NORMALIZED SOLUTIONS FOR THE GENERAL KIRCHHOFF TYPE EQUATIONS

In the present paper,we prove the existence,non-existence and multiplicity of positive normalized solutions(λ_(c),u_(c))∈R×H^(1)(R^(N))to the general Kirchhoff problem-M■,satisfying the normalization constraint f_(R)^N u^2dx=c,where M∈C([0,∞))is a given function satisfying some suitable assumptions.Our argument is not by the classical variational method,but by a global branch approach developed by Jeanjean et al.[J Math Pures Appl,2024,183:44–75]and a direct correspondence,so we can handle in a unified way the nonlinearities g(s),which are either mass subcritical,mass critical or mass supercritical.

Normalized Ground State Solutions of Nonlinear Choquard Equations with Nonconstant Potential

In this paper,we mainly focus on the following Choquard equation-{△u-V(x)(I_(a*)|u|^(p))|u|^(p-2)u=λu,x∈R^(N),u∈H^(1)(R^(N))where N≥1,λ∈R will arise as a Lagrange multiplier,0<a<N and N+a/N<p<N+a+2/N Under appropriate hypotheses on V(x),we prove that the above Choquard equation has a normalized ground state solution by utilizing variational methods.

MULTIPLICITY OF NORMALIZED SOLUTIONS FOR THE FRACTIONAL SCHR?DINGER-POISSON SYSTEM WITH DOUBLY CRITICAL GROWTH

In this paper,we are concerned with solutions to the fractional Schrodinger-Poisson system■ with prescribed mass ∫_(R^(3))|u|^(2)dx=a^(2),where a> 0 is a prescribed number,μ> 0 is a paremeter,s ∈(0,1),2 <q <2_(s)^(*),and 2_(s)^(*)=6/(3-2s) is the fractional critical Sobolev exponent.In the L2-subcritical case,we show the existence of multiple normalized solutions by using the genus theory and the truncation technique;in the L^(2)-supercritical case,we obtain a couple of normalized solutions by developing a fiber map.Under both cases,to recover the loss of compactness of the energy functional caused by the doubly critical growth,we need to adopt the concentration-compactness principle.Our results complement and improve upon some existing studies on the fractional Schrodinger-Poisson system with a nonlocal critical term.

PCR ALGORITHM FOR PARALLEL COMPUTING MINIMUM-NORM LEAST-SQUARES SOLUTION OF INCONSISTENT LINEAR EQUATIONS

This paper presents a new highly parallel algorithm for computing the minimum-norm least-squares solution of inconsistent linear equations Ax = b(A∈Rm×n,b∈R (A)). By this algorithm the solution x = A + b is obtained in T = n(log2m + log2(n - r + 1) + 5) + log2m + 1 steps with P=mn processors when m × 2(n - 1) and with P = 2n(n - 1) processors otherwise.

L1 norm optimal solution match processing in the wavelet domain

Greater attention has been paid to vintage-merge processing of seismic data and extracting more valuable information by the geophysicist. A match filter is used within many important areas such as splicing seismic data, matching seismic data with different ages and sources, 4-D seismic monitoring, and so on. The traditional match filtering method is subject to many restrictions and is usually difficult to overcome the impact of noise. Based on the traditional match filter, we propose the wavelet domain L1 norm optimal matching filter. In this paper, two different types of seismic data are decomposed to the wavelet domain, different detailed effective information is extracted for Ll-norm optimal matching, and ideal results are achieved. Based on the model test, we find that the L1 norm optimal matching filter attenuates the noise and the waveform, amplitude, and phase coherence of result signals are better than the conventional method. The field data test shows that, with our method, the seismic events in the filter results have better continuity which achieves the high precision seismic match requirements.

The Minimum Norm of Solutions of the Boolean Matrix-Equation A^k=J

Let A be an n×n primitive Boolean matrix. γ(A) is the least number k such that A k=J. σ(A) is the number of 1 entry in A . In this paper, we consider the parameter M ′(k,n)= min {σ(A)|A k=J, trace (A)=0} and obtain the values of M ′(2,n) and M ′(k,n) for k≥2n-6 . Furthermore, the characterization of solution of A 2=J with trace (A) =0 and σ(A)=3n-3 is completely determined.

Low-concentration sodium hydroxide solution injection in normal liver parenchyma of rabbits

BACKGROUND: Percutaneous ethanol injection has been widely used as a non-surgical therapy for liver cancer, but it has some shortcomings such as local diffusion and une- qual permeation. This study was designed to observe the volume, controllability and completeness of necrosis after injection of low concentration sodium hydroxide in the normal liver parenchyma so as to assess its possibility in treatment of liver cancer instead of ethanol. METHODS: Twenty-seven New Zealand rabbits were di- vided randomly into 9 groups (Aa, Ab, Ac, Ba, Bb, Bc, Ca, Cb, and Cc) by a 3 × 3 (three-by-three) factorial de- sign, each consisting of 3 rabbits. Group A was given sodi- um hydroxide solution at a concentration of 5%, while B at 2.5% and C at 1% in liver parenchyma. Each group re- ceived three doses of the solution: a (0.2 ml), b (0.5 ml) and c (1.0 ml). Then another 3 rabbits as side-effect group were dropped with sodium hydroxide solution in their liver lobe space. Liver and renal function changes in all the rab- bits were compared after injection with pre-injection. RESULTS: All the lesions were localized. At the concentra- tion of 2.5% and 5%, the lesion volume increased with the dose increased from 0.2 ml to 1.0 ml (P < 0. 05). No sig- nificant differences were found in the lesion volume of the groups receiving the same dose but different concentration. Changes in liver and renal function were not significant 7 days after injection, compared with those before injection. CONCLUSIONS: 2.5% and 5% sodium hydroxide solution could control local complete necrosis in normal liver. With regard to safety, 2.5% alkali solution is considered promis- ing as a new agent for intratumoral injection therapy in- stead of ethanol.

Exact solution of plane isolated crack normal to a bimaterial interface of infinite extent

This paper presents an exact solution for the transverse interface crack in the plane strain case. The crack is perpendicular to the interface and in one material. The exact complex stress functions are first obtained with some unknown constants. The satisfactions of all boundary conditions are then checked, the condition at infinity is considered and the unknown constants are determined. Further study may focus on the case with different shear moduli and the influence of the large deformation.

NEGATIVE NORM LEAST-SQUARES METHODS FOR THE INCOMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS

The purpose of this article is to develop and analyze least-squares approximations for the incompressible magnetohydrodynamic equations. The major advantage of the least-squares finite element method is that it is not subjected to the so-called Ladyzhenskaya-Babuska-Brezzi （LBB） condition. The authors employ least-squares functionals which involve a discrete inner product which is related to the inner product in H^-1（Ω）.

SUBNORMAL SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS

In this article, we apply the concept of hyper-order to higher order linear differential equations with periodic coefficients, investigate the existence and the form of its subnormal solution, and estimate the growth of all other solutions, and answer the question raised by Gundersen and Steinbart for more general periodic differential equations.

Multiple attenuation using λ-f domain high-order and high-resolution Radon transform based on SL0 norm

Radon transform is to use the speed difference between primary wave and multiple wave to focus the difference on different"points"or"lines"in Radon domain,so as to suppress multiple wave.However,the limited migration aperture,discrete sampling,and AVO characteristics of seismic data all will weaken the focusing characteristics of Radon transform.In addition,the traditional Radon transform does not take into account the AVO characteristics of seismic data,and uses L1 Norm,the approximate form of L0 Norm,to improve the focusing characteristics of Radon domain,which requires a lot of computation.In this paper,we combine orthogonal polynomials with the parabolic Radon transform(PRT)and find that the AVO characteristics of seismic data can be fitted with orthogonal polynomial coefficients.This allows the problem to be transformed into the frequency domain by Fourier transform and introduces a new variable,lambda,combining frequency and curvature.Through overall sampling of lambda,the PRT operator only needs to be calculated once for each frequency,yielding higher computational efficiency.The sparse solution of PRT under the constraints of the smoothed L0 Norm(SL0)obtained by the steepest descent method and the gradient projection principle.Synthetic and real examples are given to demonstrate that the proposed method has This method has advantages in improving the Radon focusing characteristics than does the PRT based on L1 norm.

EXISTENCE AND UNIQUENESS OF RENORMALIZED SOLUTIONS FOR A CLASS OF DEGENERATE PARABOLIC EQUATIONS

This article discusses the existence and uniqueness of renormalized solutions for a class of degenerate parabolic equations b（u）t - div（a（u, u）） = H（u）（f ＋ divg）.

Least Squares Solution for Discrete Time Nonlinear Stochastic Optimal Control Problem with Model-Reality Differences

In this paper, an efficient computational approach is proposed to solve the discrete time nonlinear stochastic optimal control problem. For this purpose, a linear quadratic regulator model, which is a linear dynamical system with the quadratic criterion cost function, is employed. In our approach, the model-based optimal control problem is reformulated into the input-output equations. In this way, the Hankel matrix and the observability matrix are constructed. Further, the sum squares of output error is defined. In these point of views, the least squares optimization problem is introduced, so as the differences between the real output and the model output could be calculated. Applying the first-order derivative to the sum squares of output error, the necessary condition is then derived. After some algebraic manipulations, the optimal control law is produced. By substituting this control policy into the input-output equations, the model output is updated iteratively. For illustration, an example of the direct current and alternating current converter problem is studied. As a result, the model output trajectory of the least squares solution is close to the real output with the smallest sum squares of output error. In conclusion, the efficiency and the accuracy of the approach proposed are highly presented.

A Comparative Study of Initial Basic Feasible Solution by a Least Cost Mean Method (LCMM) of Transportation Problem

Transportation Problems (TP), as is known, are a basic network problem which can be formulated as a Linear Programming Problem (LPP). Transportation networks are built up in order to save transportation cost. In the solution procedure of a TP, finding an Initial Basic Feasible Solution (IBFS) is necessary to obtain the optimal solution. Optimality gives us the optimal route that prompts either most extreme benefit or least aggregate cost whichever is required. In this research paper, a new method named Least Cost Mean Method is proposed to obtain a better IBFS where row penalty and column penalty is brought out by the mean of lowest and next lowest cost of each row and each column of the cost matrix. The method is illustrated with numerical examples. To verify the performance of the proposed method, a comparative study is also carried out and observed that it is computationally easier and yielding comparatively better solution.

The Least Square Solutions to the Quaternion Matrix Equation AX=B

A norm of a quaternion matrix is defined. The expressions of the least square solutions of the quaternion matrix equation AX = B and the equation with the constraint condition DX = E are given.

Near-Infrared Spectroscopy Combined with Absorbance Upper Optimization Partial Least Squares Applied to Rapid Analysis of Polysaccharide for Proprietary Chinese Medicine Oral Solution

Near-infrared (NIR) spectroscopy was applied to reagent-free quantitative analysis of polysaccharide of a brand product of proprietary Chinese medicine (PCM) oral solution samples. A novel method, called absorbance upper optimization partial least squares (AUO-PLS), was proposed and successfully applied to the wavelength selection. Based on varied partitioning of the calibration and prediction sample sets, the parameter optimization was performed to achieve stability. On the basis of the AUO-PLS method, the selected upper bound of appropriate absorbance was 1.53 and the corresponding wavebands combination was 400 - 1880 & 2088 - 2346 nm. With the use of random validation samples excluded from the modeling process, the root-mean-square error and correlation coefficient of prediction for polysaccharide were 27.09 mg·L-1 and 0.888, respectively. The results indicate that the NIR prediction values are close to those of the measured values. NIR spectroscopy combined with AUO-PLS method provided a promising tool for quantification of the polysaccharide for PCM oral solution and this technique is rapid and simple when compared with conventional methods.

Common least-squares solution to some matrix equations

Necessary and sufficient conditions are derived for some matrix equations that have a common least-squares solution.A general expression is provided when certain resolvable conditions are satisfied.This research extends existing work in the literature.

Robust least squares projection twin SVM and its sparse solution

Least squares projection twin support vector machine(LSPTSVM)has faster computing speed than classical least squares support vector machine(LSSVM).However,LSPTSVM is sensitive to outliers and its solution lacks sparsity.Therefore,it is difficult for LSPTSVM to process large-scale datasets with outliers.In this paper,we propose a robust LSPTSVM model(called R-LSPTSVM)by applying truncated least squares loss function.The robustness of R-LSPTSVM is proved from a weighted perspective.Furthermore,we obtain the sparse solution of R-LSPTSVM by using the pivoting Cholesky factorization method in primal space.Finally,the sparse R-LSPTSVM algorithm(SR-LSPTSVM)is proposed.Experimental results show that SR-LSPTSVM is insensitive to outliers and can deal with large-scale datasets fastly.

ON THE EXISTENCE OF SOLUTIONS FOR EQUATIONS WITH ACCRETIVE MAPPINGS IN PROBABILISTIC NORMED SPACES

The purpose of this paper is to introduce the concept of accretive mapping in probabilistic normed space (PN space, in short) and to study the existence problem of solutions for equations with accretive mappings in PN space and some existence theorems are shown.

基于最弱t-norm算法模糊贝叶斯网络土石坝渗漏风险分析

针对土石坝工程渗漏影响要素多,风险评估体系不完善的问题,提出一种基于最弱t-norm算法模糊贝叶斯网络的土石坝渗漏风险分析模型。首先基于土石坝渗漏破坏成因构建土石坝渗漏风险分析网络,再基于层次分析法和最弱t-norm聚合算法求解不同专家判断的相似性结果,最后将相似性结果转化为概率后代入贝叶斯理论计算水库渗漏破坏风险。以2005年英德尔水库渗漏破坏事件为例,计算得到该库渗漏破坏概率为0.08,属于不可接受风险,并通过反向诊断验证了该模型的可靠性与灵敏性优于传统算法。研究结果可为土石坝渗流风险分析与处置决策提供参考。