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Background: Ocular changes are traditionally associated with only a few hepatobiliary diseases. These changes are non-specific and have a low detection rate, limiting their potential use as clinically independent diagnostic features. Therefore, we aimed to engineer deep learning models to establish associations between ocular features and major hepatobiliary diseases and to advance automated screening and identification of hepatobiliary diseases from ocular images., Methods: We did a multicentre, prospective study to develop models using slit-lamp or retinal fundus images from participants in three hepatobiliary departments and two medical examination centres. Included participants were older than 18 years and had complete clinical information; participants diagnosed with acute hepatobiliary diseases were excluded. We trained seven slit-lamp models and seven fundus models (with or without hepatobiliary disease [screening model] or one specific disease type within six categories [identifying model]) using a development dataset, and we tested the models with an external test dataset. Additionally, we did a visual explanation and occlusion test. Model performances were evaluated using the area under the receiver operating characteristic curve (AUROC), sensitivity, specificity, and F1* score., Findings: Between Dec 16, 2018, and July 31, 2019, we collected data from 1252 participants (from the Department of Hepatobiliary Surgery of the Third Affiliated Hospital of Sun Yat-sen University, the Department of Infectious Diseases of the Affiliated Huadu Hospital of Southern Medical University, and the Nantian Medical Centre of Aikang Health Care [Guangzhou, China]) for the development dataset; between Aug 14, 2019, and Jan 31, 2020, we collected data from 537 participants (from the Department of Infectious Diseases of the Third Affiliated Hospital of Sun Yat-sen University and the Huanshidong Medical Centre of Aikang Health Care [Guangzhou, China]) for the test dataset. The AUROC for screening for hepatobiliary diseases of the slit-lamp model was 0·74 (95% CI 0·71-0·76), whereas that of the fundus model was 0·68 (0·65-0·71). For the identification of hepatobiliary diseases, the AUROCs were 0·93 (0·91-0·94; slit-lamp) and 0·84 (0·81-0·86; fundus) for liver cancer, 0·90 (0·88-0·91; slit-lamp) and 0·83 (0·81-0·86; fundus) for liver cirrhosis, and ranged 0·58-0·69 (0·55-0·71; slit-lamp) and 0·62-0·70 (0·58-0·73; fundus) for other hepatobiliary diseases, including chronic viral hepatitis, non-alcoholic fatty liver disease, cholelithiasis, and hepatic cyst. In addition to the conjunctiva and sclera, our deep learning model revealed that the structures of the iris and fundus also contributed to the classification., Interpretation: Our study established qualitative associations between ocular features and major hepatobiliary diseases, providing a non-invasive, convenient, and complementary method for hepatobiliary disease screening and identification, which could be applied as an opportunistic screening tool., Funding: Science and Technology Planning Projects of Guangdong Province; National Key R&D Program of China; Guangzhou Key Laboratory Project; National Natural Science Foundation of China., (Copyright © 2021 The Author(s). Published by Elsevier Ltd. This is an Open Access article under the CC BY-NC-ND 4.0 license. Published by Elsevier Ltd.. All rights reserved.)

With the rapid development of information age, various social groups and corresponding institutions are producing a large amount of information data every day. For such huge data storage and identification, in order to manage such data more efficiently and reasonably, traditional semantic similarity algorithm emerges. However, the accuracy of the traditional semantic similarity algorithm is relatively low, and the convergence of corresponding algorithm is poor. Based on this problem, this paper starts with the conceptual structure of language, analyzes the depth of language structure and the distance between nodes, and analyzes the two levels as the starting point. For the information of a specific data resource description frame type, the weight of interconnected edges is used for impact analysis so as to realize the semantic similarity impact analysis of all information data. Based on the above improvements, this paper also systematically establishes the data information modeling process based on language conceptual structure and establishes the corresponding model. In the experimental part, the improved algorithm is simulated and analyzed. The simulation results show that compared with the traditional algorithm, the algorithm has obvious accuracy improvement. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the concept of Laurent differentially essential systems and give a criterion for a Laurent differential polynomial system to be Laurent differentially essential in terms of its support matrix. Then, the sparse differential resultant for a Laurent differentially essential system is defined, and its basic properties are proved. In particular, order and degree bounds for the sparse differential resultant are given. Based on these bounds, an algorithm to compute the sparse differential resultant is proposed, which is single exponential in terms of the Jacobi number and the size of the system. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a linear univariate representation for the roots of a zero-dimensional polynomial equation system is presented, where the complex roots of the polynomial system are represented as linear combinations of the roots of several univariate polynomial equations. An algorithm is proposed to compute such a representation for a given zero-dimensional polynomial equation system based on Gröbner basis computation. The main advantage of this representation is that the precision of the roots of the system can be easily controlled. In fact, based on the linear univariate representation, we can give the exact precisions needed for isolating the roots of the univariate equations in order to obtain roots of the polynomial system with a given precision. As a consequence, a root isolating algorithm for a zero-dimensional polynomial equation system can be easily derived from its linear univariate representation. [Copyright &y& Elsevier]

Abstract: Efficient characteristic set methods for computing zeros of polynomial equation systems in a finite field are proposed. The concept of proper triangular sets is introduced and an explicit formula for the number of zeros of a proper and monic triangular set is given. An improved zero decomposition algorithm is proposed to reduce the zero set of an equation system to the union of zero sets of monic proper triangular sets. The bitsize complexity of this algorithm is shown to be for Boolean polynomials, where is the number of variables and is the number of equations. We also give a multiplication free characteristic set method for Boolean polynomials, where the sizes of the polynomials occurred during the computation do not exceed the sizes of the input polynomials and the bitsize complexity of algorithm is for input polynomials with variables and degree . The algorithms are implemented in the case of Boolean polynomials and extensive experiments show that they are quite efficient for solving certain classes of Boolean equations raising from stream ciphers. [Copyright &y& Elsevier]

Abstract: In this paper we investigate structured backward errors for three kinds of generalized saddle point systems where the matrix is not symmetric and its block is not zero and has perturbations. Computable formula of backward errors are derived. The expressions are useful for testing the stability of practical algorithms. [Copyright &y& Elsevier]

Abstract: A backward error for inverse singular value problems with respect to an approximate solution is defined, and an explicit expression for the backward error is derived by extending the approach described in [J.G. Sun, Backward errors for the inverse eigenvalue problem, Numer. Math. 82 (1999) 339–349]. The expression may be useful for testing the stability of practical algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, we present an algorithm to decompose ordinary non-linear difference polynomials with rational functions as coefficients. The algorithm provides an effective reduction of the decomposition of difference polynomials to the decomposition of linear difference polynomials over the same coefficient field. The algorithm is implemented in Maple for the constant coefficient case. Experimental results show that the algorithm is quite effective and can be used to decompose difference polynomials with thousands of terms. [Copyright &y& Elsevier]

Abstract: We present a complete numerical algorithm for isolating all the real zeros of a zero-dimensional triangular polynomial system . Our system is general, with no further assumptions. In particular, our algorithm successfully treats multiple zeros directly in such systems. A key idea is to introduce evaluation bounds and sleeve bounds. We also present a much more efficient algorithm for zero-dimensional triangular systems without multiple roots. We implemented our algorithms, and promising experimental results are shown. [Copyright &y& Elsevier]

Abstract: Given a multivariate quadratic polynomial system in a finite field , the problem MAX-MQ is to find a solution satisfying the maximal number of equations. We prove that the probability of a random assignment satisfying a non-degenerate quadratic equation is at least , where is the number of the variables in the equation. Consequently, the random assignment provides a polynomial-time approximation algorithm with approximation ratio for non-degenerate MAX-MQ. For large , the ratio is close to . According to a result by Håstad, it is NP-hard to approximate MAX-MQ with an approximation ratio of for a small positive number . Therefore, the minimal approximation ratio that can be achieved in polynomial time for MAX-MQ is . [Copyright &y& Elsevier]

Abstract: We prove several basic properties for difference ascending chains, including a necessary and sufficient condition for an ascending chain to be the characteristic set of its saturation ideal and a necessary and sufficient condition for an ascending chain to be the characteristic set of a reflexive prime ideal. Based on these properties, we propose an algorithm to decompose the zero set of a finite set of difference polynomials into the union of zero sets of certain ascending chains. This decomposition algorithm is implemented and used to solve the perfect ideal membership problem, and to prove certain difference identities automatically. [Copyright &y& Elsevier]

Abstract: We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For a first order autonomous ODE , we give an exact degree bound for its rational solutions, based on the connection between rational solutions of and rational parametrizations of the plane algebraic curve defined by . For a first order autonomous ODE, we further give a polynomial time algorithm for computing a rational general solution if it exists based on the computation of Laurent series solutions and Padé approximants. Experimental results show that the algorithm is quite efficient. [Copyright &y& Elsevier]

An algorithm is proposed to give a global approximation of an implicit real plane algebraic curve with rational quadratic B-spline curves. The algorithm consists of four steps: topology determination, curve segmentation, segment approximation and curve tracing. Due to the detailed geometric analysis, high accuracy of approximation may be achieved with a small number of quadratic segments. The final approximation keeps many important geometric features of the original curve such as the topology, convexity and sharp points. Our method is implemented and experiments show that it may achieve better approximation bound with less segments than previously known methods. We also extend the method to approximate spatial algebraic curves. [Copyright &y& Elsevier]

In this paper, we propose algorithms for the following problems in the implicitization of a set of differential rational parametric equations P. (1) To find a characteristic set for the implicit prime ideal of P. (2) To find a canonical representation for the image of P. (3) To decide whether the parameters of P are independent, and if not, to re-parameterize P so that the new parametric equations have independent parameters. (4) To compute the inversion maps of P, and as a consequence, to decide whether P is proper. If the implicit variety is of dimension one and P is not proper, to find a proper re-parameterization for P. [Copyright &y& Elsevier]

In this paper, we use two approaches to solve the Perspective-Three-Point (P3P) problem: the algebraic approach and the geometric approach. In the algebraic approach, we use Wu-Ritt's zero decomposition algorithm to give a complete triangular decomposition for the P3P equation system. This decomposition provides the first complete analytical solution to the P3P problem. We also give a complete solution classification for the P3P equation system, i.e., we give explicit criteria for the P3P problem to have one, two, three, and four solutions. Combining the analytical solutions with the criteria, we provide an algorithm, CASSC, which may be used to find complete and robust numerical solutions to the P3P problem. In the geometric approach, we give some pure geometric criteria for the number of real physical solutions. [ABSTRACT FROM AUTHOR]

In this paper, the concept of sparse difference resultant for a Laurent transformally essential system of difference polynomials is introduced and a simple criterion for the existence of sparse difference resultant is given. The concept of transformally homogenous polynomial is introduced and the sparse difference resultant is proved to be transformally homogenous. It is shown that the vanishing of the sparse difference resultant gives a necessary condition for the corresponding difference polynomial system to have non-zero solutions. Order and degree bounds for the sparse difference resultant are given. Based on these bounds, an algorithm to compute the sparse difference resultant is proposed, which is single exponential in terms of the number of variables, the Jacobi number, and the size of the Laurent transformally essential system. Furthermore, the precise order and degree, a determinant representation, and a Poisson-type product formula for the difference resultant are given. [ABSTRACT FROM AUTHOR]