entropic and conformational study of isolated double tethered chains by three dimensional lattice simulations abstract double tethered polymers are a kind of linear polymer with a peculiar topological constraint that is both of its end points are attached to a plane which the polymer segments cannot penetrate the effects of the constraint on the polymer s configurational and entropic properties were investigated by means of a three dimensional lattice simulation that combined a previously proposed idea with a very efficient chain generation algorithm in particular the value of a topology dependent critical exponent was estimated for the double tethered configurations this data is the first report on isolated and double tethered chains also two optional types of tethered polymer were investigated as asymptotes of the double and single tethered configurations

on optimal spatial subsample size for variance estimation we consider the problem of determining the optimal block or subsample size for a spatial subsampling method for spatial processes observed on regular grids we derive expansions for the mean square error of the subsampling variance estimator which yields an expression for the theoretically optimal block size the optimal block size is shown to depend in an intricate way on the geometry of the spatial sampling region as well as characteristics of the underlying random field final expressions for the optimal block size make use of some nontrivial estimates of lattice point counts in shifts of convex sets optimal block sizes are computed for sampling regions of a number of commonly encountered shapes numerical studies are performed to compare subsampling methods as well as procedures for estimating the theoretically best block size

weyl asymptotics of bisingular operators and dirichlet divisor problem we consider a class of pseudodifferential operators with crossed vector valued symbols defined on the product of two closed manifolds we study the asymptotic expansion of the counting function of positive selfadjoint operators in this class using a general theorem of aramaki we can determine the first term of the asymptotic expansion of the counting function and in a special case we are able to find the second term we give also some examples emphasizing connections with problems of analytic number theory in particular with dirichlet divisor function

a new proof of weyl s formula on the asymptotic distribution of eigenvalues one of the more perplexing problems of finite particle quantum mechanics is to find the appropriate quantum mechanical description of a given classical mechanical system unfortunately from the mathematical point of view this problem is not terribly well posed suppose the classical system in question is represented by a symplectic manifold x and a classical observable p x r suppose also for simplicity that this system is a confined system in the sense that the sets p 1 are all compact let h be a hilbert space and let p be a self adjoint operator on h representing a quantum observable without specifying anything further about p for instance its relation to other observables there is practically no way of ascertaining whether p is an appropriate quantum counterpart of p at first glance the only requirements this appears to impose on p are that the spectrum of p be discrete and that the number of eigenvalues n a less than 1 be finite for all 1 this is the analogue of the condtion that p 1 be compact by examining the statistical properties of a large aggregate of classical systems each a copy of x p one can in fact say a little bit more about the relation p to p by comparing classical and quantum partition functions one can show that for iz large

tight bounds for online class constrained packing we consider class constrained packing problems in which we are given a set of bins each having a capacity v and c compartments and n items of m different classes and the same unit size we need to fill the bins with items subject to capacity constraints such that items of different classes are placed in separate compartments thus each bin can contain items of at most c distinct classes we consider two optimization goals in the class constrained bin packing problem ccbp our goal is to pack all the items in a minimal number of bins in the class constrained multiple knapsack problem ccmk we wish to maximize the total number of items packed in m bins for m 1 the ccbp and ccmk problems model fundamental resource allocation problems in computer and manufacturing systems both are known to be strongly np hard in this paper we derive tight bounds for the online variants of these problems we first present a lower bound of 1 on the competitive ratio of any deterministic algorithm for the online ccbp where 0 1 depends on v c m and n we show that this ratio is achieved by the algorithm first fit we then consider the temporary ccbp in which items may be packed for a bounded time interval that is unknown in advance we obtain a lower bound of v c on the competitive ratio of any deterministic or randomized algorithm we show that this ratio is achieved by all any fit algorithms finally tight bounds are derived for the online ccmk and the temporary ccmk problems

bounded imaginary powers of differential operators on manifolds with conical singularities we study the minimal and maximal closed extension of a differential operator a on a manifold b with conical singularities when a acts as an unbounded operator on weighted l p spaces over b 1 p under suitable ellipticity assumptions we can define a family of complex powers a z z we also obtain sufficient information on the resolvent of a to show the boundedness of the purely imaginary powers examples concern unique solvability and maximal regularity for the solution of the cauchy problem for the laplacian on conical manifolds as well as certain quasilinear diffusion equations

sign changes of the error term in weyl s law for heisenberg manifolds let r t be the error term in weyl s law for the 2l 1 dimensional heisenberg manifold hl gl in this paper several results on the sign changes and odd moments of r t are proved in particular it is proved that for some sufficiently large constant c r t changes sign in the interval t t c t for all large t moreover for a small constant c1 there exist infinitely many subintervals in t 2t of length c1 t log 5 t such that r t c1tl 1 4 holds on each of these subintervals

efficiency of characterizing ellipses and ellipsoids by discrete moments in this paper our studies are focused on ellipses and problems related to their representation and reconstruction from the data resulting from their digitization the main result of the paper is that a finite number of discrete moments corresponded to digital ellipses is in one to one correspondence with digital ellipses which enables coding of digital ellipses with an asymptotically optimal amount of memory in addition the problem of reconstruction based on the same parameters is considered since the digitization of real shapes causes an inherent loss of information about the original objects the precision of the original shape estimation from the corresponding digital data is limited we derive a sharp upper bound for the errors in reconstruction of the center position and half axes of the ellipse in function of the applied picture resolution i e the number of pixels per unit an extension of these results to the 3d case is also given

self avoiding walks interacting with a surface discusses two models of polymer adsorption in one model a self avoiding walk on a d dimensional lattice interacts with a d 1 dimensional hyperplane in the other model this walk must also lie in or on one side of this hyperplane both models exhibit non analytic behaviour corresponding to a phase transition but these phase transitions do not occur at the same point the authors give numerical estimates of the locations of these transitions

new pathways and connections in number theory and analysis motivated by two incorrect claims of ramanujan abstract the focus of this paper commences with an examination of three not obviously related pages in ramanujan s lost notebook pages 336 335 and 332 in decreasing order of attention on page 336 ramanujan proposes two identities but the formulas are wrong each is vitiated by divergent series we concentrate on only one of the two incorrect identities which may have been devised to attack the extended divisor problem we prove here a corrected version of ramanujan s claim which contains the convergent series appearing in it the convergent series in ramanujan s faulty claim is similar to one used by g f voronoi g h hardy and others in their study of the classical dirichlet divisor problem this now brings us to page 335 which comprises two formulas featuring doubly infinite series of bessel functions the first being conjoined with the classical circle problem initiated by gauss and the second being associated with the dirichlet divisor problem the first and fourth authors along with sun kim have written several papers providing proofs of these two difficult formulas in different interpretations in this monograph we return to these two formulas and examine them in more general settings the aforementioned convergent series in ramanujan s identity is also similar to one that appears in a curious identity found in chapter 15 in ramanujan s second notebook written in a more elegant equivalent formulation on page 332 in the lost notebook this formula may be regarded as a formula for 1 2 and in 1925 s wigert obtained a generalization giving a formula for 1 k for any even integer k 2 we extend the work of ramanujan and wigert in this paper the voronoi summation formula appears prominently in our study in particular we generalize work of j r wilton and derive an analogue involving the sum of divisors function s n the modified bessel functions k s x arise in several contexts as do lommel functions we establish here new series and integral identities involving modified bessel functions and modified lommel functions among other results we establish a modular transformation for an infinite series involving s n and modified lommel functions we also discuss certain obscure related work of n s koshliakov we define and discuss two new related classes of integral transforms which we call koshliakov transforms because he first found elegant special cases of each

local average in hyperbolic lattice point counting with an appendix by niko laaksonen the hyperbolic lattice point problem asks to estimate the size of the orbit gamma z inside a hyperbolic disk of radius cosh 1 x 2 for gamma a discrete subgroup of hbox psl 2 mathbb r selberg proved the estimate o x 2 3 for the error term for cofinite or cocompact groups this has not been improved for any group and any center in this paper local averaging over the center is investigated for hbox psl 2 mathbb z the result is that the error term can be improved to o x 7 12 varepsilon the proof uses surprisingly strong input e g results on the quantum ergodicity of maas cusp forms and estimates on spectral exponential sums we also prove omega results for this averaging consistent with the conjectural best error bound o x 1 2 varepsilon in the appendix the relevant exponential sum over the spectral parameters is investigated

weighted divisor sums and bessel function series on page 335 in his lost notebook ramanujan records without proof an identity involving a finite trigonometric sum and a doubly infinite series of ordinary bessel functions we provide the first published proof of this result the identity yields as corollaries representations of weighted divisor sums in particular the summatory function for r 2 n the number of representations of the positive integer n as a sum of two squares

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

on line bin stretching we are given a sequence of items that can be packed into m unit size bins in the classical bin packing problem we fix the size of the bins and try to pack the items in the minimum number of such bins in contrast in the bin stretching problem we fix the number of bins and try to pack the items while stretching the size of the bins as least as possible we present two on line algorithms for the bin stretching problem that guarantee a stretching factor of 5 3 for any number m of bins we then combine the two algorithms and design an algorithm whose stretching factor is 1 625 for any m the analysis for the performance of this algorithm is tight the best lower bound for any algorithm is 4 3 for any m 2 we note that the bin stretching problem is also equivalent to the classical scheduling load balancing problem in which the value of the makespan maximum load is known in advance copyright 2001 elsevier science b v

weyl laws on open manifolds under suitable invertibility hypothesis the spectrum of the dirac operator on certain open spin riemannian manifolds is purely discrete and obeys a growth law depending qualitatively on the in finiteness of the volume

self interacting polymer chains terminally anchored to adsorbing surfaces of three dimensional fractal lattices abstract we have studied the adsorption problem of self attracting linear polymers modeled by self avoiding walks saws situated on three dimensional fractal structures exemplified by 3d sierpinski gasket sg family of fractals as containers of a poor solvent members of sg family are enumerated by an integer b b 2 and it is assumed that one side of each sg fractal is an impenetrable adsorbing surface we calculate the critical exponents 1 11 and s which are related to the numbers of all possible saws with one both and no ends anchored to the adsorbing boundary respectively by applying the exact renormalization group rg method for the first three members of the sg fractal family b 2 3 and 4 we have obtained specific values of these exponents for chain and globular polymer phase we discuss their mutual relations and relations with corresponding values pertinent to extended polymer chain phase

weakly parametric pseudodifferential operators and atiyah patodi singer boundary problems this paper introduces a class of pseudodifferential operators depending on a parameter in a particular way the main application is a complete expansion of the trace of the resolvent of a dirac type operator with nonlocal boundary conditions of the kind introduced by atiyah patodi and singer aps this extends the partial expansion in g2 to a complete one and extends the complete expansion in gs 1 to the case where the dirac operator does not have a product structure near the boundary a secondary application is to obtain a complete expansion of the resolvent of a bdo on a compact manifold essentially reproving a result of agranovich agr the resolvent expansion yields immediately an expansion of the trace of the heat kernel and determines the singularities of the zeta function moreover a pseudodifferential factor can be allowed a major motive for these expansions is to obtain index formulas for elliptic operators there are many such applications in the physics and geometry literature the index formula comes from one particular term in the expansion but each term is a spectral invariant and they have been used for other purposes as well as for the index in particular branson and gilkey have a number of papers e g bg and gi analyzing these invariants and drawing geometric consequences interest in the asymptotic behavior of the resolvent goes back to carleman c more recently agmon agm developed it extensively for analytic applications he introduced the fundamental idea of treating the resolvent parameter essentially as another cotangent variable this idea was developed in s1 to analyze the singularities of the zeta function of an elliptic odo on a compact manifold and in 3 to analyze the resolvent of a differential operator with differential boundary conditions the technique works smoothly for differential operators producing so called local invariants integrals over the underlying

on line load balancing in a hierarchical server topology in a hierarchical server environment jobs are to be assigned in an on line fashion to a collection of servers which form a hierarchy of capability each job requests a specific server meeting its needs but the system is free to assign it either to that server or to any other server higher in the hierarchy each job carries a certain load which it imparts to the server it is assigned to the goal is to find a competitive assignment in which the maximum total load on a server is minimized we consider the linear hierarchy in which the servers are totally ordered in terms of their capabilities we investigate several variants of the problem in the unweighted as opposed to weighted problem all jobs have unit weight in the fractional as opposed to integral model a job may be assigned to several servers each receiving some fraction of its weight finally temporary as opposed to permanent jobs may depart after being active for some finite duration of time we show an optimal e competitive algorithm for the unweighted integral permanent model the same algorithm is e 1 competitive in the weighted case its fractional version is e competitive even if temporary jobs are allowed for the integral model with temporary jobs we show an algorithm which is 4 competitive in the unweighted case and 5 competitive in the weighted case we show a lower bound of e for the unweighted case both integral and fractional this bound is valid even with respect to randomized algorithms we also show a lower bound of 3 for the unweighted integral model when temporary jobs are allowed we generalize the problem and consider hierarchies in which the servers form a tree in the tree hierarchy any job assignable to a node is also assignable to the node s ancestors we show a deterministic algorithm which is 4 competitive in the unweighted case and 5 competitive in the weighted case where only permanent jobs are allowed randomizing this algorithm improves its competitiveness to e and e 1 respectively we also show an omega sqrt n lower bound when temporary jobs are allowed

spectral form factors of rectangle billiards the berry tabor conjecture asserts that local statistical measures of the eigenvalues j of a generic integrable quantum system coincide with those of a poisson process we prove that in the case of a rectangle billiard with random ratio of sides the sum behaves for random and n large like a random walk in the complex plane with a non gaussian limit distribution the expectation value of the distribution is zero its variance which is essentially the average pair correlation function is one in accordance with the berry tabor conjecture but all higher moments 4 diverge the proof of the existence of the limit distribution uses the mixing property of a dynamical system defined on a product of hyperbolic surfaces the berry tabor conjecture and the existence of the limit distribution for a fixed generic rectangle are related to an equidistribution conjecture for long horocycles on this product space

the number of n point digital discs a digital disc is the set of all integer points inside some given disc let dn be the number of different digital discs consisting of n points different up to translation the upper bound d n o n2 was shown recently no corresponding lower bound is known in this paper we refine the upper bound to dn o n which seems to be the true order of magnitude and we show that the average dn d1 d2 dn n has upper and lower bounds which are of polynomial growth in n

exact enumeration results for self avoiding walks on the honeycomb lattice attached to a surface we consider self avoiding walks on the honeycomb lattice interacting with a surface with different energies associated between sites in contact with a linear boundary to the left of the origin and those in contact with the right of the boundary we numerically confirm recent exact results for the polymer adsorption transition and corresponding critical exponents with mixed ordinary and special boundary conditions the phase diagram is elucidated with the aid of some rigorous arguments

the average behaviour of the error term in a new kind of the divisor problem abstract we define a new kind of divisor function d 1 n by the nth coefficient of the dirichlet series s 2 and denote by 1 x the error term in the asymptotic formula for n x d 1 n then we compute the first moment of 1 x that is 1 x 1 x d x and consider discrete mean values n x 1 k n k 1 2 and deduce asymptotic formulas which are analogous to the results obtained by voronoi hardy and furuya

on the index of elliptic operators on a wedge abstract we propose an analytical approach to the index theory of pseudo differential operators on a manifold with edges it results in an intermediate algebraic index formula the latter permits much more freedom in homotopies and in particular can be transformed to the topological formula

on the recovery of a function on a circular domain we consider the problem of estimating a function f x y on the unit disk f x y x sup 2 y sup 2 spl les 1 given a discrete and noisy data recorded on a regular square grid an estimate of f x y based on a class of orthogonal and complete functions over the unit disk is proposed this class of functions has a distinctive property of being invariant to rotation of axes about the origin of coordinates yielding therefore a rotationally invariant estimate for radial functions the orthogonal set has a particularly simple form being related to the classical legendre polynomials we give the statistical accuracy analysis of the proposed estimate of f x y in the sense of the l sub 2 metric it is found that there is an inherent limitation in the precision of the estimate due to the geometric nature of a circular domain this is explained by relating the accuracy issue to the celebrated problem in the analytic number theory called the lattice points of a circle in fact the obtained bounds for the mean integrated squared error are determined by the best known result so far on the problem of lattice points within the circular domain

on line load balancing the setup for our problem consists of n servers that must complete a set of tasks each task can be handled only by a subset of the servers requires a different level of service and once assigned cannot be reassigned we make the natural assumption that the level of service is known at arrival time but that the duration of service is not the on line load balancing problem is to assign each task to an appropriate server in such a way that the maximum load on the servers is minimized in this paper we derive matching upper and lower bounds for the competitive ratio of the on line greedy algorithm for this problem namely 3n 23 2 1 o 1 and derive a lower bound n12 for any other deterministic or randomized on line algorithm