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We investigate some properties of solutions to Dirac operators with gradient potentials. Solutions to Dirac operators with gradient potentials are called monogenic functions with respect to the potential functions. We establish a Borel–Pompeiu formula, and obtain Cauchy integral formula and mean value formula about such functions. Based on integral formulas, we prove some geometric properties of monogenic functions with respect to the potential functions and construct some integral transforms. The boundedness of integral transforms in Holder space is given. We prove the Plemelj–Sokhotski formula and the Painleve theorem. As applications, we firstly prove that a kind of Riemann–Hilbert problem for monogenic functions with respect to the potential functions is solvable. Explicit representation formula of the solution is also given. We then establish solvability conditions of a kind of general Riemann–Hilbert problem for monogenic functions with respect to the potential functions. Finally, we give some examples about the above Riemann–Hilbert problem.

The pluripolar hull of a pluripolar set E in $\mathbb{P}^n$ is the intersection of all complete pluripolar sets in $\mathbb{P}^n$ that contain $E$. We prove that the pluripolar hull of each compact pluripolar set in $\mathbb{P}^n$ is $F_\sigma$. The convergence set of a divergent formal power series $f(z_{0}, \dots,z_{n})$ is the set of all "directions" $\xi \in\mathbb{P}^{n}$ along which $f$ is convergent. We prove that the union of the pluripolar hulls of a countable collection of compact pluripolar sets in $\mathbb{P}^n$ is the convergence set of some divergent series $f$. The convergence sets on $\Gamma:=\{[1:z:\psi(z)]: z\in \mathbb{C}\}\subset\mathbb{C}^2\subset\mathbb{P}^2$, where $\psi$ is a transcendental entire holomorphic function, are also studied and we obtain that a subset on $\Gamma$ is a convergence set in $\mathbb{P}^2$ if and only if it is a countable union of compact projectively convex sets, and hence the union of a countable collection of convergence sets on $\Gamma$ is a convergence set., Comment: 24 pages

A connected compact subset $E$ of $\mathbb{R}^N$ is said to be a strict Whitney set if there exists a real-valued $C^1$ function $f$ on $\mathbb{R}^N$ with $\nabla f|_E\equiv 0$ such that $f$ is constant on no non-empty relatively open subsets of $E$. We prove that each self-similar arc of Hausdorff dimension $s>1$ in $\mathbb{R}^N$ is a strict Whitney set with criticality $s$. We also study a special kind of self-similar arcs, which we call "regular" self-similar arcs. We obtain necessary and sufficient conditions for a regular self-similar arc $\Lambda$ to be a $t$-quasi-arc, and for the Hausdorff measure function on $\Lambda$ to be a strict Whitney function. We prove that if a regular self-similar arc has "minimal corner angle" $\theta_{\min}>0$, then it is a 1-quasi-arc and hence its Hausdorff measure function is a strict Whitney function. We provide an example of a one-parameter family of regular self-similar arcs with various features. For some value of the parameter $\tau$, the Hausdorff measure function of the self-similar arc is a strict Whitney function on the arc, and hence the self-similar arc is an $s$-quasi-arc, where $s$ is the Hausdorff dimension of the arc. For each $t_0\ge 1$, there is a value of $\tau$ such that the corresponding self-similar arc is a $t$-quasi-arc for each $t>t_0$, but it is not a $t_0$-quasi-arc. For each $t_0>1$, there is a value of $\tau$ such that the corresponding self-similar arc is a $t_0$-quasi-arc, but it is a $t$-quasi-arc for no $t\in [1, t_0)$., Comment: 30 pages, 3 figures

Kernels of integral transforms of the form k(xy) are called Fourier kernels. Hardy and Titchmarsh (6) and Watson (15) studied self- reciprocal transforms with Fourier kernels on the positive half-real line R+ and obtained nice characterizing conditions and other properties. Neverthe- less, the Fourier transform has an inherent complex, unitary structure and by treating it on the whole real line more interesting properties can be ex- plored. In this paper, we characterize all unitary integral transforms on L 2 (R) with Fourier kernels. Characterizing conditions are obtained through a natu- ral splitting of the kernel on R+ and R−, and the conversion to convolution integrals. A collection of concrete examples are obtained, which also include "discrete" transforms (that are not integral transforms). Furthermore, we ex- plore algebraic structures for operator-composing integral kernels and identify a certain Abelian group associated with one of such operations. Fractional powers can also be assigned a new interpretation and obtained.

A nonlinear generalization of convergence sets of formal power series, in the sense of Abhyankar–Moh [J. Reine Angew. Math. 241 (1970)], is introduced. Given a family y = φs(t, x) = sb1(x)t + b2(x)t 2 + · · · of analytic curves in C× C passing through the origin, Convφ(f) of a formal power series f(y, t, x) ∈ C[[y, t, x]] is defined to be the set of all s ∈ C for which the power series f(φs(t, x), t, x) converges as a series in (t, x). We prove that for a subset E ⊂ C there exists a divergent formal power series f(y, t, x) ∈ C[[y, t, x]] such that E = Convφ(f) if and only if E is an Fσ set of zero capacity. This generalizes the results of P. Lelong and A. Sathaye for the linear case φs(t, x) = st. We say that a formal power series f(z) = ∑∞ |α|=0 aαz α is convergent if there exists a constant C such that |aα| ≤ C |α| for all α ∈ Z+. (Here we have used multiindex notation: Z+ denotes the set of all n-tuples α := (α1, . . . , αn) of integers αi ≥ 0 if z = (z1, . . . , zn) and α ∈ Z+, then zα = z1 1 · · · zαn n and |α| := α1 + · · ·+ αn denotes the length of α ∈ Z+.) A series f is called divergent if it is not convergent. A divergent power series may still converge when restricted to a certain set of lines or planes through the origin. For example, Abhyankar and Moh [AM] considered the convergence set Conv(f) of a series f defined to be the set of all s ∈ C for which f(sz2, . . . , zn) converges as a series in (z2, . . . , zn). The convergence set of divergent series can be empty or an arbitrary countable set. Abhyankar and Moh proved that the one-dimensional Hausdorff measure of the convergence set of a divergent series is zero. In the case when n = 2, Pierre Lelong had earlier proved that if Conv(f) is not contained in an Fσ set of zero capacity then the series f is necessarily 2010 Mathematics Subject Classification: Primary 32A05, 31A15; Secondary 31C15, 32D15.

We study the convergence of a formal power series of two variables if its restrictions on curves belonging to a certain family are convergent. Also analyticity of a given C 1 function f is proved when the restriction of f on analytic curves belonging to some family is analytic. Our results generalize two known state- ments: a theorem of P. Lelong and the Bochnak-Siciak Theorem. The questions we study fall into the category of "Osgood-Hartogs- type" problems.

We study discrete fixed point sets of holomorphic self-maps of complex manifolds. The main attention is focused on the cardinality of this set and its configuration. As a consequence of one of our observations, a bounded domain in ${\Bbb C}^n$ with no non-trivial holomorphic retractions is constructed., 12 pages

We study the fixed point sets of holomorphic self- maps of a bounded domain in C n . Specifically we investigate the least number of fixed points in general position in the domain that forces any automorphism (or endomorphism) to be the identity. We have discovered that in terms of this number one can give the necessary and sufficient condition for the domain to be biholomor- phic to the unit ball. Other theorems and examples generalize and complete previous results in this area, especially the recent work of Jean-Pierre Vigue.

Let H n be the metric space of all bounded domains in C n with the metric equal to the Hausdorff distance between boundaries of domains. We prove that the dimension of the group of automorphisms of domains is an upper semicontinuous func- tion on H n . We also provide theorems and examples regarding the change in topological structure of these groups under small perturbation of a domain in H n .

It is well-known that a real analytic manifold X admits a complexification XC , a complex manifold that contains X as the fixed point set of an antiholomorphic involution. This can be seen as follows.The transition functions defining the manifold X are real-analytic local diffeomorphisms of Rn. The Taylor expansions of these transition functions can be considered as local biholomorphisms of Cn, hence they serve as transition functions of a complex manifold. The germ of the complexification XC is unique. Every sufficiently small tubular neighborhood Ω of X in the tangent bundle TX admits a real analytic diffeomorphism into XC that fixes X . Therefore a sufficiently small tubular neighborhood ofX in TX has a complex structure and can be considered as a complexification of X . In general the complex structure on the tubular neighborhood is not unique since there are manyways to embed it intoXC . There have been a lot of interest in finding canonical complex structures on tubular neighborhoods ofX in TX . With additional datum of a real analytic Riemannian metric g on X a canonical complex structure can be specified for sufficiently small tubular neighborhoods Ω of X in TX (see [GS, LS, S1]). There is a unique complex structure on Ω such that the map f(σ + iτ) = (τγ′(σ))γ(σ), σ + iτ ∈ C, is holomorphic, wherever it is defined

Results are reported of a numerical implementation of the hypcrbolic Fourier transform and the geodesic and horocyclic Radon transforms on the hyperbolic plane, and of their inverses. The study is motivated by the hyperbolic geometry approach to the linearized inverse conductivity problem, suggested by C. A. Berenstein and E. Casadio Tarabusi.

The equations which model the elastostatic shallow circular cylindrical shell (see, e.g., [6,15,23,29]) constitute an important elliptic partial differential equation (PDE) system in the study of shell structures. When the system is subjected to a concentrated point load, the response is described by a fundamental solution of the PDE system. We have found some mathematical inconsistencies in the existing literature. Therefore, in this paper, we discuss these errors, then we use partial fractions and Fourier transform techniques to determine the fundamental solution. Explicit expressions in terms of special functions and convolution integrals are derived and simplified so that the formulas are suitable for algorithmic evaluation and for application elsewhere.

In its proof, a bijective holomorphic mapping σ :Ω → B with σ(q)= 0 has been constructed. However, no explicit arguments showing the complex analyticity of σ−1 had been presented. Indeed, the main purpose of this article is to give it a concise proof. On the other hand, we present it as a separate article since we believe that the arguments here are of a separate interest, in regard to the following unsolved problem

Let $\Gamma$ be a $C^\infty$ curve in $\Bbb{C}$ containing 0; it becomes $\Gamma_\theta$ after rotation by angle $\theta$ about 0. Suppose a $C^\infty$ function $f$ can be extended holomorphically to a neighborhood of each element of the family $\{\Gamma_\theta \}$. We prove that under some conditions on $\Gamma$ the function $f$ is necessarily holomorphic in a neighborhood of the origin. In case $\Gamma$ is a straight segment the well known Bochnak-Siciak Theorem gives such a proof for real analyticity. We also provide several other results related to testing holomorphy property on a family of certain subsets of a domain in $\Bbb{C}$.

Lempert proved that the Kobayashi metric and the Caratheodory metric of smooth strongly convex domains are smooth away from the zero section of the holomorphic tangent bundle. We will show that if the strong convexity is replaced by the weaker condition of strict convexity the above smoothness result is no longer valid even if the boundary is real analytic. We study the smoothness of the Kobayashi metric of ellipsoids. We prove that the Kobayashi metric of domains of the form , where m≥3/2, is piecewise C 3 off the zero section, but not C 3.

A new boundary uniqueness result for holomorphic functions is obtained. Applications are provided.

In this paper we prove the best possible Lp estimates for the ∂-b -equation on the boundaries of real ellipsoids in Cn, and provide examples to show why they cannot be improved.

For a domain $D\subset {\Bbb{C}}^n$ we construct a continuous foliation of $D$ into one real dimensional curves such that any function $f\in {C^1(D)}$ which can be extended holomorphically into some neighborhood of each curve in the foliation will be holomorphic on $D$.

The disc-hull of a set is the union of the set arid all HiX discs whose boundaries lie in the set. We give an example of a, compact set in C2 whose disc-hull is not compact, answering a question posed by P. Ahern and W. Rudin. The polynomial hull of a compact set X c C'" is the set X of all points x E C"' at which the inequality IP(x)l : U --* C,7 is a non-constant map whose components are in H' (U), its range d>(U) is called an H?-disc, parametrized by d>. If lim.,/1(4>(rei0)) E X for almost all e'O on the unit circle T, then 4)(U) is an H'-disc whose boundary lies in X." They further define the disc-hull D(X) to be the union of X and all Hc-discs whose boundaries lie in X. Because of the maximum principle, D(X) C X. One of the questions posed in [1] (see p. 25) is whether the disc-hull D(X) is always compact for a compact set X C C". Below we answer this question negatively by constructing a counter-exaimple in c2 1. Define w {z E U: Rez > 4 Let Ro: U -> W be the Riemann map satisfying o(+i) = 1 + 23i and ,o(1) = 1. Therefore Re,o(e&0) = 2 for Ree0o ,cn(0) = 0. 2. Let X {((, ?7) E C2 ( E T, q E F(}, where the fiber F, is defined as follows. 1( = T for Re( > 0, 1T( U for ( ?i, and 1( {f01(() 'n E N} U {0} for ReX D? (0) (0, 0); therefore (0, 0) E D(X). Received by the editors August 31, 1999. 2000 Mathematics Subject Classification. Primary 32E20.

We study fixed point sets for holomorphic automorphisms (and endomorphisms) on complex manifolds. The main object of our interest is to determine the number and configuration of fixed points that forces an automorphism (endomorphism) to be the identity. These questions have been examined in a number of papers for a bounded domain in ${\Bbb C}^n$. Here we resolve the case for a general finite dimensional hyperbolic manifold. We also show that the results for non-hyperbolic manifolds are notably different., Comment: 10 pages

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We show in this paper that every domain in a separable Hilbert space, say $\cH$, which has a $C^2$ smooth strongly pseudoconvex boundary point at which an automorphism orbit accumulates is biholomorphic to the unit ball of $\cH$. This is the complete generalization of the Wong-Rosay theorem to a separable Hilbert space of infinite dimension. Our work here is an improvement from the preceding work of Kim/Krantz [KIK] and subsequent improvement of Byun/Gaussier/Kim [BGK] in the infinite dimensions., 13 pages

It is a result of classical function theory (see [FiF; Les; Mas; PeL; S]) that if f : U → U is a conformal self-mapping of a plane domain that fixes three distinct points then f(ζ) ≡ ζ. The purpose of the present paper is to put this result into a geometrically natural context and to extend it to higher-dimensional domains and manifolds. For an examination of fixed point questions from a slightly different point of view, we refer the reader to the work of Vigue (see e.g. [V1; V2]). The third-named author thanks Robert Burckel for early discussions of this topic and for basic references.

We give examples of non-homogeneous bounded domains in C which can exhaust no other domains. And we prove that every bounded domain in C with C boundary can exhaust the ball.

Many problems in structural dynamics involve stabilizing the elastic energy of partial differential equations such as the Euler-Bernoulli beam equation by boundary conditions. Exponential stability is a very desirable property such elastic systems. The energy multiplier method has been successfully applied by several people to establish exponential stability for various PDEs and boundary conditions. However, it has also been found that for certain boundary conditions the energy multiplier method is not effective in proving the exponential stability property. A recent theorem of F. L. Huang introduces a frequency domain method to study such exponential decay problems. In this paper, we derive estimates of the resolvent operator on the imaginary axis and apply Huang's theorem to establish an exponential decay result for an Euler-Bernoulli beam with rate control of the bending moment only. We also derive asymptotic limits of eigenfrequencies, which was also done earlier by P. Rideau. Finally, we indicate the realizability of these boundary feedback stabilization schemes by illustrating some mechanical designs of passive damping devices.

The (projective) convergence set of a divergent formal power series $f(x_{1},...,x_{n})$ is defined to be the image in $\PP^{n-1}$ of the set of all $x\in \mathbb{C}^{n}$ such that $f(x_{1}t,...,x_{n}t)$, as a series in $t$, converges absolutely near $t=0$. We prove that every countable union of closed complete pluripolar sets in $\PP^{n-1}$ is the convergence set of some divergent series $f$. The (affine) convergence sets of formal power series with polynomial coefficients are also studied. The higher-dimensional results of A. Sathaye, P. Lelong, N. Levenberg and R.E. Molzon, and of J. Rib\'{o}n are thus generalized., Comment: 17 pages