Our papers are the official record of our discoveries. They allow others to build on and apply our work. Each paper is the result of many months of research, so we make a special effort to make them clear, beautiful and inspirational, and publish them in leading journals.

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Number theory

### Counting recursive divisors

Three new closed-form expressions give the number of recursive divisors and ordered factorisations, which were until now hard to compute.

Number theory

### Recursive divisor properties

The recursive divisor function has a simple Dirichlet series that relates it to the divisor function and other standard arithmetic functions.

Statistical physics

### Exponential Kauffman scaling

Surprisingly, the number of attractors in the critical Kauffman model with connectivity one grows exponentially with the size of the network.

Combinatorics

### In life, there are few rules

The bipartite nature of regulatory networks means gene-gene logics are composed, which severely restricts which ones can show up in life.

Statistical physics

### Landau meets Kauffman

A new, simple approach to the critical Kauffman model with connectivity one sharpens the bounds on the number and length of attractors.

Statistical physics

### Single-input Boolean networks

A new, simpler approach to the critical Kauffman model with connectivity one reveals that it has more attractors than previously believed.

Evolvability

### Flowers of immortality

The eigenvalues of the mortality equation fall into two classes—the flower and the stem—but only the stem eigenvalues control the dynamics.

Combinatorics

### Structure of genetic computation

The structural and functional building blocks of gene regulatory networks correspond, which tell us how genetic computation is organised.

Number theory

### Recursively divisible numbers

Recursively divisible numbers are a new kind of number that are highly divisible, whose quotients are highly divisible, and so on, recursively.

Evolvability

### I want to be forever young

The mortality equation governs the dynamics of an evolving population with a given maximum age, offering a theory for programmed ageing.

Combinatorics

### Biological logics are restricted

The fraction of logics that are biologically permitted can be bounded and shown to be tiny, which makes inferring them from experiments easier.

Number theory

### Ample and pristine numbers

Parallels between the perfect and abundant numbers and their recursive analogs point to deeper structure in the recursive divisor function.

Theory of innovation

### Taming complexity

Insights from biology, physics and business shed light on the nature and costs of complexity and how to manage it in business organizations.

Theory of innovation

### Recursive structure of innovation

A theoretical model of recursive innovation suggests that new technologies are recursively built up from new combinations of existing ones.

Geometry

### Geometry of discrete space

A phase transition creates the geometry of the continuum from discrete space, but it needs disorder if it is to have the right metric.

Theory of innovation

### The rate of innovation

The distribution of product complexity helps explain why some technology sectors tend to exhibit faster innovation rates than other sectors.

Theory of innovation

### The science of strategy

The usefulness of components and the complexity of products inform the best strategy for innovation at different stages of the process.

Theory of innovation

### The secret structure of innovation

Firms can harness the shifting importance of component building blocks to build better products and services and hence increase their chances of sustained success.

Theory of innovation

### Serendipity and strategy

In systems of innovation, the relative usefulness of different components changes as the number of components we possess increases.

Graph theory

### Eigenvalues of neutral networks

The principal eigenvalue of small neutral networks determines their robustness, and is bounded by the logarithm of the number of vertices.

Discrete dynamics

### Form and function in gene networks

The structural properties of a network motif predict its functional versatility and relate to gene regulatory networks.

Network theory

### Easily repairable networks

When networks come under attack, a repairable architecture is superior to, and globally distinct from, an architecture that is robust.

Network theory

### The temperature of networks

A new concept, graph temperature, enables the prediction of distinct topological properties of real-world networks simultaneously.

Network theory

### Scales in weighted networks

Information theory fixes weighted networks’ degeneracy issues with a generalisation of binary graphs and an optimal scale of link intensities.

Information theory

### Assessing self-assembly

The information needed to self-assemble a structure quantifies its modularity and explains the prevalence of certain structures over others.

Discrete dynamics

### Random cellular automata

Of the 256 elementary cellular automata, 28 of them exhibit random behavior over time, but spatio-temporal currents still lurk underneath.

Statistical physics

### Single elimination competition

In single elimination competition the best indicator of success is a player's wealth: the accumulated wealth of all defeated players.