Strong law of large numbers for supercritical superprocesses under second moment condition

Suppose that $X=\{X_t, t\ge 0\}$ is a supercritical superprocess on a locally compact separable metric space $(E, m)$. Suppose that the spatial motion of $X$ is a Hunt process satisfying certain conditions and that the branching mechanism is of the form $$\psi(x,\lambda)=-a(x)\lambda+b(x)\lambda^2+\int_{(0,+\infty)}(e^{-\lambda y}-1+\lambda y)n(x,dy), \quad x\in E, \quad\lambda> 0,$$ where $a\in \mathcal{B}_b(E)$, $b\in \mathcal{B}_b^+(E)$ and $n$ is a kernel from $E$ to $(0,\infty)$ satisfying $$\sup_{x\in E}\int_0^\infty y^2 n(x,dy)<\infty.$$ Put $T_tf(x)=\mathbb{P}_{\delta_x}$. Let $\lambda_0>0$ be the largest eigenvalue of the generator $L$ of $T_t$, and $\phi_0$ and $\hat{\phi}_0$ be the eigenfunctions of $L$ and $\hat{L}$ (the dural of $L$) respectively associated with $\lambda_0$. Under some conditions on the spatial motion and the $\phi_0$-transformed semigroup of $T_t$, we prove that for a large class of suitable functions $f$, we have $$\lim_{t\rightarrow\infty}e^{-\lambda_0 t}< f, X_t> = W_\infty\int_E\hat{\phi}_0(y)f(y)m(dy),\quad \mathbb{P}_{\mu}{-a.s.},$$ for any finite initial measure $\mu$ on $E$ with compact support, where $W_\infty$ is the martingale limit defined by $W_\infty:=\lim_{t\to\infty}e^{-\lambda_0t}$. Moreover, the exceptional set in the above limit does not depend on the initial measure $\mu$ and the function $f$

Localization and Mobility Gap in Topological Anderson Insulator

It has been proposed that disorder may lead to a new type of topological insulator, called topological Anderson insulator (TAI). Here we examine the physical origin of this phenomenon. We calculate the topological invariants and density of states of disordered model in a super-cell of 2-dimensional HgTe/CdTe quantum well. The topologically non-trivial phase is triggered by a band touching as the disorder strength increases. The TAI is protected by a mobility gap, in contrast to the band gap in conventional quantum spin Hall systems. The mobility gap in the TAI consists of a cluster of non-trivial subgaps separated by almost flat and localized bands.Comment: 8 pages, 7 figure

Prediction of aptamer-protein interacting pairs using an ensemble classifier in combination with various protein sequence attributes

The ranked feature list given by the Relief algorithm. Within the list, a feature with a smaller index indicates that it is more important for aptamer-protein interacting pair prediction. Such a list of ranked features are used to establish the optimal feature set in the IFS procedure. (XLS 56.5 kb

Continuum field theory of 3D topological orders with emergent fermions and braiding statistics

Universal topological data of topologically ordered phases can be captured by topological quantum field theory in continuous space time by taking the limit of low energies and long wavelengths. While previous continuum field-theoretical studies of topological orders in $3$D real space focus on either self-statistics, braiding statistics, shrinking rules, fusion rules or quantum dimensions, it is yet to systematically put all topological data together in a unified continuum field-theoretical framework. Here, we construct the topological $BF$ field theory with twisted terms (e.g., $AAdA$ and $AAB$) as well as a $K$-matrix $BB$ term, in order to simultaneously explore all such topological data and reach anomaly-free topological orders. Following the spirit of the famous $K$-matrix Chern-Simons theory of $2$D topological orders, we present general formulas and systematically show how the $K$-matrix $BB$ term confines topological excitations, and how self-statistics of particles is transmuted between bosonic one and fermionic one. In order to reach anomaly-free topological orders, we explore, within the present continuum field-theoretical framework, how the principle of gauge invariance fundamentally influences possible realizations of topological data. More concretely, we present the topological actions of (i) particle-loop braidings with emergent fermions, (ii) multiloop braidings with emergent fermions, and (iii) Borromean-Rings braidings with emergent fermions, and calculate their universal topological data. Together with the previous efforts, our work paves the way toward a more systematic and complete continuum field-theoretical analysis of exotic topological properties of $3$D topological orders. Several interesting future directions are also discussed

Non-Abelian Fusion, Shrinking and Quantum Dimensions of Abelian Gauge Fluxes

Braiding and fusion rules of topological excitations are indispensable topological invariants in topological quantum computation and topological orders. While excitations in 2D are always particle-like anyons, those in 3D incorporate not only particles but also loops -- spatially nonlocal objects -- making it novel and challenging to study topological invariants in higher dimensions. While 2D fusion rules have been well understood from bulk Chern-Simons field theory and edge conformal field theory, it is yet to be thoroughly explored for 3D fusion rules from higher dimensional bulk topological field theory. Here, we perform a field-theoretical study on (i) how loops that carry Abelian gauge fluxes fuse and (ii) how loops are shrunk into particles in the path integral, which generates fusion rules, loop-shrinking rules, and descendent invariants, e.g., quantum dimensions. We first assign a gauge-invariant Wilson operator to each excitation and determine the number of distinct excitations through equivalence classes of Wilson operators. Then, we adiabatically shift two Wilson operators together to observe how they fuse and are split in the path integral; despite the Abelian nature of the gauge fluxes carried by loops, their fusions may be of non-Abelian nature. Meanwhile, we adiabatically deform world-sheets of unknotted loops into world-lines and examine the shrinking outcomes; we find that the resulting loop-shrinking rules are algebraically consistent to fusion rules. Interestingly, fusing a pair of loop and anti-loop may generate multiple vacua, but fusing a pair of anyon and anti-anyon in 2D has one vacuum only. By establishing a field-theoretical ground for fusion and shrinking in 3D, this work leaves intriguing directions, e.g., symmetry enrichment, quantum gates, and physics of braided monoidal 2-category of 2-group.Comment: Title adjusted. Abstract, Intro and Discussions revised. about 30 pages, 5 figures. 9 table

IDET: Iterative Difference-Enhanced Transformers for High-Quality Change Detection

Change detection (CD) aims to detect change regions within an image pair captured at different times, playing a significant role for diverse real-world applications. Nevertheless, most of existing works focus on designing advanced network architectures to map the feature difference to the final change map while ignoring the influence of the quality of the feature difference. In this paper, we study the CD from a new perspective, i.e., how to optimize the feature difference to highlight changes and suppress unchanged regions, and propose a novel module denoted as iterative difference-enhanced transformers (IDET). IDET contains three transformers: two transformers for extracting the long-range information of the two images and one transformer for enhancing the feature difference. In contrast to the previous transformers, the third transformer takes the outputs of the first two transformers to guide the enhancement of the feature difference iteratively. To achieve more effective refinement, we further propose the multi-scale IDET-based change detection that uses multi-scale representations of the images for multiple feature difference refinements and proposes a coarse-to-fine fusion strategy to combine all refinements. Our final CD method outperforms seven state-of-the-art methods on six large-scale datasets under diverse application scenarios, which demonstrates the importance of feature difference enhancements and the effectiveness of IDET.Comment: conferenc