Uniform Approximation by Polynomials with Integer Coefficients via the Bernstein Lattice

Let $\mathscr{C}_\mathbb{Z}([0,1])$ be the metric space of real-valued continuous functions on $[0,1]$ with integer values at $0$ and $1$, equipped with the uniform (supremum) metric $d_\infty$. It is a classical theorem in approximation theory that the ring $\mathbb{Z}[X]$ of polynomials with integer coefficients, when considered as a set of functions on $[0,1]$, is dense in $\mathscr{C}_\mathbb{Z}([0,1])$. In this paper, we offer a strengthening of this result by identifying a substantially small subset $\bigcup_n \mathscr{B}_n$ of $\mathbb{Z}[X]$ which is still dense in $\mathscr{C}_\mathbb{Z}([0,1])$. Here $\mathscr{B}_n$, which we call the ``Bernstein lattice,'' is the lattice generated by the polynomials $$p_{n,k}(x) := \binom{n}{k} x^k(1-x)^{n-k}, ~~k=0,\dots,n.$$ Quantitatively, we show that for any $f \in \mathscr{C}_\mathbb{Z}([0,1])$, $$d_\infty(f, \mathscr{B}_n) \leq \frac{9}{4} \omega_f(n^{-1/3}) + 2 n^{-1/3}, ~~n \geq 1,$$ where $\omega_f$ stands for the modulus of continuity of $f$. We also offer a more general bound which can be optimized to yield better decay of approximation error for specific classes of continuous functions.Comment: 10 pages; presented at CANT 23 (Combinatorial and Additive Number Theory 2023

Phylogenetic analysis of modularity in protein interaction networks

<p>Abstract</p> <p>Background</p> <p>In systems biology, comparative analyses of molecular interactions across diverse species indicate that conservation and divergence of networks can be used to understand functional evolution from a systems perspective. A key characteristic of these networks is their modularity, which contributes significantly to their robustness, as well as adaptability. Consequently, analysis of modular network structures from a phylogenetic perspective may be useful in understanding the emergence, conservation, and diversification of functional modularity.</p> <p>Results</p> <p>In this paper, we propose a phylogenetic framework for analyzing network modules, with applications that extend well beyond network-based phylogeny reconstruction. Our approach is based on identification of modular network components from each network separately, followed by projection of these modules onto the networks of other species to compare different networks. Subsequently, we use the conservation of various modules in each network to assess the similarity between different networks. Compared to traditional methods that rely on topological comparisons, our approach has key advantages in (<it>i</it>) avoiding intractable graph comparison problems in comparative network analysis, (<it>ii</it>) accounting for noise and missing data through flexible treatment of network conservation, and (<it>iii</it>) providing insights on the evolution of biological systems through investigation of the evolutionary trajectories of network modules. We test our method, M<smcaps>OPHY</smcaps>, on synthetic data generated by simulation of network evolution, as well as existing protein-protein interaction data for seven diverse species. Comprehensive experimental results show that M<smcaps>OPHY</smcaps> is promising in reconstructing evolutionary histories of extant networks based on conservation of modularity, it is highly robust to noise, and outperforms existing methods that quantify network similarity in terms of conservation of network topology.</p> <p>Conclusion</p> <p>These results establish modularity and network proximity as useful features in comparative network analysis and motivate detailed studies of the evolutionary histories of network modules.</p

Mixed Neural Voxels for Fast Multi-view Video Synthesis

Synthesizing high-fidelity videos from real-world multi-view input is challenging because of the complexities of real-world environments and highly dynamic motions. Previous works based on neural radiance fields have demonstrated high-quality reconstructions of dynamic scenes. However, training such models on real-world scenes is time-consuming, usually taking days or weeks. In this paper, we present a novel method named MixVoxels to better represent the dynamic scenes with fast training speed and competitive rendering qualities. The proposed MixVoxels represents the 4D dynamic scenes as a mixture of static and dynamic voxels and processes them with different networks. In this way, the computation of the required modalities for static voxels can be processed by a lightweight model, which essentially reduces the amount of computation, especially for many daily dynamic scenes dominated by the static background. To separate the two kinds of voxels, we propose a novel variation field to estimate the temporal variance of each voxel. For the dynamic voxels, we design an inner-product time query method to efficiently query multiple time steps, which is essential to recover the high-dynamic motions. As a result, with 15 minutes of training for dynamic scenes with inputs of 300-frame videos, MixVoxels achieves better PSNR than previous methods. Codes and trained models are available at https://github.com/fengres/mixvoxelsComment: ICCV 2023 (Oral

User Feedback-Informed Interface Design for Flow Management Data and Services (FMDS)

The transition to a microservices-based Flow Management Data and Services (FMDS) architecture from the existing Traffic Flow Management System (TFMS) is a critical enabler of the vision for an Information-Centric National Airspace System (NAS). The need to design a user-centric interface for FMDS is a key technical gap, as this interface connects NAS data and services to the traffic management specialists within all stakeholder groups (e.g., FAA, airlines). We provide a research-driven approach towards designing such a graphical user interface (GUI) for FMDS. Major goals include unifying the more than 50 disparate traffic management services currently hosted on TFMS, as well as streamlining the process of evaluating, modeling, and monitoring Traffic Management Initiatives (TMIs). Motivated by this, we iteratively designed a GUI leveraging human factors engineering and user experience design principles, as well as user interviews. Through user testing and interviews, we identify workflow benefits of our GUI (e.g., reduction in task completion time), along with next steps for developing a live prototype.Comment: 8 pages, 8 figure