ISSN 1439-8516 CN 11-2039/O1
We prove that the irreducible components of the moduli space of polarized Calabi-Yau pairs are projective.
In this paper, we study and characterize locally projectively flat singular square metrics with constant flag curvature. First, we obtain the sufficient and necessary conditions that singular square metrics are locally projectively flat. Furthermore, we classify locally projectively flat singular square metrics with constant flag curvature completely.
Let p ≥ 2 be a prime number and Zp be the ring of p-adic intergers. Let G be a semigroup generated by infinitely many contractive maps on pZp. It is shown that if G satisfies the open tiling conditions, then there exists a shift transformation on the limit set of G and the shift transformation is ergodic with respect to the Haar measure on pZp. As an application, we can generalize p-adic Khinchin's Theorem and p-adic Lochs' Theorem to any infinitely generated semigroup by use of the ergodicity of the shift transformation.
In this note, we consider the mappings h:X → Y between doubly connected Riemann surfaces having least ρ-Dirichlet energy. For a pair of doubly connected Riemann surfaces, in which X has finite conformal modulus, we establish the following principle:A mapping h in the class H2(X, Y) of strong limits of homeomorphisms in Sobolev space W1,2(X, Y) is ρ-energy-minimal if and only if its Hopf-differential is analytic in X and real along ∂X. It improves and extends the result of Iwaniec et al. (see Theorem 1.4 in[Arch. Ration. Mech. Anal., 209, 401-453 (2013)]). Furthermore, we give an application of the principle. Any ρ-energy minimal diffeomorphism is ρ-harmonic, however, we give a 1/|w|2-harmonic diffemorphism which is not 1/|w|2-energy minimal diffeomorphism. At last, we investigate the necessary and sufficient conditions for the existence of 1/|w|2-harmonic mapping from doubly connected domain Ω to the circular annulus A(1, R).
A proper k-edge coloring of a graph G is an assignment of one of k colors to each edge of G such that there are no two edges with the same color incident to a common vertex. Let f(v) denote the sum of colors of the edges incident to v. A k-neighbor sum distinguishing edge coloring of G is a proper k-edge coloring of G such that for each edge uv ∈ E(G), f(u) ≠ f(v). By χΣ'(G), we denote the smallest value k in such a coloring of G. Let mad (G) denote the maximum average degree of a graph G. In this paper, we prove that every normal graph with mad (G) <10/3 and Δ(G) ≥ 8 admits a (Δ(G) + 2)-neighbor sum distinguishing edge coloring. Our approach is based on the Combinatorial Nullstellensatz and discharging method.
In this paper, we define the generalized diffusion operator L=d/dM d/dS for two suitable measures on the line, which includes the generators of the birth-death processes, the one-dimensional diffusion and the gap diffusion among others. Via the standard resolvent approach, the associated generalized diffusion processes are constructed.
Let k be a positive integer. A graph G is k-weight choosable if, for any assignment L(e) of k real numbers to each e ∈ E(G), there is a mapping f:E(G) → R such that f(uv) ∈ L(uv) and Σe∈∂(u) f(e) ≠ Σe∈∂(v) f(e) for each uv ∈ E(G), where ∂(v) is the set of edges incident with v. As a strengthening of the famous 1-2-3-conjecture, Bartnicki, Grytczuk and Niwcyk[Weight choosability of graphs. J. Graph Theory, 60, 242-256 (2009)] conjecture that every graph without isolated edge is 3-weight choosable. This conjecture is wildly open and it is even unknown whether there is a constant k such that every graph without isolated edge is k-weight choosable. In this paper, we show that every connected graph of maximum degree 4 is 4-weight choosable.
