On singular value distribution of large dimensional auto-covariance matrices

Let $(\varepsilon_j)_{j\geq 0}$ be a sequence of independent $p-$dimensional random vectors and $\tau\geq1$ a given integer. From a sample $\varepsilon_1,\cdots,\varepsilon_{T+\tau-1},\varepsilon_{T+\tau}$ of the sequence, the so-called lag $-\tau$ auto-covariance matrix is $C_{\tau}=T^{-1}\sum_{j=1}^T\varepsilon_{\tau+j}\varepsilon_{j}^t$. When the dimension $p$ is large compared to the sample size $T$, this paper establishes the limit of the singular value distribution of $C_\tau$ assuming that $p$ and $T$ grow to infinity proportionally and the sequence satisfies a Lindeberg condition on fourth order moments. Compared to existing asymptotic results on sample covariance matrices developed in random matrix theory, the case of an auto-covariance matrix is much more involved due to the fact that the summands are dependent and the matrix $C_\tau$ is not symmetric. Several new techniques are introduced for the derivation of the main theorem

Local Radial Basis Function Methods for Solving Partial Differential Equations

Meshless methods are relatively new numerical methods which have gained popularity in computational and engineering sciences during the last two decades. This dissertation develops two new localized meshless methods for solving a variety partial differential equations. Recently, some localized meshless methods have been introduced in order to handle large-scale problems, or to avoid ill-conditioned problems involving global radial basis function approximations. This dissertation explains two new localized meshelss methods, each derived from the global Method of Approximate Particular Solutions (MAPS). One method, the Localized Method of Approximate Particular Solutions (LMAPS), is used for elliptic and parabolic partial differential equations (PDEs) using a global sparse linear system of equations. The second method, the Explicit Localized Method of Approximate Particular Solutions (ELMAPS), is constructed for solving parabolic types of partial differential equations by inverting a finite number of small linear systems. For both methods, the only information that is needed in constructing the approximating solution to PDEs, consists of the local nodes that fall within the domain of influence of the data. Since the methods are completely mesh free, they can be used for irregularly shaped domains. Both methods are tested and compared with existing global and local meshless methods. The results illustrate the accuracy and efficiency of our proposed methods

Semantic-Aware Local-Global Vision Transformer

Vision Transformers have achieved remarkable progresses, among which Swin Transformer has demonstrated the tremendous potential of Transformer for vision tasks. It surmounts the key challenge of high computational complexity by performing local self-attention within shifted windows. In this work we propose the Semantic-Aware Local-Global Vision Transformer (SALG), to further investigate two potential improvements towards Swin Transformer. First, unlike Swin Transformer that performs uniform partition to produce equal size of regular windows for local self-attention, our SALG performs semantic segmentation in an unsupervised way to explore the underlying semantic priors in the image. As a result, each segmented region can correspond to a semantically meaningful part in the image, potentially leading to more effective features within each of segmented regions. Second, instead of only performing local self-attention within local windows as Swin Transformer does, the proposed SALG performs both 1) local intra-region self-attention for learning fine-grained features within each region and 2) global inter-region feature propagation for modeling global dependencies among all regions. Consequently, our model is able to obtain the global view when learning features for each token, which is the essential advantage of Transformer. Owing to the explicit modeling of the semantic priors and the proposed local-global modeling mechanism, our SALG is particularly advantageous for small-scale models when the modeling capacity is not sufficient for other models to learn semantics implicitly. Extensive experiments across various vision tasks demonstrates the merit of our model over other vision Transformers, especially in the small-scale modeling scenarios

Decoupling Recognition from Detection: Single Shot Self-Reliant Scene Text Spotter

Typical text spotters follow the two-stage spotting strategy: detect the precise boundary for a text instance first and then perform text recognition within the located text region. While such strategy has achieved substantial progress, there are two underlying limitations. 1) The performance of text recognition depends heavily on the precision of text detection, resulting in the potential error propagation from detection to recognition. 2) The RoI cropping which bridges the detection and recognition brings noise from background and leads to information loss when pooling or interpolating from feature maps. In this work we propose the single shot Self-Reliant Scene Text Spotter (SRSTS), which circumvents these limitations by decoupling recognition from detection. Specifically, we conduct text detection and recognition in parallel and bridge them by the shared positive anchor point. Consequently, our method is able to recognize the text instances correctly even though the precise text boundaries are challenging to detect. Additionally, our method reduces the annotation cost for text detection substantially. Extensive experiments on regular-shaped benchmark and arbitrary-shaped benchmark demonstrate that our SRSTS compares favorably to previous state-of-the-art spotters in terms of both accuracy and efficiency.Comment: To be appeared in the Proceedings of the ACM International Conference on Multimedia (ACM MM), 202

A reproducing kernel method for solving singularly perturbed delay parabolic partial differential equations

In this article, we put forward an efficient method on the foundation of a few reproducing kernel spaces(RK-spaces) and the collocation method to seek the solution of delay parabolic partial differential equations(PDEs) with singular perturbation. The approximated solution  to the equations is formulated and proved the exact solution is uniformly convergent by the solution. Furthermore, the partial differentiation of the approximated solution is also proved the partial derivatives of the exact solution is uniformly convergent by the solution. Meanwhile, we show that the accuracy of our method is in the order of T/n where T is the final time and n is the number of spatial (and time) discretization in the domain of interests. Three numerical examples are put forward to demonstrate the effectiveness of our presented scheme

Neural, electrophysiological and anatomical basis of brain-network variability and its characteristic changes in mental disorders

Functional brain networks demonstrate significant temporal variability and dynamic reconfiguration even in the resting state. Currently, most studies investigate temporal variability of brain networks at the scale of single (micro) or whole-brain (macro) connectivity. However, the mechanism underlying time-varying properties remains unclear, as the coupling between brain network variability and neural activity is not readily apparent when analysed at either micro or macroscales. We propose an intermediate 15 (meso) scale analysis and characterize temporal variability of the functional architecture associated with a particular region. This yields a topography of variability that reflects the whole-brain and, most importantly, creates an analytical framework to establish the fundamental relationship between variability of regional functional architecture and its neural activity or structural connectivity. We find that temporal variability reflects the dynamical reconfiguration of a brain region into distinct functional modules at different times and may be indicative of brain flexibility and adaptability. Primary and unimodal sensory-motor cortices demon- 20 strate low temporal variability, while transmodal areas, including heteromodal association areas and limbic system, demonstrate the high variability. In particular, regions with highest variability such as hippocampus/parahippocampus, inferior and middle temporal gyrus, olfactory gyrus and caudate are all related to learning, suggesting that the temporal variability may indicate the level of brain adaptability. With simultaneously recorded electroencephalography/functional magnetic resonance imaging and functional magnetic resonance imaging/diffusion tensor imaging data, we also find that variability of regional functional architec- 25 ture is modulated by local blood oxygen level-dependent activity and a-band oscillation, and is governed by the ratio of intra- to inter-community structural connectivity. Application of the mesoscale variability measure to multicentre datasets of three mental disorders and matched controls involving 1180 subjects reveals that those regions demonstrating extreme, i.e. highest/lowest variability in controls are most liable to change in mental disorders. Specifically, we draw attention to the identification of diametrically opposing patterns of variability changes between schizophrenia and attention deficit hyperactivity disorder/autism. 30 Regions of the default-mode network demonstrate lower variability in patients with schizophrenia, but high variability in patients with autism/attention deficit hyperactivity disorder, compared with respective controls. In contrast, subcortical regions, especially the thalamus, show higher variability in schizophrenia patients, but lower variability in patients with attention deficit hyperactivity disorder. The changes in variability of these regions are also closely related to symptom scores. Our work provides insights into the dynamic organization of the resting brain and how it changes in brain disorders. The nodal variability measure may also be 35 potentially useful as a predictor for learning and neural rehabilitation

Atomically Sharp Internal Interface in a Chiral Weyl Semimetal Nanowire

Internal interfaces in Weyl semimetals (WSMs) are predicted to host distinct topological features that are different from the commonly studied external interfaces (crystal-to-vacuum boundaries). However, the lack of atomically sharp and crystallographically oriented internal interfaces in WSMs makes it difficult to experimentally investigate hidden topological states buried inside the material. Here, we study a unique internal interface known as merohedral twin boundary in chemically synthesized single-crystal nanowires (NWs) of CoSi, a chiral WSM of space group P213 (No. 198). High resolution scanning transmission electron microscopy reveals that this internal interface is (001) twin plane and connects two enantiomeric counterparts at an atomically sharp interface with inversion twinning. Ab-initio calculations show localized internal Fermi arcs at the (001) twin boundary that can be clearly distinguished from both external Fermi arcs and bulk states. These merohedrally twinned CoSi NWs provide an ideal material system to probe unexplored topological properties associated with internal interfaces in WSMs.Comment: 19 pages, 4 figure

Surface-Confined Two-Dimensional Crystal Growth on a Monolayer

Conventional vapor deposition or epitaxial growth of two-dimensional (2D) materials and heterostructures is conducted in a large chamber in which masses transport from the source to the substrate. Here we report a chamber-free, on-chip approach for growing a 2D crystalline structures directly in a nanoscale surface-confined 2D space. The method is based on a surprising discovery of a rapid, long-distance, non-Fickian transport of a uniform layer of atomically thin palladium (Pd) on a monolayer crystal of tungsten ditelluride (WTe2), at temperatures well below the known melting points of all materials involved. The resulting nanoconfined growth realizes a controlled formation of a stable new 2D crystalline material, Pd7WTe2 , when the monolayer seed is either free-standing or fully encapsulated in a van der Waals stack. The approach is generalizable and highly compatible with nanodevice fabrication, promising to expand the library of 2D materials and their functionalities