Caterpillars with maximum degree 3 are antimagic

Let $G=(V,E)$ be a connected graph with $m$ edges. An antimagic labeling of $G$ is a one-to-one mapping from $E$ to $\{1,2,\dots ,m\}$ such that the vertex sum (i.e., sum of the labels assigned to edges incident to a vertex) for distinct vertices are different. A graph $G$ is called antimagic if $G$ has an antimagic labeling. It was conjectured by Hartsfield and Ringel that every tree other than $K_2$ is antimagic. The conjecture remains open though it was verified for trees with some constrains. Caterpillars are an important subclass of trees. This paper shows caterpillars with maximum degree 3 are antimagic, which gives an affirmative answer to an open problem of Lozano et al. (2019).     

Variability in the noise-induced modes of climate dynamics

New features of noise-induced climate variability are revealed on the basis of the three-dimensional model derived by Saltzman and Maasch. It is shown that the climate system can be highly noise excitable and it possesses the large-amplitude fluctuations even in those regions where its akin deterministic model does not contain any self-sustained oscillations. Intermittency in small- and large amplitude climate fluctuations between different basins of attraction of a limit cycle and stable equilibria substantially influencing the climate state (from warm to cold and vice versa) are found at various noise intensities. Suddenly occurring jumps between the basins of attraction of two stable equilibria corresponding to the warm and cold climate states are statistically confirmed under a certain diapason of noise intensities. The climate system undergoes transitions between its equilibria in the presence of noise in its prognostic variables. In addition, such transitions become more likely with increasing the noise intensity.     

RADIATIVE FORCING AND GREENHOUSE EFFECT DUE TO THE ATMOSPHERIC TRACE GASES

The radiative forcing and greenhouse effect due to the atmospheric trace gases have been calculated by using an advanced radiative-convective model developed in this paper. The relationship between radiative forcing and concentration is given for each trace gas. The resuhs show that (ⅰ) the radiative forcing and then the greenhouse effect are significantly affected by the overlapping of atmospheric absorption bands; (ⅱ) the increasing concentration of trace gases other than CO_2, such as CH_4, N_2O, CFCs, etc., may potentially play an important role in the future global warming;(ⅲ) the proposed substitutes, such as HCFC124 and HFC125, for the chlorofluorocarbons which are considered to destroy the ozone layer have still considerable greenhouse effect even though their ozone depletion potentials are much smaller than CFCs; and (ⅳ) the feedback processes within the earth-atmosphere system have important effect on the surface temperature change due to the radiative forcing to the system.     

Matching subsequences in trees

Given two rooted, labeled trees P and T the tree path subsequence problem is to determine which paths in P are subsequences of which paths in T. Here a path begins at the root and ends at a leaf. In this paper we propose this problem as a useful query primitive for XML data, and provide new algorithms improving the previously best known time and space bounds.     

Development of a multi‐scale ocean model by using particle Laplacian method for anisotropic mass transfer

The direct injection of CO₂ into the deep ocean is one of the feasible ways for the mitigation of the global warming, although there is a concern about its environmental impact near the injection point. To minimize its biological impact, it is necessary to make CO₂ disperse as quickly as possible, and it is said that injection with a pipe towed by a moving ship is effective for this purpose. Because the injection ship moves over a spatial scale of O(10²km), a mesoscale model is necessary to analyse the dispersion of CO₂. At the same time, since it is important to investigate high CO₂ concentration near the injection point, a small‐scale model is also required. Therefore, in this study, a numerical model was developed to analyse CO₂ dispersion in the deep ocean by using a fixed mesoscale and a moving small‐scale grid systems, the latter of which is nested and moves in the former along the trajectory of the moving ship. To overcome the artificial diffusion of mass concentration at the interface of the two different grid systems and to keep its spatial accuracy almost the same as that in the small‐scale, a particle Laplacian method was adopted and newly modified for anisotropic diffusion in the ocean. Copyright © 2010 John Wiley & Sons, Ltd.     

New facts

We use ``iterated square sequences' to show that there is an -definable partition such that if is an inner model not containing :

(a)

For some is stationary.

(b)

For each there is a generic extension of in which does not exist and is non-stationary.

This result is then applied to show that if is an inner model without , then some sentence not true in can be forced over .

     

The Forcing Convexity Number of a Graph

For two vertices u and v of a connected graph G, the set I(u,v) consists of all those vertices lying on a u-v geodesic in G. For a set S of vertices of G, the union of all sets I(u,v) for u, v S is denoted by I(S). A set S is a convex set if I(S) = S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. A convex set S in G with |S| = con(G) is called a maximum convex set. A subset T of a maximum convex set S of a connected graph G is called a forcing subset for S if S is the unique maximum convex set containing T. The forcing convexity number f(S, con) of S is the minimum cardinality among the forcing subsets for S, and the forcing convexity number f(G, con) of G is the minimum forcing convexity number among all maximum convex sets of G. The forcing convexity numbers of several classes of graphs are presented, including complete bipartite graphs, trees, and cycles. For every graph G, f(G, con) con(G). It is shown that every pair a, b of integers with 0 a b and b is realizable as the forcing convexity number and convexity number, respectively, of some connected graph. The forcing convexity number of the Cartesian product of H × K ₂ for a nontrivial connected graph H is studied.     

On the number of -equivalent non-isomorphic models

We prove that if is consistent then is consistent with the following statement: There is for every a model of cardinality which is -equivalent to exactly non-isomorphic models of cardinality . In order to get this result we introduce ladder systems and colourings different from the ``standard' counterparts, and prove the following purely combinatorial result: For each prime number and positive integer it is consistent with that there is a ``good' ladder system having exactly pairwise nonequivalent colourings.

     

具强迫力的奇数阶中立型微分方程的振动性

Abstract. In this paper, the forced odd order neutral differential equations of the form are con-sidered