an h-magic labeling in a h-decomposable graph g is a bijection f : v (g) ∪ e(g) → {1, 2, ..., p + q} such that for every copy h in the decomposition, σνεv(h) f(v) +  σeεe(h) f(e) is constant. f is said to be h-e-super magic if f(e(g)) = {1, 2, · · · , q}. a family of subgraphs h1,h2, · · · ,hh of g is a mixed cycle-decomposition of g if every subgraph hi is isomorphic to some cycle ck, for k ≥ ...

Let p and q be arbitrary positive numbers. It is shown that if q < p, then all solutions to the difference equation xn+1 = p+ q xn 1 + xn−1 , n = 0, 1, 2, . . . , x−1 > 0, x0 > 0 (E) converge to the positive equilibrium x = 12 ( q − 1 + √ (q − 1)2 + 4 p ) . The above result, taken together with the 1993 result of Kocić and Ladas for equation (E) with q ≥ p, gives global attractivity of the posi...

We provide the proofs for the theorems in the main paper. 1 Proofs for Planted Clustering In this section, Theorems 1–6 refer to the theorems in the main paper. Equations are numbered continuously from the main paper. We let n1 := rK and n2 := n − rK be the numbers of nonisolated and isolated nodes, respectively. 1.1 Proof of Theorem 1 The proof relies on information theoretical arguments and t...

1.2. Let E be a Euclidean vector space, Φ ⊂ E∗ a root system. Denote Q ⊂ h∗ the root lattice, and P ⊂ h∗ the weight lattice. Let Q∨ ⊂ E be the lattice generated by the coroots α∨, α ∈ Φ, the coroot lattice is dual to the weight lattice P ⊂ h∗, and P∨ ⊂ E the dual weight lattice, which is dual to the root lattice Q. Let H = HomZ(P,C) = Q∨ ⊗Z C∗ be the complex algebraic torus with Lie algebra h =...

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1. When the asset pays continuous dividend yield at the rate q, the expected rate of return of the asset is r − q under the risk neutral measure (see Chap 3 of Kwok's text for justification). Under the continuous Geometric Brownian process model, the logarithm of the asset price ratio over ∆t interval is normally distributed with mean r − q − σ 2 2 ∆t and variance σ 2 ∆t. Accordingly, the mean ...

where v ranges over all places of Q and Qv is the completion of Q at v, denote its TateShafarevich group. As usual, L(E/Q, s) is the complex L-function of E over Q. Since E is now known to be modular, Kolyvagin’s work [11] shows that X(E/Q) is finite if L(E/Q, s) has a zero at s = 1 of order ≤ 1, and that gE/Q is equal to the order of the zero of L(E/Q, s) at s = 1. His proof relies heavily on ...

We introduce the notion of a cosheaf on a fan Σ and define the Z2 Hodge spaces Hpq(Σ), which are the homology groups Hp(∧E ) of the qth exterior power of the cosheaf E on Σ. Geometrically, for σ ∈ Σ the stalk Eσ of the cosheaf E is the compact real torus in the real orbit Oσ(R) of the real toric variety XΣ(R). The Z2 Hodge spaces Hpq(Σ) are indexed by pairs p, q with 0 ≤ q ≤ p ≤ d, where d = di...