On a Non-Definite Sturm-Liouville Problem in the Two-Turning Point Case—Analysis and Numerical Results

In this paper, we study the non-definite Sturm-Liouville problem comprising of a regular Sturm-Liouville equation and Dirichlet boundary conditions on a closed interval. We consider the case in which the weight function changes sign twice in the given interval of definition. We give detailed numerical results on the spectrum of the problem, from which we verify various results on general non definite Sturm-Liouville problems. We also present some theoretical results which support the numerical results. Some numerical results seem to be in contrast with the results that are so far obtained in the case where the weight function changes sign once. This leads to more open questions for future studies in this particular area.

Hermite Positive Definite Solution of the Quaternion Matrix Equation Xm + B*XB = C

This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation Xm+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .

Comparative efficacy and safety between endoscopic submucosal dissection,surgery and definitive chemoradiotherapy in patients with cT1N0M0 esophageal cancer

BACKGROUND Endoscopic submucosal dissection(ESD)and surgical resection are the standard of care for cT1N0M0 esophageal cancer(EC),whereas definitive chemoradiotherapy(d-CRT)is a treatment option.Nevertheless,the comparative efficiency and safety of ESD,surgery and d-CRT for cT1N0M0 EC remain unclear.AIM To compare the efficiency and safety of ESD,surgery and d-CRT for cT1N0M0 EC.METHODS We retrospectively analyzed the hospitalized data of a total of 472 consecutive patients with cT1N0M0 EC treated at Sun Yat-sen University Cancer center between 2017-2019 and followed up until October 30th,2022.We analyzed demographic,medical recorded,histopathologic characteristics,imaging and endoscopic,and follow-up data.The Kaplan-Meier method and Cox proportional hazards modeling were used to analyze the difference of survival outcome by treatments.Inverse probability of treatment weighting(IPTW)was used to minimize potential confounding factors.RESULTS We retrospectively analyzed patients who underwent ESD(n=99)or surgery(n=220)or d-CRT(n=16)at the Sun Yat-sen University Cancer Center from 2017 to 2019.The median follow-up time for the ESD group,the surgery group,and the d-CRT group was 42.0 mo(95%CI:35.0-60.2),45.0 mo(95%CI:34.0-61.75)and 32.5 mo(95%CI:28.3-40.0),respectively.After adjusting for background factors using IPTW,the highest 3-year overall survival(OS)rate and 3-year recurrence-free survival(RFS)rate were observed in the ESD group(3-year OS:99.7% and 94.7% and 79.1%;and 3-year RFS:98.3%,87.4% and 79.1%,in the ESD,surgical,and d-CRT groups,respectively).There was no difference of severe complications occurring between the three groups(P≥0.05).Multivariate analysis showed that treatment method,histology and depth of infiltration were independently associated with OS and RFS.CONCLUSION For cT1N0M0 EC,ESD had better long-term survival and lower hospitalization costs than those who underwent d-CRT and surgery,with a similar rate of severe complications occurring.

The Njiki’s Fundamental Theorem-Definition on Fractions in the Mathematical Set ℚ and by Extension in ℝ and ℂ, for the Purpose of Leading to the Construction of Some Algebraic Structures as Its Theoretical Applications and for the Practical Ones

The purpose of the research in the NJIKI’s fundamental THEOREM-DEFINITION on fractions in the mathematical set ℚand by extension in ℝand ℂand in order to construct some algebraic structures is about the proved EXISTENCE and the DEFINITION by NJIKI of two INNOVATIVE, IMPORTANT and TEACHABLE operations of addition or additive operations, in ℚ, marked ⊕and +α,β, and taken as VECTORIAL, TRIANGULAR, of THREE or PROPORTIONAL operations and in order to make THEM not be different from the RATIONAL ONE, +, but to bring much more and new information on fractions, and, by extension in ℝand ℂ. And the very NJIKI’s fundamental THEOREM-DEFINITION having many APPLICATIONS in the everyday life of the HUMAN BEINGS and without talking about computer sciences, henceforth being supplied with very interesting new ALGORITHMS. And as for the work done in the research, it will be waiting for its extension to be done after publication and along with the research results concerned.

POSITIVE DEFINITE KERNEL IN SUPPORT VECTOR MACHINE(SVM)

The relationship among Mercer kernel, reproducing kernel and positive definite kernel in support vector machine （SVM） is proved and their roles in SVM are discussed. The quadratic form of the kernel matrix is used to confirm the positive definiteness and their construction. Based on the Bochner theorem, some translation invariant kernels are checked in their Fourier domain. Some rotation invariant radial kernels are inspected according to the Schoenberg theorem. Finally, the construction of discrete scaling and wavelet kernels, the kernel selection and the kernel parameter learning are discussed.

SYMMETRIC POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATIONS (AX,XB)=(C,D) AND AXB=C

The symmetric positive definite solutions of matrix equations (AX,XB)=(C,D) and AXB=C are considered in this paper. Necessary and sufficient conditions for the matrix equations to have symmetric positive definite solutions are derived using the singular value and the generalized singular value decompositions. The expressions for the general symmetric positive definite solutions are given when certain conditions hold.

The Necessary and Sufficient Conditions for Two Kinds of Matrix Equtions Having Semipositive Subdefinite Solutions

Finding solutions of matrix equations in given set SR n×n is an active research field. Lots of investigation have done for these cases, where S are the sets of general or symmetric matrices and symmetric positive definite or sysmmetric semiposite definite matrices respectively . Recently, however, attentions are been paying to the situation for S to be the set of general(semi) positive definite matrices(called as semipositive subdefinite matrices below) . In this paper the necessary and sufficient conditions for the following two kinds of matrix equations having semipositive, subdefinite solutions are obtained. General solutions and symmetric solutions of the equations (Ⅰ) and (Ⅱ) have been considered in in detail.

The Classification of Positive Definite Unimodular Lattices Over Z[(1+21^(1/2))/2]

In this paper, the author applies adjacent lattice method and Siegel mass formula to determine the classes of positive definite unimodular lattices of rank 4 over Z , and obtains that the class number of unit genus gen( I 4 ) is nine and the class number of even unimodular lattices is three, and also gives the representative lattices of each class.

The Classification of Positive Definite Unimodular Lattices Over Z[(1+211/2)/2]

In this paper, the author applies adjacent lattice method and Siegel mass formula to determine the classes of positive definite unimodular lattices of rank 4 over Z , and obtains that the class number of unit genus gen( I 4 ) is nine and the class number of even unimodular lattices is three, and also gives the representative lattices of each class.

A Remark on the Positive Definite Problem of a Binary Quartic Form

In this paper,we discussed the positive definite problem of binary quartic forms and obtained a necessary and sufficient conditions.

The Metapositive Definite Self－Conjugate Solution of the Matrix Equation AXB＝C over a Skew Field

Let F be the strong p-division ring [4]. This paper is sequel to [1]. Metapositive definite self-conjugate matrix over F is defined and the necessary and sufficient conditions for determining whether a partitioned matrix over F is metapositive definite self-conjugate are given.Moreover,a decomposition of pairwise matrices over F with the same numbers of columns is also presented. Whence some necessary and sufficient conditions for the existence of and the explicit expression for the metapositive definite self-conjugate solution of the matrix equation AXB=C over F are derived.

Some new bound estimates of the Hermitian positive definite solutions of the nonlinear matrix equation X^s+ A~*X^(-t) A = Q

The range and existence conditions of the Hermitian positive definite solutions of nonlinear matrix equations Xs+A*X-tA=Q are studied, where A is an n×n non-singular complex matrix and Q is an n×n Hermitian positive definite matrix and parameters s,t>0. Based on the matrix geometry theory, relevant matrix inequality and linear algebra technology, according to the different value ranges of the parameters s,t, the existence intervals of the Hermitian positive definite solution and the necessary conditions for equation solvability are presented, respectively. Comparing the existing correlation results, the proposed upper and lower bounds of the Hermitian positive definite solution are more accurate and applicable.

Two Structure-Preserving-Doubling Like Algorithms to Solve the Positive Definite Solution of the Equation X-A^(H)X^(-1)A=Q

In this paper,we study the nonlinear matrix equation X-A^(H)X^(-1)A=Q,where A,Q∈C^(n×n),Q is a Hermitian positive definite matrix and X∈C^(n×n)is an unknown matrix.We prove that the equation always has a unique Hermitian positive definite solution.We present two structure-preserving-doubling like algorithms to find the Hermitian positive definite solution of the equation,and the convergence theories are established.Finally,we show the effectiveness of the algorithms by numerical experiments.

Spectrum of a class of fourth order left-definite differential operators

The spectrum of a class of fourth order left-definite differential operators is studied. By using the theory of indefinite differential operators in Krein space and the relationship between left-definite and right-definite operators, the following conclusions are obtained： if a fourth order differential operator with a self-adjoint boundary condition that is left-definite and right-indefinite, then all its eigenvalues are real, and there exist countably infinitely many positive and negative eigenvalues which are unbounded from below and above, have no finite cluster point and can be indexed to satisfy the inequality …≤λ-2≤λ-1≤λ-0〈0〈λ0≤λ1≤λ2≤…

Siemens Definition AS128 CT维修5例

介绍了Siemens Definition AS128 CT的硬件构成,分析并总结了该设备在日常使用中出现的检查床水平移动后机架掉电、图像重建系统(image reconstruction system,IRS)计算机硬盘故障、开机时F1开关脱开、“Exam”按钮灰化等故障产生的原因及维修思路,为临床工程技术人员排除相似故障提供了参考。

DIAGONALLY COMPENSATED REDUCTION AND MULTISPLITTING OF A SYMMETRIC POSITIVE DEFINITE MATRIX

To solve the symmetric positive definite linear system Ax = b on parallel and vector machines, multisplitting methods are considered. Here the s.p.d. (symmetric positive definite) matrix A need not be assumed in a special form (e.g. the dissection form [11]). The main tool for deriving our methods is the diagonally compensated reduction (cf. [1]). The convergence of such methods is also discussed by using this tool. [WT5,5”HZ]

Thermal Sense Prediction Model for Definite Human Group in Indoor Case

In order to build a prediction model of the indoor thermal comfort for a given human group, the original predicted mean vote （PMV） equation is reconstructed and simplified, the modified PMV equation is named PMVR （PMV for region） , where five variables are needed to be fitted with the dataset of actual thermal sense of a definite human group. As the fitting algorithm, the particle swarm optimization algorithm is used to optimize the solution, and a fixed PMVR can be finally determined. Experiment results indicate that for a definite human group, PMVR is more accurate on the prediction of thermal sense compared with some other models.

ON RUBIN'S HARMONIC ANALYSIS AND ITS RELATED POSITIVE DEFINITE FUNCTIONS

In this paper,a new formulation of the Rubin’s q-translation is given,which leads to a reliable q-harmonic analysis.Next,related q-positive definite functions are introduced and studied,and a Bochner’s theorem is proved.

SEMI-DEFINITE RELAXATION ALGORITHM OF MULTIPLE KNAPSACK PROBLEM

The multiple knapsack problem denoted by MKP (B,S,m,n) can be defined as fol- lows.A set B of n items and a set Sof m knapsacks are given such thateach item j has a profit pjand weightwj,and each knapsack i has a capacity Ci.The goal is to find a subset of items of maximum profit such that they have a feasible packing in the knapsacks.MKP(B,S,m,n) is strongly NP- Complete and no polynomial- time approximation algorithm can have an approxima- tion ratio better than0 .5 .In the last ten years,semi- definite programming has been empolyed to solve some combinatorial problems successfully.This paper firstly presents a semi- definite re- laxation algorithm (MKPS) for MKP (B,S,m,n) .It is proved that MKPS have a approxima- tion ratio better than 0 .5 for a subclass of MKP (B,S,m,n) with n≤ 1 0 0 ,m≤ 5 and maxnj=1{ wj} minmi=1{ Ci} ≤ 2 3 .

Study on the Relation of Definite Integral and Indefinite Integral

Relation of definite integral and indefinite integral was discussed and an important result was gotten. If f(x) is bounded and has primary function, the formal definite integral x s f(t)dt is the indefinite integral of f(x), where x is a self-variable, s is a parameter,~f(x) is a function defined in(-∞, +∞), which comes from f(x) by restriction and extension. In other words, the indefinite integral is a special form of definite integral, its lower integral limit and upper integral limit are all indefinite.