International Journal of Mathematics

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.

We study the classification of the pairs (N, X) where N is a Stein surface and X is a ℂ-complete holomorphic vector field with isolated singularities on N. We describe the role of transverse sections in the classification of X and give necessary and sufficient conditions on X in order to have N biholomorphic to ℂ². As a sample of our results, we prove that N is biholomorphic to ℂ² if H²(N, ℤ) = 0, X has a finite number of singularities and exhibits a singularity with three separatrices or, equivalently, a singularity with first jet of the form where λ₁/λ₂ ∈ ℚ₊. We also study flows with many periodic orbits (i.e. orbits diffeomorphic to ℂ*), in a sense we will make clear, proving they admit a meromorphic first integral or they exhibit some special periodic orbit, whose holonomy map is a non-resonant nonlinearizable diffeomorphism map.

In this paper, we consider a new class 𝒞(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 𝒞(ϕ, ψ, η) are investigated. A class of multiple close-to-convex functions is also considered.

By using the Milnor ring structure of the cohomology of Grassmannian, we prove a vanishing theorem of Witten genus for generalized string complete intersections in products of Grassmannian.

Let Φ be a positive unital Tr-preserving map on M_n(ℂ). We give various characterizations for Φ to preserve the von Neumann entropy of a state ρ on M_n(ℂ). Among others, it is given by that Φ behaves to ρ as a *-automorphism. This is also equivalent to that the entropy of the stochastic matrix arising from {ρ, Φ} is zero.

We extend the topological field theory (itsy bitsy topological field theory) of our previous work from mod 2 to twisted coefficients. This topological field theory is derived from sutured Floer homology (SFH) but described purely in terms of surfaces with signed points on their boundary (occupied surfaces) and curves on those surfaces respecting signs (sutures). It has information-theoretic (itsy) and quantum-field-theoretic (bitsy) aspects. In the process we extend some results of SFH, consider associated ribbon graph structures, and construct explicit admissible Heegaard decompositions.

We show that an exact Lagrangian cobordism L ⊂ ℝ × P × ℝ from a Legendrian submanifold Λ ⊂ P × ℝ to itself satisfies H_i(L; 𝔽) = H_i(Λ; 𝔽) for any field 𝔽, given that Λ admits a spin exact Lagrangian filling and that the concatenation of any spin exact Lagrangian filling of Λ and L is also spin. The main tool used is Seidel's isomorphism in wrapped Floer homology. In contrast to that, for loose Legendrian submanifolds of ℂⁿ × ℝ, we construct examples of such cobordisms whose homology groups have arbitrarily high ranks. In addition, we prove that the front S^m-spinning construction preserves looseness, which implies certain forgetfulness properties of it.