We call a subgraph of an edge-colored graph rainbow, if all its edges have different colors. The anti-Ramsey number G in complete K n , denoted by r ( ) is the maximum colors edge-coloring with no rainbow copy . In this paper, we determine exact value for star forests and double stars approximate linear forests.

The forcing number of a perfect matching M of a graph G is the smallest cardinality of subsets of M that are contained in no other perfect matchings of G. The forcing spectrum of G is the collection of forcing numbers of all perfect matchings of G. In this paper, we classify the perfect matchings of a generalized Petersen graph P (n, 2) in two types, and show that the forcing spectrum is the un...

The quasigeostrophic equation is a prototypical geophysical fluid model. In this paper, we consider time-periodic motions of this model under dissipation and time-dependent wind forcing. We show that when the wind forcing is time-periodic and the spatial square-integral of the wind forcing is bounded in time, the full nonlinear quasigeostrophic model has time-periodic motions, under some condit...

Using the theory of rudimentary recursion and provident sets developed in a previous paper, we give a treatment of set forcing appropriate for working over models of a theory PROVI which may plausibly claim to be the weakest set theory supporting a smooth theory of set forcing, and of which the minimal model is Jensen’s Jω. Much of the development is rudimentary or at worst given by rudimentary...

In this paper I will communicate some new consequences of the Proper Forcing Axiom. First, the Bounded Proper Forcing Axiom implies that there is a well ordering of R which is Σ1-definable in (H(ω2),∈). Second, the Proper Forcing Axiom implies that the class of uncountable linear orders has a five element basis. The elements are X, ω1, ω∗ 1 , C, C ∗ where X is any suborder of the reals of size ...

It is well known to generalize the meagre ideal replacing ℵ 0 by a (regular) cardinal λ > ℵ 0 and requiring the ideal to be λ +-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing ℵ 0 by λ, so requiring it to be (< λ)-complete. Of course, we would welcome additional properties generalizing the one...

We demonstrate that there is an optimal forcing length scale for low Prandtl number dynamo flows that can significantly reduce the required energy injection rate. The investigation is based on simulations of the induction equation in a periodic box of size 2πL. The flows considered are the laminar and turbulent ABC flows forced at different forcing wave numbers k_{f}, where the turbulent case i...

The regional waters in Singapore Strait are characterized by complex hydrodynamic phenomena as a result of the combined effect of three large water bodies viz. the South China Sea, the Andaman Sea, and the Java Sea. This leads to anomalies in water levels and generates residual currents. Numerical hydrodynamic models are generally used for predicting water levels in the ocean and seas. But thei...