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Let f : (ℝn, 0) → (ℝ, 0) be a nonconstant analytic function defined in a neighborhood of the origin 0 ∈ ℝn. The classical Łojasiewicz inequality states that there exist positive constants δ, c and l such that |f(x)| ≥ cd(x, f-1(0))l for |x| ≤ δ, where d(x, f-1(0)) denotes the distance from x to the set f-1(0). The Łojasiewicz exponent of f at the origin 0 ∈ ℝn, denoted by , is the infimum of the exponents l satisfying the Łojasiewicz inequality. In this paper, we establish a formula for computing the Łojasiewicz exponent of f in terms of the Newton polyhedron of f in the case where f is nonnegative and nondegenerate. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. The first is to show a second main theorem with truncated counting function for holomorphic curves from punctured disks into semi-Abelian varieties. The second is to give an alternative proof of a Big Picard's theorem for algebraically non-degenerate holomorphic curves by Nevanlinna theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new class 풞(ϕ, ψ, η) of analytic functions defined by means of subordination. Coefficient bounds, Fekete-Szegö problem and norm estimates of the pre-Schwarzian derivatives of functions belonging to the class 풞(ϕ, ψ, η) are investigated. A class of multiple close-to-convex functions is also considered. [ABSTRACT FROM AUTHOR]

We study the local dynamics of holomorphic maps f in Cn tangent to the identity at a fixed point p with a non-degenerate characteristic direction [v]. In [M. Hakim, Analytic transformation of (Cp,0) tangent to the identity, Duke Math. J. 92 (1998) 403-428], n -- 1 invariants αj, 1 < j < n -- 1, called the directors, were associated to [u] and it was shown that if Re αj > 0 for all j then f has an attracting domain at p tangent to [v]. In this paper, we study the case Reαj = 0 for some j. With the help of a new invariant ¡j. called the non-dicritical order, we show that / has an attracting domain at p tangent to [v] if n > 1. We also study the "spiral domains" when μ = 0. For n = 2, we show that / has an attracting domain at p tangent to [v] if and only if either the director α > 0 or μ > 1. [ABSTRACT FROM AUTHOR]

This paper looks at subordination properties involving 1+βzp'(z) and the class of Sokól- Stankiewicz starlike functions where condition on β is determined so that the relation 1 + βzp'(z) -< ... would imply p(z) -< ... Similar results are obtained for expressions of the form 1 + βz p' (z)/p and 1 + .... These results are then applied to obtain other relations involving similar expressions. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the generalized Hyers-Ulam stability for fractional differential equations of the form: $$ D^{\alpha}_{z}f(z)= G(f(z),zf\prime(z),z^{2}f\Prime(z);z), \quad 2 < \alpha \leq 3 $$ in a complex Banach space. Furthermore, applications are illustrated. [ABSTRACT FROM AUTHOR]

Let Y be a pure dimensional analytic variety in ℂn with an isolated singularity at the origin such that the exceptional set X of a desingularization of Y is regular. The main objective of the present paper is to present a technique which allows us to determine obstructions to the solvability of the $\bar{\partial}$ equation in the L2, respectively L∞, sense on Y* = Y\{0} in terms of certain cohomology classes on X. More precisely, let Ω ⊂⊂ Y be a Stein domain with 0 ∈ Ω, Ω* = Ω\{0}. We give a sufficient condition for the solvability of the $\bar{\partial}$ equation in the L2-sense on Ω*; and in the L∞ sense, if Ω is in addition strongly pseudoconvex. If Y is an irreducible cone, we also give some necessary conditions and obtain optimal Hölder estimates for solutions of the $\bar{\partial}$ equation. [ABSTRACT FROM AUTHOR]

Let Ω be a B-regular domain in ℂn and let V be a locally irreducible analytic variety in Ω. Given a continuous function $\phi \in C(\bar{V} \cap \partial \Omega)$, we prove that there is a unique maximal plurisubharmonic function u on V with boundary values given by ϕ and furthermore that u is continuous on $\bar{V}$. [ABSTRACT FROM AUTHOR]

We explicitly write down all eigenvalues of the Rumin Laplacian on the standard contact spheres, and express the analytic torsion functions associated with the Rumin complex in terms of the Riemann zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions. [ABSTRACT FROM AUTHOR]

We introduce a join construction, as a way of completing the description of the relative conormal space of an analytic function on a complex analytic space that has a non-vanishing derivative at the origin. Then we show how to obtain a numerical criterion for Thom's A f condition. [ABSTRACT FROM AUTHOR]

We prove a result on the relationship between the relative conormal space of an analytic function f on affine space and the relative conormal space of f restricted to a hyperplane slice, at a point where the relative conormal space of f is "microlocally trivial". This result is intended to be used as a lemma to prove other conormal results by induction via hyperplane slicing. We present two proofs, one algebraic and one topological, since the methods may lead to other results in this area. [ABSTRACT FROM AUTHOR]

Let n∈ℤ with n≥3. Let Sn and An denote, respectively, the symmetric group and alternating group on n letters. Let m be an indeterminate, and define fm(x):=xn+a(m,n)x+b(m,n), where a(m,n),b(m,n) are certain prescribed forms in m. For a certain set of these forms, we show unconditionally that there exist infinitely many primes p such that fp(x) is irreducible over ℚ, Galℚ(fp)=Sn, and the fields K=ℚ(𝜃) are distinct and monogenic, where fp(𝜃)=0. Using a different set of forms, we establish a similar result for all square-free values of n≡1(mod4), with 5≤n≤401, and any positive integer value of m for which a(m,n) is square-free. Additionally, in this case, we prove that Galℚ(fp)=An. Finally, we show that these results can be extended under the assumption of the abc-conjecture. Our methods make use of recent results of Helfgott and Pasten. [ABSTRACT FROM AUTHOR]

During the development of the theory of slice regular functions over the real algebra of quaternions in the last decade, some natural questions arose about slice regular functions on the open unit ball in . This work establishes several new results in this context. Along with some useful estimates for slice regular self-maps of fixing the origin, it establishes two variants of the quaternionic Schwarz-Pick lemma, specialized to maps that are not injective. These results allow a full generalization to quaternions of two theorems proven by Landau for holomorphic self-maps of the complex unit disk with . Landau had computed, in terms of , a radius such that is injective at least in the disk and such that the inclusion holds. The analogous result proven here for slice regular functions allows a new approach to the study of Bloch-Landau-type properties of slice regular functions . [ABSTRACT FROM AUTHOR]

Let j be a nonzero integer and let U be a bounded domain. We construct a Fredholm symbol calculus for the C*-algebra generated by the poly-Bergman projection and the operators of multiplication by continuous functions. We define the j-removal boundary in Hilbert spaces of polyanalytic functions and prove that the quotient poly-Toeplitz C*-algebra generated by cosets of poly-Toeplitz operators with continuous symbols is •-isomorphic to the C*-algebra of continuous functions over the j-essential boundary. Unlike the Bergman case, we also show that if j ≠ ±1 then the j-essential boundary coincides with the set of non-isolated points on the boundary. [ABSTRACT FROM AUTHOR]

In this note, we consider locally invertible analytic mappings of a two-dimensional space over a non-archimedean field. Such a map is called semi-hyperbolic if its Jacobian has eigenvalues λ1 and λ2 so that λ1 = 1 and |λ2| ≠ 1. We prove that two analytic semi-hyperbolic maps are analytically equivalent if and only if they are formally equivalent, applying a generalized version of an estimation scheme from our earlier work [A. Jenkins and S. Spallone, A p-adic approach to local dynamics: Analytic flows and analytic maps tangent to the identity, Ann. Fac. Sci. Toulouse Math. (6) 18(3) (2009) 611-634]. [ABSTRACT FROM AUTHOR]

The splice quotients are an interesting class of normal surface singularities with rational homology sphere links. In general, it can be difficult to determine whether or not a singularity is a splice quotient (an analytic condition). We consider splice quotient deformations of splice quotients of the form z2 = xa + yb, and show that in general not all equisingular deformations are splice quotients. [ABSTRACT FROM AUTHOR]

We show that any proper rigid analytic surface admits a relatively minimal regular model. Further, we give a criterion for relative minimality of proper regular rigid analytic surfaces. [ABSTRACT FROM AUTHOR]

We say that a complex analytic space, X, is an intersection cohomology manifold if and only if the shifted constant sheaf on X is isomorphic to intersection cohomology; with field coefficients, this is quickly seen to be equivalent to X being a homology manifold. Given an analytic function f on an intersection cohomology manifold, we describe a simple relation between V(f) being an intersection cohomology manifold and the vanishing cycle Milnor monodromy of f. We then describe how the Sebastiani–Thom isomorphism allows us to easily produce intersection cohomology manifolds with arbitrary singular sets. Finally, as an easy application, we obtain restrictions on the cohomology of the Milnor fiber of a hypersurface with a special type of one-dimensional critical locus. [ABSTRACT FROM AUTHOR]

Let N = V ⊕ Z be a step-two nilpotent group. We study the Radon transform on functions on N defined by integration over the orbits {(z, t)(v, 0); v ∈ V} of the points (z, t) by the action of the subspace V ⊕ 0 of N. For N being of H-type, Siegel-type defined as the Shilov boundary of a complex Siegel domain, or Type F2r, b, a real analogue of the Siegel type, we find an inversion and a Plancherel formula for the Radon transform. We prove certain Lp - Lq boundedness properties for the transform. This generalizes earlier works of Geller–Stein and Strichartz for the Heisenberg groups. [ABSTRACT FROM AUTHOR]

We show that moments on vertical and horizontal affine slices of certain perturbations of the boundary of the ball are sufficient to detect the existence of a holomorphic extension to the interior. For one domain, the result is proved for L∞ functions. [ABSTRACT FROM AUTHOR]

Let Z be a smooth projective rational surface. A condition that implies the polyhedrality of the cone of curves of Z is given. This one depends only on the configuration of infinitely near points associated with the morphism which provides Z from a relatively minimal model X and it holds for a wide range of surfaces whose anticanonical bundle is not ample. When the above configuration is a chain, the condition consists uniquely on deciding whether certain datum is positive. Furthermore, we study polyhedrality and regularity of the characteristic cone and of the cone of curves of Z for some particular cases. [ABSTRACT FROM AUTHOR]

Examines the interpolating sequences on the algebra of bounded analytic functions. Determination of the Hilbert space of holomorphic functions; Equivalence of the space to existence of vector-valued interpolants.