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The number of ordered factorizations and the number of recursive divisors are two related arithmetic functions that are recursively defined. But it is hard to construct explicit representations of these functions. Taking advantage of their recursive definition and a geometric interpretation, we derive three closed-form expressions for them both. These expressions shed light on the structure of these functions and their number-theoretic properties. Surprisingly, both functions can be expressed as simple generalized hypergeometric functions.

We recently introduced the recursive divisor function $\kappa_x(n)$, a recursive analogue of the usual divisor function. Here we calculate its Dirichlet series, which is ${\zeta(s-x)}/(2 - \zeta(s))$. We show that $\kappa_x(n)$ is related to the ordinary divisor function by $\kappa_x * \sigma_y = \kappa_y * \sigma_x$, where * denotes the Dirichlet convolution. Using this, we derive several identities relating $\kappa_x$ and some standard arithmetic functions. We also clarify the relation between $\kappa_0$ and the much-studied number of ordered factorizations $K(n)$, namely, $\kappa_0 = {\bf 1} * K$.

The Kauffman model is the archetypal model of genetic computation. It highlights the importance of criticality, at which many biological systems seem poised. In a series of advances, researchers have honed in on how the number of attractors in the critical regime grows with network size. But a definitive answer has proved elusive. We prove that, for the critical Kauffman model with connectivity one, the number of attractors grows at least, and at most, as $(2/\!\sqrt{e})^N$. This is the first proof that the number of attractors in a critical Kauffman model grows exponentially., Comment: 5 pages, 3 figures

The Kauffman model of genetic computation highlights the importance of criticality at the border of order and chaos. The model with connectivity one is of special interest because it is exactly solvable. But our understanding of its behavior is incomplete, and much of what we do know relies on heuristic arguments. Here, we show that the key quantities in the model are intimately related to aspects of number theory. Using these links, we derive improved bounds for the number of attractors as well as the mean attractor length, which is harder to compute. Our work suggests that number theory is the natural language for deducing many properties of the critical Kauffman model with connectivity one, and opens the door to further insight into this deceptively simple model., Comment: 15 pages, 3 figures

The critical Kauffman model with connectivity one is the simplest class of critical Boolean networks. Nevertheless, it exhibits intricate behavior at the boundary of order and chaos. We introduce a formalism for expressing the dynamics of multiple loops as a product of the dynamics of individual loops. Using it, we prove that the number of attractors scales as $2^m$, where $m$ is the number of nodes in loops - as fast as possible, and much faster than previously believed.

We report on the fabrication and characterization of a Fabry-Perot microcavity enclosing a thin diamond membrane at cryogenic temperatures. The cavity is designed to enhance resonant emission of single nitrogen-vacancy centers by allowing spectral and spatial tuning while preserving the optical properties observed in bulk diamond. We demonstrate cavity finesse at cryogenic temperatures within the range of F = 4,000-12,000 and find a sub-nanometer cavity stability. Modeling shows that coupling nitrogen-vacancy centers to these cavities could lead to an increase of remote entanglement success rates by three orders of magnitude.

Innovation is to organizations what evolution is to organisms: it is how organisations adapt to changes in the environment and improve. Governments, institutions and firms that innovate are more likely to prosper and stand the test of time; those that fail to do so fall behind their competitors and succumb to market and environmental change. Yet despite steady advances in our understanding of evolution, what drives innovation remains elusive. On the one hand, organizations invest heavily in systematic strategies to drive innovation. On the other, historical analysis and individual experience suggest that serendipity plays a significant role in the discovery process. To unify these two perspectives, we analyzed the mathematics of innovation as a search process for viable designs across a universe of building blocks. We then tested our insights using historical data from language, gastronomy and technology. By measuring the number of makeable designs as we acquire more components, we observed that the relative usefulness of different components is not fixed, but cross each other over time. When these crossovers are unanticipated, they appear to be the result of serendipity. But when we can predict crossovers ahead of time, they offer an opportunity to strategically increase the growth of our product space. Thus we find that the serendipitous and strategic visions of innovation can be viewed as different manifestations of the same thing: the changing importance of component building blocks over time., Comment: 7 pages, 6 figures

A neutral network is a subgraph of a Hamming graph, and its principal eigenvalue determines its robustness: the ability of a population evolving on it to withstand errors. Here we consider the most robust small neutral networks: the graphs that interpolate pointwise between hypercube graphs of consecutive dimension (the point, line, line and point in the square, square, square and point in the cube, and so on). We prove that the principal eigenvalue of the adjacency matrix of these graphs is bounded by the logarithm of the number of vertices, and we conjecture an analogous result for Hamming graphs of alphabet size greater than two.

We study the relative importance of "top-speed" (long-term growth rate) and "acceleration" (how quickly the long-term growth rate can be reached) in the evolutionary race to increase population size. We observe that fitness alone does not capture growth rate: robustness, a property of neutral network shape, combines with fitness to include the effect of deleterious mutations, giving growth rate. Similarly, we show that growth rate alone does not capture population size: regularity, a different property of neutral network shape, combines with growth rate to include the effect of higher depletion rates early on, giving size. Whereas robustness is a function of the principal eigenvalue of the neutral network adjacency matrix, regularity is a function of the principal eigenvector. We show that robustness is not correlated with regularity, and observe in silico the selection for regularity by evolving RNA ribozymes. Despite having smaller growth rates, the more regular ribozymes have the biggest populations.

We present a quantitative measure of physical complexity, based on the amount of information required to build a given physical structure through self-assembly. Our procedure can be adapted to any given geometry, and thus to any given type of physical system. We illustrate our approach using self-assembling polyominoes, and demonstrate the breadth of its potential applications by quantifying the physical complexity of molecules and protein complexes. This measure is particularly well suited for the detection of symmetry and modularity in the underlying structure, and allows for a quantitative definition of structural modularity. Furthermore we use our approach to show that symmetric and modular structures are favoured in biological self-assembly, for example of protein complexes. Lastly, we also introduce the notions of joint, mutual and conditional complexity, which provide a useful distance measure between physical structures., Comment: 9 pages, submitted for publication

In recent work we presented a new approach to the analysis of weighted networks, by providing a straightforward generalization of any network measure defined on unweighted networks. This approach is based on the translation of a weighted network into an ensemble of edges, and is particularly suited to the analysis of fully connected weighted networks. Here we apply our method to several such networks including distance matrices, and show that the clustering coefficient, constructed by using the ensemble approach, provides meaningful insights into the systems studied. In the particular case of two data sets from microarray experiments the clustering coefficient identifies a number of biologically significant genes, outperforming existing identification approaches., Comment: Accepted for publication in J. Phys. A

Let $\sigma=(\sigma_1,..., \sigma_N)$, where $\sigma_i =\pm 1$, and let $C(\sigma)$ denote the number of permutations $\pi$ of $1,2,..., N+1,$ whose up-down signature $\mathrm{sign}(\pi(i+1)-\pi(i))=\sigma_i$, for $i=1,...,N$. We prove that the set of all up-down numbers $C(\sigma)$ can be expressed by a single universal polynomial $\Phi$, whose coefficients are products of numbers from the Taylor series of the hyperbolic tangent function. We prove that $\Phi$ is a modified exponential, and deduce some remarkable congruence properties for the set of all numbers $C(\sigma)$, for fixed $N$. We prove a concise upper-bound for $C(\sigma)$, which describes the asymptotic behaviour of the up-down function $C(\sigma)$ in the limit $C(\sigma) \ll (N+1)!$., Comment: Recommended for publication in Discrete Mathematics subject to revisions

We present a new approach to the calculation of measures in weighted networks, based on the translation of a weighted network into an ensemble of edges. This leads to a straightforward generalization of any measure defined on unweighted networks, such as the average degree of the nearest neighbours, the clustering coefficient, the `betweenness', the distance between two nodes and the diameter of a network. All these measures are well established for unweighted networks but have hitherto proven difficult to define for weighted networks. Further to introducing this approach we demonstrate its advantages by applying the clustering coefficient constructed in this way to two real-world weighted networks., Comment: 4 pages 3 figures