Solvability and Construction of a Solution to the Fredholm Integral Equation of the First Kind

The issues of solvability and construction of a solution of the Fredholm integral equation of the first kind are considered. It is done by immersing the original problem into solving an extremal problem in Hilbert space. Necessary and sufficient conditions for the existence of a solution are obtained. A method of constructing a solution of the Fredholm integral equation of the first kind is developed. A constructive theory of solvability and construction of a solution to a boundary value problem of a linear integrodifferential equation with a distributed delay in control, generated by the Fredholm integral equation of the first kind, has been created.

MULTIPLE RECIPROCITY METHOD WITH TWO SERIES OF SEQUENCES OF HIGH-ORDER FUNDAMENTAL SOLUTION FOR THIN PLATE BENDING

The boundary value problem of plate bending problem on two_parameter foundation was discussed.Using two series of the high_order fundamental solution sequences, namely, the fundamental solution sequences for the multi_harmonic operator and Laplace operator, applying the multiple reciprocity method(MRM), the MRM boundary integral equation for plate bending problem was constructed. It proves that the boundary integral equation derived from MRM is essentially identical to the conventional boundary integral equation. Hence the convergence analysis of MRM for plate bending problem can be obtained by the error estimation for the conventional boundary integral equation. In addition, this method can extend to the case of more series of the high_order fundamental solution sequences.

A priliminary study on the sequence characteristics and focal mechanism solution of the Jiashi strong earthquake swarm

The sequence characteristics and focal mechanism solution of the Jiashi, Xinjiang strong earthquake swarm are analyzed and studied in this paper. The result shows that before the M S=6.6 earthquake, value h of sequence frequency attenuation coefficient was less than 1, then value h was more than 1. Before occurrence of M S6.0 earthquakes the energy is released either in a continuously strengthened way or a sharply strengthened way, and before M S5.0 earthquakes the sequence frequency shows calm. The study on the focal mechanism solution of the strong earthquake swarm shows that the source faults are mainly in a right lateral, strike slip way and the faults have characteristics of tensor shear.

Constructing infinite sequence exact solutions of nonlinear evolution equations

To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.

De novel heterozygous copy number deletion on 7q31.31-7q31.32 involving TSPAN12 gene with familial exudative vitreoretinopathy in a Chinese family

AIM:To investigate the genetic and clinical characteristics of patients with a large heterozygous copy number deletion on 7q31.31-7q31.32.METHODS:A family with familial exudative vitreoretinopathy(FEVR)phenotype was included in the study.Whole-exome sequencing(WES)was initially used to locate copy number variations(CNVs)on 7q31.31-31.32,but failed to detect the precise breakpoint.The long-read sequencing,Oxford Nanopore sequencing Technology(ONT)was used to get the accurate breakpoint which is verified by quantitative real-time polymerase chain reaction(QPCR)and Sanger Sequencing.RESULTS:The proband,along with her father and younger brother,were found to have a heterozygous 4.5 Mb CNV deletion located on 7q31.31-31.32,which included the FEVRrelated gene TSPAN12.The specific deletion was confirmed as del(7)(q31.31q31.32)chr7:g.119451239_123956818del.The proband exhibited a phase 2A FEVR phenotype,characterized by a falciform retinal fold,macular dragging,and peripheral neovascularization with leaking of fluorescence.These symptoms led to a significant decrease in visual acuity in both eyes.On the other hand,the affected father and younger brother showed a milder phenotype.CONCLUSION:The heterozygous CNV deletion located on 7q31.31-7q31.32 is associated with the FEVR phenotype.The use of long-read sequencing techniques is essential for accurate molecular diagnosis of genetic disorders.

Identification of a LMNA (c.646C>T) variant by whole-exome sequencing in combination with a dilated cardiomyopathy (DCM) related gene filter in a family with familiar DCM

Dilated cardiomyopathy（DCM）is characterized by the dilated heart chambers and reduced systolic function in the absence of specific aetiology[1].Approximately one third of DCM cases are hereditary.In recent years,DCM concomitant with arrhythmias and sudden death resulting from gene mutation has been widely

Existence of Solutions for a Non-Autonomous Evolution Equations with Nonlocal Conditions

The existence of mild solutions for non-autonomous evolution equations with nonlocal conditions in Banach space is studied in this article. We obtained the existence of at least one mild solution to the evolution equations by using Krasnoselskii’s fixed point theorem as well as the theory of the evolution family. The interest of this paper is that any assumptions are not imposed on the nonlocal terms and Green’s functions and a new alternative method is applied to prove the existence of mild solutions. The results obtained in this paper may improve some related conclusions on this topic. An example is given as an application of the results.

Infinite Sequence Soliton-Like Exact Solutions of (2 + 1)-Dimensional Breaking Soliton Equation

寻求新无限的顺序象 soliton 一样非线性的进化方程的准确答案(旧姓) 由在辅助方程方法上开发建设和机械化的二个特征，秒种椭圆形的方程高度被学习并且新类型答案和 B ? cklund 转变被获得。( 2+1 )然后，维的碎 soliton 方程作为一个例子和它的无限的顺序被选象soliton一样准确解决方案在符号的计算系统 Mathematica 的帮助下被构造，它包括无限的顺序 Jacobi 椭圆形的类型的光滑的象soliton一样解决方案， Jacobi 椭圆形的类型的无限的顺序协议 soliton 解决方案和指数的函数类型和三角形的功能的无限的顺序山峰 soliton 解决方案打字。

Inverse scattering method and soliton solution family for the Einstein-Maxwell theory with multiple Abelian gauge fields

A Hauser-Ernst-type extended hyperbolic complex linear system given in our previous paper [Gao Y J 2004 Chin. Phys. 13 602] is slightly modified and used to develop a new inverse scattering method for the stationary axisymmetric Einstein-Maxwell theory with multiple Abelian gauge fields. The reduction procedures in this inverse scattering method are found to be fairly simple, which makes the inverse scattering method be fine and effective in practical application. As an example, a concrete family of soliton solutions for the considered theory is obtained.

INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM

Invariant subspace method is exploited to obtain exact solutions of the two- component b-family system. It is shown that the two-component b-family system admits the generalized functional separable solutions. Furthermore, blow up and behavior of those exact solutions are also investigated.

The New Infinite Sequence Solutions of Multiple Sine-Gordon Equations

By the function transformation and the first integral of the ordinary differential equations, the problem of solving the solutions of the double sine-Gordon equation and the treble sine-Gordon equation is researched, and the new solutions are obtained. First, the problem of solving the solutions of the double sine-Gordon equation and the treble sine-Gordon equation is changed to the problem of solving the solutions of the nonlinear ordinary differential equation. Second, with the help of the B?cklund transformation and the nonlinear superposition formula of solutions of the first kind of elliptic equation and the Riccati equation, the new infinite sequence soliton-like solutions of two kinds of sine-Gordon equations are constructed.

Infinite Sequence of Conservation Laws and Analytic Solutions for a Generalized Variable-Coefficient Fifth-Order Korteweg-de Vries Equation in Fluids

在这份报纸，为在液体的一个概括可变系数的第五顺序的 Korteweg-de Vries 方程的能量守恒定律的一个无限的序列基于 B 被构造 ? cklund 转变。Hirota 双线性的形式和符号的计算被使用获得三种答案。可变系数能影响典型的线的保存密度，联系流动，和外观。soliton 结构上的波浪数字的效果也被讨论并且 soliton 结构打字，例如，双 periodic soliton，平行 soliton 和 soliton 建筑群，被介绍。

ON EXISTENCE OF SOLUTIONS OF DIFFERENCE RICCATI EQUATION

Consider the difference Riccati equation f(z+1) =(A(z)f(z)+B(z))/(C(z)f(z)+D(z)),where A,B, C,D are meromorphic functions, we give its solution family with one-parameter H(f(z))={f_0(z),f(z)=((f_1(z)-f_0(z))(f_2(z)-f_0(z)))/(Q(z)(f_2(z)-f_1(z))+(f_2(z)-f_0(z)))}, where Q(z) is any constant in C or any periodic meromorphic function with period 1, and f_0(z),f_1(z),f_2(z) are its three distinct meromorphic solutions.

Entire solutions of a certain type of functional-differential equations

In this paper, we shall utilize Nevanlinna value distribution theory and normality theory to study the solvability of a certain type of functional-differential equations. We also consider the solutions of some nonlinear differential equations.

Solute carrier family 2 members 1 and 2 as prognostic biomarkers in hepatocellular carcinoma associated with immune infiltration

BACKGROUND Metabolic reprogramming has been identified as a core hallmark of cancer.Solute carrier family 2 is a major glucose carrier family.It consists of 14 members,and we mainly study solute carrier family 2 member 1(SLC2A1)and solute carrier family 2 member 2(SLC2A2)here.SLC2A1,mainly existing in human erythrocytes,brain endothelial cells,and normal placenta,was found to be increased in hepatocellular carcinoma(HCC),while SLC2A2,the major transporter of the normal liver,was decreased in HCC.AIM To identify if SLC2A1 and SLC2A2 were associated with immune infiltration in addition to participating in the metabolic reprogramming in HCC.METHODS The expression levels of SLC2A1 and SLC2A2 were tested in HepG2 cells,HepG215 cells,and multiple databases.The clinical characteristics and survival data of SLC2A1 and SLC2A2 were examined by multiple databases.The correlation between SLC2A1 and SLC2A2 was analyzed by multiple databases.The functions and pathways in which SLC2A1,SLC2A2,and frequently altered neighbor genes were involved were discussed in String.Immune infiltration levels and immune marker genes associated with SLC2A1 and SLC2A2 were discussed from multiple databases.RESULTS The expression level of SLC2A1 was up-regulated,but the expression level of SLC2A2 was down-regulated in HepG2 cells,HepG215 cells,and liver cancer patients.The expression levels of SLC2A1 and SLC2A2 were related to tumor volume,grade,and stage in HCC.Interestingly,the expression levels of SLC2A1 and SLC2A2 were negatively correlated.Further,high SLC2A1 expression and low SLC2A2 expression were linked to poor overall survival and relapse-free survival.SLC2A1,SLC2A2,and frequently altered neighbor genes played a major role in the occurrence and development of tumors.Notably,SLC2A1 was positively correlated with tumor immune infiltration,while SLC2A2 was negatively correlated with tumor immune infiltration.Particularly,SLC2A2 methylation was positively correlated with lymphocytes.CONCLUSION SLC2A1 and SLC2A2 are independent therapeutic targets for HCC,and they are quintessential marker molecules for predicting and regulating the number and status of immune cells in HCC.

GROWTH OF MEROMORPHIC SOLUTIONS OF SOME ALGEBRAIC DIFFERENTIAL EQUATIONS

In this paper, by means of the normal family theory, we estimate the growth order of meromorphic solutions of some algebraic differential equations and improve the related result of Barsegian et al. [6]. We also give some examples to show that our results occur in some special cases.

MAXIMAL SUBSPACES FOR SOLUTIONS OF THE SECOND ORDER ABSTRACT CAUCHY PROBLEM

For a continuous, increasing function ω ： R^＋ →R^＋／{0} of finite exponential type, this paper introduces the set Z（A, ω） of all x in a Banach space X for which the second order abstract differential equation （2） has a mild solution such that [ω（t）]^-1u（t,x） is uniformly continues on R^＋, and show that Z（A, ω） is a maximal Banach subspace continuously embedded in X, where A ∈ B（X） is closed. Moreover, A[z（A,ω） generates an O（ω（t）） strongly continuous cosine operator function family.

Family with sequence similarity 134 member B-mediated reticulophagy ameliorates hepatocyte apoptosis induced by dithiothreitol

BACKGROUND Endoplasmic reticulum(ER)stress-related hepatocyte apoptosis is responsible for multiple hepatic diseases.Previous studies have revealed that endoplasmic reticulophagy(ER-phagy)promotes the selective clearance of damaged ER fragments during ER stress,playing a crucial role in maintaining ER homeostasis and inhibiting apoptosis.Family with sequence similarity 134 member B(FAM134B)is a receptor involved in ER-phagy that can form a complex with calnexin(CNX)and microtubule-associated protein 1 light chain 3(LC3).The complex can mediate the selective isolation of ER fragments to attenuate hepatocyte apoptosis.However,the precise regulatory mechanisms remain unclear.AIM To elucidate the effect of FAM134B-mediated ER-phagy on ER stress-induced apoptosis in buffalo rat liver 3A(BRL-3A)rat hepatocytes and the potential regulatory mechanisms.METHODS ER stress-related hepatocyte apoptosis was induced using dithiothreitol(DTT).Proteins related to ER stress and autophagy were measured with western blotting.Protein complex interactions with FAM134B were isolated by co-immunoprecipitation.ER-phagy was evaluated in immunofluorescence experiments.Cell cycle distribution and apoptosis were measured by flow cytometry.Mitochondrial Ca^(2+) levels were evaluated by the co-localization of intracellular Ca^(2+)-tracker and Mitotracker.The small interfering RNA against FAM134B was used to knockdown FAM134B in BRL-3A cells.RESULTS ER stress-related and autophagy-related proteins in BRL-3A cells were elevated by both short and long-term DTT treatment.Furthermore,co-immunoprecipitation confirmed an interaction between FAM134B,CNX,FAM134B,and LC3 in BRL-3A cells.Immunofluorescence assays revealed that autolysosomes significantly decreased following short-term DTT treatment,but increased after long-term treatment.Mitochondrial Ca2+levels and apoptotic rates were dramatically elevated,and more cells were arrested in the G1 stage after short-term DTT treatment;however,these decreased 48 h later.Moreover,FAM134B downregulation accelerated mitochondrial apoptotic pathway activation and aggravated hepatocyte apoptosis under ER stress.CONCLUSION FAM134B-mediated ER-phagy attenuates hepatocyte apoptosis by suppressing the mitochondrial apoptotic pathway.Our findings provide new evidence highlighting the importance of FAM134Bmediated ER-phagy in attenuating hepatocyte apoptosis.

The Long-Term Dynamic Behavior of Solutions to a Class of Generalized Higher-Order Kirchhoff-Type Coupled Wave Equations

In this paper, we study the long-term dynamic behavior of a class of generalized high-order Kirchhoff-type coupled wave equations. Firstly, the existence of uniqueness global solution of this kind of equations in Ek space is proved by prior estimation and Galerkin method;Then, through using Rellich-Kondrachov compact embedding theorem, it is proved that the solution semigroup S(t) has the family of the global attractors Ak in space Ek;Finally, through linearization method, proves that the operator semigroup S(t) Frechet differentiable and the attenuation of linearization problem volume element. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractors Ak.

Existence of Ordered Solutions to Quasilinear Schrodinger Equations with General Nonlinear Term

In this paper, the existence of a pair of ordered solutions for the following class of equations in ?(1)?was studied. A bounded (PS) (Palais-Smale) sequence was constructed and the related variational principle was used to prove the existence of the positive solution. The existence of the ordered solutions is finally found.