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
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
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
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
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
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
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.
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
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
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
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
In this work we try to give a new contraction type in multi-valued mapping on complete metric spaces. We prove the existence of fixed point for (<i>r</i>,<i>φ</i>,<i>ψ</i>)-Suzuki...In this work we try to give a new contraction type in multi-valued mapping on complete metric spaces. We prove the existence of fixed point for (<i>r</i>,<i>φ</i>,<i>ψ</i>)-Suzuki contraction in such spaces. Around our paper, the function <i>ψ</i> is absolutely continuous, and in this case, the contraction proposed by as has a fixed point.展开更多
In this paper, we investigate the continuous dependence of solutions of the functional dif-ferential equation with infinite delay x'(t) = f(t, x_t) on initial functions. Endowing the phasespare a g-norm as well as...In this paper, we investigate the continuous dependence of solutions of the functional dif-ferential equation with infinite delay x'(t) = f(t, x_t) on initial functions. Endowing the phasespare a g-norm as well as a supremum norm. we show that if the equation satifies a mild fadingmemory dondition, then the continuity of f in respect to the topology induced by the supremumnorm can yield the continuity of solutions of the equation in respect to the topology induced bythe g-norm which is stronger than the ahead one.展开更多
In this paper, we introduce the notion of intuitionistic fuzzy α-generalized closed sets in intuitionistic fuzzy minimal structure spaces and investigate some of their properties. Further, we introduce and study the ...In this paper, we introduce the notion of intuitionistic fuzzy α-generalized closed sets in intuitionistic fuzzy minimal structure spaces and investigate some of their properties. Further, we introduce and study the concept of intuitionistic fuzzy α-generalized minimal continuous functions.展开更多
In this paper, the strictly weak major efficient point of set is introduced. A functional as a separate function is constructed, therefore, a necessary and sufficient condition for the strictly weak major efficient po...In this paper, the strictly weak major efficient point of set is introduced. A functional as a separate function is constructed, therefore, a necessary and sufficient condition for the strictly weak major efficient point of set is established.展开更多
Using known operator-valued Fourier multiplier results on vectorvalued HSlder continuous function spaces, we completely characterize the wellposedness of the degenerate differential equations (Mu)'(t) = Au(t) ...Using known operator-valued Fourier multiplier results on vectorvalued HSlder continuous function spaces, we completely characterize the wellposedness of the degenerate differential equations (Mu)'(t) = Au(t) + f(t) for t ∈ R in HSlder continuous function spaces C^α(R; X) by the boundedness of the M-resolvent of A, where A and M are closed operators on a Banach space X satisfying D(A) D(M).展开更多
The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the...The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e. g., if D = F 2 + c, then h(D) ≥ degF/degP; if D = (SG)2 + cS, then h(D) ≥ degS/degP; if D = (A m + a)2 + A, then h(D) ≥ degA/degP, where P is an irreducible polynomial splitting in K, c ∈ F q . In addition, three types of quadratic function fields K are found to have ideal class numbers bigger than one.展开更多
Objective To assess renal function in patients with obstructive sleep apnea syndrome (OSAS) during two nights when they were untreated and treated with continuous positive airway pressure (CPAP). Methods Tubular fu...Objective To assess renal function in patients with obstructive sleep apnea syndrome (OSAS) during two nights when they were untreated and treated with continuous positive airway pressure (CPAP). Methods Tubular function was estimated on the basis of the renal lithium clearance and its derived formulae, urinary osmolality, osmolal clearance and free water clearance; while glomerular function was evaluated by endogenous creatinine clearance. Twenty patients with OSAS and sixteen normal controls were studied. Results The untreated patients with OSAS had significantly lower fractional proximal tubular sodium reabsorption (59.1%±7.1%,P<0.001 vs controls), lower fractional distal tubular sodium reabsorption (93.8%±1.7%, P<0.05 vs controls) and urinary osmolality (573.0±107.9 mosm·kg 1 , P<0.05 vs controls). The absolute distal tubular reabsorption rate of sodium and osmolal clearance in the untreated patients were higher 47.0±26.0 mmol·L 1 and 25.0±4.1 ml·L 1 ; P< 0.01 and P<0.05 respectively vs controls). During CPAP treated nights all abnormal renal function indexes in the OSAS patients restored to normal control values (P>0.05). Conclusions The natriuresis and diuresis of OSAS patients were due to the decrease in proximal and distal tubular sodium reabsorption and in tubular concentration ability during their nocturnal sleep and were returned to normal by CPAP therapy.展开更多
Using known Ca-multiplier result, we give necessary and sufficient conditions for the second order delay equations:u″(t)=Au(t)+Fut+Gu′+f(t),t∈Rto have maximal regularity in HSlder continuous function spac...Using known Ca-multiplier result, we give necessary and sufficient conditions for the second order delay equations:u″(t)=Au(t)+Fut+Gu′+f(t),t∈Rto have maximal regularity in HSlder continuous function spaces C^α (R, X), where X is a Banach space, A is a closed operator in X, F, G ∈L(C([-r, 0], X), X) are delay operators for some fixed r 〉 0.展开更多
In this paper, iterative or successive approximation methods for the Hamilton-Jacobi-Bellman-lsaacs equations (HJBIEs) arising in both deterministic and stochastic optimal control for affine nonlinear systems are de...In this paper, iterative or successive approximation methods for the Hamilton-Jacobi-Bellman-lsaacs equations (HJBIEs) arising in both deterministic and stochastic optimal control for affine nonlinear systems are developed. Convergence of the methods are established under fairly mild assumptions, and examples are solved to demonstrate the effectiveness of the methods. However, the results presented in the paper are preliminary, and do not yet imply in anyway that the solutions computed will be stabilizing. More improvements and experimentation will be required before a satisfactory algorithm is developed.展开更多
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
基金The work is supported by Project 69 with Ministry of ScienceEducation, Bulgaria.
文摘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.
文摘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.
文摘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.
基金Supported by the National Natural Science Foundation of China (10771049, 10801043)the Hebei Natural Science Foundation (A2007000225, A2010000346)
文摘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.
文摘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.
文摘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.
文摘In this work we try to give a new contraction type in multi-valued mapping on complete metric spaces. We prove the existence of fixed point for (<i>r</i>,<i>φ</i>,<i>ψ</i>)-Suzuki contraction in such spaces. Around our paper, the function <i>ψ</i> is absolutely continuous, and in this case, the contraction proposed by as has a fixed point.
基金This research was supported in part by an NSF grant with number NSY-DMS-8521408. On leave from South China Normal University, Guangshou, PRC. This research was supported in part by the National Science Foundation of PRC
文摘In this paper, we investigate the continuous dependence of solutions of the functional dif-ferential equation with infinite delay x'(t) = f(t, x_t) on initial functions. Endowing the phasespare a g-norm as well as a supremum norm. we show that if the equation satifies a mild fadingmemory dondition, then the continuity of f in respect to the topology induced by the supremumnorm can yield the continuity of solutions of the equation in respect to the topology induced bythe g-norm which is stronger than the ahead one.
文摘In this paper, we introduce the notion of intuitionistic fuzzy α-generalized closed sets in intuitionistic fuzzy minimal structure spaces and investigate some of their properties. Further, we introduce and study the concept of intuitionistic fuzzy α-generalized minimal continuous functions.
文摘In this paper, the strictly weak major efficient point of set is introduced. A functional as a separate function is constructed, therefore, a necessary and sufficient condition for the strictly weak major efficient point of set is established.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 11171172) and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20120002110044).
文摘Using known operator-valued Fourier multiplier results on vectorvalued HSlder continuous function spaces, we completely characterize the wellposedness of the degenerate differential equations (Mu)'(t) = Au(t) + f(t) for t ∈ R in HSlder continuous function spaces C^α(R; X) by the boundedness of the M-resolvent of A, where A and M are closed operators on a Banach space X satisfying D(A) D(M).
文摘The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e. g., if D = F 2 + c, then h(D) ≥ degF/degP; if D = (SG)2 + cS, then h(D) ≥ degS/degP; if D = (A m + a)2 + A, then h(D) ≥ degA/degP, where P is an irreducible polynomial splitting in K, c ∈ F q . In addition, three types of quadratic function fields K are found to have ideal class numbers bigger than one.
文摘Objective To assess renal function in patients with obstructive sleep apnea syndrome (OSAS) during two nights when they were untreated and treated with continuous positive airway pressure (CPAP). Methods Tubular function was estimated on the basis of the renal lithium clearance and its derived formulae, urinary osmolality, osmolal clearance and free water clearance; while glomerular function was evaluated by endogenous creatinine clearance. Twenty patients with OSAS and sixteen normal controls were studied. Results The untreated patients with OSAS had significantly lower fractional proximal tubular sodium reabsorption (59.1%±7.1%,P<0.001 vs controls), lower fractional distal tubular sodium reabsorption (93.8%±1.7%, P<0.05 vs controls) and urinary osmolality (573.0±107.9 mosm·kg 1 , P<0.05 vs controls). The absolute distal tubular reabsorption rate of sodium and osmolal clearance in the untreated patients were higher 47.0±26.0 mmol·L 1 and 25.0±4.1 ml·L 1 ; P< 0.01 and P<0.05 respectively vs controls). During CPAP treated nights all abnormal renal function indexes in the OSAS patients restored to normal control values (P>0.05). Conclusions The natriuresis and diuresis of OSAS patients were due to the decrease in proximal and distal tubular sodium reabsorption and in tubular concentration ability during their nocturnal sleep and were returned to normal by CPAP therapy.
基金supported by the NSF of China (No. 10571099)Specialized Research Fund for the Doctoral Program of Higher Educationthe Tsinghua Basic Research Foundation (JCpy2005056)
文摘Using known Ca-multiplier result, we give necessary and sufficient conditions for the second order delay equations:u″(t)=Au(t)+Fut+Gu′+f(t),t∈Rto have maximal regularity in HSlder continuous function spaces C^α (R, X), where X is a Banach space, A is a closed operator in X, F, G ∈L(C([-r, 0], X), X) are delay operators for some fixed r 〉 0.
文摘In this paper, iterative or successive approximation methods for the Hamilton-Jacobi-Bellman-lsaacs equations (HJBIEs) arising in both deterministic and stochastic optimal control for affine nonlinear systems are developed. Convergence of the methods are established under fairly mild assumptions, and examples are solved to demonstrate the effectiveness of the methods. However, the results presented in the paper are preliminary, and do not yet imply in anyway that the solutions computed will be stabilizing. More improvements and experimentation will be required before a satisfactory algorithm is developed.