Abstract:
The biological pathways and functional properties by which misexpressed microRNAs (miRNAs) contribute to liver carcinogenesis have been intensively investigated. However, little is known about the upstream mechanisms that deregulate miRNA expressions in this process. In hepatocellular carcinoma (HCC), hepatitis B virus (HBV) X protein (HBx), a transcriptional trans-activator, is frequently expressed in truncated form without carboxyl-terminus but its role in miRNA expression and HCC development is unclear.

Abstract:
It has long been suggested that solutions to linear scalar wave equation $$\Box_g\phi=0$$ on a fixed subextremal Reissner-Nordstr\"om spacetime with non-vanishing charge are generically singular at the Cauchy horizon. We prove that generic smooth and compactly supported initial data on a Cauchy hypersurface indeed give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole. In particular, the solution generically does not belong to $W^{1,2}_{loc}$. This instability is related to the celebrated blue shift effect in the interior of the black hole. The problem is motivated by the strong cosmic censorship conjecture and it is expected that for the full nonlinear Einstein-Maxwell system, this instability leads to a singular Cauchy horizon for generic small perturbations of Reissner-Nordstr\"om spacetime. Moreover, in addition to the instability result, we also show as a consequence of the proof that Price's law decay is generically sharp along the event horizon.

Abstract:
In this paper, we study the future causally geodesically complete solutions of the spherically symmetric Einstein-scalar field system. Under the a priori assumption that the scalar field $\phi$ scatters locally in the scale-invariant bounded-variation (BV) norm, we prove that $\phi$ and its derivatives decay polynomially. Moreover, we show that the decay rates are sharp. In particular, we obtain sharp quantitative decay for the class of global solutions with small BV norms constructed by Christodoulou. As a consequence of our results, for every future causally geodesically complete solution with sufficiently regular initial data, we show the dichotomy that either the sharp power law tail holds or that the spacetime blows up at infinity in the sense that some scale invariant spacetime norms blow up.

Abstract:
A statistical distinguishability based on relative entropy characterises the fitness of quantum states for phase estimation. This criterion is employed in the context of a Mach-Zehnder interferometer and used to interpolate between two regimes, of local and global phase distinguishability. The scaling of distinguishability in these regimes with photon number is explored for various quantum states. It emerges that local distinguishability is dependent on a discrepancy between quantum and classical rotational energy. Our analysis demonstrates that the Heisenberg limit is the true upper limit for local phase sensitivity. Only the `NOON' states share this bound, but other states exhibit a better trade-off when comparing local and global phase regimes.

Abstract:
We investigate one-point reduction methods of finite topological spaces. These methods allow one to study homotopy theory of cell complexes by means of elementary moves of their finite models. We also introduce the notion of h-regular CW-complex, generalizing the concept of regular CW-complex, and prove that the h-regular CW-complexes, which are a sort of combinatorial-up-to-homotopy objects, are modeled (up to homotopy) by their associated finite spaces. This is accomplished by generalizing a classical result of McCord on simplicial complexes.

Abstract:
We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of using strong collapses is the existence and uniqueness of cores and their relationship with the nerves of the complexes. From this theory we derive new results for studying simplicial collapsibility with a different point of view. We analyze vertex-transitive simplicial $G$-actions and prove a particular case of the Evasiveness conjecture for simplicial complexes. Moreover, we reduce the general conjecture to the class of minimal complexes. We also strengthen a result of V. Welker on the barycentric subdivision of collapsible complexes. We obtain this and other results on collapsibility of polyhedra by means of the characterization of the different notions of collapses in terms of finite topological spaces.

Abstract:
We introduce the notion of a coloring of a poset, which consists of a labeling of the edges in its Hasse diagram by elements in a given group $G$. We use $G$-colorings to describe the covering maps of posets and present a new method based on colorings to obtain concrete and simple presentations of the fundamental group of polyhedra.

Abstract:
In this paper we study the 2-dimension of a finite poset from the topological point of view. We use homotopy theory of finite topological spaces and the concept of a beat point to improve the classical results on 2-dimension, giving a more complete answer to the problem of all possible 2-dimensions of an n-point poset.

Abstract:
We characterize the smallest finite spaces with the same homotopy groups of the spheres. Similarly, we describe the minimal finite models of any finite graph. We also develop new combinatorial techniques based on finite spaces to study classical invariants of general topological spaces.

Abstract:
We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse of finite spaces induces a simplicial collapse of their associated simplicial complexes. Moreover, a simplicial collapse induces a collapse of the associated finite spaces. This establishes a one-to-one correspondence between simple homotopy types of finite simplicial complexes and simple equivalence classes of finite spaces. We also prove a similar result for maps: We give a complete characterization of the class of maps between finite spaces which induce simple homotopy equivalences between the associated polyhedra. Furthermore, this class describes all maps coming from simple homotopy equivalences at the level of complexes. The advantage of this theory is that the elementary move of finite spaces is much simpler than the elementary move of simplicial complexes: It consists of removing (or adding) just a single point of the space.