasoro algebra at central charge 1 admits a braided tensor category structure equivalent to mathrm rep su 2 up to modification by an abelian 3 cocycle

on functors between module categories for associative algebras and for n graded vertex algebras abstract we prove that the weak associativity for modules for vertex algebras is equivalent to a residue formula for iterates of vertex operators obtained using the weak associativity and the lower truncation property of vertex operators together with a known formula expressing products of components of vertex operators as linear combinations of iterates of components of vertex operators by requiring that these two formulas instead of the commutator formula hold we construct a functor s from the category of modules for zhu s algebra of a vertex operator algebra v to the category of n gradable weak v modules we prove that s has a universal property and the functor t of taking top levels of n gradable weak v modules is a left inverse of s in particular s is equal to a functor implicitly given by zhu and explicitly constructed by dong li and mason and we obtain a new construction without using relations corresponding to the commutator formula the hard part of this new construction is a technical theorem stating roughly that in a module for zhu s algebra the relation corresponding to the residue formula mentioned above can in fact be obtained from the relations corresponding to the action of zhu s algebra