1. Problem 15-2 on Page 364 of [CLRS01]. Solution: Assume that the optimal solution packs the n words in r lines; this solution can be decomposed into two parts, viz., the words packed on the first line and the words packed in the remaining (r − 1) lines. Regardless of the choice for the last word on line 1, the remaining words must be packed optimally in the lines 2 through (r − 1). (Why?) We ...

and Applied Analysis 3 Proof. Assume that 1.1 has a nonoscillatory solution x t . Without loss of generality, suppose that it is an eventually positive solution if it is an eventually negative solution, the proof is similar , that is, x t > 0 for all t ≥ t0. We consider the following three cases. Case 1. Suppose that x′ t is oscillatory. Then there exists t1 ≥ t0 such that x′ t1 0. From 1.1 , w...

In this paper we extend our previously discovered exact solution for an SU(2) gauge theory coupled to a massless, non-interacting scalar field, to the general group SU(N+1). Using the first-order formalism of Bogomolny, an exact, spherically symmetric solution for the gauge and scalar fields is found. This solution is similiar to the Schwarzschild solution of general relativity, in that the gau...

Abstract: In this paper, we study a pessimistic linear bilevel multi-follower programming problem in a referential-uncooperative situation (for short, PLBMF-RU) where the followers are uncooperative while having cross reference to decision making information of the other followers. We first give a model and the solution definitions of PLBMF-RU, and then use a simple example to illustrate it. Fu...

and Applied Analysis 3 Remark 1.5. Condition A3 in Theorem 1.4 can be replaced by A3’ φ t, x ψ t, x is monotone in x and left lower semicontinuous at any t ∈ 0, 1 . 2. Main Results A real T -periodic continuous function y : R → R is said to be a T -periodic solution of 1.1 if substitution of it into 1.1 yields an identity for all t ∈ R. It is well known see, e.g., 1, 2 that 1.1 has a T -periodi...

The paper concerns with the computation with high accuracy of the non-negative deenite stabilizing solution of the matrix Riccati equation A T X + XA + XQX + R = 0, where A is stable, Q = Q T 0 and R = R T 0. The limiting accuracy solution can be computed by using an iterative reenement technique based on the Newton's method. A proof of global convergence of the iterative process, with nal quad...

In this article, we study the stochastic wave equation on entire space $\mathbb{R}^d$, driven by a space-time L\'evy white noise with possibly infinite variance (such as $\alpha$-stable noise). equation, is multiplied Lipschitz function $\sigma(u)$ of solution. We assume that spatial dimension $d=1$ or $d=2$. Under general conditions measure noise, prove existence solution, and show that, funct...

where f is an operator mapping from some domain D in a real or complex Banach space X to another Y. Ostrowski and Kantorovich once gave a popular formulation which does not presume the existence of the solution (see Wang, 1999). However, it implies the existence of the solution because their hypothesis can result in the existence of x∗ = lim xn . Therefore, the existence of the solution is a ve...

An explicit representation formula for a solution is given in Theorem 1, when g is a bounded smooth vector field. The case of a complete vector field g ∈ C1(Rn,Rn) is analyzed in Theorem 2, introducing adequate stopping times. The main support in writing a solution comes from the solution yλ(τ1, τ2) = G (F (τ1, τ2)) [λ], (τ1, τ2) ∈ R, λ ∈ R, satisfying a deterministic gradient system  ∂τ1yλ (...

Let q ≥ 1 + 2 N . We prove that any positive solution of (E) ∂tu − ∆u + u = 0 in R × (0,∞) admits an initial trace which is a nonnegative Borel measure, outer regular with respect to the fine topology associated to the Bessel capacity C 2 q ,q in R N (q = q/q − 1)) and absolutely continuous with respect to this capacity. If ν is a nonnegative Borel measure in R with the above properties we cons...