This paper consists of two components - a computational part and a theoretical part. The former targets the computer-aided geometric design of tubular surfaces. The latter focuses on the algebraic geometry of a family of conic curves. At the application level, we provide a straightforward and easy to implement computational algorithm to rationally parametrize generalized real tubular surfaces via moving lines. We discover that syzygies, i.e., moving lines, can be calculated directly from a given implicit equation of a projective conic. Specifically, we describe two linear polynomial vectors in 3-space whose entries are formulated in terms of the coefficients of the given implicit equation of the conic. We then prove that these two vectors are, in fact, a μ-basis, the generators for the syzygy module of the given conic, and furnish the rational parametrization of the given conic. At the theoretical level, we first briefly review the classical projection method for a rational parametrization of a generic non-degenerate conic. This is compared to the syzygy method, i.e., moving lines. We conclude the paper with an illustrative figure that depicts and compares the classical projection method and our moving line method.

Let R = K [x] be a univariate polynomial ring over an algebraically closed field K of characteristic zero. Let A is an element of M (m,m) (R) be an m x m matrix over R with non -zero determinate det(A) is an element of R. In this paper, utilizing linear -algebraic techniques, we investigate the relationship between a basis for the syzygy module of f( 1) , ... , f (m) and a basis for the syzygy module of g (1) , ... , g( m) , where [g( 1) ,... , g (m) ] = [f (1) ,..., f( m )]A.

This paper consists of two components - an application part and a theoretical part, where the former targets the applications of computer aided geometric designs in generating parametric curves, and the latter focuses on the algebraic analysis of rational space curves. At the application level, we construct a family of rational space curves via quaternion products of two generating curves. At the theoretical level, we use algebraic methods to extract a mu -basis for this family of curves, and describe a basis for a special submodule of the syzygy module in terms of a mu -basis for the syzygy module of this family of curves. A commutative diagram is provided to summarize these results.

In this paper, we prove that the family of binomials $x_1^{a_1} \cdots x_m^{a_m}-y_1^{b_1}\cdots y_n^{b_n}$ with $\gcd(a_1, \ldots, a_m, b_1, \ldots, b_n)=1$ is irreducible by identifying the connection between the irreducibility of a binomial in ${\mathbb C}[x_1, \ldots, x_m, y_1, \ldots, y_n]$ and ${\mathbb C}(x_2, \ldots, x_m, y_1, \ldots, y_n)[x_1]$. Then we show that the necessary and sufficient conditions for the irreducibility of this family of binomials is equivalent to the existence of a unimodular matrix $U_i$ with integer entries such that $(a_1, \ldots, a_m, b_1, \ldots, b_n)^T=U_i \be_i$ for $i\in \{1, \ldots, m+n\}$, where $\be_i$ is the standard basis vector.

Suppose $S$ is a parametrized surface in complex projective 3-space $mathbf{P}^3$ given as the image of $phi: mathbf{P}^1 imes mathbf{P}^1 o mathbf{P}^3$. The implicitization problem is to compute an implicit equation $F=0$ of $S$ using the parametrization $phi$. An algorithm using syzygies exists for computing $F$ if $phi$ has no base points, i.e. $phi$ is everywhere defined. This work is an extension of this algorithm to the case of a surface with multiple base points of total multiplicity k. We accomplish this in three chapters. In Chapter 2, we develop the concept and properties of Castelnuovo-Mumford regularity in biprojective spaces. In Chapter 3, we give a criterion for regularity in biprojective spaces. These results are applied to the implicitization problem in Chapter 4.

BACKGROUND Hepatocellular carcinoma (HCC) is one of the most commonly diagnosed malignant tumors in the world. In recent years, more and more inhibitors against gene targets have been found to be beneficial to survival. However, the function of homo-sapiens histone H3 associated protein kinase (GSG2) in HCC has not been completely understood. METHODS The expression of GSG2 in HCC tissues was detected by immunohistochemical staining. The lentivirus-mediated short hairpin RNA (shRNA) was used to knockdown GSG2 expression in HCC cell lines Hep3B2.1-7 and SK-HEP-1. Cell proliferation and colony formation were detected by MTT assay and colony formation assay, respectively, and flow cytometry assay was used to investigate the cell apoptosis in vitro. Mice xenograft model was constructed to detect the functions of GSG2 on tumor growth in vivo. Human Apoptosis Antibody Array was conducted to find the possible mechanism. RESULTS GSG2 was overexpressed in HCC tissues compared with adjacent normal tissues. The knockdown of GSG2 had the functions of inhibiting the progression of HCC, including inhibiting cell proliferation and colony formation and promoting cell apoptosis. Compared with shCtrl group, the shGSG2 group expressed higher apoptotic genes such as caspase 3, caspase 8, Fas and FasL, while lower IGF1, Bcl2 and Bcl-w. CONCLUSIONS Our study showed that knockdown of GSG2 suppresses the tumor growth in vitro and vivo. Therefore, GSG2 might play an oncogenic role in HCC.

In this paper, we first generate a family of rational surfaces in affine 3-space from three rational space curves by dual quaternion multiplication utilizing dual quaternions as a tool to represent rigid transformations. We provide an algorithm to compute all the base points of the homogeneous tensor product parametrization of this family of surfaces. Our main focus is the syzygies of these surfaces. We discover two sets of special syzygies, and show that the syzygy module and a μ-basis of this surface can be extracted from either set of special syzygies. Finally, we describe the structure of a free resolution of the module generated by these special syzygies, and use this free resolution to classify the minimal free resolutions of this module.

A surface of revolution is a surface that can be generated by rotating a planar curve, the directrix, around a straight line, the axis, in the same plane. Using the mathematics of quaternions, we provide a parametric equation of a surface of revolution generated by rotating a directrix about an axis by quaternion multiplication of the parametric representations of the directrix curve and the line of axis. Then, we describe an algorithm to determine whether a parametric surface is a surface of revolution, and identify the axis and the directrix. Examples are provided to illustrate our algorithm.

Background: Hepatocellular carcinoma (HCC) is the main type of primary liver cancer and shows a heavy burden worldwide. Its recurrence and mortality rate are still uncontrolled by the usage of present treatments. More attention has been focused on exploring specific genes that play important roles in HCC procession, and the function of DEP domain containing 1B (DEPDC1B) in HCC has not been researched. Methods: Immunohistochemical staining was used to detect the expression level of DEPDC1B in tumor tissues and adjacent normal tissues. After DEPDC1B and CDK1 knockdown in cell lines HEP3B2.1-7 and SK-HEP-1, MTT assay and colony formation assay was used to detect cell growth, flow cytometry assay was used to investigate cell apoptosis and cell cycle, wound-healing assay and Transwell assay were used to examine the tumor cell migration. Moreover, a xenograft model was constructed to research functions of DEPDC1B in tumor growth in vivo. Results: The results show that DEPDC1B knockdown inhibit the progression of HCC, through inhibiting cell proliferation, migration, colony formation, leading to G2 phase arrest, and promoting cell apoptosis in vitro, and CDK1 was selected for further mechanic research according to the results of Human GeneChip prime view. The results of recovery experiment displayed that the functions of DEPDC1B on HCC progression were mediated by CDK1. DEPDC1B knockdown can also inhibit tumor growth in vivo. Conclusions: The study confirmed that DEPDC1B knockdown restrains the tumor growth in vitro and vivo, and it can interact with CDK1 and rescued by CDK1. The study suggested that DEPDC1B was as a potential therapeutic target involved in HCC growth and progression.

In this paper, we study a family of rational monomial parametrizations. We investigate a few structural properties related to the corresponding monomial ideal J generated by the parametrization. We first find the implicit equation of the closure of the image of the parametrization. Then we provide a minimal graded free resolution of the monomial ideal J, and describe the minimal graded free resolution of the symmetric algebra of J. Finally, we provide a method to compute the defining equations of the Rees algebra of J using three moving planes that follow the parametrization.

BACKGROUND Primary hepatic neuroendocrine tumors (PHNETs), a group of neuroendocrine neoplasms, are extremely rare. There are only few case reports about PHNETs in the literature. The lack of large samples and multicenter research results in poor diagnostic and therapeutic approaches. AIM To discuss the clinical characteristics, diagnosis, and treatment of PHNETs and risk factors related to survival. METHODS We retrospectively analyzed the clinical data, imaging features, immunohistochemistry data, and treatment efficacy of 40 patients who were pathologically diagnosed with PHNETs and admitted to The First Affiliated Hospital of Zhengzhou University from January 1, 2014 to November 15, 2019. Finally, survival analysis was performed to identify the risk factors for survival. RESULTS The main symptoms and signs included intermittent abdominal pain (19 patients, 47.5%) and bloating (8 patients, 20.0%). The positive rates of tested tumor markers were recorded as follows: Carbohydrate antigen 19-9 (CA19-9) (6 patients, 15.0%), CA72-4 (3 patients, 7.5%), carcinoembryonic antigen (7 patients, 17.5%), and alpha-fetoprotein (6 patients, 15.0%). Immunohistochemical staining results showed positivity for Syn in 38 (97.4%) of 39 patients, for chromogranin A in 17 (65.4%) of 26 patients, for CD56 in 35 (94.6%) of 37 patients, for AE1/AE3 in 28 (87.5%) of 32 patients, and for Ki-67 in all 40 (100.0%) patients. The overall survival rate was significantly related to the tumor grade, AE1/AE3, and Ki-67. No significant correlation was found between other parameters (age, gender, tumor number, tumor size, metastasis, and treatment) and overall survival. CONCLUSION Higher grade, negative AE1/AE3, and higher Ki-67 are associated with a worse survival rate. Kinds of treatment and other parameters have no significant influence on overall survival.

A quaternion rational surface is a rational surface generated by two rational space curves via quaternion multiplication. In general, the structure of the graded minimal free resolution of a rational surface is unknown. The goal of this paper is to construct the graded minimal free resolution of a quaternion rational surface generated by two rational space curves. We will provide the explicit formulas for the maps of these graded minimal free resolutions. The approach we take is to utilize the information of the mu-bases of the generating rational curves, and create the generating sets for the first and second syzygy modules in the graded minimal free resolutions. In addition, we show that the ideal generated by the first syzygy module expressed in terms of moving planes is exactly the same as the ideal generated by the parametrization in the affine ring.

In this work we consider constructions of genus three curves Y such that End(Jac(Y))circle times Q contains the totally real cubic number field Q(zeta(7) + (zeta) over bar (7)). We construct explicit three-dimensional families whose general member is a nonhyperelliptic genus 3 curve with this property. The case when Y is hyperelliptic was studied in J. W. HOFFMAN, H. WANG, 7-gons and genus 3 hyperelliptic curves, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales., Serie A. Matematicas 107 (2013), 35-52, and some nonhyperelliptic curves were constructed in J. W. HOFFMAN, Z. LIANG, Y. SAKAI, H. WANG, Genus 3 curves whose Jacobians have endomorphisms by Q(zeta(7) + (zeta) over bar (7)), J. Symb. Comp. 74 (2016), 561-577.

布 - 加综合征(Budd-Chiari syndrome,BCS)是由主肝静脉和(或)肝后段下腔静脉血流受阻引起的复杂肝脏血管疾病,其梗阻部位可位于主肝静脉至下腔静脉与右心房交汇处的任一部位,从而引起肝后型门静脉高压和(或)下腔静脉高压综合征 [1].BCS 患者多因持续的肝脏淤血、缺氧而引起肝硬化、门静脉高压症,甚至发生肝细胞肝癌,临床上早期干预可在一定程度上延缓病情进展 [2].对于 BCS 的治疗,传统外科手术曾经占据主导地位并取得了满意的临床效果,但近年来随着介入技术的快速发展,介入治疗在 BCS 中的应用使传统外科手术面临严峻的挑战.虽然介入治疗的效果已得到广泛证实,但目前仍不能完全取代外科手术的作用.另外,在医学技术与理念快速发展的当下,积极促进技术创新、推动制定相关诊疗指南与共识,并发展多学科团队协作(multidisciplinary teamwork,MDT)模式,将为 BCS 的外科治疗带来新的发展动力.

WHAT IS KNOWN AND OBJECTIVE:Terlipressin has been shown to be effective in controlling variceal bleeding and decreasing associated mortality. Terlipressin is a synthetic analogue of vasopressin and is safer than arginine vasopressin; it induces selective vasoconstriction by stimulating the vasopressin V1 receptors that are predominantly located in the splanchnic tissues. However, severe hyponatraemia may occur during terlipressin treatment, resulting in neurological manifestations.CASE SUMMARY:We describe two patients who presented a marked decrease in serum sodium levels and developed obvious neurological manifestations after receiving terlipressin therapy. Although the two patients were given sodium supplementation, their serum sodium levels continually declined. After the discontinuation of terlipressin, their serum sodium levels rapidly recovered to normal limits, and the neurological manifestations subsequently disappeared in both patients.WHAT IS NEW AND CONCLUSION:Some studies have reported hyponatraemia as a side effect of terlipressin; however, severe hyponatraemia with neurological manifestations has rarely been reported. We presented the cases of 2 patients with obvious neurological manifestations after receiving terlipressin therapy. Severe hyponatraemia may develop in patients treated with terlipressin, resulting in associated neurological symptoms. Therefore, the close monitoring of serum sodium is necessary.

A ruled surface of revolution with moving axes and angles is a rational tensor product surface generated from a line and a rational space curve by rotating the line (the directrix) around vectors and angles generated by the rational space curve (the director). Only right circular cylinders and right circular cones are ruled surfaces that are also surfaces of revolution, but we show that a rich collection of other ruled surfaces such as hyperboloids of one sheet, 2-fold Whitney umbrellas, and a wide variety other interesting ruled shapes are ruled surfaces of revolution with moving axes and angles. We present a fast way to compute the implicit equation of a ruled surface of revolution with moving axes and angles from two linearly independent vectors that are perpendicular to the directrix of the surface. We also provide an algorithm for determining whether or not a given rational ruled surface is a ruled surface of revolution with moving axes and angles.

A ruled translational surface is a rational tensor product surface generated by translating a rational space curve along a straight line or equivalently translating a straight line along a rational space curve. We show how to compute the implicit equation of a ruled translational surface from two linearly independent vectors that are perpendicular to the generating line of the surface.

A translational surface is a rational tensor product surface generated from two rational space curves by translating one curve along the other curve. Translational surfaces can also be generated from two rational space curves by dual quaternion multiplication. Using the mathematics of dual quaternions, we provide a necessary and sufficient condition for a rational tensor product surface to be a translational surface. Examples are provided to illustrate our theorems and flesh out our algorithms.