Existence of Perfect 4-Deletion-Correcting Codes with Length Six

By a T ^*(2, k, v)-code we mean a perfect4-deletion-correcting code of length 6 over an alphabet of size v, which is capable of correcting anycombination of up to 4 deletions and/or insertions of letters that occur in transmission of codewords. Thethird author (DCC Vol. 23, No. 1) presented a combinatorial construction for such codes and prove thata T ^*(2, 6, v)-code exists for all positive integers v 3 (mod 5), with 12 possible exceptions of v. In this paper, the notion of a directedgroup divisible quasidesign is introduced and used to show that a T ^*(2, 6,v)-code exists for all positive integers v 3 (mod 5), except possiblyfor v {173, 178, 203, 208}. The 12 missing cases for T ^*(2,6, v)-codes with v 3 (mod 5) are also provided, thereby the existenceproblem for T ^*(2, 6, v)-codes is almost complete.     

Some combinatorial constructions for optimal perfect deletion-correcting codes

There are two kinds of perfect t-deletion-correcting codes of length k over an alphabet of size v, those where the coordinates may be equal and those where all coordinates must be different. We call these two kinds of codes T*(k − t, k, v)-codes and T(k − t, k, v)-codes respectively. The cardinality of a T(k − t, k, v)-code is determined by its parameters, while T*(k − t, k, v)-codes do not necessarily have a fixed size. Let N(k − t, k, v) denote the maximum number of codewords in any T*(k − t, k, v)-code. A T*(k − t, k, v)-code with N(k − t, k, v) codewords is said to be optimal. In this paper, some combinatorial constructions for optimal T*(2, k, v)-codes are developed. Using these constructions, we are able to determine the values of N(2, 4, v) for all positive integers v. The values of N(2, 5, v) are also determined for almost all positive integers v, except for v = 13, 15, 19, 27 and 34.      

On the spanning connectivity of graphs

A k-containerC(u,v) of G between u and v is a set of k internally disjoint paths between u and v. A k-container C(u,v) of G is a k^*-container if it contains all vertices of G. A graph G is k^*-connected if there exists a k^*-container between any two distinct vertices. The spanning connectivity of G, κ^*(G), is defined to be the largest integer k such that G is w^*-connected for all 1?w?k if G is a 1^*-connected graph. In this paper, we prove that κ^*(G)?2δ(G)-n(G)+2 if (n(G)/2)+1?δ(G)?n(G)-2. Furthermore, we prove that κ^*(G-T)?2δ(G)-n(G)+2-|T| if T is a vertex subset with |T|?2δ(G)-n(G)-1.     

A Class of Iterative Algorithms and Solvability of Nonlinear Variational Inequalities Involving Multivalued Mappings

The solvability of the following class of nonlinear variational inequality (NVI) problems based on a class of iterative procedures, which possess an equivalence to a class of projection formulas, is presented.Determine an element x ^* K and u ^* T(x ^*) such that u ^*, x – x ^* 0 for all x K where T: K P(H) is a multivalued mapping from a real Hilbert space H into P(H), the power set of H, and K is a nonempty closed convex subset of H. The iterative procedure adopted here is represented by a nonlinear variational inequality: for arbitrarily chosen initial points x ⁰, y ⁰ K, u ⁰ T(y ⁰) and v ⁰ T(x ⁰), we have u ^k + x ^k+1 – y ^k , x – x ^k+1 0, x K, for u ^k T(y ^k ) and for k 0where v ^k + y ^k – x ^k , x – y ^k 0, x K and for v ^k T(x ^k ).     

A bound for Wilson's theorem (III)

In this article we prove the following statement. For any positive integers k ≥ 3 and λ, let c(k, λ) = exp{exp{k;rcub;}. If λv(v − 1) ≡ 0 (mod k(k − 1)) and λ(v − 1) ≡ 0 (mod k − 1) and v > c(k, λ), then a B(v, k, λ) exists. © 1996 John Wiley & Sons, Inc.     

Existence of T*(3, 4,v)‐codes

A word of length k over an alphabet Q of size v is a vector of length k with coordinates taken from Q. Let Q^*₄ be the set of all words of length 4 over Q. A T^*(3, 4, v)‐code over Q is a subset C^*? Q^*₄ such that every word of length 3 over Q occurs as a subword in exactly one word of C^*. Levenshtein has proved that a T^*(3, 4, vv)‐code exists for all even v. In this paper, the notion of a generalized candelabra t‐system is introduced and used to show that a T^*(3, 4, v)‐code exists for all odd v. Combining this with Levenshtein's result, the existence problem for a T^*(3,4, v)‐code is solved completely. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 42–53, 2005.     

A general descent framework for the monotone variational inequality problem

We present a framework for descent algorithms that solve the monotone variational inequality problem VIP _v which consists in finding a solutionv ^* _v satisfyings(v ^*)^T(v–v ^*)0, for allv _v. This unified framework includes, as special cases, some well known iterative methods and equivalent optimization formulations. A descent method is developed for an equivalent general optimization formulation and a proof of its convergence is given. Based on this unified logarithmic framework, we show that a variant of the descent method where each subproblem is only solved approximately is globally convergent under certain conditions.This research was supported in part by individual operating grants from NSERC.     

Speeding up the Dreyfus–Wagner algorithm for minimum Steiner trees

The Dreyfus–Wagner algorithm is a well-known dynamic programming method for computing minimum Steiner trees in general weighted graphs in time O ^*(3 ^k ), where k is the number of terminal nodes to be connected. We improve its running time to O ^*(2.684 ^k ) by showing that the optimum Steiner tree T can be partitioned into T = T ₁∪ T ₂ ∪ T ₃ in a certain way such that each T _i is a minimum Steiner tree in a suitable contracted graph G _i with less than terminals. In the rectilinear case, there exists a variant of the dynamic programming method that runs in O ^*(2.386 ^k ). In this case, our splitting technique yields an improvement to O ^*(2.335 ^k ).     

Decreasing Subsequences in Permutations and Wilf Equivalence for Involutions

In a recent paper, Backelin, West and Xin describe a map φ^* that recursively replaces all occurrences of the pattern k... 21 in a permutation σ by occurrences of the pattern (k−1)... 21 k. The resulting permutation φ^*(σ) contains no decreasing subsequence of length k. We prove that, rather unexpectedly, the map φ^* commutes with taking the inverse of a permutation. In the BWX paper, the definition of φ^* is actually extended to full rook placements on a Ferrers board (the permutations correspond to square boards), and the construction of the map φ^* is the key step in proving the following result. Let T be a set of patterns starting with the prefix 12... k. Let T′ be the set of patterns obtained by replacing this prefix by k... 21 in every pattern of T. Then for all n, the number of permutations of the symmetric group _n that avoid T equals the number of permutations of _n that avoid T′. Our commutation result, generalized to Ferrers boards, implies that the number of involutions of _n that avoid T is equal to the number of involutions of _n avoiding T′, as recently conjectured by Jaggard. Both authors were partially supported by the European Commission's IHRP Programme, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe”     

Random walks on the torus with several generators

Given n vectors {_i} ∈ [0, 1)^d, consider a random walk on the d‐dimensional torus ??^d = ?^d/?^d generated by these vectors by successive addition and subtraction. For certain sets of vectors, this walk converges to Haar (uniform) measure on the torus. We show that the discrepancy distance D(Q*^k) between the kth step distribution of the walk and Haar measure is bounded below by D(Q*^k) ≥ C₁k^?n/2, where C₁ = C(n, d) is a constant. If the vectors are badly approximated by rationals (in a sense we will define), then D(Q*^k) ≤ C₂k^?n/2d for C₂ = C(n, d, _j) a constant. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Graph Connectivity After Path Removal

Let G be a graph and u, v be two distinct vertices of G. A u—v path P is called nonseparating if G—V(P) is connected. The purpose of this paper is to study the number of nonseparating u—v path for two arbitrary vertices u and v of a given graph. For a positive integer k, we will show that there is a minimum integer (k) so that if G is an (k)-connected graph and u and v are two arbitrary vertices in G, then there exist k vertex disjoint paths P ₁[u,v], P ₂[u,v], . . ., P _k [u,v], such that G—V (P _i [u,v]) is connected for every i (i = 1, 2, ..., k). In fact, we will prove that (k) 22k+2. It is known that (1) = 3.. A result of Tutte showed that (2) = 3. We show that (3) = 6. In addition, we prove that if G is a 5-connected graph, then for every pair of vertices u and v there exists a path P[u, v] such that G—V(P[u, v]) is 2-connected.* Supported by NSF grant No. DMS-0070059 Supported by ONR grant N00014-97-1-0499 Supported by NSF grant No. 9531824     

A spectral mapping theorem for scalar-type spectral operators in locally convex spaces

LetT be a continuous scalar-type spectral operator defined on a quasi-complete locally convex spaceX, that is,T=fdP whereP is an equicontinuous spectral measure inX andf is aP-integrable function. It is shown that (T) is precisely the closedP-essential range of the functionf or equivalently, that (T) is equal to the support of the (unique) equicontinuous spectral measureQ ^* defined on the Borel sets of the extended complex plane ^* such thatQ ^*({})=0 andT=zdQ _*(z). This result is then used to prove a spectral mapping theorem; namely, thatg((T))=(g(T)) for anyQ ^*-integrable functiong: ^{* *} which is continuous on (T). This is an improvement on previous results of this type since it covers the case wheng((T))/{} is an unbounded set in a phenomenon which occurs often for continuous operatorsT defined in non-normable spacesX.     

Resolvable Mendelsohn designs with block size 4

Summary Letv andK be positive integers. A (v, k, 1)-Mendelsohn design (briefly (v, k, 1)-MD) is a pair (X,B) whereX is av-set (ofpoints) andB is a collection of cyclically orderedk-subsets ofX (calledblocks) such that every ordered pair of points ofX are consecutive in exactly one block ofB. A necessary condition for the existence of a (v, k, 1)-MD isv(v–1) 0 (modk). If the blocks of a (v, k, 1)-MD can be partitioned into parallel classes each containingv/k blocks wherev ) (modk) or (v – 1)/k blocks wherev 1 (modk), then the design is calledresolvable and denoted briefly by (v, k, 1)-RMD. It is known that a (v, 3,1)-RMD exists if and only ifv 0 or 1 (mod 3) andv 6. In this paper, it is shown that the necessary condition for the existence of a (v, 4, 1)-RMD, namelyv 0 or 1 (mod 4), is also sufficient, except forv = 4 and possibly exceptingv = 12. These constructions are equivalent to a resolvable decomposition of the complete symmetric directed graphK _v ^* onv vertices into 4-circuits.Research supported by the Natural Sciences and Engineering Research Council of Canada under Grant A-5320.     

Upper bounds on the general covering number Cλ(v,k, t,m)

A collection of k‐subsets (called blocks) of a v‐set X (v) = {1, 2,…, v} (with elements called points) is called a t‐(v, k, m, λ) covering if for every m‐subset M of X (v) there is a subcollection of with such that every block K ∈ has at least t points in common with M. It is required that v ≥ k ≥ t and v ≥ m ≥ t. The minimum number of blocks in a t‐(v, k, m, λ) covering is denoted by C_λ(v, k, t, m). We present some constructions producing the best known upper bounds on C_λ(v, k, t, m) for k = 6, a parameter of interest to lottery players. © 2004 Wiley Periodicals, Inc.     

Building Symmetric Designs With Building Sets

We introduce a uniform technique for constructing a family of symmetric designs with parameters (v(q ^m+1-1)/(q-1), kq ^m ,q ^m), where m is any positive integer, (v, k, ) are parameters of an abelian difference set, and q = k ²/(k - ) is a prime power. We utilize the Davis and Jedwab approach to constructing difference sets to show that our construction works whenever (v, k, ) are parameters of a McFarland difference set or its complement, a Spence difference set or its complement, a Davis–Jedwab difference set or its complement, or a Hadamard difference set of order 9 · 4 ^d , thus obtaining seven infinite families of symmetric designs.     

Inequalities of Putnam and Berger-Shaw forp-quasihyponormal operators

For an operatorT satisfying thatT ^*(T ^* T–TT ^*)T0, we shall show that and, moreover, tr itT isn-multicyclic.For an operatorT satisfying thatT ^* {(T ^* T) ^p –(TT ^*) ^p }T0 for somep (0, 1], we shall show that and, moreover, ifT isn-multicyclic.     

Designs on vector spaces constructed using quadratic forms

Let N be the set of nonnegative integers, let , t, v be in N and let K be a subset of N, let V be a v-dimensional vector space over the finite field GF(q), and let W _Kbe the set of subspaces of V whose dimensions belong to K. A t-[v, K, ; q]-design on V is a mapping : W _K N such that for every t-dimensional subspace, T, of V, we have (B)=. We construct t-[v, {t, t+1}, ; q-designs on the vector space GF(q ^v) over GF(q) for t2, v odd, and q ^t(q–1)² equal to the number of nondegenerate quadratic forms in t+1 variables over GF(q). Moreover, the vast majority of blocks of these designs have dimension t+1. We also construct nontrivial 2-[v, k, ; q]-designs for v odd and 3kv–3 and 3-[v, 4, q ⁶+q ⁵+q ⁴; q]-designs for v even. The distribution of subspaces in the designs is determined by the distribution of the pairs (Q, a) where Q is a nondegenerate quadratic form in k variables with coefficients in GF(q) and a is a vector with elements in GF(q ^v) such that Q(a)=0.This research was partly supported by NSA grant #MDA 904-88-H-2034.     

Reconstructing graphs as subsumed graphs of hypergraphs,and some self-complementary triple systems

LetV be a set ofn elements. The set of allk-subsets ofV is denoted . Ak-hypergraph G consists of avertex-set V(G) and anedgeset , wherek≥2. IfG is a 3-hypergraph, then the set of edges containing a given vertexvεV(G) define a graphG _v . The graphs {G _v νvεV(G)} aresubsumed byG. Each subsumed graphG _v is a graph with vertex-setV(G) − v. They can form the set of vertex-deleted subgraphs of a graphH, that is, eachG _v ≽H −v, whereV(H)=V(G). In this case,G is a hypergraphic reconstruction ofH. We show that certain families of self-complementary graphsH can be reconstructed in this way by a hypergraphG, and thatG can be extended to a hypergraphG ^*, all of whose subsumed graphs are isomorphic toH, whereG andG ^* are self-complementary hypergraphs. In particular, the Paley graphs can be reconstructed in this way. This work was supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada.     

On k-ordered Hamiltonian graphs

A Hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices of G, there exists a Hamiltonian cycle that encounters v₁, v₂, …, v_k in this order. Define f(k, n) as the smallest integer m for which any graph on n vertices with minimum degree at least m is a k-ordered Hamiltonian graph. In this article, answering a question of Ng and Schultz, we determine f(k, n) if n is sufficiently large in terms of k. Let g(k, n) = − 1. More precisely, we show that f(k, n) = g(k, n) if n ≥ 11k − 3. Furthermore, we show that f(k, n) ≥ g(k, n) for any n ≥ 2k. Finally we show that f(k, n) > g(k, n) if 2k ≤ n ≤ 3k − 6. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 17–25, 1999     

Global existence for one-dimensional motion of non-isentropic viscous fluids

We study the p-system with viscosity given by v_t ? u_x = 0, u_t + p(v)_x = (k(v)u_x)_x + f(∫ vdx, t), with the initial and the boundary conditions (v(x, 0), u(x,0)) = (v₀, u₀(x)), u(0,t) = u(X,t) = 0. To describe the motion of the fluid more realistically, many equations of state, namely the function p(v) have been proposed. In this paper, we adopt Planck's equation, which is defined only for v > b(> 0) and not a monotonic function of v, and prove the global existence of the smooth solution. The essential point of the proof is to obtain the bound of v of the form b < h(T) ? v(x, t) ? H(T) < ∞ for some constants h(T) and H(T).