optimal mock jacobi theta functions abstract we classify the optimal mock jacobi forms of weight one with rational coefficients the space they span is thirty four dimensional and admits a distinguished basis parameterized by genus zero groups of isometries of the hyperbolic plane we show that their fourier coefficients can be expressed explicitly in terms of singular moduli and obtain positivity conditions which distinguish the optimal mock jacobi forms that appear in umbral moonshine we find that many of ramanujan s mock theta functions can be expressed simply in terms of the optimal mock jacobi forms with rational coefficients

mock modular forms as p adic modular forms in this paper we consider the question of correcting mock modular forms in order to obtain p adic modular forms in certain cases we show that a mock modular form m is a p adic modular form furthermore we prove that otherwise the unique correction of m is intimately related to the shadow of m

the f q mock theta function conjecture and partition ranks in 1944 freeman dyson initiated the study of ranks of integer partitions here we solve the classical problem of obtaining formulas for ne n resp no n the number of partitions of n with even resp odd rank thanks to rademacher s celebrated formula for the partition function this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f q a problem with its own long history dating to ramanujan s last letter to hardy little was known about this problem until dragonette in 1952 obtained asymptotic results in 1966 g e andrews refined dragonette s results and conjectured an exact formula for the coefficients of f q by constructing a weak maass poincare series whose holomorphic part is q 1f q24 we prove the andrews dragonette conjecture and as a consequence obtain the desired formulas for ne n and no n

hilbert space frames containing a riesz basis and banach spaces which have no subspace isomorphic toc0 abstract we prove that a hilbert space frame fi i icontains a riesz basis if every subfamily fi i j j i is a frame for its closed span secondly we give a new characterization of banach spaces which do not have any subspace isomorphic toc0 this result immediately leads to an improvement of a recent theorem of holub concerning frames consisting of a riesz basis plus finitely many elements

tenth order mock theta functions in ramanujan s lost notebook iii ramanujan s lost notebook contains many results on mock theta functions in particular the lost notebook contains eight identities for tenth order mock theta functions previously the author proved six of the eight tenth order mock theta function identities it is the purpose of this paper to prove the fifth and sixth identities of ramanujan s tenth order mock theta functions the properties of modular forms are used for the proofs of theta function identities

some hecke rogers type identities abstract recently a new partition function associated with ramanujan s third order mock theta function q was discovered and subsequently its overpartition analogue was introduced in this paper we prove an intriguing identity arising from the study of that analogue which involves a double series of hecke rogers type in addition two further identities will be given our proofs heavily rely on formulas from the work of liu 12

dyson s crank of a partition holds he was thus led to conjecture the existence of some other partition statistic which he called the crank this unknown statistic should provide a combinatorial interpretation of p lln 6 in the same way that 1 1 and 1 2 treat the primes 5 and 7 in 4 5 one of us was able to find a crank relative to vector partitions as follows for a partition 7r let 7r be the number of parts of ir and cr n be the sum of the parts of ir or the number ir is partitioning with the convention 0 for the empty partition 0 of 0 let

dyson s ranks and appell lerch sums denote by p n the number of partitions of n and by n a m n the number of partitions of n with rank congruent to a modulo m we find and prove a general formula for dyson s ranks by considering the deviation of the ranks from the average begin aligned d a m sum n 0 infty left n a m n frac p n m right q n end aligned using appell lerch sum properties we decompose d a m into modular and mock modular parts so that the mock modular component is supported on certain arithmetic progressions whose modulus we can control using our decomposition we show how our formula gives as a straightforward consequence atkin and swinnerton dyer s results on ranks as well as celebrated work of bringmann ono and rhoades s on maass forms we also give an example of how our methods apply to work of lovejoy and osburn on rank differences

a robust numerical solution to reconstruct a globally relative shear modulus distribution from strain measurements to noninvasively quantify tissue elasticity for differentiating malignancy of soft tissue the authors previously proposed a two dimensional 2 d mechanical inverse problem in which simultaneous partial differential equations pde s represented the target distribution globally of relative shear moduli with respect to reference shear moduli such that the relative values could be determined from strain distributions obtained by conventional ultrasound us or nuclear magnetic resonance nmr imaging based analysis here the authors further consider the analytic solution in the region of interest subsequently demonstrating that the problem is inevitably ill conditioned in real world applications i e noise in measurement data and improper configurations of mechanical sources reference regions make it impossible to guarantee the existence of a stable and unique target global distribution next based on clarification of the inherent problematic conditions the authors describe a newly developed numerical based implicit integration approach that novelly incorporates a computationally efficient regularization method designed to solve this differential inverse problem using just low pass filtered spectra derived from strain measurements to evaluate method effectiveness reconstructions of the global distribution are carried out using intentionally created ill conditioned models the resultant reconstructions indicate the robust solution is highly suitable while also showing it has high potential to be applied in the development of an effective yet versatile diagnostic tool for quantifying the distribution of elasticity in various soft tissues

representations of mock theta functions abstract motivated by the works of liu we provide a unified approach to find appell lerch series and hecke type series representations for mock theta functions we establish a number of parameterized identities with two parameters a and b specializing the choices of a b we not only give various known and new representations for the mock theta functions of orders 2 3 5 6 and 8 but also present many other interesting identities we find that some mock theta functions of different orders are related to each other in the sense that their representations can be deduced from the same a b parameterized identity furthermore we introduce the concept of false appell lerch series we then express the appell lerch series false appell lerch series and hecke type series in this paper using the building blocks m x q z and f a b c x y q introduced by hickerson and mortenson as well as m x q z and f a b c x y q introduced in this paper we also show the equivalences of our new representations for several mock theta functions and the known representations

bailey pairs and indefinite quadratic forms we construct classes of bailey pairs where the exponent of q in n is an indefinite quadratic form as an application we obtain families of q hypergeometric mock theta multisums

on the rank and the crank modulo 4 and 8 in this paper we prove some identities conjectured by lewis for the rank and crank of partitions concerning the modulo 4 and 8 these identities are similar to dyson s identities for the rank modulo 5 and 7 which give a combinatorial interpretation to ramanujan s partition congruences for this we use multisection of series and some of the results that watson established for the third order mock theta functions

from sheaves on p 2 to a generalization of the rademacher expansion moduli spaces of stable coherent sheaves on a surface are of much interest for both mathe matics and physics yoshioka computed generating functions of poincare polynomials of such moduli spaces if the surface is p 2 and the rank of the sheaves is 2 motivated by physical arguments this pa per investigates the modular properties of these generating functions it is shown that these functions can be written in terms of the lerch sum and theta function based on this we prove a conjecture by vafa and witten which expresses the generating functions of euler numbers as a mixed mock mod ular form moreover we derive an exact formula of rademacher type for the fourier coefficients of this function this formula requires a generalization of the classical circle method this is the first example of an exact formula for the fourier coefficients of mixed mock modular forms which is of independent mathematical interest 1 introduction and statement of results in the past interactions between physics and mathematics have led to many interesting results motivated by strong weak coupling duality or s duality in physics this article considers various gen erating functions which appear in the study of moduli spaces of stable coherent sheaves on the projective plane p 2 we express the generating functions of poincare polynomials of moduli spaces of rank 2 sheaves in terms of the lerch sum and theta function which we will recall later using these expressions we prove a conjec ture by vafa and witten 45 for the generating functions of euler numbers these functions appear to be related to ramanujan s mock theta functions and therefore transform almost as weakly holomorphic modular forms i e meromorphic mod ular forms whose poles if there are any may only lie in cusps our second main result is an exact formula for the fourier coefficients of these generating functions that formally resembles the rademacher expansion for the coefficients of weakly holomorphic modular forms moduli spaces of coherent sheaves on a complex surface s receive much attention see for example 28 for an extensive work on such moduli spaces more specifically one is interested in the moduli space m r c1 c 2 of semi stable sheaves of rank r with first chern class c1 and second chern class c2 w e will consider topological invariants of m in particular the poincare polynomial p m s 2d imc m i 0 bi m s i and the euler number m p m 1 w here

tenth order mock theta functions in ramanujan s lost notebook ramanujan s lost notebook contains many results on mock theta functions in particular the lost notebook contains eight identities for tenth order mock theta functions previously the author proved the first six of ramanujan s tenth order mock theta function identities it is the purpose of this paper to prove the seventh and eighth identities of ramanujan s tenth order mock theta function identities which are expressed by mock theta functions and a definite integral l j mordell s transformation formula for the definite integral plays a key role in the proofs of these identities also the properties of modular forms are used for the proofs of theta function identities

optimal frames for erasures we study frames from the viewpoint of coding theory we introduce a numerical measure of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting

a computationally efficient approach for inverse material characterization combining gappy pod with direct inversion abstract an approach for computationally efficient inverse material characterization from partial field response measurements that combines the gappy proper orthogonal decomposition pod machine learning technique with a physics based direct inversion strategy is presented and evaluated gappy pod is used to derive a data reconstruction tool from a set of potential system response fields that are generated from available a priori information regarding the potential distribution of the unknown material properties then the gappy pod technique is applied to reconstruct the full spatial distribution of the system response from whatever portion of the response field has been measured with the chosen system testing method lastly a direct inversion strategy is presented that is derived from the equations governing the system response i e physics of the system which utilizes the full field response reconstructed by gappy pod to produce an estimate of the spatial distribution of the unknown material properties the direct inversion technique is a particularly computationally efficient inversion technique requiring a cost equivalent to a single numerical analysis therefore the majority of the computational expense of the presented approach is the one time potential response generation for the gappy pod technique which leads to an approach that is substantially computationally efficient overall two numerically simulated examples are shown in which the elastic modulus distribution was characterized based on partial field displacement response measurements both static and dynamic the inversion procedure was shown to have the capability to efficiently provide accurate estimates to material property distributions from partial field response measurements the direct inversion with gappy pod response estimation was also shown to be substantially tolerant to noise in comparison to the direct inversion given measured full field response lastly a physical example regarding elastography of an arterial construct from ultrasound imaging response measurements is shown to validate the practical applicability of the direct inversion approach with gappy pod response reconstruction

ramanujan s lost notebook vi the mock theta conjectures 1 introduction in this paper we shall consider only ramanujan s two families of fifth order mock theta functions these functions were briefly described in ramanujan s last letter to g h hardy 11 pp 354 3551 and g n watson subsequently 12 131 proved all the assertions about these functions contained in the letter subsequently the identities watson proved were greatly generalized in 2 also the fifth order mock theta functions were shown to have double series expansions involving indefinite quadratic forms 7 however there remains a profound mystery about these functions namely no one including ramanujan has ever proved that these functions are indeed mock theta functions and not just some clever combination of theta functions this problem was described in detail by watson 12 p 2741 and redescribed in 7 pp 113 l 141 in s p 97 eqs 3 5 and 3 6 a formula from the lost notebook was pointed out that would if valid lead to the establishment of at least some of the fifth order mock theta functions as truly mock theta functions in the sense of watson 12 p 78 footnote there are indeed ten such identities live for each of the two families we have two objects in this paper first we wish to show that the identities in each family are equivalent i e if one is true all five are true and if one is false all five are false to introduce our second objective we recall some basic notions from partition theory 4 p 142 8 p 841 the rank of a partition is the largest

an extension of the hardy ramanujan circle method and applications to partitions without sequences we develop a generalized version of the hardy ramanujan circle method in order to derive asymptotic series expansions for the products of modular forms and mock theta functions classical asymptotic methods including the circle method do not work in this situation because such products are not modular and in fact the error integrals that occur in the transformations of the mock theta functions can and often do make a significant contribution to the asymptotic series the resulting series include principal part integrals of bessel functions whereby the main asymptotic term can also be identified to illustrate the application of our method we calculate the asymptotic series expansion for the number of partitions without sequences andrews showed that the generating function for such partitions is the product of the third order mock theta function chi and a modular infinite product series the resulting asymptotic expansion for this example is particularly interesting because the error integrals in the modular transformation of the mock theta function component appear in the exponential main term

positivity preserving transformations for q binomial coefficients several new transformations for q binomial coefficients are found which have the special feature that the kernel is a polynomial with nonnegative coefficients by studying the group like properties of these positivity preserving transformations as well as their connection with the bailey lemma many new summation and transformation formulas for basic hypergeometric series are found the new q binomial transformations are also applied to obtain multisum rogers ramanujan identities to find new representations for the rogers szego polynomials and to make some progress on bressoud s generalized borwein conjecture for the original borwein conjecture we formulate a refinement based on new triple sum representations of the borwein polynomials

angle criteria for frame sequences and frames containing a riesz basis we obtain a condition implying that the union of two frame sequences is also a frame sequence christensen found a condition for this in terms of orthogonal projections we phrase our condition by use of the angle between closed subspaces also a lower bound formula is obtained we then apply the results to find conditions for a frame containing a riesz basis to be a riesz frame

dyson s ranks and maass forms which were defined by ramanujan and watson decades ago in his last letter to hardy dated january 1920 see pages 127 131 of 27 ramanujan lists 17 such functions and he gives 2 more in his lost notebook 27 in his paper the final problem an account of the mock theta functions 32 watson defines 3 further functions surprisingly much remains unknown about these enigmatic series ramanujan s claims about their analytic properties remain open and there is even debate concerning the rigorous definition of such a function despite these seemingly problematic issues ramanujan s mock theta functions indeed possess many striking properties and they have been the subject of an astonishing number of important works for example see 5 6 7 8 12 13 14 18 19 20 23 27 28 32 33 35 36 to name a few watson predicted this high level of activity in his 1936 presidential address to the london mathematical society with his prophetic words see page 80 of 32

symmetric approximation of frames and bases in hilbert spaces we introduce the symmetric approximation of frames by normalized tight frames extending the concept of the symmetric orthogonalization of bases by orthonormal bases in hilbert spaces we prove existence and uniqueness results for the symmetric approximation of frames by normalized tight frames even in the case of the symmetric orthogonalization of bases our techniques and results are new a crucial role is played by whether or not a certain operator related to the initial frame or basis is hilbert schmidt

hecke type double sums appell lerch sums and mock theta functions i by introducing a dual notion between partial theta functions and appell lerch sums we find and prove a formula which expresses hecke type double sums in terms of appell lerch sums not only does our formula prove classical hecke type sum identities such as those in work of kac and peterson on affine lie algebras and hecke modular forms but once we have the hecke type forms for ramanujan s mock theta functions our formula gives straightforward proofs of many of the classical mock theta identities indeed we obtain a new proof of the mock theta conjectures our formula also applies to positive level string functions associated with admissible representations of the affine lie algebra a 1 1 of kac and wakimoto

on the explicit construction of higher deformations of partition statistics the modularity of the partition generating function has many important consequences for example asymptotics and congruences for p n in a series of papers the author and ono 12 13 connected the rank a partition statistic introduced by dyson to weak maass forms a new class of functions which are related to modular forms and which were rst considered in 11 here we do a further step towards understanding how weak maass forms arise from interesting partition statistics by placing certain 2 marked durfee symbols introduced by andrews 1 into the framework of weak maass forms to do this we construct a new class of functions which we call quasiweak maass forms because they have quasimodular forms as components as an application we prove two conjectures of andrews it seems that this new class of functions will play an important role in better understanding weak maass forms of higher weight themselves and also their derivatives as a side product we introduce a new method which enables us to prove transformation laws for generating functions over incomplete lattices

boundary vector field for parametric active contours snakes are widely used in the fields of computer vision and image processing there are a number of shortcomings of the traditional snakes including the capture range problem and the concave object extraction problem gradient vector flow snake enhances the concave object extraction capability however it suffers from high computational requirement in this paper we present a new external force which is generated by a novel interpolation scheme which reduces the computational requirement significantly and at the same time improves the capture range and concave object extraction capability

on the seventh order mock theta functions in a recent paper h we proved the mock theta conjectures these are identities stated by ramanujan in his lost notebook jr2 pp 19 20 involving two of the 5th order mock 0 functions in fact ramanujan gave one such identity for each of the ten 5th order functions as shown in a g1 these ten identities are all equivalent to the two proved in i hi in this paper we prove the corresponding identities for ramanujan s 7th order mock 0 functions and show how they are related to rank differences of partitions to state our results we first introduce some standard notation if q and x are complex numbers with ql 1 and n is a nonnegative integer then

on ramanujan s definition of mock theta function in his famous deathbed letter ramanujan defined the notion of a mock theta function and offered some examples of functions he believed satisfied his definition very recently griffin et al established for the first time that ramanujan s mock theta functions actually satisfy his own definition on the other hand zwegers 2002 doctoral thesis zwegers s 2002 mock theta functions phd thesis univ utrecht utrecht the netherlands showed that all of ramanujan s examples are holomorphic parts of harmonic maass forms this has led to an alternate definition of a mock theta function this paper shows that ramanujan s definition of mock theta function is not equivalent to the modern definition

sixth order mock theta functions two new mock theta functions of the sixth order are defined the main theorem in this paper theorem 1 1 provides four transformation formulas relating the new mock theta functions with ramanujan s mock theta functions of the sixth order two further representations of the new mock theta functions are established lastly a hitherto unproved entry from ramanujan s lost notebook related to sixth order mock theta functions is proved

on the dual nature of partial theta functions and appell lerch sums abstract in recent work hickerson and the author demonstrated that it is useful to think of appell lerch sums as partial theta functions this notion can be used to relate identities involving partial theta functions with identities involving appell lerch sums in this sense appell lerch sums and partial theta functions appear to be dual to each other this duality theory is not unlike that found by andrews between various sets of identities of rogers ramanujan type with respect to baxter s solution to the hard hexagon model of statistical mechanics as an application we construct bilateral q series with mixed mock modular behaviour in subsequent work we see that our bilateral series are well suited for computing radial limits of ramanujan s mock theta functions

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

new combinatorial interpretations of ramanujan s partition congruences mod 5 7 and 11 soit p n le nombre de partitions non restreintes de n on donne de nouvelles interpretations combinatoires des congruences p 5n 4 p 7n 5 et p 11n 6 0 mod 5 7 et 11 respectivement

tissue elasticity reconstruction using linear perturbation method a new method to reconstruct the elastic modulus of soft tissue subjected to an external static compression is presented in this approach the newton raphson method is used to vary a finite element fe model of the elasticity equations to fit in a least squared sense a set of axial tissue displacement fields estimated using a correlation technique applied to ultrasound signals the ill conditioning of the hessian matrix is eliminated using the tikhonov regularization technique this regularization provides a compromise between fidelity to the observed data and a priori information of the solution using an echographic image formation model it is shown that the method converges within a few iterations 8 10 and that strain images artifacts which are common in elastography are significantly reduced after the resolution of the inverse problem

numerical study of multigrid implementations of some iterative image reconstruction algorithms the numerical behavior of multigrid implementations of the landweber generalized landweber art and mlem iterative image reconstruction algorithms is investigated comparisons between these algorithms and with their single grid implementations are made on two small scale synthetic pet systems for phantom objects exhibiting different characteristics and on one full scale synthetic system for a shepp logan phantom the authors also show analytically the effects of noise and initial condition on the generalized landweber iteration and note how to choose the shaping operator to filter out noise in the data or to enhance features of interest in the reconstructed image original contributions include 1 numerical studies of the convergence rates of single grid and multigrid implementations of the landweber generalized landweber art and mlem iterations and 2 effects of noise and initial condition on the generalized landweber iteration with procedures for filtering out noise or enhancing image features

frames containing a riesz basis and approximation of the frame coefficients using finite dimensional methods abstract we consider frames which contain a riesz basis among those we find riesz frames i e frames with the property that every subfamily is a frame for its closed linear span with a common lower bound we point out the connection to the projection method which can be used to approximate the frame coefficients for a riesz frame using finitely many elements of the frame under different conditions we show that the projection method even approximates the frame coefficient in l 2 sense

tenth order mock theta functions in ramanujan s lost notebook iv ramanujan s lost notebook contains many results on mock theta functions in particular the lost notebook contains eight identities for tenth order mock theta functions previously the author proved the first six of ramanujan s tenth order mock theta function identities it is the purpose of this paper to prove the seventh and eighth identities of ramanujan s tenth order mock theta function identities which are expressed by mock theta functions and a definite integral l j mordell s transformation formula for the definite integral plays a key role in the proofs of these identities also the properties of modular forms are used for the proofs of theta function identities

the bailey chain and mock theta functions standard applications of the bailey chain preserve mixed mock modularity but not mock modularity after illustrating this with some examples we show how to use a change of base in bailey pairs due to bressoud ismail and stanton to explicitly construct families of q hypergeometric multisums which are mock theta functions we also prove identities involving some of these multisums and certain classical mock theta functions

acceleration of landweber type algorithms by suppression of projection on the maximum singular vector a procedure that speeds up convergence during the initial stage the first 100 forward and backward projections of landweber type algorithms for iterative image reconstruction for positron emission tomography pet which include the landweber generalized landweber and steepest descent algorithms is discussed the procedure first identifies the singular vector associated with the maximum singular value of the pet system matrix and then suppresses projection of the data on this singular vector after a single landweber iteration it is shown that typical pet system matrices have a significant gap between their two largest singular values hence this suppression allows larger gains in subsequent iterations speeding up convergence by roughly a factor of three

the spt function of andrews recently andrews introduced the function s n spt n which counts the number of smallest parts among the integer partitions of n we show that its generating function satisfies an identity analogous to ramanujan s mock theta identities as a consequence we are able to completely determine the parity of s n using another type of identity one based on hecke operators we obtain a complete multiplicative theory for s n modulo 3 these congruences confirm unpublished conjectures of garvan and sellers our methods generalize to all integral moduli

on the connection between exponential bases and certain related sequences in l2 π π a sequence of vectors fn in a separable hilbert space h is a frame if there are positive constants a b such that a f 2 n f fn 2 b f 2 for all f h and fn is a riesz sequence if it is a riesz basis in the closure of the space spanned by the vectors fn the latter of the following two questions has been raised by khrushchev nikolskii and pavlov can every frame of complex exponentials ei nx in l2 be made into a riesz basis by removing from ei nx a suitable collection of the functions ei nx can every riesz sequence ei nx in l2 be made into a riesz basis by adjoining to ei nx a suitable collection of exponentials ei nx ei nx we show that this is indeed so for all frames and riesz sequences ei nx studied so far and we prove that we can always solve both problems in a certain weak sense however our main conclusion is that the answer to both questions is no

nullspaces and frames in this paper we give new characterizations of riesz and conditional riesz frames in terms of the properties of the nullspace of their synthesis operators on the other hand we also study the oblique dual frames whose coefficients in the reconstruction formula minimize different weighted norms

multiplicative q hypergeometric series arising from real quadratic fields andrews dyson and hickerson showed that 2 q hypergeometric series going back to ramanujan are related to real quadratic fields which explains interesting properties of their fourier coefficients there is also an interesting relation of such series to automorphic forms here we construct more such examples arising from interesting combinatorial statistics

pairs of eta quotients with dual weights and their applications abstract let d be the differential operator defined by d 1 2 i d d z this induces a map d k 1 m k 0 n m k 2 0 n where m k 0 n is the space of weakly holomorphic modular forms of weight k on 0 n the operator d k 1 plays important roles in the theory of eichler shimura cohomology and harmonic weak maass forms on the other hand eta quotients are fundamental objects in the theory of modular forms and partition functions in this paper we show that the structure of eta quotients is very rarely preserved under the map d k 1 between dual spaces m k 0 n and m k 2 0 n more precisely we classify dual pairs f d k 1 f under the map d k 1 such that f is an eta quotient and d k 1 f is a non zero constant multiple of an eta quotient when the levels are square free we give the complete classification of such pairs in general we find a necessary condition for such pairs the weight of the primitive eta quotient f z d i 1 z b 1 d i t z b t is less than or equal to 4 and every prime divisor of each d i is less than 11 we also give various applications of these classifications in particular we find all eta quotients of weight 2 and square free level n such that they are in the eisenstein space for 0 n to prove our main theorems we use various combinatorial properties of a latin square matrix whose rows and columns are exactly divisors of n

constrained iterative reconstruction by the conjugate gradient method the conjugate gradient method incorporating the object extent constraint is applied to image reconstruction of a three dimensional object using an incomplete projection data set the missing information is recovered by constraining the solution with the knowledge of the outer boundary of the object extent which may be a priori measured or known the algorithm is derived from the least squares criterion as an advanced version of conventional iterative reconstruction algorithms such as sirt simultaneous iterative reconstruction technique and ilst iterative least squares technique in the case of reconstruction from noisy projection data a method based on the minimum mean square error criterion is also proposed computer simulated reconstruction images of a phantom using limited angle and number of views are presented the result shows that the conjugate gradient method incorporating the object extent constraining provides the fastest convergence and the least error

overpartitions and class numbers of binary quadratic forms we show that the zagier eisenstein series shares its nonholomorphic part with certain weak maass forms whose holomorphic parts are generating functions for overpartition rank differences this has a number of consequences including exact formulas asymptotics and congruences for the rank differences as well as q series identities of the mock theta type

eulerian series as modular forms in 1988 hickerson proved the celebrated mock theta conjectures a collection of ten identities from ramanujan s lost notebook which express certain modular forms as linear combinations of mock theta functions in the context of maass forms these identities arise from the peculiar phenomenon that two different harmonic maass forms may have the same non holomorphic parts using this perspective we construct several infinite families of modular forms which are differences of mock theta functions

some characters of kac and wakimoto and nonholomorphic modular functions answering a question of kac we relate the character formulas for certain sl m 1 modules to automorphic forms we show that these q series are the holomorphic parts of nonholomorphic modular functions

iterative reconstruction methods using regularization and optimal current patterns in electrical impedance tomography an iterative reconstruction method which minimizes the effects of ill conditioning is discussed based on the modified newton raphson algorithm a regularization method which integrates prior information into the image reconstruction was developed this improves the conditioning of the information matrix in the modified newton raphson algorithm optimal current patterns were used to obtain voltages with maximal signal to noise ratio snr a complete finite element model fem was used for both the internal and the boundary electric fields reconstructed images from phantom data show that the use of regularization optimal current patterns and a complete fem model improves image accuracy the authors also investigated factors affecting the image quality of the iterative algorithm such as the initial guess image iteration and optimal current updating

partitions with short sequences and mock theta functions p a macmahon was the first to examine integer partitions in which consecutive integers were not allowed as parts such partitions may be described as having sequences of length 1 recently it was shown that partitions containing no sequences of consecutive integers of length k are of interest in seemingly unrelated problems concerning threshold growth models the object now is to develop the subject intrinsically to both provide deeper understanding of the theory and application of partitions and reveal the surprising role of ramanujan s mock theta functions