Let T (G) be the number of spanning trees in graph G. In this note we explore the asymptotics of T (G) for circulant graphs. The circulant graph Cs1,s2,···,sk n is the 2k regular graph with n vertices labelled 0, 1, 2, · · · , n − 1, where node i has the 2k neighbors, (0 ≤ i ≤ n − 1) adjacent to vertices i + s1, i + s2, · · · , i + sk mod n. In this note we give a closed formula for the asympto...

and Applied Analysis 3 (d) If p = q, x = y, |c| = |d| = 1, k = l = 1, m = n = 2, θ0(t) = θ1(t) ≡ 0, θ0(t) = δ(t), F = F (t, xθ0(t), x 󸀠 θ1(t) , xθ0(t)) = f (x (t)) x 󸀠 (t) + g (t, x (t − δ (t))) + e (t) , (15) then system (10) reduces to (7). The main purpose of this paper is to establish sufficient conditions for the existence of π-antiperiodic solutions to system (10) by using the method of c...

The present study in pot experiments was conducted on “Influence of phosphorus and sulphur application sesame yield high P soils Telangana”. experiment two different levels status with 67.29 kg P2O5 ha-1 soil 1 (S1) 83.46 2 (S2). treatments were taken factorial completely randomized design combination five (P0-0, P25- 5, P50-10,P75-15 P100-20 soil) four (S0-0,S1-10,S2-20 S3-30 soil). A signific...

and the two values in (1, 1) are both nonnegative. Note that for each functional constraint, there is a nonnegative “slack” between the evaluation on the left-hand side and the constant on the right-hand side. Explicitly, these slacks are given by 1 (= 6 − 5), 4 (= 3 − (−1)), 3 (= 5 − 2), and 1 (= 4 − 3), respectively. Now, if we let s1 = 1, s2 = 4, s3 = 3, and s4 = 1, then the augmented soluti...

It is easy to deal with a Venn Diagram for 1 ≤ n ≤ 3 sets. When n gets larger, the picture becomes more complicated, that's why we thought at the following codification. That’s why we propose an easy and systematic algebraic way of dealing with the representation of intersections and unions of many sets. Introduction. Let's first consider 1 ≤ n ≤ 9, and the sets S1, S2, ..., Sn. Then one gets 2...

The study was conducted to assess the native and alien plant species in one of the highly disturbed forest (S1) and less disturbed forest (S2) in Mt. Manunggal, Cebu Island, Philippines. Twenty-four quadrats with a size of 20mx20 m were established using a quadrat sampling technique to identify and record all plant species. Diversity indices were utilized to determine species abundance, richnes...

Exercise 1. Show that distances between sets do not necessarily satisfy the triangle inequality. That is, it is possible that d(S1, S2) + d(S2, S3) > d(S1, S3) for some sets S1, S2 and S3. Exercise 2. Prove that d(x, y) ≥ d(S, x)− d(S, y) and thus d(x, y) ≥ |d(S, x)− d(S, y)|. Proof. Fix ε > 0. Let y′ ∈ S be such that d(y′, y) ≤ d(S, y) + ε (if S is a finite set, there is y′ ∈ S s.t. d(y, y′) =...

Hepatocyte expression of pre-S1 and pre-S2 in relation to hepatitis B virus replication (hepatitis B virus-DNA in serum and HBcAg in the liver), histological activity and hepatitis delta virus superinfection was studied by indirect immunofluorescence on frozen sections of liver specimens from 68 patients with chronic hepatitis B virus infection. All 44 patients with chronic type B hepatitis had...

The sequent calculus sL for the Lambek calculus L ([2]) has no structural rules. Interestingly, sL is equivalent to a multimodal calculus mL, which consists of the nonassociative Lambek calculus with the structural rule of associativity. This paper proves that the sequent calculus or hypersequent calculus hD of the discontinuous Lambek calculus1 ([7], [4] and [8]), which like sL has no structur...

When two successive, spatially overlapping, targets (S1 and S2) are presented on a blank background, S2 typically dominates in explicit perception. We tested whether S2 dominance is also found for the conditions of presenting S1 and S2 in a stream of irrelevant objects. Successive target letters were presented within a stream of invariant stimulus items (capital Is). The stream items were prese...