Some convergence theorems of fuzzy concave integral on fuzzyσ-algebra

In this paper,we consider the extension of the concave integral from classical crispσ-algebra to fuzzyσ-algebra of fuzzy sets.Firstly,the concept of fuzzy concave integral on a fuzzy set is introduced.Secondly,some important properties of such integral are discussed.Finally,various kinds of convergence theorems of a sequence of fuzzy concave integrals are proved.

EXISTENCE OF MULTIPLE SOLUTIONS FOR SINGULAR QUASILINEAR ELLIPTIC SYSTEM WITH CRITICAL SOBOLEV-HARDY EXPONENTS AND CONCAVE-CONVEX TERMS

The main purpose of this paper is to establish the existence of multiple solutions for singular elliptic system involving the critical Sobolev-Hardy exponents and concave-convex nonlinearities. It is shown, by means of variational methods, that under certain conditions, the system has at least two positive solutions.

MULTIPLE POSITIVE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEMS INVOLVING CONCAVE-CONVEX NONLINEARITIES AND MULTIPLE HARDY-TYPE TERMS

In this paper, we deal with the existence and multiplicity of positive solutions for the quasilinear elliptic problem -△pu-∑i=1^kμi｜u｜^p-2/｜x-ai｜p^u=｜u｜^p^＊-2u＋λ｜u｜^q-2u,x∈Ω,where Ω belong to R^N（N ≥ 3） is a smooth bounded domain such that the different points ai∈Ω,i= 1,2,...,k,0≤μi〈μ^-=（N-p/p）^p,λ〉0,1≤q〈p,and p^＊=p^N/N-p.The results depend crucially cn the parameters λ,q and μi for i=1,2,...,k.

Convex decomposition of concave clouds for the ultra-short-term power prediction of distributed photovoltaic system

Concave clouds will cause miscalculation by the power prediction model based on cloud ieatures for distributed photovoltaic （PV） plant. The algorithm for decomposing concave cloud into convex images is proposed. Adopting minimum polygonal approximation （MPP） to demonstrate the contour of concave cloud, cloud features are described and the subdivision lines of convex decomposition for the concave clouds are determined by the centroid point scattering model and centroid angle func- tion, which realizes the convex decomposition of concave cloud. The result of MATLAB simulation indicates that the proposed algorithm can accurately detect cloud contour comers and recognize the concave points. The proposed decomposition algorithm has advantages of less time complexity and decomposition part numbers compared to traditional algorithms. So the established model can make the convex decomposition of complex concave clouds completely and quickly, which is available for the existing prediction algorithm for the ultra-short-term power output of distributed PV system based on the cloud features.

ON SEMILINEAR ELLIPTIC EQUATIONS WITH CONCAVE AND CONVEX NONLINEARITIES IN R^N

In this paper, a class of semilinear elliptic equations with sublinear and superlinear nonlinearities in R-N is studied. By making use of variational method and L-infinity estimation, the authors obtain some results about existence of multiple positive solutions and asymptotic behavior of the solutions.

CONVEXITY OF THE FREE BOUNDARY FOR AN AXISYMMETRIC INCOMPRESSIBLE IMPINGING JET

This paper is devoted to the study of the shape of the free boundary for a threedimensional axisymmetric incompressible impinging jet.To be more precise,we will show that the free boundary is convex to the fluid,provided that the uneven ground is concave to the fluid.

On the Application of Mixed Models of Probability and Convex Set for Time-Variant Reliability Analysis

In time-variant reliability problems,there are a lot of uncertain variables from different sources.Therefore,it is important to consider these uncertainties in engineering.In addition,time-variant reliability problems typically involve a complexmultilevel nested optimization problem,which can result in an enormous amount of computation.To this end,this paper studies the time-variant reliability evaluation of structures with stochastic and bounded uncertainties using a mixed probability and convex set model.In this method,the stochastic process of a limit-state function with mixed uncertain parameters is first discretized and then converted into a timeindependent reliability problem.Further,to solve the double nested optimization problem in hybrid reliability calculation,an efficient iterative scheme is designed in standard uncertainty space to determine the most probable point(MPP).The limit state function is linearized at these points,and an innovative random variable is defined to solve the equivalent static reliability analysis model.The effectiveness of the proposed method is verified by two benchmark numerical examples and a practical engineering problem.

Distributed Stochastic Optimization with Compression for Non-Strongly Convex Objectives

We are investigating the distributed optimization problem,where a network of nodes works together to minimize a global objective that is a finite sum of their stored local functions.Since nodes exchange optimization parameters through the wireless network,large-scale training models can create communication bottlenecks,resulting in slower training times.To address this issue,CHOCO-SGD was proposed,which allows compressing information with arbitrary precision without reducing the convergence rate for strongly convex objective functions.Nevertheless,most convex functions are not strongly convex(such as logistic regression or Lasso),which raises the question of whether this algorithm can be applied to non-strongly convex functions.In this paper,we provide the first theoretical analysis of the convergence rate of CHOCO-SGD on non-strongly convex objectives.We derive a sufficient condition,which limits the fidelity of compression,to guarantee convergence.Moreover,our analysis demonstrates that within the fidelity threshold,this algorithm can significantly reduce transmission burden while maintaining the same convergence rate order as its no-compression equivalent.Numerical experiments further validate the theoretical findings by demonstrating that CHOCO-SGD improves communication efficiency and keeps the same convergence rate order simultaneously.And experiments also show that the algorithm fails to converge with low compression fidelity and in time-varying topologies.Overall,our study offers valuable insights into the potential applicability of CHOCO-SGD for non-strongly convex objectives.Additionally,we provide practical guidelines for researchers seeking to utilize this algorithm in real-world scenarios.

Relaxed Stability Criteria for Time-Delay Systems:A Novel Quadratic Function Convex Approximation Approach

This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays.By introducing two adjustable parameters and two free variables,a novel convex function greater than or equal to the quadratic function is constructed,regardless of the sign of the coefficient in the quadratic term.The developed lemma can also be degenerated into the existing quadratic function negative-determination(QFND)lemma and relaxed QFND lemma respectively,by setting two adjustable parameters and two free variables as some particular values.Moreover,for a linear system with time-varying delays,a relaxed stability criterion is established via our developed lemma,together with the quivalent reciprocal combination technique and the Bessel-Legendre inequality.As a result,the conservatism can be reduced via the proposed approach in the context of constructing Lyapunov-Krasovskii functionals for the stability analysis of linear time-varying delay systems.Finally,the superiority of our results is illustrated through three numerical examples.

Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave and Convex Nonlinearities

In this paper, we studied the combined effect of concave and convex nonlinearities on the number of positive solutions for a semilinear elliptic system. We prove the existence of at least four positive solutions for a semilinear elliptic system involving concave and convex nonlinearities by using the Nehari manifold and the center mass function.

Accelerated Primal-Dual Projection Neurodynamic Approach With Time Scaling for Linear and Set Constrained Convex Optimization Problems

The Nesterov accelerated dynamical approach serves as an essential tool for addressing convex optimization problems with accelerated convergence rates.Most previous studies in this field have primarily concentrated on unconstrained smooth con-vex optimization problems.In this paper,on the basis of primal-dual dynamical approach,Nesterov accelerated dynamical approach,projection operator and directional gradient,we present two accelerated primal-dual projection neurodynamic approaches with time scaling to address convex optimization problems with smooth and nonsmooth objective functions subject to linear and set constraints,which consist of a second-order ODE(ordinary differential equation)or differential conclusion system for the primal variables and a first-order ODE for the dual vari-ables.By satisfying specific conditions for time scaling,we demonstrate that the proposed approaches have a faster conver-gence rate.This only requires assuming convexity of the objective function.We validate the effectiveness of our proposed two accel-erated primal-dual projection neurodynamic approaches through numerical experiments.

Multiple Solutions for a Class of Variable-Order Fractional Laplacian Equations with Concave-Convex Nonlinearity

This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in RN, s(⋅) ∈ C (RN × RN, (0,1)), (-Δ)s(⋅) is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.

MULTIPLICITY OF POSITIVE SOLUTIONS FOR SINGULAR ELLIPTIC SYSTEMS WITH CRITICAL SOBOLEV-HARDY AND CONCAVE EXPONENTS

In this paper,we consider a singular elliptic system with both concave non-linearities and critical Sobolev-Hardy growth terms in bounded domains.By means of variational methods,the multiplicity of positive solutions to this problem is obtained.

CONCAVE BIOLOGICAL SURFACES FOR STRONG WET ADHESION

Plant leaves, insects and geckos are masters of adhesion or anti-adhesion by smartly designed refined surface structures with micro- and nano- ＇technologies＇. Understanding the basic principles in the design of the unique surface structures is of great importance in the manufacture or synthesis of micro- and nano- devices in MEMS or NEMS. This study is right inspired by this effort, focusing on the mechanics of wet adhesion between fibers having concave tips and a flat substrate via capillary forces. We show that the concave surface can effectively enhance the wet adhesion by reducing the effective contact angle of the fiber, firmly pinning the liquid bridge at its circumferential edge. A critical contact angle is identified below which the adhesion strength can achieve its maximum, being insensitive to the contact angle between the fiber and liquid. The analytical expression for the critical angle is derived. Then a tentative design for the profile of concave surfaces is proposed, considering the effects of chamfering size, deformation and buckling, etc. The effect of liquid volume on the wet adhesion of multiple-fiber system is also discussed.

MECHANICAL ANALYSIS OF CYLINDERS BEING UPSET BETWEEN SPHERICAL CONCAVE PLATEN AND CONCAVE SUPPORTING PLATE

Mechanical analysis of cylinders being upset between spherical concave platen and concave supporting plate is conducted. Rigid-plastic mechanical models for cylinders are presented. When the ratio of height to diameter, is larger than 1, there exists two-dimensional tensile stress in the deformed body, when the ratio is smaller than 1, there exists shear stress in static hydraulic zone. The former breaks through the theory that there is three-dimensional compressive stress irrespective of any ratio of height to diameter. The latter satisfactorily explains the mechanism of layer-like cracks in disk-shaped forgings and the flanges of forged gear axles. The representation of the two models makes the upsetting, theory into correct and perfect stage.

Concave Minimization for Sparse Solutions of Absolute Value Equations

Based on concave function, the problem of finding the sparse solution of absolute value equations is relaxed to a concave programming, and its corresponding algorithm is proposed, whose main part is solving a series of linear programming. It is proved that a sparse solution can be found under the assumption that the connected matrixes have range space property(RSP). Numerical experiments are also conducted to verify the efficiency of the proposed algorithm.

Simulation of thermal field induced by concave spherical transducer in multi-layer media

High intensity focused ultrasound(HIFU)therapy is an effective method in clinical treatment of tumors,in order to explore the bio-heat conduction mechanism of in multi-layer media by concave spherical transducer,temperature field induced by this kind of transducer in multi-layer media will be simulated through solving Pennes equation with finite difference method,and the influence of initial sound pressure,absorption coefficient,and thickness of different layers of biological tissue as well as thermal conductivity parameter on sound focus and temperature distribution will be analyzed,respectively.The results show that the temperature in focus area increases faster while the initial sound pressure and thermal conductivity increase.The absorption coefficient is smaller,the ultrasound intensity in the focus area is bigger,and the size of the focus area is increasing.When the thicknesses of different layers of tissue change,the focus position changes slightly,but the sound intensity of the focus area will change obviously.The temperature in focus area will rise quickly before reaching a threshold,and then the temperature will keep in the threshold range.

APPROXIMATE CONVEXITY AND CONCAVITY OFGENERALIZED GROTZSCH RING FUNCTION

In this paper, the so-called approximate convexity and concavity properties of generalized Groetzsch ring function μa （r） by studying the monotonieity,convexity or concavity of certain composites of μa（r） are obtained.

Error compensation on precision machine tool servo control system based on digital concave filter

It is concluded from the results of testing the frequency characteristics of the sub micron precision machine tool servo control system, that the existence of several oscillating modalities is the main factor that affects the performance of the control system. To compensate for this effect,several concave filters are utilized in the system to improve the control accuracy. The feasibility of compensating for several oscillating modalities with a single concave filter is also studied. By applying a modified Butterworth concave filter to the practical system, the maximum stable state output error remains under ±10 nm in the closed loop positioning system.

The Fixed Point Theorems of α-concave Covex Operators with Application

The coceptions of two element α-concave convex and mixed α-concave convex operators are introduced. The fixed point theorems of the two type operators are obtained. By these theorems,the existence and uniquence of solution of three type nonlinear integral equations is studied.