We present a new short proof of the explicit formula for the group of links (and also link maps) in the ’quadruple point free’ dimension. Denote by Lp,q (respectively, C m−p p ) the group of smooth embeddings S p ⊔S → S (respectively, S → S) up to smooth isotopy. Denote by LM p,q the group of link maps S ⊔ S → S up to link homotopy. Theorem 1. If p ≤ q ≤ m− 3 and 2p+ 2q ≤ 3m− 6 then Lmp,q ∼= πp...

A graph G(p, q) is said to be odd harmonious if there exists an injection f : V (G) → {0, 1, 2, · · · , 2q − 1} such that the induced function f∗ : E(G) → {1, 3, · · · , 2q − 1} defined by f∗(uv) = f(u) + f(v) is a bijection. A graph that admits odd harmonious labeling is called odd harmonious graph. In this paper, we prove that shadow and splitting of graph K2,n, Cn for n ≡ 0 (mod 4), the grap...

Let G = (V, E) be a finite, simple and undirected graph. A graph G with q edges is said to be odd-graceful if there is an injection f : V (G) → {0, 1, 2, . . . , 2q− 1} such that, when each edge xy is assigned the label |f (x)− f (y)| , the resulting edge labels are {1, 3, 5, . . . , 2q− 1} and f is called an odd graceful labeling of G. Motivated by the work of Z. Gao [6] in which he studied th...

Parkinson disease (PD) is the second most common neurodegenerative disorder, surpassed in frequency only by Alzheimer disease. Elsewhere we have reported linkage to chromosome 2q in a sample of sibling pairs with PD. We have now expanded our sample to include 150 families meeting our strictest diagnostic definition of verified PD. To further delineate the chromosome 2q linkage, we have performe...

(1) pg(X) < 2q(X)− 3 admits a fibration (i.e. a surjective morphism with connected fibres) onto a smooth curve of genus at least 2 (cf. also the modern age references [BPV], Proposition 4.2 and [Be1], C.8). This basic result in surface theory concerns in fact surfaces whose Albanese map is generically finite: otherwise the Albanese map of X automatically provides a fibration onto a smooth curve...

gastric duplication cysts comprise 2-7% of gastrointestinal duplications, rare congenital malformations that can be present at almost any part of the alimentary tract. they mostly present with gastrointestinal obstruction symptoms, ulceration, and painless hemorrhage. symptoms include nausea, vomiting, and fullness sensation. gastric duplications are mostly cystic in shape. herein, we present a...