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.

We study differentiability properties of a potential of the type K⋆μ, where μ is a finite Radon measure in RN and the kernel K satisfies |∇jK(x)|≤C|x|−(N−1+j),j=0,1,2. We introduce a notion of differentiability in the capacity sense, where capacity is classical capacity in the de la Vallée Poussin sense associated with the kernel |x|−(N−1). We require that the first order remainder at a point is small when measured by means of a normalized weak capacity “norm” in balls of small radii centered at the point. This implies weak LN/(N−1) differentiability and thus Lp differentiability in the Calderón–Zygmund sense for 1≤p<N/(N−1). We show that K⋆μ is a.e. differentiable in the capacity sense, thus strengthening a recent result by Ambrosio, Ponce and Rodiac. We also present an alternative proof of a quantitative theorem of the authors just mentioned, giving pointwise Lipschitz estimates for K⋆μ. As an application, we study level sets of newtonian potentials of finite Radon measures.