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Consider an age-dependent, single-species branching process defined by a progeny number distribution, and a lifetime distribution associated with each independent particle. In this paper, we focus on the associated inverse problem where one wishes to formally solve for the progeny number distribution or the lifetime distribution that defines the Bellman-Harris branching process. We derive results for the existence and uniqueness (the identifiability) of these two distributions given one of two types of information: the extinction time probability of the entire process (extinction time distribution), or the distribution of the total number of particles at one fixed time. We demonstrate that perfect knowledge of the distribution of extinction times allows us to formally determine either the progeny number distribution or the lifetime distribution. Furthermore, we show that these constructions are unique. We then consider “data” consisting of a perfectly known total number distribution given at one specific time. For a process with known progeny number distribution and exponentially distributed lifetimes, we show that the rate parameter is identifiable. For general lifetime distributions, we also show that the progeny distribution is globally unique. Our results are presented through four theorems, each describing the constructions in the four distinct cases.

We apply quantum gravitational results to spatially unbounded Friedmann universes and try to answer some questions related to dark energy, dark matter, inflation and the missing antimatter.

We explore the fine structure of the holographic entanglement entropy proposal (the Ryu-Takayanagi formula) in AdS$_3$/CFT$_{2}$. With the guidance from the boundary and bulk modular flows we find a natural slicing of the entanglement wedge with the modular planes, which are co-dimension one bulk surfaces tangent to the modular flow everywhere. This gives an one-to-one correspondence between the points on the boundary interval $\mathcal{A}$ and the points on the Ryu-Takayanagi (RT) surface $\mathcal{E}_{\mathcal{A}}$. In the same sense an arbitrary subinterval $\mathcal{A}_2$ of $\mathcal{A}$ will correspond to a subinterval $\mathcal{E}_2$ of $\mathcal{E}_{\mathcal{A}}$. This fine correspondence indicates that the length of $\mathcal{E}_2$ captures the contribution $s_{\mathcal{A}}(\mathcal{A}_2)$ from $\mathcal{A}_2$ to the entanglement entropy $S_{\mathcal{A}}$, hence gives the contour function for entanglement entropy. Furthermore we propose that $s_{\mathcal{A}}(\mathcal{A}_2)$ in general can be written as a simple linear combination of entanglement entropies of single intervals inside $\mathcal{A}$. This proposal passes several non-trivial tests.

Symmetry-Protected Topological (SPT) phases are gapped phases of quantum matter protected by global symmetries that cannot be adiabatically deformed to a trivial phase without breaking symmetry. In this work, we show that, for several SPT phases that are short range entangled (SRE), enlarging symmetries may effectively achieve the consequences of explicitly breaking symmetries. In other words, we demonstrate that non-trivial SPT phases can be unwound to trivial ones by symmetry extension- through a path where the Hilbert space is enlarged and the Hamiltonian is invariant under an extended symmetry group applying the idea of Wang, Wen and Witten in arXiv:1705.06728. We show examples of both bosonic and fermionic SPT phases in 1+1 dimensions, including Haldane's bosonic spin chain and layers of Kitaev's fermionic Majorana chains. By adding degrees of freedom into the boundary/bulk, we can lift the zero mode degeneracy, or unwind the whole system. Furthermore, based on properties of Schur cover, we sketch a general picture of unwinding applicable to any 1+1 D bosonic SPT phase protected by on-site finite symmetry. Altogether we show that SRE states can be unwound by symmetry breaking, inversion and symmetry extension.