We study Turing instabilities in 3-component reaction-diffusion systems. The existence of a complementary pair of stable-unstable subsystems always gives rise to Turing instability for suitable diagonal diffusion matrices. There are two types of Turing instability, one called steady instability and the other wave instability. To determine which of the two types of instability actually occurs, easily verifiable conditions on unstable subsystems are given. A complementary pair of unstable-unstable subsystems in a stable full system also leads to steady instability. Our results give a perspective to the rich variety and complexity of pattern dynamics in 3-component systems of reaction-diffusion equations at the onset.

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We consider two mixed curves C, C′ ⊂ C² which are defined by mixed functions of two variables z = (z₁, z₂). We have shown in [4], that they have canonical orientations. If C and C′ are smooth and intersect transversely at P, the intersection number I_top(C, C′; P) is topologically defined. We will generalize this definition to the case when the intersection is not necessarily transversal or either C or C′ may be singular at P using the defining mixed polynomials.

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We study the problem of uniqueness of complete hypersurfaces immersed in a semi-Riemannian warped product whose warping function has convex logarithm. By applying a maximum principle at the infinity due to S. T. Yau and supposing a natural comparison inequality between the mean curvature of the hypersurface and that of the slices of the region where the hypersurface is contained, we obtain rigidity theorems in such ambient spaces. Applications to the hyperbolic and the steady state spaces are given.

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In this paper we calculate the collection of limit functions obtained by applying an extension of Zalcman's Lemma, due to X. C. Pang to the non-normal family {f(nz): n ∈ N} in C, where f = Re^P. Here R and P are an arbitrary rational function and a polynomial, respectively, where P is a non-constant polynomial.

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Let f : Rⁿ → R be a polynomial function of degree d with f(0) = 0. The classical Łojasiewiz inequality states that there exist c > 0 and α > 0 such that |f(x)| ≥ cd(x, f^–1(0))^α in a neighbourhod of the origin 0 ∈ Rⁿ, where d(x, f^–1 (0)) denotes the distance from x to the set f^–1(0). We prove that the smallest such exponent α is not greater than R(n, d) with R(n, d) := max{d(3d – 4)^n–1, 2d(3d – 3)^n–2}.

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Let X be a smooth complex projective variety of dimension n and let L₁, ..., L_n–i be ample line bundles on X, where i is an integer with 0 ≤ i ≤ n – 1. In the first part, we defined the ith sectional geometric genus g_i(X, L₁, ..., L_n–i) and the ith sectional H-arithmetic genus χ_i^H(X, L₁, ..., L_n–i) of (X, L₁, ..., L_n–i). In this third part, we will investigate g₂(X, L₁, ..., L_n–2) and χ₂^H(X, L₁, ..., L_n–2). Moreover we will give some applications of the sectional invariants of multi-polarized manifolds.

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We consider the dynamics of polynomial skew products that are semiconjugate to some polynomial products on C². We show that the fiberwise Green functions exist outside thin sets, whose upper semicontinuous regularizations are defined, continuous and plurisubharmonic on C². This result is obtained from the existence of Green functions of polynomials outside thin sets.

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We consider the Neumann Sturm-Liouville problem defined on trees such that the ratios of lengths of edges are not necessarily rational. It is shown that the potential function of the Sturm-Liouville operator must be zero if the spectrum is equal to that for zero potential. This extends previous results and gives an Ambarzumyan theorem for the Neumann Sturm-Liouville problem on trees. To prove this, we compute approximated eigenvalues for zero potential by using a generalized pigeon hole argument, and make use of recursive formulas for characteristic functions.

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Let (Mⁿ, g, e^–f dvol_g) be a smooth metric measure space of dimension n. In this note, we first prove a nonexistence result for Mⁿ with the Bakry-Émery Ricci tensor is bounded from below. Then we show that f ∈ L^∞ (Mⁿ, e^–f dvol) and |∇f| ∈ L^∞ (Mⁿ, e^–f dvol) are equivalent for complete gradient shrinking Ricci solitons. Furthermore, we prove that there is no non-Einstein shrinking soliton when the normalized function $\tilde f$ is non-positive.

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We prove that a null 2-type hypersurface in the Euclidean (n + 1)-space is an open portion of a spherical cylinder S^n–1 × R if and only if it is δ(2)-ideal.

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In this paper some properties of weakly first countable spaces and sequence-covering images of metric spaces are studied. Strictly Fréchet spaces are characterized as the spaces in which every sequence-covering mapping onto them is strictly countably bi-quotient. Strict accessibility spaces are introduced, in which a T₁-space X is strict accessibility if and only if every quotient mapping onto X is strictly countably bi-quotient. For a T₂, k-space X every quotient mapping onto X is strictly countably bi-quotient or bi-quotient if and only if X is discrete. They partially answer some questions posed by F. Siwiec in [16, 17].

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In this paper we give a criterion of irreducibility for a complex power series in two variables, using the notion of jacobian Newton diagrams, defined with respect to any direction. Then we apply our method to study the branches of plane algebraic curves. For an affine plane curve with one point at infinity, we also obtain a criterion for an analytical irreducibility in terms of the Newton diagram of a discriminant, without using coordinates centered at the point at infinity.

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