In this present paper, by utilizing ij−δ semi open sets, we introduce ij−∧δ, ij−∨δ, g− ij−∧δ, g− ij−∨δ sets and investigate some of their properties in bitopological spaces. Also, we present and study the notions of ij − T∨sδ space, ij−∧δ continuous, ij−∧δ irresolute, ij−∧δ open and ij−∧δ homeomorphism functions.

Assume that n and δ are positive integers with 3 ≤ δ < n. Let hc(n, δ) be the minimum number of edges required to guarantee ann-vertex graphGwithminimumdegree δ(G) ≥ δ to be hamiltonian connected. Any n-vertex graphGwith δ(G) ≥ δ is hamiltonian connected if |E(G)| ≥ hc(n, δ). We prove that hc(n, δ) = C(n − δ + 1, 2) + δ2 − δ + 1 if δ ≤ b n+3×(n mod 2) 6 c+1,hc(n, δ) = C(n−b n 2 c+1, 2)+b n 2 c ...

a r t i c l e i n f o Keywords: Lithium isotopes A-type granite Mafic enclave Continental crust Lithium concentrations and isotopic compositions of 39 A-type granites and mafic enclaves from 11 plutons in Northeast China and the North China craton are used to constrain their genesis and to characterize further the average composition of the continental crust. Lithium concentrations (2.8 to 80 p...

Let γ(n, δ) denote the maximum possible domination number of a graph with n vertices and minimum degree δ. Using known results we determine γ(n, δ) for δ = 0, 1, 2, 3, n ≥ δ +1 and for all n, δ where δ = n − k and n is sufficiently large relative to k. We also obtain γ(n, δ) for all remaining values of (n, δ) when n ≤ 14 and all but 6 values of (n, δ) when n = 15 or 16.

If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ+,∞)-distributive and λ is 2λ supercompact, then by [3, Theorem 5], {δ < κ | δ is δ+ strongly compact yet δ isn’t δ+ supercompact} must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in whic...

A generalized star-triangle transformation and a concept of triangle-duality have been introduced recently in the physics literature to predict exact percolation threshold values of several lattices. We investigate the mathematical conditions for the solution of bond percolation models, and identify an infinite class of lattice graphs for which exact bond percolation thresholds may be rigorousl...

Caro, Davila, and Pepper recently proved δ(G)α(G)≤Δ(G)μ(G) for every graph G with minimum degree δ(G), maximum Δ(G), independence number α(G), matching μ(G). Answering some problems they posed, we characterize the extremal graphs δ(G)<Δ(G) as well δ(G)=Δ(G)=3.

environmental variables, including soil water content (swc), act as constraints on crop growth and productivity. therefore, open air (e0), perforated (e1) and non-perforated (e2) plastic housings were used with well-watered (w0), moderately-watered (w1) and water-stressed (w2) bush bean plants to explore the relationships between water-use efficiency (wue), carbon isotope discrimination (δ) and...

We study a standard model of exchange economies with individual endowments. It is well known that no rule is individually rational, e cient, and strategyproof. In order to quantify the extent of this impossibility, we parametrize axioms on allocation rules. Given an axiom A, a parametrization of A is a continuum of axioms {δ-A}δ∈[0,1] such that (i) δ-A is equivalent to A only if δ = 1; (ii) δ-A...