The hallmark of a major evolutionary transition, whereby independently replicating units became grouped together into larger wholes, is the “decoupling” of fitness of the collective from the individual fitnesses of the units. It is widely believed that the key element making such decoupling possible is extensive cooperation between the lower-level entities. Here, it is demonstrated that cohesive multicellular behavior can arise in a purely competitive setting as a generic consequence of division of labor. A minimal theoretical model of competitive coexistence on multiple resources provides an explicit manifestation of fitness decoupling in a rigorous mathematical sense. This form of multicellular behavior is not vulnerable to “cheaters” and can be expected to be widespread in microbial ecosystems.

We consider a two-dimensional particle of charge e interacting with a homogeneous magnetic eld perpendicular to the plane and a potential well which is transported along a closed loop in the plane. We show that a bound state corresponding to a simple isolated eigenvalue acquires at that Berry's phase equal to 2 sgne times the number of ux quanta through the oriented area encircled by the loop. We also argue that this is a purely quantum e ect since the corresponding Hannay angle is zero.

As> iVinf^ b^ where dx denotes the degree of the vertex x. In particular, we derive lower bounds of eigenvalues for convex subgraphs which consist of lattice points in an d-dimensional Riemannian manifolds M with convex boundary. The techniques involve both the (discrete) heat kernels of graphs and improved estimates of the (continuous) heat kernels of Riemannian manifolds. We prove eigenvalue lower bounds for convex subgraphs of the form ce2/(dD (M)) 2 where e denotes the distance between two closest lattice points, D (M) denotes the diameter of the manifold M and c is a constant (independent of the dimension d and the number of vertices in 5, but depending on the how" dense" the lattice points are). This eigenvalue bound is useful for bounding the rates of convergence for various random walk problems. Since many enumeration problems can be approximated by considering random walks in convex subgraphs of some appropriate host graph, the eigenvalue inequalities here have many applications.

We provide a non conformal generalization of the Compère-Song-Strominger (CSS) boundary conditions for AdS3 gravity that breaks the \hat{u}(1) Kac-Moody-Virasoro symmetry to two u(1)s. The holographic dual specified by the new boundary conditions can be understood as an irrelevant deformation of a warped conformal field theory (WCFT). Upon consistent reduction to two dimensions, AdS3 gravity results in a deformed Jackiw-Teitelboim dilaton gravity model coupled to a Maxwell field. We show that near extremality the boundary conditions inherited from generalized CSS boundary conditions in three dimensions give rise to an effective action exhibiting the same symmetry breaking pattern as the complex Sachdev-Ye-Kitaev models. Besides the Schwarzian term reflecting the breaking of conformal symmetry, the effective action contains an additional term that captures the breaking of the \hat{u}(1) Kac-Moody symmetry.

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