For two vertices u and v of a graph G, the set I(u, v) consists of all vertices lying on some u − v geodesic in G. If S is a set of vertices of G, then I(S) is the union of all sets I(u, v) for u, v ∈ S. A set S is a geodetic set if I(S) = V (G). A minimum geodetic set is a geodetic set of minimum cardinality and this cardinality is the geodetic number g(G). A subset T of a minimum geodetic set...

We provide an exposition of supercompact Radin forcing and present several methods for iterating Radin forcing. In this paper we give an exposition of supercompact Radin forcing using coherent sequences of ultrafilters. This version of Radin forcing includes as special cases the Prikry forcing and Magidor forcing, both the measurable and supercompact versions. We also introduce some methods for...

Lei, Yeh and Zhang put forward the anti-forcing number af(G,M) for a perfect matching M in graph G, which is minimum of edges G not whose deletion results subgraph with unique M. The numbers all matchings form spectrum G. polynomial Af(G,x) counting classifying possessing same In this paper, we deduce recurrence formula continuity catacondensed hexagonal systems.

This paper, a contribution to “micro set theory”, is the study promised by the first author in [M4], but improved and extended by work of the second. We use the rudimentarily recursive (set theoretic) functions and the slightly larger collection of gentle functions to develop the theory of provident sets, which are transitive models of PROVI, a very weak subsystem of KP which nevertheless suppo...

In this article we study the strength of Σ 1 3 absoluteness (with real parameters) in various types of generic extensions, correcting and improving some results from [2]. We shall also make some comments relating this work to the bounded forcing axioms BMM, BPFA and BSPFA. The statement " Σ 1 3 absoluteness holds for ccc forcing " means that if a Σ 1 3 formula with real parameters has a solutio...

Zero forcing is an iterative graph coloring process where at each discrete time step, a colored vertex with a single uncolored neighbor forces that neighbor to become colored. The zero forcing number of a graph is the cardinality of the smallest set of initially colored vertices which forces the entire graph to eventually become colored. Connected forcing is a variant of zero forcing in which t...

In this paper we study the notion of forcing for Lukasiewicz predicate logic ( L∀, for short), along the lines of Robinson’s forcing in classical model theory. We deal with both finite and infinite forcing. As regard to the former we prove a Generic Model Theorem for L∀, while for the latter, we study the generic and existentially complete standard models of L∀.

For a connected graph G = (V,E) of order at least two, a total detour monophonic set of a graph G is a detour monophonic set S such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total detour monophonic set of G is the total detour monophonic number of G and is denoted by dmt(G). A subset T of a minimum total detour monophonic set S of G is a forcing total...

Understanding the past significant changes of the East Asia Summer Monsoon (EASM) and Winter Monsoon (EAWM) is critical for improving the projections of future climate over East Asia. One key issue that has remained outstanding from the paleo-climatic records is whether the evolution of the EASM and EAWM are correlated. Here, using a set of long-term transient simulations of the climate evoluti...

In 1962 Paul Cohen invented set-theoretic forcing to solve the independence problem of continuum hypothesis. It turns out that forcing is quite powerful tool and it has applications in many branches of mathematics. In 1970s Abraham Robinson extended Cohen’s forcing to model theory and developed finite forcing and infinite forcing. In this term paper we study Robinson’s finite forcing and relate...