Some Infinite Classes of Optimal (v, {3, 4}, 1, Q)-OOCs with {Q \in \{(\frac {1}{3}, \frac {2}{3}), (\frac {2}{3}, \frac{1}{3})\}}

Variable-weight optical orthogonal code (OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service requirements. It is proved that optimal (v, {3, 4}, 1, (1/2, 1/2))-OOCs exist for some complete congruence classes of v. In this paper, for ${Q \in \{(1/3, 2/3), (2/3, 1/3)\}}$ , by using skew starters, it is also proved that optimal (v, {3, 4}, 1, Q)-OOCs exist for some complete congruence classes of v.     

Anderson model with decaying randomness existence of extended states

We study the Anderson model with decaying randomness inv ≥ 3 dimensions and show that there is absolutely continuous spectrum in [?2v, 2v]. The distribution of the potentials is assumed to have finite variance and the coupling constants decay at infinity at a rate α > 1.     

Optimal broadcast domination in polynomial time

Broadcast domination was introduced by Erwin in 2002, and it is a variant of the standard dominating set problem, such that different vertices can be assigned different domination powers. Broadcast domination assigns an integer power f(v)?0 to each vertex v of a given graph, such that every vertex of the graph is within distance f(v) from some vertex v having f(v)?1. The optimal broadcast domination problem seeks to minimize the sum of the powers assigned to the vertices of the graph. Since the presentation of this problem its computational complexity has been open, and the general belief has been that it might be NP-hard. In this paper, we show that optimal broadcast domination is actually in P, and we give a polynomial time algorithm for solving the problem on arbitrary graphs, using a non-standard approach.     

On optimal (v, 5, 2, 1) optical orthogonal codes

The size of a (v, 5, 2, 1) optical orthogonal code (OOC) is shown to be at most equal to ${\lceil{\frac{v}{12}}\rceil}$ when v ≡ 11 (mod 132) or v ≡ 154 (mod 924), and at most equal to ${\lfloor{\frac{v}{12}}\rfloor}$ in all the other cases. Thus a (v, 5, 2, 1)-OOC is naturally said to be optimal when its size reaches the above bound. Many direct and recursive constructions for infinite classes of optimal (v, 5, 2, 1)-OOCs are presented giving, in particular, a very strong indication about the existence of an optimal (p, 5, 2, 1)-OOC for every prime p ≡ 1 (mod 12).     

Banach lattices with the fatou property and optimal domains of kernel operators

New features of the Banach function space L¹_w(v), that is, the space of all v-scalarly integrable functions (with v any vector measure), are exposed. The Fatou property plays an essential role and leads to a new representation theorem for a large class of abstract Banach lattices. Applications are also given to the optimal domain of kernel operators taking their values in a Banach function space.     

Periodic solutions for wave equations with variable coefficients with nonlinear localized damping

We discuss the existence of periodic solutions to the wave equation with variable coefficients utt−div(A(x)∇u)+ρ(x,ut)=f(x,t) with Dirichlet boundary condition. Here ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g^′(v)?0 where a(x) is nonnegative, being positive only in a neighborhood of a part of the domain.     

Zero-free regions of derivatives of Riemann zeta function

Zero-free regions of thekth derivative of the Riemann zeta function ζ^(k)(s) are investigated. It is proved that fork≥3, ζ^(k)(s) has no zero in the region Res≥(1·1358826...)k+2. This result is an improvement upon the hitherto known zero-free region Res≥(7/4)k+2 on the right of the imaginary axis. The known zero-free region on the left of the imaginary axis is also improved by proving that ζ ^k)(s) may have at the most a finite number of non-real zeros on the left of the imaginary axis which are confined to a semicircle of finite radiusr _k centred at the origin.     

Long zero-free sequences in finite cyclic groups

A sequence in an additively written abelian group is called zero-free if each of its nonempty subsequences has sum different from the zero element of the group. The article determines the structure of the zero-free sequences with lengths greater than n/2 in the additive group Z_n of integers modulo n. The main result states that for each zero-free sequence  of length ?>n/2 in Z_n there is an integer g coprime to n such that if denotes the least positive integer in the congruence class ga_i (modulo n), then . The answers to a number of frequently asked zero-sum questions for cyclic groups follow as immediate consequences. Among other applications, best possible lower bounds are established for the maximum multiplicity of a term in a zero-free sequence with length greater than n/2, as well as for the maximum multiplicity of a generator. The approach is combinatorial and does not appeal to previously known nontrivial facts.     

Existence of resolvable optimal strong partially balanced designs

We shall refer to a strong partially balanced design SPBD(v,b,k;λ,0) whose b is the maximum number of blocks in all SPBD(v,b,k;λ,0), as an optimal strong partially balanced design, briefly OSPBD(v,k,λ). Resolvable strong partially balanced design was first formulated by Wang, Safavi-Naini and Pei [Combinatorial characterization of l-optimal authentication codes with arbitration, J. Combin. Math. Combin. Comput. 37 (2001) 205-224] in investigation of l-optimal authentication codes. This article investigates the existence of resolvable optimal strong partially balanced design ROSPBD(v,3,1). We show that there exists an ROSPBD(v,3,1) for any v?3 except v=6,12.     

Bi-Sobolev mappings and elliptic equations in the plane

Suppose that f=(u,v) is a homeomorphism in the plane of the Sobolev class such that its inverse is of the same Sobolev class. We prove that u and v have the same set of critical points. As an application we show that u and v are distributional solutions to the same non-trivial degenerate elliptic equation in divergence form. We study similar properties also in higher dimensions.     

An existence theory for pairwise balanced designs,III: Proof of the existence conjectures

Given positive integers k and λ, balanced incomplete block designs on v points with block size k and index λ exist for all sufficiently large integers v satisfying the congruences λ(v ? 1) ≡ 0 (mod k ? 1) and λv(v ? 1) ≡ 0 (mod k(k ? 1)). Analogous results hold for pairwise balanced designs where the block sizes come from a given set K of positive integers. We also observe that the number of nonisomorphic designs on v points with given block size k > 2 and index λ tends to infinity as v increases (subject to the above congruences).     

A theorem on the maximum number of disjoint steiner triple systems

We prove that D(2v + 1) ? v + 1 + D(v) for v > 3 where D(v) denotes the maximum number of pairwise disjoint Steiner triple systems of order v. Since D(v) ? v ? 2 it follows that for v > 3, D(2v + 1) = 2v ?1 whenever D(v) = v ? 2.     

Degree bounded matroids and submodular flows

We consider two related problems, the Minimum Bounded Degree Matroid Basis problem and the Minimum Bounded Degree Submodular Flow problem. The first problem is a generalization of the Minimum Bounded Degree Spanning Tree problem: We are given a matroid and a hypergraph on its ground set with lower and upper bounds f(e)≤g(e) for each hyperedge e. The task is to find a minimum cost basis which contains at least f(e) and at most g(e) elements from each hyperedge e. In the second problem we have a submodular flow problem, a lower bound f(v) and an upper bound g(v) for each node v, and the task is to find a minimum cost 0–1 submodular flow with the additional constraint that the sum of the incoming and outgoing flow at each node v is between f(v) and g(v). Both of these problems are NP-hard (even the feasibility problems are NP-complete), but we show that they can be approximated in the following sense. Let opt be the value of the optimal solution. For the first problem we give an algorithm that finds a basis B of cost no more than opt such that f(e)?2Δ+1≤|B∩e|≤g(e)+2Δ?1 for every hyperedge e, where Δ is the maximum degree of the hypergraph. If there are only upper bounds (or only lower bounds), then the violation can be decreased to Δ?1. For the second problem we can find a 0–1 submodular flow of cost at most opt where the sum of the incoming and outgoing flow at each node v is between f(v)?1 and g(v)+1. These results can be applied to obtain approximation algorithms for several combinatorial optimization problems with degree constraints, including the Minimum Crossing Spanning Tree problem, the Minimum Bounded Degree Spanning Tree Union problem, the Minimum Bounded Degree Directed Cut Cover problem, and the Minimum Bounded Degree Graph Orientation problem.     

The optimal strategy structure of an intermittently operated service channel

An analysis is made of the policy space structure for an intermittently operated M/M/1 queuing system. When the queue length builds up to limit N, the service channel is opened and service rendered until the queue length is reduced to some lower number, v, whereupon the channel is closed. Certain costs, namely these of lost customers, of giving service and of opening and closing the channel are imposed. A technique for dividing the policy space into regions of optimal {v, N} is developed so that the best strategy for any combination of costs is immediately visible to the decision maker.     

Scheduling of nonresumable jobs and flexible maintenance activities on a single machine to minimize makespan

This study addresses a single machine scheduling problem with periodic maintenance, where the machine is assumed to be stopped periodically for maintenance for a constant time w during the scheduling period. Meanwhile, the maintenance period [u, v] is assumed to have been previously arranged and the time w is assumed not to exceed the available maintenance period [u, v] (i.e. w ? v − u). The time u(v) is the earliest (latest) time at which the machine starts (stops) its maintenance. The objective is to minimize the makespan. Two mixed binary integer programming (BIP) models are provided for deriving the optimal solution. Additionally, an efficient heuristic is proposed for finding the near-optimal solution for large-sized problems. Finally, computational results are provided to demonstrate the efficiency of the models and the effectiveness of the heuristics. The mixed BIP model can optimally solve up to 100-job instances, while the average percentage error of the heuristic is below 1%.     

On 2-designs

Denote by M_v the set of integers b for which there exists a 2-design (linear space) with v points and b lines. M_v is determined as accurately as possible. On one hand, it is shown for v > v₀ that M_v contains the interval [v + p + 1, v + p + q ? 1]. On the other hand for v of the form p² + p + 1 it is shown that the interval [v + 1, v + p ? 1] is disjoint from M_v; and if v > v₀ and p is of the form q² + q, then an additional interval [v + p + 1, v + p + q ? 1] is disjoint from M_v.     

New Effects of an Old Z. Zalcwasser's Theorem

It is well known that the strong (C, 1)-summability of an orthogonal series does not imply its very strong (C, 1)-summability, generally. For a given index-sequence {v _n }, first, Z. Zalcwasser gave an interesting condition implying the strong (C, 1)-summability of these partial sums s _{v n} (x). We show that Zalcwasser's condition on {v _n } holds if and only if the subsequence {v ₂ n} is quasi geometrically increasing. Utilizing this fact and known theorems several strong summability results are presented for given index-sequences {v _n }.     

On the maximum number of disjoint Steiner triple systems

Let D(v) be the maximum number of pairwise disjoint Steiner triple systems of order v. We prove that D(3v)≥2v+D(v) for every v ≡ 1 or 3 (mod 6), v≥3. As a corollary, we have D(3ⁿ)=3ⁿ-2 for every n≥1.     

Domination theorems in non-commutativeC *-Algebras

LetA be a Banach algebra with unite. Ifu, v ₁, …,v _n are elements ofA thenu is said to be dominated byv ₁, …,v _n if there is γ>O such that for everyx∈A we have ‖ux‖≤γ(‖v ₁ x‖+…+‖v _n x‖). It is shown that for everyx∈A we have ‖ux‖≤γ(‖v ₁ x‖+…+‖v _n x‖). it is shown for finite-dimensional algebras that the above condition is sufficient foru to belong to the left ideal generated byv ₁, …,v _n in some superalgebra ofA. A similar result is proved for subalgebras of the algebra of all bounded operators on a Hilbert space.