In this work,we apply the Brzdȩk and Ciepliński's fixed point theorem to investigate new stability results for the generalized Cauchy functional equation of the form f(ax+by)=af(x)+bf(y),where a,b∈N and f is a m...In this work,we apply the Brzdȩk and Ciepliński's fixed point theorem to investigate new stability results for the generalized Cauchy functional equation of the form f(ax+by)=af(x)+bf(y),where a,b∈N and f is a mapping from a commutative group(G,+)to a 2-Banach space(Y,||·,·||).Our results are generalizations of main results of Brzdȩk and Ciepliński[J Brzdȩk,K Ciepliński.On a fixed point theorem in 2-normed spaces and some of its applications.Acta Mathematica Scientia,2018,38B(2):377-390].展开更多
This is a series of studies on Wu＇s conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy＇s function f（z） （z = x ＋ iy） and its integral ...This is a series of studies on Wu＇s conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy＇s function f（z） （z = x ＋ iy） and its integral J[f（z）]≡（2πi）-∮cf（t）（t-z）-1dt taken along the unit circle as contour C,inside which（the open domain D＋） f（z） is regular but has singularities distributed in open domain Doutside C. Resolution is given to the inverse problem that the singularities of f（z） can be determined in analytical form in terms of the values f（t） of f（z） numerically prescribed on C（｜t｜ = 1） ,as so enunciated by Wu＇s conjecture. The case of a single singularity is solved using complex algebra and analysis to acquire the solution structure for a standard reference. Multiple singularities are resolved by reducing them to a single one by elimination in principle,for which purpose a general asymptotic method is developed here for resolution to the conjecture by induction,and essential singularities are treated with employing the generalized Hilbert transforms. These new methods are applicable to relevant problems in mathematics,engineering and technology in analogy with resolving the inverse problem presented here.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
In characterizing the semistable law, Shimizu reduced the problem to solving the equationH(x)=integral from n=1 to∞(H(x+y)d(μ-ν)(y), x≥0) where μ andτ are given positive measures on [0,∞). In thisnote, we obtai...In characterizing the semistable law, Shimizu reduced the problem to solving the equationH(x)=integral from n=1 to∞(H(x+y)d(μ-ν)(y), x≥0) where μ andτ are given positive measures on [0,∞). In thisnote, we obtain a simple proof and show that some of his conditions can be weakened.展开更多
In this paper, the Cauchy type integral for M-analytic function is investigated which is by definition the regular solution of the elliptic system f_x+Mf_y=0, where M is a constant m×m matrix without any real eig...In this paper, the Cauchy type integral for M-analytic function is investigated which is by definition the regular solution of the elliptic system f_x+Mf_y=0, where M is a constant m×m matrix without any real eigenvalues and f is an m×q matrix.展开更多
Generally,the classic iterative learning control(ILC)methods focus on finding design conditions for repetitive systems to achieve the perfect tracking of any specified trajectory,whereas they ignore a fundamental prob...Generally,the classic iterative learning control(ILC)methods focus on finding design conditions for repetitive systems to achieve the perfect tracking of any specified trajectory,whereas they ignore a fundamental problem of ILC:whether the specified trajectory is trackable,or equivalently,whether there exist some inputs for the repetitive systems under consideration to generate the specified trajectory?The current paper contributes to dealing with this problem.Not only is a concept of trackability introduced formally for any specified trajectory in ILC,but also some related trackability criteria are established.Further,the relation between the trackability and the perfect tracking tasks for ILC is bridged,based on which a new convergence analysis approach is developed for ILC by leveraging properties of a functional Cauchy sequence(FCS).Simulation examples are given to verify the effectiveness of the presented trackability criteria and FCS-induced convergence analysis method for ILC.展开更多
In this paper, the author proves the Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras. This is used to investigate isomorphisms between quasi-Banach algebras.
We consider a strictly pathwise setting for Delta hedging exotic options,based on Follmer’s pathwise It¨o calculus.Price trajectories areˆd-dimensional continuous functions whose pathwise quadratic variations an...We consider a strictly pathwise setting for Delta hedging exotic options,based on Follmer’s pathwise It¨o calculus.Price trajectories areˆd-dimensional continuous functions whose pathwise quadratic variations and covariations are determined by a given local volatility matrix.The existence of Delta hedging strategies in this pathwise setting is established via existence results for recursive schemes of parabolic Cauchy problems and via the existence of functional Cauchy problems on path space.Our main results establish the nonexistence of pathwise arbitrage opportunities in classes of strategies containing these Delta hedging strategies and under relatively mild conditions on the local volatility matrix.展开更多
基金This work was supported by Research Professional Development Project under the Science Achievement Scholarship of Thailand(SAST)and Thammasat University Research Fund,Contract No.TUGG 33/2562The second author would like to thank the Thailand Research Fund and Office of the Higher Education Commission under grant no.MRG6180283 for financial support during the preparation of this manuscript.
文摘In this work,we apply the Brzdȩk and Ciepliński's fixed point theorem to investigate new stability results for the generalized Cauchy functional equation of the form f(ax+by)=af(x)+bf(y),where a,b∈N and f is a mapping from a commutative group(G,+)to a 2-Banach space(Y,||·,·||).Our results are generalizations of main results of Brzdȩk and Ciepliński[J Brzdȩk,K Ciepliński.On a fixed point theorem in 2-normed spaces and some of its applications.Acta Mathematica Scientia,2018,38B(2):377-390].
文摘This is a series of studies on Wu＇s conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy＇s function f（z） （z = x ＋ iy） and its integral J[f（z）]≡（2πi）-∮cf（t）（t-z）-1dt taken along the unit circle as contour C,inside which（the open domain D＋） f（z） is regular but has singularities distributed in open domain Doutside C. Resolution is given to the inverse problem that the singularities of f（z） can be determined in analytical form in terms of the values f（t） of f（z） numerically prescribed on C（｜t｜ = 1） ,as so enunciated by Wu＇s conjecture. The case of a single singularity is solved using complex algebra and analysis to acquire the solution structure for a standard reference. Multiple singularities are resolved by reducing them to a single one by elimination in principle,for which purpose a general asymptotic method is developed here for resolution to the conjecture by induction,and essential singularities are treated with employing the generalized Hilbert transforms. These new methods are applicable to relevant problems in mathematics,engineering and technology in analogy with resolving the inverse problem presented here.
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.
文摘In characterizing the semistable law, Shimizu reduced the problem to solving the equationH(x)=integral from n=1 to∞(H(x+y)d(μ-ν)(y), x≥0) where μ andτ are given positive measures on [0,∞). In thisnote, we obtain a simple proof and show that some of his conditions can be weakened.
基金This work is supported in part by the National Natural Science Foundation of China.
文摘In this paper, the Cauchy type integral for M-analytic function is investigated which is by definition the regular solution of the elliptic system f_x+Mf_y=0, where M is a constant m×m matrix without any real eigenvalues and f is an m×q matrix.
基金supported in part by the National Natural Science Foundation of China (62273018)in part by the Science and Technology on Space Intelligent Control Laboratory (HTKJ2022KL502006)。
文摘Generally,the classic iterative learning control(ILC)methods focus on finding design conditions for repetitive systems to achieve the perfect tracking of any specified trajectory,whereas they ignore a fundamental problem of ILC:whether the specified trajectory is trackable,or equivalently,whether there exist some inputs for the repetitive systems under consideration to generate the specified trajectory?The current paper contributes to dealing with this problem.Not only is a concept of trackability introduced formally for any specified trajectory in ILC,but also some related trackability criteria are established.Further,the relation between the trackability and the perfect tracking tasks for ILC is bridged,based on which a new convergence analysis approach is developed for ILC by leveraging properties of a functional Cauchy sequence(FCS).Simulation examples are given to verify the effectiveness of the presented trackability criteria and FCS-induced convergence analysis method for ILC.
基金the Korea Research Foundation (No. KRF-2005-041-C00027).
文摘In this paper, the author proves the Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras. This is used to investigate isomorphisms between quasi-Banach algebras.
基金support by Deutsche Forschungsgemeinschaft through the Research Training Group RTG 1953.
文摘We consider a strictly pathwise setting for Delta hedging exotic options,based on Follmer’s pathwise It¨o calculus.Price trajectories areˆd-dimensional continuous functions whose pathwise quadratic variations and covariations are determined by a given local volatility matrix.The existence of Delta hedging strategies in this pathwise setting is established via existence results for recursive schemes of parabolic Cauchy problems and via the existence of functional Cauchy problems on path space.Our main results establish the nonexistence of pathwise arbitrage opportunities in classes of strategies containing these Delta hedging strategies and under relatively mild conditions on the local volatility matrix.
