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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]

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 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]