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For convex univalent functions we give instances where the sharp bound for various coefficient functionals are identical to those for the corresponding bound for the inverse function. We give instances where the sharp bounds differ and also suggest some significant open problems., Comment: 9 pages

Relation Extraction is a way of obtaining the semantic relationship between entities in text. The state-of-the-art methods use linguistic tools to build a graph for the text in which the entities appear and then a Graph Convolutional Network (GCN) is employed to encode the pre-built graphs. Although their performance is promising, the reliance on linguistic tools results in a non end-to-end process. In this work, we propose a novel model, the Self-determined Graph Convolutional Network (SGCN), which determines a weighted graph using a self-attention mechanism, rather using any linguistic tool. Then, the self-determined graph is encoded using a GCN. We test our model on the TACRED dataset and achieve the state-of-the-art result. Our experiments show that SGCN outperforms the traditional GCN, which uses dependency parsing tools to build the graph., Comment: CIKM-2020

For $f\in \mathcal{S}$, the class of normalized functions, analytic and univalent in the unit disk $\mathbb{D}$ and given by $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$ for $z\in \mathbb{D}$, we give an upper bound for the coefficient difference $|a_4|-|a_3|$ when $f\in \mathcal{S}$. This provides an improved bound in the case $n=3$ of Grispan's 1976 general bound $||a_{n+1}|-|a_n||\le 3.61\dots .$ Other coefficients bounds, and bounds for the second and third Hankel determinants when $f\in \mathcal{S}$ are found when either $a_2=0,$ or $a_3=0$.

In this paper we develop an isogeometric B\'ezier dual mortar method for the biharmonic problem on multi-patch domains. The well-posedness of the discrete biharmonic problem requires a discretization with $C^1$ continuous basis functions. Hence, two Lagrange multipliers are required to apply both $C^0$ and $C^1$ continuity constraints on each intersection. The dual mortar method utilizes dual basis functions to discretize the Lagrange multiplier spaces. In order to preserve the sparsity of the coupled problem, we develop a dual mortar suitable $C^1$ constraint and utilize the B\'ezier dual basis to discretize the Lagrange multiplier spaces. The B\'ezier dual basis functions are constructed through B\'ezier projection and possess the same support size as the corresponding B-spline basis functions. We prove that this approach leads to a well-posed discrete problem and specify requirements to achieve optimal convergence. Although the B\'ezier dual basis is sub-optimal due to the lack of polynomial reproduction, our formulation successfully postpones the domination of the consistency error for practical problems. We verify the theoretical results and demonstrate the performance of the proposed formulation through several benchmark problems., Comment: 35 pages

In this paper we demonstrate the use of B\'{e}zier projection to alleviate locking phenomena in structural mechanics applications of isogeometric analysis. Interpreting the well-known $\bar{B}$ projection in two different ways we develop two formulations for locking problems in beams and nearly incompressible elastic solids. One formulation leads to a sparse symmetric symmetric system and the other leads to a sparse non-symmetric system. To demonstrate the utility of B\'{e}zier projection for both geometry and material locking phenomena we focus on transverse shear locking in Timoshenko beams and volumetric locking in nearly compressible linear elasticity although the approach can be applied generally to other types of locking phenemona as well. B\'{e}zier projection is a local projection technique with optimal approximation properties, which in many cases produces solutions that are comparable to global $L^2$ projection. In the context of $\bar{B}$ methods, the use of B\'ezier projection produces sparse stiffness matrices with only a slight increase in bandwidth when compared to standard displacement-based methods. Of particular importance is that the approach is applicable to any spline representation that can be written in B\'ezier form like NURBS, T-splines, LR-splines, etc. We discuss in detail how to integrate this approach into an existing finite element framework with minimal disruption through the use of B\'ezier extraction operators and a newly introduced dual basis for the B\'{e}zierprojection operator. We then demonstrate the behavior of the two proposed formulations through several challenging benchmark problems., Comment: 26 pages

In this paper, we introduce the hierarchical B-spline complex of discrete differential forms for arbitrary spatial dimension. This complex may be applied to the adaptive isogeometric solution of problems arising in electromagnetics and fluid mechanics. We derive a sufficient and necessary condition guaranteeing exactness of the hierarchical B-spline complex for arbitrary spatial dimension, and we derive a set of local, easy-to-compute, and sufficient exactness conditions for the two-dimensional setting. We examine the stability properties of the hierarchical B-spline complex, and we find that it yields stable approximations of both the Maxwell eigenproblem and Stokes problem provided that the local exactness conditions are satisfied. We conclude by providing numerical results showing the promise of the hierarchical B-spline complex in an adaptive isogeometric solution framework.

In this paper we present a method for direct evaluation of generalized B-splines (GB-splines) via the local representation of these curves as piecewise functions. To accomplish this we introduce a local structure that makes GB-spline curves more amenable to the techniques used in constructing bases of higher degree. This basis is used to perform direct computation of piecewise representation of GB-spline bases and curves.

It is shown that for $f$ analytic and convex in $z\in D=\{z:|z|<1\}$ and given by $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}$, the difference of coefficients $||a_{3}|-|a_{2}||\le 25/48$ and $||a_{4}|-|a_{3}||\le 25/48$ . Both inequalities are sharp., Comment: Withdrawn by the author, since the results were found not to be best possible

We introduce B\'{e}zier projection as an element-based local projection methodology for B-splines, NURBS, and T-splines. This new approach relies on the concept of B\'{e}zier extraction and an associated operation introduced here, spline reconstruction, enabling the use of B\'{e}zier projection in standard finite element codes. B\'{e}zier projection exhibits provably optimal convergence and yields projections that are virtually indistinguishable from global $L^2$ projection. B\'{e}zier projection is used to develop a unified framework for spline operations including cell subdivision and merging, degree elevation and reduction, basis roughening and smoothing, and spline reparameterization. In fact, B\'{e}zier projection provides a \emph{quadrature-free} approach to refinement and coarsening of splines. In this sense, B\'{e}zier projection provides the fundamental building block for $hpkr$-adaptivity in isogeometric analysis., Comment: 56 pages, 28 figures

In this paper hierarchical analysis-suitable T-splines (HASTS) are developed. The resulting spaces are a superset of both analysis-suitable T-splines and hierarchical B-splines. The additional flexibility provided by the hierarchy of T-spline spaces results in simple, highly localized refinement algorithms which can be utilized in a design or analysis context. A detailed theoretical formulation is presented including a proof of local linear independence for analysis-suitable T-splines, a requisite theoretical ingredient for HASTS. B\'{e}zier extraction is extended to HASTS simplifying the implementation of HASTS in existing finite element codes. The behavior of a simple HASTS refinement algorithm is compared to the local refinement algorithm for analysis-suitable T-splines demonstrating the superior efficiency and locality of the HASTS algorithm. Finally, HASTS are utilized as a basis for adaptive isogeometric analysis.

Various models for interacting spherical bubbles in a compressible liquid based on delay differential equations are considered. It is shown that most previously proposed models for interacting spherical bubbles in a compressible liquid based on the Keller-Miksis and Gilmore-Akulichev models are unstable for closely spaced bubbles. A new model for a single spherical bubble in a compressible liquid is proposed and used to derive a stable model for interacting bubbles. A qualitative comparison to the results of direct numerical integration of the fluid equations of motion suggests that the new model provides more accurate results than the standard Keller-Miksis or Gilmore-Akulichev models for single bubble dynamics., Comment: 13 pages, 4 figures