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A Numerical Method for Solving Ill-Conditioned Equation Systems Arising from Radial Basis Functions
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作者 Edward J. Kansa 《American Journal of Computational Mathematics》 2023年第2期356-370,共15页
Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are ... Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers. 展开更多
关键词 Continuously Differentiable Radial Basis functions Global Maxima and Minima Solutions of Ill-Conditioned Linear Equations Block Gaussian Elimination Arbitrary Precision Arithmetic
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From Hölder Continuous Solutions of 3D Incompressible Navier-Stokes Equations to No-Finite Time Blowup on [ 0,∞ ]
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作者 Terry E. Moschandreou 《Advances in Pure Mathematics》 2024年第9期695-743,共49页
This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through... This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of arbitrary r balls of the 3-Torus. The continuity equation is satisfied together with spatially periodic boundary conditions. The yicomponent forcing terms consist of a function F as part of its expression that is arbitrarily small in an r ball where it is associated with a singular forcing expression both for inviscid and viscous cases. As a result, a significant simplification occurs with a v3(vifor all velocity components) only governing PDE resulting. The extension of three restricted subspaces in each of the principal directions in the Cartesian plane is shown as the Cartesian product ℋ=Jx,t×Jy,t×Jz,t. On each of these subspaces vi,i=1,2,3is continuous and there exists a linear independent subspace associated with the argument of the W function. Here the 3-Torus is built up from each compact segment of length 2R on each of the axes on the 3 principal directions x, y, and z. The form of the scaled velocities for non zero scaled δis related to the definition of the W function such that e−W(ξ)=W(ξ)ξwhere ξdepends on t and proportional to δ→0for infinite time t. The ratio Wξis equal to 1, making the limit δ→0finite and well defined. Considering r balls where the function F=(x−ai)2+(y−bi)2+(z−ci)2−ηset equal to −1e+rwhere r>0. is such that the forcing is singular at every distance r of centres of cubes each containing an r-ball. At the centre of the balls, the forcing is infinite. The main idea is that a system of singular initial value problems with infinite forcing is to be solved for where the velocities are shown to be locally Hölder continuous. It is proven that the limit of these singular problems shifts the finite time blowup time ti∗for first and higher derivatives to t=∞thereby indicating that there is no finite time blowup. Results in the literature can provide a systematic approach to study both large space and time behaviour for singular solutions to the Navier Stokes equations. Among the references, it has been shown that mathematical tools can be applied to study the asymptotic properties of solutions. 展开更多
关键词 Navier-Stokes Periodic Navier-Stokes Equations 3-Torus PERIODIC Ball Sphere Hölder Continuous functions Uniqueness Angular Velocity Velocity in Terms of Vorticity
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On the Method of Multiplier-enlargement and Approximation of Unbounded Continuous Functions 被引量:1
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作者 郑成德 王仁宏 《Northeastern Mathematical Journal》 CSCD 2001年第2期231-235,共5页
By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive... By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive linear operators. As an example, Hermite Fejér interpolation polynomial operators are analysed and studied, and a general conclusion is obtained. 展开更多
关键词 positive linear operator approximation of unbounded continuous function method of multiplier enlargement Hermite Fejér interpolation polynomial operator
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CONSTRUCTION OF SOME KIESSWETTER-LIKE FUNCTIONS-THE CONTINUOUS BUT NON-DIFFERENT-IABLE FUNCTION DEFINED BY QUINARY DECIMAL 被引量:1
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作者 TieYong YangGuangjun 《Analysis in Theory and Applications》 2004年第1期58-68,共11页
In this paper, we construct some continuous but non-differentiable functions defined by quinary dec-imal, that are Kiesswetter-like functions. We discuss their properties, then investigate the Hausdorff dimensions of ... In this paper, we construct some continuous but non-differentiable functions defined by quinary dec-imal, that are Kiesswetter-like functions. We discuss their properties, then investigate the Hausdorff dimensions of graphs of these functions and give a detailed proof. 展开更多
关键词 Kiesswetter-like functions continuous but non-differentiable quinary decimal iterated function system inequality Hausdorff dimension
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On Approximation of Nonbounded Continuous Functions
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作者 ZHENGCheng-de WANGRen-hong 《Chinese Quarterly Journal of Mathematics》 CSCD 2003年第1期44-48,共5页
This paper generalizes the basic principle of multiplier-enlargement approach to approximating any nonbounded continuous functions with positive linear operators, and as an example, Bernstein polynomial operators are ... This paper generalizes the basic principle of multiplier-enlargement approach to approximating any nonbounded continuous functions with positive linear operators, and as an example, Bernstein polynomial operators are analysed and studied. This paper gives a certain theorem as a general rule to approximate any nonbounded continuous functions. 展开更多
关键词 positive linear operator approximation of nonbounded continuous function method of multiplier-enlargement Bernstein polynomial operator
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On the generalized Cauchy function and new Conjecture on its exterior singularities 被引量:1
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作者 Theodore Yaotsu Wu 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2011年第2期135-151,共17页
This article studies on Cauchy’s function f (z) and its integral, (2πi)J[ f (z)] ≡ ■C f (t)dt/(t z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z =... This article studies on Cauchy’s function f (z) and its integral, (2πi)J[ f (z)] ≡ ■C f (t)dt/(t z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x + iy plane consisted of the simply connected open domain D + bounded by C and the open domain D outside C. (1) With f (z) assumed to be C n (n ∞-times continuously differentiable) z ∈ D + and in a neighborhood of C, f (z) and its derivatives f (n) (z) are proved uniformly continuous in the closed domain D + = [D + + C]. (2) Cauchy’s integral formulas and their derivatives z ∈ D + (or z ∈ D ) are proved to converge uniformly in D + (or in D = [D +C]), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) The same claims (as for f (z) and J[ f (z)]) are shown extended to hold for the complement function F(z), defined to be C n z ∈ D and about C. (4) The uniform convergence theorems for f (z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four general- ized Hilbert-type integral transforms are proved. (5) Further, the singularity distribution of f (z) in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . (6) A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. (7) Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f (z) in domain D , based on the continuous numerical value of f (z) z ∈ D + = [D + + C], is presented for resolution as a conjecture. 展开更多
关键词 Uniform continuity of Cauchy’s function · Uni- form convergence of Cauchy’s integral formula · Generalized Hilbert-type integral transforms · functional properties and singularity distributions
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APPROXIMATION OF CONTINUOUS FUNCTIONS BY GENERALIZED KARAMATA MEANS
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作者 S.A.SETTU 《Analysis in Theory and Applications》 1994年第1期88-98,共11页
A sufficient condition for the order of approximation of a continuous 2π periodic function with a given majorant for the modulus of continuity by the [F, d_n] means of its Fourier serier to be of Jackson order is obt... A sufficient condition for the order of approximation of a continuous 2π periodic function with a given majorant for the modulus of continuity by the [F, d_n] means of its Fourier serier to be of Jackson order is obtained. This sufficient condition is shown to be not enough for the order of approximation by partial sums of their Fourier series to be of Jackson order. The error estimate is shown to be the best possible. 展开更多
关键词 exp APPROXIMATION OF CONTINUOUS functionS BY GENERALIZED KARAMATA MEANS
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RATIONAL FUNCTIONS OF BEST UNIFORM APPROXIMATION AND HOLOMORPHIC CONTINUATION OF FUNCTIONS
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作者 I.V.Ivanov R.K.Kovacheva J.Gilewicz 《Analysis in Theory and Applications》 1996年第1期1-9,共9页
Let f be a function, continuous and real valued on the segment △,△ (-∞,∞) and {Rn} be the sequence of the rational functions of best uniform approximation to fon △ of order (n,n). In the present work, the converg... Let f be a function, continuous and real valued on the segment △,△ (-∞,∞) and {Rn} be the sequence of the rational functions of best uniform approximation to fon △ of order (n,n). In the present work, the convergence of {Rn} in the complex plane is considered for the special caseswhen the poles (or the zeros, respectively) of {Rn} accumulate in the terms of weak convergence of measures to acompact set of zera capacity.As a consequence, sufficient conditions for the holomorphic and the meromorphic continuability of fare given. 展开更多
关键词 RATIONAL functionS OF BEST UNIFORM APPROXIMATION AND HOLOMORPHIC CONTINUATION OF functionS
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A Basic Method of Constructing Production Functions
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作者 Guo Yaohuang Wang Yaping Zhang Wel(School of Economics and Management,Soultwest Jiaotong University), Chengdu 610031,China 《Journal of Modern Transportation》 1995年第2期127-132,共6页
In this paper,a new basic method of constructing smooth production functions isgiven,and many ordinary production functions are drawn. It is obvious thatsome more production functions can be obtained from the method.
关键词 production function continuously differentiable function homogeneousfunction
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A Technique for Estimation of Box Dimension about Fractional Integral
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作者 Ruhua Zhang 《Advances in Pure Mathematics》 2023年第10期714-724,共11页
This paper discusses further the roughness of Riemann-Liouville fractional integral on an arbitrary fractal continuous functions that follows Rfs. [1]. A novel method is used to reach a similar result for an arbitrary... This paper discusses further the roughness of Riemann-Liouville fractional integral on an arbitrary fractal continuous functions that follows Rfs. [1]. A novel method is used to reach a similar result for an arbitrary fractal function , where is the Riemann-Liouville fractional integral. Furthermore, a general resultis arrived at for 1-dimensional fractal functions such as with unbounded variation and(or) infinite lengths, which can infer all previous studies such as [2] [3]. This paper’s estimation reveals that the fractional integral does not increase the fractal dimension of f(x), i.e. fractional integration does not increase at least the fractal roughness. And the result has partly answered the fractal calculus conjecture and completely answered this conjecture for all 1-dimensional fractal function (Xiao has not answered). It is significant with a comparison to the past researches that the box dimension connection between a fractal function and its Riemann-Liouville integral has been carried out only for Weierstrass type and Besicovitch type functions, and at most Hlder continuous. Here the proof technique for Riemann-Liouville fractional integral is possibly of methodology to other fractional integrals. 展开更多
关键词 Upper Box Dimension Riemann-Liouville Fractional Integral Fractal Continuous function Box Dimension
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Global Convergence of a New Restarting Conjugate Gradient Method for Nonlinear Optimizations 被引量:1
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作者 SUN Qing-ying(Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China Department of Applied Mathematics, University of Petroleum , Dongying 257061, China) 《Chinese Quarterly Journal of Mathematics》 CSCD 2003年第2期154-162,共9页
Conjugate gradient optimization algorithms depend on the search directions with different choices for the parameters in the search directions. In this note, by combining the nice numerical performance of PR and HS met... Conjugate gradient optimization algorithms depend on the search directions with different choices for the parameters in the search directions. In this note, by combining the nice numerical performance of PR and HS methods with the global convergence property of the class of conjugate gradient methods presented by HU and STOREY(1991), a class of new restarting conjugate gradient methods is presented. Global convergences of the new method with two kinds of common line searches, are proved. Firstly, it is shown that, using reverse modulus of continuity function and forcing function, the new method for solving unconstrained optimization can work for a continously dif ferentiable function with Curry-Altman's step size rule and a bounded level set. Secondly, by using comparing technique, some general convergence properties of the new method with other kind of step size rule are established. Numerical experiments show that the new method is efficient by comparing with FR conjugate gradient method. 展开更多
关键词 nonlinear programming restarting conjugate gradient method forcing function reverse modulus of continuity function CONVERGENCE
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Global Convergence of a New Restarting Three Terms Conjugate Gradient Method for Non-linear Optimizations 被引量:1
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作者 SUN Qing-ying SANG Zhao-yang TIAN Feng-ting 《Chinese Quarterly Journal of Mathematics》 CSCD 2011年第1期69-76,共8页
In this note,by combining the nice numerical performance of PR and HS methods with the global convergence property of FR method,a class of new restarting three terms conjugate gradient methods is presented.Global conv... In this note,by combining the nice numerical performance of PR and HS methods with the global convergence property of FR method,a class of new restarting three terms conjugate gradient methods is presented.Global convergence properties of the new method with two kinds of common line searches are proved. 展开更多
关键词 nonlinear programming restarting three terms conjugate gradient method forcing function reverse modulus of continuity function convergence
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Global Convergence of a New Conjugate Gradient Methods with a Modification of the Curry-Altman Step-size Rule
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作者 CAOLi-hua SUNQing-ying 《Chinese Quarterly Journal of Mathematics》 CSCD 2004年第2期198-204,共7页
Conjugate gradient optimization algorithms depend on the search directions with different choices for the parameter in the search directions. In this note, conditions are given on the parameter in the conjugate gradie... Conjugate gradient optimization algorithms depend on the search directions with different choices for the parameter in the search directions. In this note, conditions are given on the parameter in the conjugate gradient directions to ensure the descent property of the search directions. Global convergence of such a class of methods is discussed. It is shown that, using reverse modulus of continuity function and forcing function, the new method for solving unconstrained optimization can work for a continuously differentiable function with a modification of the Curry-Altman's step-size rule and a bounded level set. Combining PR method with our new method, PR method is modified to have global convergence property.Numerical experiments show that the new methods are efficient by comparing with FR conjugate gradient method. 展开更多
关键词 nonlinear programming forcing function reverse modulus of continuity function convergence
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Exact Convergence Rates of Functional Modulus of Continuity of a Wiener Process
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作者 王文胜 《Journal of Mathematical Research and Exposition》 CSCD 北大核心 2002年第4期507-514,共8页
Let {W(t),t > 0} be a standard Wiener process and S be the set of Strassen's functions. In this paper we investigate the exact rates of convergence to zero of the variables supp<t<1-h inff∈s sup0<x<... Let {W(t),t > 0} be a standard Wiener process and S be the set of Strassen's functions. In this paper we investigate the exact rates of convergence to zero of the variables supp<t<1-h inff∈s sup0<x<1 |(W(t + hx) - W(t))(2hlogh-1)-1/2 - f(x)| and inf0<t<1-h sup0<x<1|(W(t + hx) -W(t))(2hlogh-1)-1/2 - f(x)| for any f ∈ S. As a consequence, a relation between the modulus of non-differentiability and the functional modulus of continuity for a Wiener process is established. 展开更多
关键词 Wiener process functional modulus of continuity modulus of non-differentiability
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Boundary properties for several singular integral operators in real Clifford analysis 被引量:7
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作者 YANG He-ju XIE Yong-hong 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2010年第3期349-358,共10页
On the basis of the Cauchy integral formulas for regular and biregular functions, we define some Cauchy-type singular integral operators. Then we discuss the Holder continuous property of some singular integral operat... On the basis of the Cauchy integral formulas for regular and biregular functions, we define some Cauchy-type singular integral operators. Then we discuss the Holder continuous property of some singular integral operators with one integral variable. Then we divide a singular integral operator with two variables into three parts and prove its Holder continuous property on the boundary. 展开更多
关键词 Singular integral operator Holder continuous function boundary property.
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FUNCTIONAL MODULUS OF CONTINUITY FOR BROWNIAN MOTION IN HLDER NORM 被引量:10
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作者 WEI QICAI School of Economics, Zhejiang University, Hangzhou 310028, China. 《Chinese Annals of Mathematics,Series B》 SCIE CSCD 2001年第2期223-232,共10页
The author establishes a large deviation for k-dimensional Brownian motion B in stronger topology, by which the functional modulus of continuity for B in Holder norm can be obtained.
关键词 Large deviations functional modulus of continuity k-dimensional Brownian motion Holder norm
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Development of slope mass rating system using K-means and fuzzy c-means clustering algorithms 被引量:1
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作者 Jalali Zakaria 《International Journal of Mining Science and Technology》 SCIE EI CSCD 2016年第6期959-966,共8页
Classification systems such as Slope Mass Rating(SMR) are currently being used to undertake slope stability analysis. In SMR classification system, data is allocated to certain classes based on linguistic and experien... Classification systems such as Slope Mass Rating(SMR) are currently being used to undertake slope stability analysis. In SMR classification system, data is allocated to certain classes based on linguistic and experience-based criteria. In order to eliminate linguistic criteria resulted from experience-based judgments and account for uncertainties in determining class boundaries developed by SMR system,the system classification results were corrected using two clustering algorithms, namely K-means and fuzzy c-means(FCM), for the ratings obtained via continuous and discrete functions. By applying clustering algorithms in SMR classification system, no in-advance experience-based judgment was made on the number of extracted classes in this system, and it was only after all steps of the clustering algorithms were accomplished that new classification scheme was proposed for SMR system under different failure modes based on the ratings obtained via continuous and discrete functions. The results of this study showed that, engineers can achieve more reliable and objective evaluations over slope stability by using SMR system based on the ratings calculated via continuous and discrete functions. 展开更多
关键词 SMR based on continuous functions Slope stability analysis K-means and FCM clustering algorithms Validation of clustering algorithms Sangan iron ore mines
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An Iterative Relaxation Approach to the Solution of the Hamilton-Jacobi-Bellman-Isaacs Equation in Nonlinear Optimal Control
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作者 M.D.S.Aliyu 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI CSCD 2018年第1期360-366,共7页
In this paper, we propose an iterative relaxation method for solving the Hamilton-Jacobi-Bellman-Isaacs equation(HJBIE) arising in deterministic optimal control of affine nonlinear systems. Local convergence of the me... In this paper, we propose an iterative relaxation method for solving the Hamilton-Jacobi-Bellman-Isaacs equation(HJBIE) arising in deterministic optimal control of affine nonlinear systems. Local convergence of the method is established under fairly mild assumptions, and examples are solved to demonstrate the effectiveness of the method. An extension of the approach to Lyapunov equations is also discussed. The preliminary results presented are promising, and it is hoped that the approach will ultimately develop into an efficient computational tool for solving the HJBIEs. 展开更多
关键词 Affine nonlinear system bounded continuous function CONVERGENCE Hamilton-Jacobi-Bellman-Isaacs equation Lyapunov equation relaxation method Riccati equation
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Weighted ebyev-Ostrowski type inequalities
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作者 Arif Rafiq Nazir Ahmad Mir Farooq Ahmad 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2007年第7期901-906,共6页
In account of the famous Cebysev inequality, a rich theory has appeared in the literature. We establish some new weighted Cebysev type integral inequalities. Our proofs are of independent interest and provide new esti... In account of the famous Cebysev inequality, a rich theory has appeared in the literature. We establish some new weighted Cebysev type integral inequalities. Our proofs are of independent interest and provide new estimates on these types of inequalities. 展开更多
关键词 Cebysev type inequalities L∞ (a b) spaces absolutely continuous functions weight functions
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Some Remarks on an Integral Inequality
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作者 HU Jian-jun 《Chinese Quarterly Journal of Mathematics》 CSCD 北大核心 2008年第4期554-558,共5页
Given two positive constants α and β, we prove that the integral inequality ∫_0^1f^α+β(x)dx≥∫_0^1∫^α(x)x^β dx holds for all non-negative valued continuous functions ∫ satisfying ∫_x^1f(t)dt≥∫_x^1t... Given two positive constants α and β, we prove that the integral inequality ∫_0^1f^α+β(x)dx≥∫_0^1∫^α(x)x^β dx holds for all non-negative valued continuous functions ∫ satisfying ∫_x^1f(t)dt≥∫_x^1tdt for x∈[0,1] if and only if α+β≥1.This solves an open problem proposed recently by Ngo, Thang, Dat, and Tuan. 展开更多
关键词 INTEGRAL INEQUALITY continuous function
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