ON THE SINGULAR DIRECTIONS OF VALUE DISTRIBUTION OF HOLOMORPHIC CURVES IN P^n(C)

This article proves the existence of singular directions of value distribution theory for some transcendental holomorphic curves in the n-dimensional complex projective space P^n（C）.. An example is given to complement these results.

UNIQUENESS OF SOLUTION OF FIELD POINT OF SINGULAR SOURCE OUTSIDE-REGION-DISTRIBUTION METHOD

The uniqueness of solution of field point, inside a convex region due to singular source(s) with kernel function decreasing with distance increasing, outside-region-distribution(s) such that the boundary condition expressed by the response of the source(s) is satisfied, is proved by using the condition of kernel function decreasing with distance increasing anal an integral inequality. Examples of part of these singular sources such as Kelvin's point force, Point-Ring-Couple (PRC) etc. are given. The proof of uniqueness of solution of field point in a twisted shaft of revolution due to PRC distribution is given as an example of application.

NUMERICAL ANALYSIS OF FLUID FLOW AND ADDED MASS INDUCED BY VIBRATION OF STRUCTURE

The fluid flow induced by light-density, low-stiffness structures was treated as inviscid, incompressible irrotational and steady plane flow. On the basis of the dipole configuration method, a singularity distribution method of distributing sources/sinks and dipoles on interfaces of the structure and fluid was developed to solve the problem of fluid flow induced by the vibration of common structures, such as columns and columns with fins, deduce the expression of kinetic energy of the fluid flow, and obtain the added mass finally. The calculational instances with analytical solutions prove the reliability of this method.

A Lifting Line Theory for a Three-dimensional Hydrofoil

Prandtl’s lifting line theory was generalized to the lifting problem of a three-dimensional hydrofoil in the presence of a free surface. Similar to the classical lifting theory, the singularity distribution method was utilized to solve two-dimensional lifting problems for the hydrofoil beneath the free surface at the air-water interface, and a lifting line theory was developed to correct three-dimensional effects of the hydrofoil with a large aspect ratio. Differing from the classical lifting theory, the main focus was on finding the three-dimensional Green function of the free surface induced by the steady motion of a system of horseshoe vortices under the free surface. Finally, numerical examples were given to show the relationship between the lift coefficient and submergence Froude numbers for 2-D and 3-D hydrofoils. If the submergence Froude number is small free surface effect will be significant registered as the increase of lift coefficient. The validity of these approaches was examined in comparison with the results calculated by other methods.

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.

Dynamics of a self-propelled particle under different driving modes in a channel flow

In this paper, a model that combines the lattice Boltzmann method with the singularity distribution method is proposed to simulate a self-propelled particle swimming（exhibiting translation and rotation） in a channel flow. The results show that the velocity distribution for a self-propelled particle swimming deviates from a Maxwellian distribution and exhibits highvelocity tails. The influence of an eccentric potential doublet on the translation velocity of the particle is significant. The velocity decay process can be described using a double exponential model form. No large differences in the velocity distribution were observed for different translation Reynolds numbers, rotation Reynolds numbers, or regular intervals.

On resolution to Wu's conjecture on Cauchy function's exterior singularities

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.

ON SINGULAR MATRIX VARIATE BETA DISTRIBUTION

In this paper the density of the matrix variate beta distribution of rank lower than itsdimensionality is obtained with respect to a suitably defined differential form under the condi-tion that the difference between the identity and this matrix has full rank. As preliminaries,the Jacobian of a transformation related to decomposing a nonnegative-definite matrix into theproduct of a matrix of full column rank and its transpose and that of the transformation of anonnegative-definite matrix into its congruent matrix are established.

DEGENERATE SEMI-GROUP METHODS FOR THE EXPONENTIAL STABILITY OF THE FIRST ORDER SINGULAR DISTRIBUTED PARAMETER SYSTEMS

Exponential stability of the first order singular distributed parameter systems is discussedin the light of degenerate semi-group methods,which is described by the abstract developing equationin Hilbert space.The necessary and sufficient conditions concerning the exponential stability of thefirst order singular distributed parameter systems are given.

Exact controllability for singular distributed parameter system in Hilbert space

Exact controllability of singular distributed parameter control system is discussed via functional analysis and the theory of generalized operator semi-group in Hilbert space, Necessary and sufficient conditions concerning the exact controllability are given. Relations between exact controllability and stability of singular distributed parameter system are specified.

Controllability of Singular Distributed Parameter Systems in the Sense of Mild Solution

Necessary and sufficient conditions for the exact controllability and approximate controllability of a singular distributed parameter system are obtained.These general results are used to examine the exact controllability and approximate controllability of the Dzektser equation in the theory of seepage.

Feedback Stabilization for a Class of Second Order Singular Distributed Parameter System in Hilbert Space

Feedback stabilization for a class of second order singular distributed parameter system with multi- inputs is discussed via functional analysis and operator theory in Hilbert space, the solutions of the problem and the constructive expressions of the solutions are given by the generalized inverse of bounded linear operator. This research is theoretically important for studying the stability of the singular distributed parameter system.

ON STATE FEEDBACK AND POLE ASSIGNMENT OF THE SECOND ORDER COUPLED SINGULAR DISTRIBUTED PARAMETER SYSTEMS IN HILBERT SPACE

State feedback and pole assignment of the second order coupled singular distributed parameter systems are discussed via functional analysis and operator theory in Hilbert space, in which infinite many poles are changed. The solutions of the problem and the constructive expression of the solutions are given by the generalized inverse of bounded linear operator. This research is theoretically important for studying the pole assignment and stabilization of the singular distributed parameter systems.