Quantitatively identifying direct dependencies between variables is an important task in data analysis, in particular for reconstructing various types of networks and causal relations in science and engineering. One of the most widely used criteria is partial correlation, but it can only measure linearly direct association and miss nonlinear associations. However, based on conditional independence, conditional mutual information (CMI) is able to quantify nonlinearly direct relationships among variables from the observed data, superior to linear measures, but suffers from a serious problem of underestimation, in particular for those variables with tight associations in a network, which severely limits its applications. In this work, we propose a new concept, “partial independence,” with a new measure, “part mutual information” (PMI), which not only can overcome the problem of CMI but also retains the quantification properties of both mutual information (MI) and CMI. Specifically, we first defined PMI to measure nonlinearly direct dependencies between variables and then derived its relations with MI and CMI. Finally, we used a number of simulated data as benchmark examples to numerically demonstrate PMI features and further real gene expression data from Escherichia coli and yeast to reconstruct gene regulatory networks, which all validated the advantages of PMI for accurately quantifying nonlinearly direct associations in networks.

Recent understandings of molecular evolution, together with the fossil records, have established that there are both linear and nonlinear processes in the creation of novel species, which is strikingly similar to the generation of prime numbers and human creativity. Each creation of a more complex species is like a prime number, unpredictable, discontinuous, and yet can be modeled by a smooth curve in relation to time. The mystery behind the complexity increases in nature and human civilizations might well turn out to be similar to that behind the appearances of prime numbers. Here we show that an algorithm for the creative process of humans can create prime numbers in a lawful and yet unpredictable fashion. The essence of primes is the duality of uniqueness and uniformity together with the creation algorithm. The algorithm consists of the non-linear process of uniformity selection to create the unique and the linear process of uniqueness selection to form the uniformity. The iterations of this algorithm can create an infinite number of primes. The algorithm appears to have been hardwired in the human brain as shown by recent experimental studies. This new understanding can deduce some of the best-known properties of primes and may explain the nearly constant and yet seemingly random creation of novelty in relation to time.

The DK Flip Conjecture of Bondal-Orlov and Kawamata states that there should be an embedding of derived categories for any flip, which is known to be true for toroidal flips. In this paper, we construct new examples of Grassmannian flips that satisfy the DK Flip Conjecture.

In a recent series of articles, the authors have studied the transition behavior of partial Bergman kernels $\Pi_{k, [E_1, E_2]} (z,w)$ and the associated DOS (density of states) $\Pi_{k, [E_1, E_2]} (z)$ across the interface $\mathcal{C}$ between the allowed and forbidden regions. Partial Bergman kernels are Toeplitz Hamiltonians quantizing Morse functions $H : M \to \mathbb{R}$ on a Kähler manifold. The allowed region is $H^{-1} ([E_1, E_2])$ and the interface $\mathcal{C}$ is its boundary. In prior articles it was assumed that the endpoints $E_j$ were regular values of $H$. This article completes the series by giving parallel results when an endpoint is a critical value of $H$. In place of the Erf scaling asymptotics in a $k^{-\frac{1}{2}}$ tube around $\mathcal{C}$ for regular interfaces, one obtains $\delta$-asymptotics in $k^{-\frac{1}{4}}$ tubes around singular points of a critical interface. In $k^{-\frac{1}{2}}$ tubes, the transition law is given by the osculating metaplectic propagator.

We give a partial converse to [8, Theorem 1.3] (as a resolution of [2, Problem 8.4] for the quasiconformal Q-composition) for Q0<α<2−1(Rn≥2), and yet demonstrate that if f:R2→R2 is a homeomorphism then the boundedness of u↦u∘f on Q2−1<α<1(R2)⊂BMO(R2) yields the quasiconformality of f.