On the generalized Cauchy function and new Conjecture on its exterior singularities

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.

Dependence on forming parameters of an integral panel during the electromagnetic incremental forming process

Nowadays, more and more attentions are paid to electromagnetic incremental forming（EMIF）, especially for a part with a large-scale size, e.g., an integral panel with stiffened ribs. In this work, the bending of a panel into a double-curvature profile via EMIF is carried out experimentally and evaluated by comparing the formed profile with the desired profile. During the process,discharges at four positions along different discharge paths are designed. The effects of forming parameters on the die-fittingness of the workpiece are discussed, for which two evaluation indices are used to judge forming results. The results show that a discharge voltage in an incremental mode is helpful to improve the fittingness and avoid the collision rebound against the die at the same time.Discharging at the diagonal positions with the ‘‘X＂ discharge path exhibits the minimal shape deviation and the best forming uniformity. On the contrary, discharging at the parallel positions with the ‘‘Z＂ discharge path obtains the worst forming quality. Overlap of the coil at different positions should be given during EMIF; however, a lower overlap rate of the coil helps improve the forming quality. The results obtained in this work are useful for forming integral panels with stiffened ribs via the EMIF process.

Quadratic stabilization of switched nonlinear systems

In this paper, the problem of quadratic stabilization of multi-input multi-output switched nonlinear systems under an arbitrary switching law is investigated.When switched nonlinear systems have uniform normal form and the zero dynamics of uniform normal form is asymptotically stable under an arbitrary switching law, state feedbacks are designed and a common quadratic Lyapunov function of all the closed-loop subsystems is constructed to realize quadratic stabilizability of the class of switched nonlinear systems under an arbitrary switching law.The results of this paper are also applied to switched linear systems.