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We introduce two convergent series expansions (direct and recursive) in terms of Bessel functions and the number of representations of an integer as a sum of squares for N -dimensional Madelung constants, MN(s) , where s is the exponent of the Madelung series (usually chosen as s=1/2). The convergence of the Bessel function expansion is discussed in detail. Values for MN(s) for s=12,32,3 and 6 for dimension up to N=20 are presented. This work extends Zucker's original analysis on N -dimensional Madelung constants for even dimensions up to N=8. [ABSTRACT FROM AUTHOR]

Fast convergent series are presented for lattice sums associated with the simple cubic, face-centered cubic, body-centered cubic, and hexagonal close-packed structures for interactions described by an inverse power expansion in terms of the distances between the lattice points, such as the extended Lennard-Jones potential. These lattice sums belong to a class of slowly convergent series, and their exact evaluation is related to the well-known number-theoretical problem of finding the number of representations of an integer as a sum of three squares. We review and analyze this field in some detail and use various techniques such as the decomposition of the Epstein zeta function introduced by Terras or the van der Hoff–Benson expansion to evaluate lattice sums in three dimensions to computer precision. [ABSTRACT FROM AUTHOR]

We record $\binom{42}{2}+\binom{23}{2}+\binom{13}{2}=1192$ functional identities that, apart from being amazingly amusing in themselves, find application in the derivation of Ramanujan-type formulas for $1/\unicode[STIX]{x1D70B}$ and in the computation of mathematical constants. [ABSTRACT FROM AUTHOR]

The Ramanujan–Mordell Theorem for sums of an even number of squares is extended to other quadratic forms and quadratic polynomials. [ABSTRACT FROM AUTHOR]

In Ramanujan's lost notebook, infinite product formulas are recorded for each of the functions φ(q) + φ(q5), φ(q) - φ(q5), φ(q) + √5φ(q5) and φ(q) - √5φ(q5) where φ(q) = ∑n=-∞∞ qn2 is the generating function for squares. Ramanujan also gave similar results that involve the Rogers-Ramanujan continued fraction. We provide a survey of these and other identities. We state and prove cubic analogues of Ramanujan's results, many of which are new. That is, we provide factorizations for the eight functions φ(q3) ± φ(q), φ(q3) ± iφ(q), √3φ(q3) ± iφ(q) and √3φ(q3) ± φ(q) as well as corresponding results for the generating function of the triangular numbers. [ABSTRACT FROM AUTHOR]

We briefly review Ramanujan's theories of elliptic functions to alternative bases, describe their analogues for levels 5 and 7, and develop new theories for levels 14 and 15. This gives rise to a rich interplay between theta functions, eta-products and Eisenstein series. Transformation formulas of degrees five and seven for hypergeometric functions are obtained, and the paper ends with some series for 1/π similar to ones found by Ramanujan. [ABSTRACT FROM AUTHOR]

One of the properties of the Rogers–Ramanujan continued fraction is its representation as an infinite product given by ( q ) = q 1 / 5 ∏ j = 1 ∞ ( 1 − q j ) ( j 5 ) where ( j p ) is the Legendre symbol. In this work we study the level 13 function R ( q ) = q ∏ j = 1 ∞ ( 1 − q j ) ( j 13 ) and establish many properties analogous to those for the fifth power of the Rogers–Ramanujan continued fraction. Many of the properties extend to other levels ℓ for which ℓ − 1 divides 24, and a brief account of these results is included. [ABSTRACT FROM AUTHOR]

We employ a modular method to establish the new result that two types of Eisenstein series to the tredecic base may be parametrised in terms of the eta quotients ${\it\eta}^{13}({\it\tau})/{\it\eta}(13{\it\tau})$ and ${\it\eta}^{2}(13{\it\tau})/{\it\eta}^{2}({\it\tau})$. The method can also be used to give short and simple proofs for the analogous cubic, quintic and septic theories. [ABSTRACT FROM PUBLISHER]

The theory of quasimodular forms is used to evaluate the convolution sums ... σ(l) σ(m), ... σ(l) σ(m) and ... σ(l) σ(m) for all positive integers n. As a consequence, the number of representations of a positive integer n by the octonary quadratic form x12 + x22 + x32 + x42 + 5(x52 + x62 + x72 + x82 is determined. [ABSTRACT FROM AUTHOR]

Abstract: A novel metal-based chelating method has been used to provide an order of magnitude increase in immunoassay performance on cyclic olefin copolymer (COC) plastics compared with passive binding. COCs are hydrophobic, and without surface modification they are often unsuitable for applications where protein adhesion is desired. When interacting with the bare plastic, the majority of the bound proteins will be denatured and become nonfunctional. Many of the surface modification techniques reported to date require costly equipment setup or the use of harsh reaction conditions. Here, we have successfully demonstrated the use of a simple and quick metal chelation method to increase the sensitivity, activity, and efficiency of protein binding to COC surfaces. A detailed analysis of the COC surfaces after activation with the metal complexes is presented, and the immunoassay performance was studied using three different antibody pairs. [Copyright &y& Elsevier]

Abstract: The Rogers–Ramanujan continued fraction has a representa-tion as an infinite product given by where and is the Legendre symbol. In his letters to Hardy and in his notebooks, Ramanujan recorded some exact numerical values of the Rogers–Ramanujan continued fraction for specific values of q. In this work, we give explicit evaluations of the level 13 analogue defined by [Copyright &y& Elsevier]

Abstract: For any positive integer n we state and prove formulas for the number of solutions, in integers, of , , and . Some conjectures are listed at the end of the paper. [Copyright &y& Elsevier]

A general theorem is stated that unifies 93 rational Ramanujan-type series for 1/π, 40 of which are believed to be new. Moreover, each series is shown to have a companion identity, thereby giving another 93 series, the majority of which are new. [ABSTRACT FROM PUBLISHER]

In the 'lost notebook', Ramanujan recorded infinite product expansions for , where r = r( q) is the Rogers-Ramanujan continued fraction. We shall give analogues of these results that involve Ramanujan's function k = k( q) = r( q) r( q). [ABSTRACT FROM AUTHOR]

Abstract: We generalize two identities involving Eisenstein series given in Chapter 19 of Ramanujanʼs second notebook. Four infinite families of Eisenstein series are obtained and their properties are investigated. The generating function of each infinite family is shown to be an elliptic function which has a simple infinite product expansion. Transformation properties of the Eisenstein series are investigated from first principles without using the transformation properties of Dedekindʼs eta-function. [Copyright &y& Elsevier]

In this article, we investigate congruences satisfied by Apéry-like numbers. [ABSTRACT FROM AUTHOR]

A simple construction of Eisenstein series for the congruence subgroup Γ0(p) is given. The construction makes use of the Jacobi triple product identity and Gauss sums, but does not use the modular transformation for the Dedekind eta-function. All positive integral weights are handled in the same way, and the conditionally convergent cases of weights 1 and 2 present no extra difficulty. [ABSTRACT FROM AUTHOR]

Abstract: Two proofs of a theta function identity of R.W. Gosper and R. Schroeppel are given. A cubic analogue is presented, and several interesting special cases are noted. [Copyright &y& Elsevier]

Let rk(n) and tk(n) denote the number of representations of an integer n as a sum of k squares, and as a sum of k triangular numbers, respectively. We prove that $$ t_8(n) = \frac{1}{2^{10}\times 3^{2}} (r_8(8n+8) - 16r_8(2n+2)), $$ and therefore the study of the sequence t8(n) is reduced to the study of subsequences of r8(n). We give an additional 21 analogous results for sums of squares and sums of triangular numbers induced by partitions of 8. We give a brief indication of what happens for the case k ≥ 9. [ABSTRACT FROM AUTHOR]

Abstract: We develop a theory for Eisenstein series to the septic base, which was started by S. Ramanujan in his “Lost Notebook.” We show that two types of septic Eisenstein series may be parameterized in terms of the septic theta function and the eta quotient . This is accomplished by constructing elliptic functions which have the septic Eisenstein series as Taylor coefficients. The elliptic functions are shown to be solutions of a differential equation, and this leads to a recurrence relation for the septic Eisenstein series. [Copyright &y& Elsevier]

In this article, we use the theory of elliptic functions to construct theta function identities which are equivalent to Macdonald's identities for A2, B2 and G2. Using these identities, we express, for d = 8, 10 or 14, certain theta functions in the form ηd(τ)F(P, Q, R), where η(τ) is Dedekind's eta-function, and F(P, Q, R) is a polynomial in Ramanujan's Eisenstein series P, Q and R. We also derive identities in the case when d = 26. These lead to a new expression for η26(τ). This work generalizes the results for d = 1 and d = 3 which were given by Ramanujan on page 369 of ‘The Lost Notebook’. [ABSTRACT FROM AUTHOR]

Abstract: We prove a simple and explicit formula, which expresses the 26th power of Dedekind''s η-function as a double series. The proof relies on properties of Ramanujan''s Eisenstein series P, Q and R, and parameters from the theory of elliptic functions. The formula reveals a number of properties of the product , for example its lacunarity, the action of the Hecke operator, and sufficient conditions for a coefficient to be zero. [Copyright &y& Elsevier]

The quintuple product identity was first discovered about 90 years ago. It has been published in many different forms, and at least 29 proofs have been given. We shall give a comprehensive survey of the work on the quintuple product identity, and a detailed analysis of the many proofs. [ABSTRACT FROM AUTHOR]

Let rk(n) and tk(n) denote the number of representations of n as a sum of k squares, and as a sum of k triangular numbers, respectively. We give a generalization of the result rk(8n + k) = cktk(n), which holds for 1 ≤ k ≤ 7, where ck is a constant that depends only on k. Two proofs are provided. One involves generating functions and the other is combinatorial. [ABSTRACT FROM AUTHOR]

Let rk(n) denote the number of representations of an integer n as a sum of k squares. We prove that for odd primes p,r11( p2)=330/31(p9+1)−22(−1)(p−1)/2p4+352/31H(p),where H(p) is the coefficient of qp in the expansion ofq∏j=1

∞

Some new identities for the four cubic theta functions a′(q,z), a(q,z), b(q,z) and c(q,z) are given. For example, we show thata′(q,z)3=b(q,z)3+c(q)2c(q,z).This is a counterpart of the identitya(q,z)3=b(q)2b(q,z3)+c(q,z)3,which was found by Hirschhorn et al.The Laurent series expansions of the four cubic theta functions are given. Their transformation properties are established using an elementary approach due to K. Venkatachaliengar. By applying the modular transformation to the identities given by Hirschhorn et al., several new identities in which a′(q,z) plays the role of a(q,z) are obtained. [Copyright &y& Elsevier]

The 3φ2 transformations are used to derive q-analogues of Kummer''s 20 relations. [Copyright &y& Elsevier]

Let Bn be the multivariate Bernstein operator of degree n for a simplex in Rs. In this paper, we show that Bn is diagonalisable with the same eigenvalues as the univariate Bernstein operator, i.e.,λ(n)k≔n!/(n−k)!1/nk1/nk, k=1,…,n, 1=λ(n)1>λ(n)2>···>λ(n)n>0 , and we describe the corresponding eigenfuctions and their properties. Since Bn reproduces only the linear polynomials, these are the eigenspace for λ(n)1=1. For k>1, the λ(n)k-eigenspace consists of polynomials of exact degree k, which are uniquely determined by their leading term. These are described in terms of the substitution of the barycentric coordinates (for the underlying simplex) into elementary eigenfunctions. It turns out that there are eigenfunctions of every degree k which are common to each Bn, n⩾k, for sufficiently large s. The limiting eigenfunctions and their connection with orthogonal polynomials of several variables is also considered. [Copyright &y& Elsevier]

The third version of the Regional Atmosphere and Land (RAL3) science configuration is documented. Developed through international partnership, RAL configurations define settings for the Unified Model atmosphere and Joint UK Land Environment Simulator when applied across timescales with kilometre and sub-km scale model grids. The RAL3 configuration represents a major advance compared to previous versions by delivering a common science definition suitable for application to tropical and mid-latitude regions. Developments within RAL3 include the introduction of a double-moment microphysics scheme and a bi-modal cloud scheme, replacing use of a single-moment scheme and different cloud schemes for mid-latitudes and tropics in previous versions. Updates have been implemented to the boundary layer scheme and a consolidation of land model settings to be more consistent with Global Atmosphere and Land (GAL) science configurations. Physics developments aimed to address priorities for model performance improvement identified by users. This paper documents the RAL3 science configuration, including a series of iterative revisions delivered since its first release, and their characteristics. Evidence is provided from the variety of assessments of RAL3, relative to the previous version (RAL2). Collaborative development and evaluation across organisations has enabled evaluation across a range of domains, grid-spacing and timescales. The analysis indicates more realistic precipitation distributions; improved representation of clouds and of visibility; a continued trend to more realistic representation of convection; and reduced near-surface wind speeds, but a persistent cold temperature bias. Overall the convective-scale verification scores and climatological model distributions relative to observations improve for the majority of variables. Ensemble results show improvements to the spread-error relationship. User feedback from subjective assessment activities has also been positive. Differences between RAL3 revisions and RAL2 are further illustrated through process-based analysis of a convective system over the UK. The latest RAL3 configuration (RAL3.3) is therefore recommended for research, operational numerical weather prediction and climate production at km and sub-km scales. [ABSTRACT FROM AUTHOR]

Recidivism, costs, and psychosocial outcomes are reported for a post-arrest diversion program in Wayne County (Detroit), MI. Program features included: rapid, standardized assessment of psychosocial functioning with the Juvenile Inventory For Functioning®, an individualized plan for addressing needs, engagement of caregivers, service provision by youth assistance programs in the youth's community, and access to mental health and substance use services as needed. The adjudication rate for new offenses one-year post services was 7.7%, for a program that costs $1,500 per youth. Significant improvement in functioning was observed for youth with an exit assessment. Functioning at entry predicted recidivism. [ABSTRACT FROM PUBLISHER]