On unimodality problems in Pascal's triangle

Many sequences of binomial coefficients share various unimodality properties. In this paper we consider the unimodality problem of a sequence of binomial coefficients located in a ray or a transversal of the Pascal triangle. Our results give in particular an affirmative answer to a conjecture of Belbachir et al which asserts that such a sequence of binomial coefficients must be unimodal. We also propose two more general conjectures.Comment: 12 pages, 2 figure

Towards Real-World Test-Time Adaptation: Tri-Net Self-Training with Balanced Normalization

Test-Time Adaptation aims to adapt source domain model to testing data at inference stage with success demonstrated in adapting to unseen corruptions. However, these attempts may fail under more challenging real-world scenarios. Existing works mainly consider real-world test-time adaptation under non-i.i.d. data stream and continual domain shift. In this work, we first complement the existing real-world TTA protocol with a globally class imbalanced testing set. We demonstrate that combining all settings together poses new challenges to existing methods. We argue the failure of state-of-the-art methods is first caused by indiscriminately adapting normalization layers to imbalanced testing data. To remedy this shortcoming, we propose a balanced batchnorm layer to swap out the regular batchnorm at inference stage. The new batchnorm layer is capable of adapting without biasing towards majority classes. We are further inspired by the success of self-training~(ST) in learning from unlabeled data and adapt ST for test-time adaptation. However, ST alone is prone to over adaption which is responsible for the poor performance under continual domain shift. Hence, we propose to improve self-training under continual domain shift by regularizing model updates with an anchored loss. The final TTA model, termed as TRIBE, is built upon a tri-net architecture with balanced batchnorm layers. We evaluate TRIBE on four datasets representing real-world TTA settings. TRIBE consistently achieves the state-of-the-art performance across multiple evaluation protocols. The code is available at \url{https://github.com/Gorilla-Lab-SCUT/TRIBE}.Comment: 23 pages, 7 figures and 22 table

Proof of a conjecture of Lundow and Rosengren on the bimodality of p,q-binomial coefficients

AbstractLundow and Rosengren observed that the magnetization distributions of Ising model are remarkably well-fitted by the p,q-binomial distributions based on an assumption that the p,q-binomial distribution is unimodal or bimodal. In the present paper we show that this assumption holds for all positive numbers p and q

NMR analysis of the dynamic exchange of the NS2B cofactor between open and closed conformations of the West Nile Virus NS2B-NS3 protease

BACKGROUND The two-component NS2B-NS3 proteases of West Nile and dengue viruses are essential for viral replication and established targets for drug development. In all crystal structures of the proteases to date, the NS2B cofactor is located far from the substrate binding site (open conformation) in the absence of inhibitor and lining the substrate binding site (closed conformation) in the presence of an inhibitor. METHODS In this work, nuclear magnetic resonance (NMR) spectroscopy of isotope and spin-labeled samples of the West Nile virus protease was used to investigate the occurrence of equilibria between open and closed conformations in solution. FINDINGS In solution, the closed form of the West Nile virus protease is the predominant conformation irrespective of the presence or absence of inhibitors. Nonetheless, dissociation of the C-terminal part of the NS2B cofactor from the NS3 protease (open conformation) occurs in both the presence and the absence of inhibitors. Low-molecular-weight inhibitors can shift the conformational exchange equilibria so that over 90% of the West Nile virus protease molecules assume the closed conformation. The West Nile virus protease differs from the dengue virus protease, where the open conformation is the predominant form in the absence of inhibitors. CONCLUSION Partial dissociation of NS2B from NS3 has implications for the way in which the NS3 protease can be positioned with respect to the host cell membrane when NS2B is membrane associated via N- and C-terminal segments present in the polyprotein. In the case of the West Nile virus protease, discovery of low-molecular-weight inhibitors that act by breaking the association of the NS2B cofactor with the NS3 protease is impeded by the natural affinity of the cofactor to the NS3 protease. The same strategy can be more successful in the case of the dengue virus NS2B-NS3 protease.The project was funded by the Australian Research Council (http://www.arc.gov.au), grant DP0877540

Anchor Space Optimal Transport: Accelerating Batch Processing of Multiple OT Problems

The optimal transport (OT) theory provides an effective way to compare probability distributions on a defined metric space, but it suffers from cubic computational complexity. Although the Sinkhorn's algorithm greatly reduces the computational complexity of OT solutions, the solutions of multiple OT problems are still time-consuming and memory-comsuming in practice. However, many works on the computational acceleration of OT are usually based on the premise of a single OT problem, ignoring the potential common characteristics of the distributions in a mini-batch. Therefore, we propose a translated OT problem designated as the anchor space optimal transport (ASOT) problem, which is specially designed for batch processing of multiple OT problem solutions. For the proposed ASOT problem, the distributions will be mapped into a shared anchor point space, which learns the potential common characteristics and thus help accelerate OT batch processing. Based on the proposed ASOT, the Wasserstein distance error to the original OT problem is proven to be bounded by ground cost errors. Building upon this, we propose three methods to learn an anchor space minimizing the distance error, each of which has its application background. Numerical experiments on real-world datasets show that our proposed methods can greatly reduce computational time while maintaining reasonable approximation performance.Comment: 26 pages, 4 figures, 6 table

Secondary sulfur metabolism in cellular signalling and oxidative stress responses

The sulfur metabolism pathway in plants produces a variety of compounds that are central to the acclimation response to oxidative stresses such as drought and high light. Primary sulfur assimilation provides the amino acid cysteine, which is utilized in protein synthesis and as a precursor for the cellular redox buffer glutathione. In contrast, the secondary sulfur metabolism pathway produces sulfated compounds such as glucosinolates and sulfated peptides, as well as a corresponding by-product 3'-phosphoadenosine 5'-phosphate (PAP). Emerging evidence over the past decade has shown that secondary sulfur metabolism also has a crucial engagement during oxidative stress. This occurs across various cellular, tissue and organismal levels including chloroplast-to-nucleus retrograde signalling events mediated by PAP, modulation of hormonal signalling by sulfated compounds and PAP, control of physiological responses such as stomatal closure, and potential regulation of plant growth. In this review, we examine the contribution of the different components of plant secondary metabolism to oxidative stress homeostasis, and how this pathway is metabolically regulated. We further outline the key outstanding questions in the field that are necessary to understand how and why this 'specialized' metabolic pathway plays significant roles in plant oxidative stress tolerance

Two-Dimensional Controlled Syntheses of Polypeptide Molecular Brushes via N-Carboxyanhydride Ring-Opening Polymerization and Ring-Opening Metathesis Polymerization.

Well-defined molecular brushes bearing polypeptides as side chains were prepared by a "grafting through" synthetic strategy with two-dimensional control over the brush molecular architectures. By integrating N-carboxyanhydride ring-opening polymerizations (NCA ROPs) and ring-opening metathesis polymerizations (ROMPs), desirable segment lengths of polypeptide side chains and polynorbornene brush backbones were independently constructed in controlled manners. The N2 flow accelerated NCA ROP was utilized to prepare polypeptide macromonomers with different lengths initiated from a norbornene-based primary amine, and those macromonomers were then polymerized via ROMP. It was found that a mixture of dichloromethane and an ionic liquid were required as the solvent system to allow for construction of molecular brush polymers having densely-grafted peptide chains emanating from a polynorbornene backbone, poly(norbornene-graft-poly(β-benzyl-l-aspartate)) (P(NB-g-PBLA)). Highly efficient postpolymerization modification was achieved by aminolysis of PBLA side chains for facile installment of functional moieties onto the molecular brushes