On arcs sharing the maximum number of points with ovals in cyclic affine planes of odd order

The sporadic complete 12‐arc in PG(2, 13) contains eight points from a conic. In PG(2,q) with q>13 odd, all known complete k‐arcs sharing exactly ½(q+3) points with a conic 𝒞 have size at most ½(q+3)+2, with only two exceptions, both due to Pellegrino, which are complete (½(q+3)+3) arcs, one in PG(2, 19) and another in PG(2, 43). Here, three further exceptions are exhibited, namely a complete (½(q+3)+4)‐arc in PG(2, 17), and two complete (½(q+3)+3)‐arcs, one in PG(2, 27) and another in PG(2, 59). The main result is Theorem 6.1 which shows the existence of a (½(q^r+3)+3)‐arc in PG(2,q^r) with r odd and q≡3 (mod 4) sharing ½(q^r+3) points with a conic, whenever PG(2,q) has a (½(q^r+3)+3)‐arc sharing ½(q^r+3) points with a conic. A survey of results for smaller q obtained with the use of the MAGMA package is also presented. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 25–47, 2010     

S‐essential spectra and application to an example of transport operators

In this article, we give some results on the S‐essential spectra of a linear operator defined on a Banach space. Furthermore, we apply the obtained results to determine the S‐essential spectra of an integro‐differential operator with abstract boundary conditions in the Banach space L_p([?a,a] × [?1,1]),p ≥ 1 and a > 0. Copyright © 2012 John Wiley & Sons, Ltd.     

Solution of large‐scale weighted least‐squares problems

A sequence of least‐squares problems of the form min_y∥G^1/2(A^T y?h)∥₂, where G is an n×n positive‐definite diagonal weight matrix, and A an m×n (m?n) sparse matrix with some dense columns; has many applications in linear programming, electrical networks, elliptic boundary value problems, and structural analysis. We suggest low‐rank correction preconditioners for such problems, and a mixed solver (a combination of a direct solver and an iterative solver). The numerical results show that our technique for selecting the low‐rank correction matrix is very effective. Copyright © 2002 John Wiley & Sons, Ltd.     

Resolvable packings of Kv with K2 × Kc's

The study of resolvable packings of K_v with K_r × K_c's is motivated by the use of DNA library screening. We call such a packing a (v, K_r × K_c, 1)‐RP. As usual, a (v, K_r × K_c, 1)‐RP with the largest possible number of parallel classes (or, equivalently, the largest possible number of blocks) is called optimal. The resolvability implies v ≡ 0 (mod rc). Let ρ be the number of parallel classes of a (v, K_r × K_c, 1)‐RP. Then we have ρ ≤ ?(v‐1)/(r + c ? 2)?. In this article, we present a number of constructive methods to obtain optimal (v, K₂ × K_c, 1)‐RPs meeting the aforementioned bound and establish some existence results. In particular, we show that an optimal (v, K₂ × K₃, 1)‐RP meeting the bound exists if and only if v ≡ 0 (mod 6). © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 177–189, 2009     

On some subsets of Schechter's essential spectrum of a matrix operator and application to transport operator

This paper is devoted to the investigation of the essential approximate point spectrum and the essential defect spectrum of a 2 × 2 block operator matrix on a product of Banach spaces. The obtained results are applied to a two‐group transport operators with general boundary conditions in the Banach space L_p ([–a, a ] × [–1, 1]) × L_p ([–a, a ] × [–1, 1]), a > 0, p ≥ 1 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimally (k,k)‐edge‐connected graphs

For an integer l > 1, the l‐edge‐connectivity of a connected graph with at least l vertices is the smallest number of edges whose removal results in a graph with l components. A connected graph G is (k, l)‐edge‐connected if the l‐edge‐connectivity of G is at least k. In this paper, we present a structural characterization of minimally (k, k)‐edge‐connected graphs. As a result, former characterizations of minimally (2, 2)‐edge‐connected graphs in [J of Graph Theory 3 (1979), 15–22] are extended. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 116–131, 2003     

All (k;g)‐cages are k‐edge‐connected

A (k;g)‐cage is a k‐regular graph with girth g and with the least possible number of vertices. In this paper, we prove that (k;g)‐cages are k‐edge‐connected if g is even. Earlier, Wang, Xu, and Wang proved that (k;g)‐cages are k‐edge‐connected if g is odd. Combining our results, we conclude that the (k;g)‐cages are k‐edge‐connected. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 219–227, 2005     

Hamilton connectivity of line graphs and claw‐free graphs

Let G be a graph and let V₀ = {ν∈ V(G): d_G(ν) = 6}. We show in this paper that: (i) if G is a 6‐connected line graph and if |V₀| ≤ 29 or G[V₀] contains at most 5 vertex disjoint K₄'s, then G is Hamilton‐connected; (ii) every 8‐connected claw‐free graph is Hamilton‐connected. Several related results known before are generalized. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Some new results on superimposed codes

A (w,r) cover‐free family is a family of subsets of a finite set such that no intersection of w members of the family is covered by a union of r others. A (w,r) superimposed code is the incidence matrix of such a family. Such a family also arises in cryptography as the concept of key distribution pattern. In the present paper, we give some new results on superimposed codes. First we construct superimposed codes from super‐simple designs which give us results better than superimposed codes constructed by other known methods. Next we prove the uniqueness of the (1,2) superimposed code of size 9 × 12, the (2,2) superimposed code of size 14 × 8, and the (2,3) superimposed code of size 30 × 10. Finally, we improve numerical values of upper bounds for the asymptotic rate of some (w,r) superimposed codes. © 2004 Wiley Periodicals, Inc.     

Regular reals

Say that α is an n‐strongly c. e. (n‐strongly computably enumerable) real if α is a sum of n many strongly c. e. reals, and that α is regular if α is n‐strongly c. e. for some n. Let Sⁿ be the set of all n‐strongly c. e. reals, Reg be the set of regular reals and CE be the set of c. e. reals. Then we have: S¹ ? S² ? · · · ? Sⁿ ? · · · ? ? Reg ? CE . This gives a hierarchy of the c. e. reals. We also study the regularity of the d. c. e. reals. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

New large sets of t‐designs

Abstact: We introduce generalizations of earlier direct methods for constructing large sets of t‐designs. These are based on assembling systematically orbits of t‐homogeneous permutation groups in their induced actions on k‐subsets. By means of these techniques and the known recursive methods we construct an extensive number of new large sets, including new infinite families. In particular, a new series of LS[3](2(2 + m), 8·3^m ? 2, 16·3^m ? 3) is obtained. This also provides the smallest known ν for a t‐(ν, k, λ) design when t ≥ 16. We present our results compactly for ν ≤ 61, in tables derived from Pascal's triangle modulo appropriate primes. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 40–59, 2001     

Boundary value problems with eigenvalue depending boundary conditions

We investigate some classes of eigenvalue dependent boundary value problems of the form where A ? A⁺ is a symmetric operator or relation in a Krein space K, τ is a matrix function and Γ₀, Γ₁ are abstract boundary mappings. It is assumed that A admits a self‐adjoint extension in K which locally has the same spectral properties as a definitizable relation, and that τ is a matrix function which locally can be represented with the resolvent of a self‐adjoint definitizable relation. The strict part of τ is realized as the Weyl function of a symmetric operator T in a Krein space H, a self‐adjoint extension à of A × T in K × H with the property that the compressed resolvent P_K (à – λ)^–1|_K k yields the unique solution of the boundary value problem is constructed, and the local spectral properties of this so‐called linearization à are studied. The general results are applied to indefinite Sturm–Liouville operators with eigenvalue dependent boundary conditions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the critical exponents of random k‐SAT

There has been much recent interest in the satisfiability of random Boolean formulas. A random k‐SAT formula is the conjunction of m random clauses, each of which is the disjunction of k literals (a variable or its negation). It is known that when the number of variables n is large, there is a sharp transition from satisfiability to unsatisfiability; in the case of 2‐SAT this happens when m/n → 1, for 3‐SAT the critical ratio is thought to be m/n ≈ 4.2. The sharpness of this transition is characterized by a critical exponent, sometimes called ν = ν_k (the smaller the value of ν the sharper the transition). Experiments have suggested that ν₃ = 1.5 ± 0.1. ν₄ = 1.25 ± 0.05, ν₅ = 1.1 ± 0.05, ν₆ = 1.05 ± 0.05, and heuristics have suggested that ν_k → 1 as k → ∞. We give here a simple proof that each of these exponents is at least 2 (provided the exponent is well defined). This result holds for each of the three standard ensembles of random k‐SAT formulas: m clauses selected uniformly at random without replacement, m clauses selected uniformly at random with replacement, and each clause selected with probability p independent of the other clauses. We also obtain similar results for q‐colorability and the appearance of a q‐core in a random graph. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 182–195, 2002     

A characterization of weakly four‐connected graphs

A graph G = (V, E) is called weakly four‐connected if G is 4‐edge‐connected and G ? x is 2‐edge‐connected for all x ∈ V. We give sufficient conditions for the existence of ‘splittable’ vertices of degree four in weakly four‐connected graphs. By using these results we prove that every minimally weakly four‐connected graph on at least four vertices contains at least three ‘splittable’ vertices of degree four, which gives rise to an inductive construction of weakly four‐connected graphs. Our results can also be applied in the problem of finding 2‐connected orientations of graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 217–229, 2006     

Positive solutions of non‐positone Dirichlet boundary value problems with singularities in the phase variables

The paper presents existence results for positive solutions of the differential equations x ″ + μh (x) = 0 and x ″ + μf (t, x) = 0 satisfying the Dirichlet boundary conditions. Here μ is a positive parameter and h and f are singular functions of non‐positone type. Examples are given to illustrate the main results. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some observations on the substructure lattice of a Δ1 ultrapower

Given a (nontrivial) Δ₁ ultrapower ?/??, let ??_U denote the set of all Π₂‐correct substructures of ?/??; i.e., ??_U is the collection of all those subsets of |?/??| that are closed under computable (in the sense of ?/??) functions. Defining in the obvious way the lattice ??(?/??)) with domain ??_U, we obtain some preliminary results about lattice embeddings into – or realization as – an ??(?/??). The basis for these results, as far as we take the matter, consists of (1) the well‐known class of (non‐trivial) minimal ?/??'s, which function as atoms, and (2) the class of minimalfree ?/??'s, to whose nonemptiness a substantial section of the paper is devoted. It is shown that an infinite, convergent monotone sequence together with its limit point is embeddable in an ??(?/??), and that the initial segment lattices {0, 1,…, n } are not just embeddable in (as is trivial), but in fact realizable as, lattices ??(?/??). Finally, the diamond is (easily) embeddable; and if it is not realizable, then either the 1 ‐ 3 ‐ 1 lattice or the pentagon is at least embeddable (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

New Ore‐Type Conditions for H‐Linked Graphs

For a fixed (multi)graph H, a graph G is H‐linked if any injection f: V(H)→V(G) can be extended to an H‐subdivision in G. The notion of an H ‐linked graph encompasses several familiar graph classes, including k‐linked, k‐ordered and k‐connected graphs. In this article, we give two sharp Ore‐type degree sum conditions that assure a graph G is H ‐linked for arbitrary H. These results extend and refine several previous results on H ‐linked, k‐linked, and k‐ordered graphs. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:69–77, 2012     

Optimal (4up, 5, 1) optical orthogonal codes

By a (ν, k, 1)‐OOC we mean an optical orthogonal code. In this paper, it is proved that an optimal (4p, 5, 1)‐OOC exists for prime p ≡ 1 (mod 10), and that an optimal (4up, 5, 1)‐OOC exists for u = 2, 3 and prime p ≡ 11 (mod 20). These results are obtained by applying Weil's theorem. © 2004 Wiley Periodicals, Inc.     

Computing the conditioning of the components of a linear least‐squares solution

In this paper, we address the accuracy of the results for the overdetermined full rank linear least‐squares problem. We recall theoretical results obtained in (SIAM J. Matrix Anal. Appl. 2007; 29 (2):413–433) on conditioning of the least‐squares solution and the components of the solution when the matrix perturbations are measured in Frobenius or spectral norms. Then we define computable estimates for these condition numbers and we interpret them in terms of statistical quantities when the regression matrix and the right‐hand side are perturbed. In particular, we show that in the classical linear statistical model, the ratio of the variance of one component of the solution by the variance of the right‐hand side is exactly the condition number of this solution component when only perturbations on the right‐hand side are considered. We explain how to compute the variance–covariance matrix and the least‐squares conditioning using the libraries LAPACK (LAPACK Users' Guide (3rd edn). SIAM: Philadelphia, 1999) and ScaLAPACK (ScaLAPACK Users' Guide. SIAM: Philadelphia, 1997) and we give the corresponding computational cost. Finally we present a small historical numerical example that was used by Laplace (Théorie Analytique des Probabilités. Mme Ve Courcier, 1820; 497–530) for computing the mass of Jupiter and a physical application if the area of space geodesy. Copyright © 2008 John Wiley & Sons, Ltd.     

Spaces of Bessel‐potential type and embeddings: the super‐limiting case

We consider Bessel‐potential spaces modelled upon Lorentz‐Karamata spaces and establish embedding theorems in the super‐limiting case. In addition, we refine a result due to Triebel, in the context of Bessel‐potential spaces, itself an improvement of the Brézis‐Wainger result (super‐limiting case) about the “almost Lipschitz continuity” of elements of H^1+n/p_p (?ⁿ). These results improve and extend results due to Edmunds, Gurka and Opic in the context of logarithmic Bessel potential spaces. We also give examples of embeddings of Besselpotential type spaces which are not of logarithmic type. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)