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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.

We provide a comprehensive study of the function $h=h(q)$defined by $$\begin{eqnarray}h=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{12j-1})(1-q^{12j-11})}{(1-q^{12j-5})(1-q^{12j-7})}\end{eqnarray}$$and show that it has many properties that are analogues of corresponding results for Ramanujan’s function $k=k(q)$defined by $$\begin{eqnarray}k=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{10j-1})(1-q^{10j-2})(1-q^{10j-8})(1-q^{10j-9})}{(1-q^{10j-3})(1-q^{10j-4})(1-q^{10j-6})(1-q^{10j-7})}.\end{eqnarray}$$

AbstractWe prove an observation associated with ?3(?)?3(7?) which is found on page?54 of Ramanujan?s Lost Notebook (S. Ramanujan, The Lost Notebook and Other Unpublished Papers(Narosa, New Delhi, 1988)). We then study functions of the type ?3(a?)?3(b?) with a+b=8.

Building state-of-the-art processors is expensive and time consuming. Once the design is finalized and implemented, simulations are used to evaluate functionality and performance of the system. The Sim-alpha processor simulator is one of the most important tools for performance evaluations. Enhancing processor simulators is one of the major research field and many studies are underway related to this area. Current, open source, processor simulators do not account for the influences caused by multi-processing. In this study, we had shown that most processor simulations only test one program at a time on a virtual processor. The goal of the project was to demonstrate how processor simulators work when external influences are incorporated. Hardware or software interrupts are events that alter sequence of instructions executed by a processor. A context switch occurs when a multitasking operating system suspends the currently running process, and starts executing another. An additional code was added to the Sim-alpha program to allow for context switch. Benchmarks were executed with and without time slice context switch as well as different time slices. The results had shown that when the number of cycles before flushing the cache increases, the miss rate will decrease. For example if we are flushing the cache every 150 cycles, the cache miss rate is 48% compare to 2% without flushing the cache. The effect of flushing the cache is significant on the cache performance of processor simulators. In real life environments, processor must support multiple processes. We demonstrated with a simple change in the code that these simulators can have a more realistic workload. The effect of flushing the cache is significant on the cache performance of processor simulators. Current models do not account for this and may over estimate the performance gains of a particular processor design.

The function $$\displaylines{\displaystyle \Phi(\theta;q) = \theta + 3\sum_{k=1}^\infty\frac{\sin(2k\theta)q^k}{k(1+q^k+q^{2k})}}$$occurs in one of Ramanujan’s inversion formulas for elliptic integrals. In this article, a common generalization of the cubic elliptic functions $$\displaylines{g_1(\theta;q) = \frac{1}{6} + \sum_{k=1}^\infty \frac{q^k}{1+q^k+q^{2k}}\cos k\theta,\cr g_2(\theta;q) = \frac{1}{2} \frac{\sin \frac{\theta}{2}}{\sin \frac{3\theta}{2}} + \sum_{k=1}^\infty \frac{\chi_3(k)q^k}{1-q^k} \cos k\theta,} $$is given. The function g1is the derivative of Ramanujan’s function Φ (after rescaling), and χ3(n) = 0, 1 or −1 according as n≡ 0, 1 or 2 (mod 3), respectively, and |q| < 1. Many properties of the common generalization, as well as the functions g1and g2, are proved.The function $$\displaylines{\displaystyle \Phi(\theta;q) = \theta + 3\sum_{k=1}^\infty\frac{\sin(2k\theta)q^k}{k(1+q^k+q^{2k})}}$$occurs in one of Ramanujan’s inversion formulas for elliptic integrals. In this article, a common generalization of the cubic elliptic functions $$\displaylines{g_1(\theta;q) = \frac{1}{6} + \sum_{k=1}^\infty \frac{q^k}{1+q^k+q^{2k}}\cos k\theta,\cr g_2(\theta;q) = \frac{1}{2} \frac{\sin \frac{\theta}{2}}{\sin \frac{3\theta}{2}} + \sum_{k=1}^\infty \frac{\chi_3(k)q^k}{1-q^k} \cos k\theta,} $$is given. The function g1is the derivative of Ramanujan’s function Φ (after rescaling), and χ3(n) = 0, 1 or −1 according as n≡ 0, 1 or 2 (mod 3), respectively, and |q| < 1. Many properties of the common generalization, as well as the functions g1and g2, are proved.

Let rk(n) denote the number of representations of an integer nas a sum of ksquares. We prove that $$\begin{gathered} r_5 \left( n \right) = r_5 \left( {n\prime } \right)\left[ {\frac{{2^{3\left\lfloor {\lambda /2} \right\rfloor + 3} - 1}}{{2^3 - 1}} - \varepsilon _5 \left( {n\prime } \right)\frac{{2^{3\left\lfloor {\lambda /2} \right\rfloor } - 1}}{{2^3 - 1}}} \right] \hfill \\ {\text{ }} \times \mathop \prod \limits_p \left[ {\frac{{p^{3\left\lfloor {\lambda p/2} \right\rfloor + 3} - 1}}{{p^3 - 1}} - p\left( {\frac{{n\prime }}{p}} \right)\frac{{p^{3\left\lfloor {\lambda p/2} \right\rfloor } - 1}}{{p^3 - 1}}} \right], \hfill \\ \end{gathered}$$ where $$\varepsilon _5 \left( {n\prime } \right) = \left\{ \begin{gathered} {\text{0 if }}n\prime \equiv 1 \left( {\bmod {\text{ }}8} \right) \hfill \\ 4{\text{ if }}n\prime \equiv 2{\text{ or 3 }}\left( {\bmod {\text{ 4}}} \right) \hfill \\ 16/7{\text{ if }}n\prime \equiv 5{\text{ }}\left( {\bmod {\text{ 8}}} \right). \hfill \\ \end{gathered} \right.$$ Here n= 2λ∏ppλpis the prime factorisation of n, n′ is the square-free part of n, the products are taken over the odd primes p, and ( $$\frac{n}{p}$$ ) is the Legendre symbol. Some similar formulas for r7(n) and r9(n) are also proved.

The Bernstein operator Bn reproduces the linear polynomials, which are therefore eigenfunctions corresponding to the eigenvalue 1. We determine the rest of the eigenstructure of Bn. Its eigenvalues areλ(n)k≔n!(n−k)!1nk,k=0, 1, …, n,and the corresponding monic eigenfunctions p(n)k are polynomials of degree k, which have k simple zeros in [0, 1]. By using an explicit formula, it is shown that p(n)k converges as n→∞ to a polynomial related to a Jacobi polynomial. Similarly, the dual functionals to p(n)k converge as n→∞ to measures that we identity. This diagonal form of the Bernstein operator and its limit, the identity (Weierstrass density theorem), is applied to a number of questions. These include the convergence of iterates of the Bernstein operator and why Lagrange interpolation (at n+1 equally spaced points) fails to converge for all continuous functions whilst the Bernstein approximants do. We also give the eigenstructure of the Kantorovich operator. Previously, the only member of the Bernstein family for which the eigenfunctions were known explicitly was the Bernstein–Durrmeyer operator, which is self adjoint.

Ramanujan's partition congruences can be proved by first showing that the coefficients in the expansions of (q; q)r∞satisfy certain divisibility properties when r= 4, 6 and 10. We show that much more is true. For these and other values of r, the coefficients in the expansions of (q; q)r∞satisfy arithmetic relations, and these arithmetic relations imply the divisibility properties referred to above. We also obtain arithmetic relations for the coefficients in the expansions of (q; q)r∞(qt; qt)s∞, for t= 2, 3, 4 and various values of rand s. Our proofs are explicit and elementary, and make use of the Macdonald identities of ranks 1 and 2 (which include the Jacobi triple product, quintuple product and Winquist's identities). The paper concludes with a list of conjectures.

AbstractA new, elementary proof of the Macdonald identities for An?1using induction on nis given. Specifically, the Macdonald identity for Anis deduced by multiplying the Macdonald identity for An?1and nJacobi triple product identities together.