A note on 3-blocked designs

A blocking set of a design different from a 2-(λ + 2, λ + 1, λ) design has at least 3 points. The aim of this note is to establish which 2-(v, k, λ) designs D with r ≥ 2λ may contain a blocking 3-set. The main results are the following. If D contains a blocking 3-set, then D is one of the following designs: a 2-(2λ + 3, λ + 1, λ), a 2-(2λ + 1), λ + 1, λ), a 2-(2λ - 1, λ, λ), a 2-(4λ + 3, 2λ + 1, λ) Hadamard design with λ odd, or a 2-(4λ - 1, 2λ, λ) Hadamard design. Moreover a blocking 3-set in a 2-(4λ + 3, 2λ + 1, λ) Hadamard design exists if and only if there is a line with three points. In the case of 2- (4λ - 1, 2λ, λ) Hadamard design with λ odd, we give necessary and sufficient conditions for the existence of a blocking 3-set, while in the case λ even, a necessary condition is given. © 1997 John Wiley & Sons, Inc.     

On c‐Bhaskar Rao designs with block size 4

Several new families of c‐Bhaskar Rao designs with block size 4 are constructed. The necessary conditions for the existence of a c‐BRD (υ,4,λ) are that: (1)λ_min=?λ/3 ≤ c ≤ λ and (2a) c≡λ (mod 2), if υ > 4 or (2b) c≡ λ (mod 4), if υ = 4 or (2c) c≠ λ ? 2, if υ = 5. It is proved that these conditions are necessary, and are sufficient for most pairs of c and λ; in particular, they are sufficient whenever λ?c ≠ 2 for c > 0 and whenever c ? λ_min≠ 2 for c < 0. For c < 0, the necessary conditions are sufficient for υ> 101; for the classic Bhaskar Rao designs, i.e., c = 0, we show the necessary conditions are sufficient with the possible exception of 0‐BRD (υ,4,2)'s for υ≡ 4 (mod 6). © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 361–386, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10009     

Application of a Mickens finite‐difference scheme to the cylindrical Bratu‐Gelfand problem

The boundary value problem Δu + λe^u = 0 where u = 0 on the boundary is often referred to as “the Bratu problem.” The Bratu problem with cylindrical radial operators, also known as the cylindrical Bratu‐Gelfand problem, is considered here. It is a nonlinear eigenvalue problem with two known bifurcated solutions for λ < λ_c, no solutions for λ > λ_c and a unique solution when λ = λ_c. Numerical solutions to the Bratu‐Gelfand problem at the critical value of λ_c = 2 are computed using nonstandard finite‐difference schemes known as Mickens finite differences. Comparison of numerical results obtained by solving the Bratu‐Gelfand problem using a Mickens discretization with results obtained using standard finite differences for λ < 2 are given, which illustrate the superiority of the nonstandard scheme. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 327–337, 2004     

Existence of entire solutions for a class of quasilinear elliptic equations

The paper deals with the existence of entire solutions for a quasilinear equation ${(\mathcal E)_\lambda}$ in ${\mathbb{R}^N}$ , depending on a real parameter λ, which involves a general elliptic operator in divergence form A and two main nonlinearities. The competing nonlinear terms combine each other, being the first subcritical and the latter supercritical. We prove the existence of a critical value λ* > 0 with the property that ${(\mathcal E)_\lambda}$ admits nontrivial non-negative entire solutions if and only if λ ≥ λ*. Furthermore, when ${\lambda > \overline{\lambda} \ge \lambda^*}$ , the existence of a second independent nontrivial non-negative entire solution of ${(\mathcal{E})_\lambda}$ is proved under a further natural assumption on A.     

Local bifurcation from characteristic values with finite multiplicity and its application to axisymmetric buckled states of a thin spherical shell

The purpose of this paper is to study bifurcation points of the equation T(v) = L(λ,v) + M(λ,v), (λ,v) ? Λ × D in Banach spaces, where for any fixed λ ? Λ, T, L(λ,·) are linear mappings and M(λ,·) is a nonlinear mapping of higher order, M(λ,0) = 0 for all λ ? Λ. We assume that λ is a characteristic value of the pair (T, L) such that the mapping T – L(λ ,·) is Fredholm with nullity p and index s, p > s ? 0. We shall find some sufficient conditions to show that (λ ,0) is a bifurcation point of the above equation. The results obtained will be used to consider bifurcation points of the axisymmetric buckling of a thin spherical shell subjected to a uniform compressive force consisting of a pair of coupled non-linear ordinary differential equations of second order.     

Balanced incomplete block designs with block size 8

The necessary conditions for the existence of a balanced incomplete block design on υ ≥ k points, with index λ and block size k, are that: For k = 8, these conditions are known to be sufficient when λ = 1, with 38 possible exceptions, the largest of which is υ = 3,753. For these 38 values of υ, we show (υ, 8, λ ) BIBDs exist whenever λ > 1 for all but five possible values of υ, the largest of which is υ = 1,177, and these five υ's are the only values for which more than one value of λ is open. For λ>1, we show the necessary conditions are sufficient with the definite exception of two further values of υ, and the possible exception of 7 further values of υ, the largest of which is υ=589. In particular, we show the necessary conditions are sufficient for all λ> 5 and for λ = 4 when υ ≠ 22. We also look at (8, λ) GDDs of type 7^m. Our grouplet divisible design construction is also refined, and we construct and exploit α ‐ frames in constructing several other BIBDs. In addition, we give a PBD basis result for {n: n ≡ 0, 1; mod 8, n ≥ 8}, and construct a few new TDs with index > 1. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 233–268, 2001     

The Eigenvalues of the Angular Spheroidal Wave Equation

Approximate relations are obtained between the eigenvalues λ and the ellipticity parameter c² of the angular spheroidal wave equation. Although based on WKBJ methods and the assumption that λ is large, the relations are useful throughout the complex c²-plane. They are exact at c² = 0, and reproduce the standard asymptotic formulas for λ when c² is large. At intermediate values of c², they provide approximations for the square-root branch points of the multivalued function λ(c²) in the complex c²-plane at which adjacent eigenvalues of the same class become equal in pairs. These branch points lie on an infinite sequence of distorted circular rings. Their exact locations have been computed for the first four rings for angular wavenumbers m = 0,…,4.     

Generalized Hopf bifurcation in Hilbert space

We consider a family of semilinear evolution equations in Hilbert space of the form with, in general, unbounded operators *A(λ), F(λ·) depending analytically on a real parameter λ. We assume that the origin is a stationary solution, i.e. F(λ,0) = 0, for all λ ε R and that the linearization (with respect to u) at the origin is given by du/dt + A(λ)u = 0. Our essential assumption is the following: A(λ) possesses one pair of simple complex conjugate eigenvalues μ(λ) = Re μ(λ) ± i Im μ(λ) such that Im μ(0) > 0 and for some m ε N or If m = 1 the curves of eigenvalues μ(λ) cross the imaginary axis transversally at ±i Im μ(0). In this case a unique branch of periodic solutions emanates from the origin at λ = 0 which is commonly called Hopf bifurcation. If μ(λ) and the imaginary axis are no longer transversal, i.e. m > 1, we call a bifurcation of periodic solutions, if it occurs, a generalized Hopf bifurcation. It is remarkable that up to m such branches may exist. Our approach gives the number of bifurcating solutions, their direction of bifurcation, and its asymptotic expansion. We regain the results of D. Flockerzi who established them in a completely different way for ordinary differential equations.     

Local bifurcations of plane running waves for the generalized cubic Schrödinger equation

For the generalized cubic Schrödinger equation, we consider a periodic boundary value problem in the case of n independent space variables. For this boundary value problem, there exists a countable set of plane running waves periodic with respect to the time variable. We analyze their stability and local bifurcations under the change of stability. We show that invariant tori of dimension 2, ..., n + 1 can bifurcate from each of them. We obtain asymptotic formulas for the solutions on invariant tori and stability conditions for bifurcating tori as well as parameter ranges in which, starting from n = 3, a subcritical (stiff) bifurcation of invariant tori is possible.     

Further results on the maximum size of a hole in an incomplete t‐wise balanced design with specified minimum block size*

Kreher and Rees 3 proved that if h is the size of a hole in an incomplete balanced design of order υ and index λ having minimum block size , then, They showed that when t = 2 or 3, this bound is sharp infinitely often in that for each h ≥ t and each k ≥ t + 1, (t,h,k) ≠(3,3,4), there exists an ItBD meeting the bound. In this article, we show that this bound is sharp infinitely often for every t, viz., for each t ≥ 4 there exists a constant C_t > 0 such that whenever (h ? t)(k ? t ? 1) ≥ C_t there exists an ItBD meeting the bound for some λ = λ(t,h,k). We then describe an algorithm by which it appears that one can obtain a reasonable upper bound on C_t for any given value of t. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 256–281, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10014     

Ground states for Kirchhoff‐type equations with critical or supercritical growth

In this paper, we study the following Kirchhoff‐type equation with critical or supercritical growth where a>0, b>0, λ>0, p≥6 and f is a continuous superlinear but subcritical nonlinearity. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for small λ>0 by Nehari method. Moreover, we regard b as a parameter and obtain a convergence property of the ground state solution as b↘0. Our main contribution is related to the fact that we are able to deal with the case p>6.     

Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I

We consider the blow-up of the solution to a semilinear heat equation with nonlinear boundary condition. We establish conditions on nonlinearities sufficient to guarantee that u(x, t) exists for all time t > 0 as well as conditions on data forcing the solution u(x, t) to blow up at some finite time t*. Moreover, an upper bound for t* is derived. Under somewhat more restrictive conditions, lower bounds for t* are also derived.     

Complete classification of the positive solutions of −Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">u</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We study the equation −Δu + u ^q = 0, q > 1, in a bounded C ² domain Ω ⊂ ℝ ^N . A positive solution of the equation is moderate if it is dominated by a harmonic function and σ-moderate if it is the limit of an increasing sequence of moderate solutions. It is known that in the subcritical case, 1 < q <, q _c = (N + 1)/(N − 1), every positive solution is σ-moderate [32]. More recently, Dynkin proved, by probabilistic methods, that this remains valid in the supercritical case for q ≤ 2, [15]. The question remained open for q > 2. In this paper, we prove that for all q ≥ q _c , every positive solution is σ-moderate. We use purely analytic techniques, which apply to the full supercritical range. The main tools come from linear and non-linear potential theory. Combined with previous results, our result establishes a one-to-one correspondence between positive solutions and their boundary traces in the sense of [36].     

On the limiting behaviour of solutions to boundary integral equations associated with time harmonic wave equations for small frequencies

The treatment of boundary value problems for Helmholtz equation and for the time harmonic Maxwell's equations by boundary integral equations leads to integral equations of the second kind which are uniquely solvable for small positive frequencies λ. However, the integral equations obtained in the limiting case λ = 0 which are related to boundary value problems of potential theory in general are not uniquely solvable since the corresponding boundary value problems are not. By first considering in a general setting of a Banach space X the limiting behaviour of solutions ?_λ to the equation ?_λ – K λ ? λ = fλ as λ → 0 where {K_λ: X → X, λ ∈ (0,α)}, α > 0, denotes a family of compact linear operators such that I - K_λ (I identity) is bijective for λ∈(0,α) whilst I - K₀ is not and ‖ K_λ – K₀‖ →, 0, ‖f_λ – f0‖ → 0, λ → 0, and then applying the results to the boundary integral operators, the limiting behaviour of the integral equations is considered. Thus, the results obtained by Mac Camey for the Helmholtz equation are extended to the case of non-connected boundaries and Werner's results on the integral equations for the Maxwell's equations are extended to the case of multiply connected boundaries.     

Bifurcation of periodic solutions for a semilinear wave equation

Bifurcation of time periodic solutions and their regularity are proved for a semilinear wave equation, u_tt?u_xx?λu=f(λ,x,u),x?(0,π), t?R, together with Dirichlet or Neumann boundary conditions at x = 0 and x = π. The set of values of the real parameter λ where bifurcation from the trivial solution u = 0 occurs is dense in $\text{R}$.     

Co‐existence of Poisson and non‐Poisson processes in ordered parallel multilane pedestrian traffic

We show that non‐Poisson and Poisson processes can coexist in ordered parallel multilane pedestrian traffic, in the presence of lane switching which asymmetrically benefits the switchers and nonswitchers. Pedestrians join at the tail end of a queue and transact at the opposite front end. Their aim is to complete a transaction within the shortest possible time, and they can transfer to a shorter queue with probability p_s. Traffic is described by the utilization parameter U = λ〈t_s〉/N, where λ is the average rate of pedestrians entering the system, 〈t_s〉 is the average transaction time, and N is the number of lanes. Using an agent‐based model, we investigate the dependence of the average completion time 〈t_c〉 with variable K = 1 + (1 ? U)^?1 for different N and 〈t_s〉 values. In the absence of switching (p_s = 0), we found that 〈t_c〉 ∝ K^τ, where τ ≈ 1 regardless of N and 〈t_s〉. Lane switching (p_s = 1) reduces 〈t_c〉 for a given K, but its characteristic dependence with K differs for nonswitchers and switchers in the same traffic system. For the nonswitchers, 〈t_c〉 ∝ K^τ, where τ < 1. At low K values, switchers have a larger 〈t_c〉 that also increases more rapidly with K. At large K, the increase rates become equal for both. For nonswitchers, the possible t_c values obey an exponentially decaying probability density function p(t_c). The switchers on the other hand, are described by a fat‐tailed p(t_c) implying that a few are penalized with t_c values that are considerably longer than any of those experienced by nonswitchers. © 2006 Wiley Periodicals, Inc. Complexity 11: 35–42, 2006     

Symmetry and bifurcation near families of solutions

Let X, Z and Λ be Banach spaces, M: X × Λ → Z a C¹-function, and assume that the equation M(x, λ) = 0 has a family of solutions for λ = 0. In this paper we consider the bifurcation of solutions from this family, for ¦λ¦ small, under the condition that both the unperturbed (λ = 0) and the perturbed (λ ≠ 0) equations have certain symmetry properties. The problem is reduced by the Liapunov-Schmidt method, and the bifurcation equations are solved by a straightforward use of the symmetry. As an application we obtain existence of certain periodic solutions for the undamped Duffing equation, a result recently obtained by Schmitt and Mazzanti using different methods.     

On the enumeration of chains in regular chain-groups

Let N be a regular chain-group on E (see W. T. Tutte, Canad. J. Math.8 (1956), 13–28); for instance, N may be the group of integer flows or tensions of a directed graph with edge-set E). It is known that the number of proper Z_λ-chains of N (λ ∈ Z, λ ≥ 2) is given by a polynomial in λ, P(N, λ) (when N is the chain-group of integer tensions of the connected graph G, λP(N, λ) is the usual chromatic polynomial of G). We prove the formula: P(N, λ) = Σ_{[E′]∈O(N)⁺/～}Q(R_[E′](N), λ), where O(N)⁺ is the set of orientations of N with a proper positive chain, ～ is a simple equivalence relation on O(N)⁺ (sequence of reversals of positive primitive chains), and Q(R_[E′](N), λ) is the number of chains with values in [1, λ ? 1] in any reorientation of N associated to an element of [E′]. Moreover, each term Q(R_[E′](N), λ) is a polynomial in λ. As applications we obtain: P(N, 0) = (?1)^r(N)∥O(N)⁺/～∥; P(N, ?1) = (?1)^r(N)∥O(N)⁺∥ (a result first proved by Brylawski and Lucas); P(N, λ + 1) ≥ P(N, λ) for λ ≥ 2, λ ∈ Z. Our result can also be considered as a refinement of the following known fact: A regular chain-group N has a proper Z_λ-chain iff it has a proper chain in [?λ + 1, λ ? 1].     

TWO POSITIVE SOLUTIONS TO THREE-POINT SINGULAR BOUNDARY VALUE PROBLEMS

In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem: -u′′(t) = λf(t, u(t)) for all t ∈ (0, 1) subjecting to u(0) = 0 and αu(η) = u(1), where η∈ (0, 1), α∈ [0, 1), and λ is a positive parameter. The nonlinear term f(t, u) is nonnegative, and may be singular at t = 0, t = 1, and u = 0. By the fixed point index theory and approximation method, we establish that there exists λ* ∈ (0, +∞], such that the above problem has at least two positive solutions for any λ∈ (0, λ*) under certain conditions on the nonlinear term f.     

Existential closure of block intersection graphs of infinite designs having finite block size and index

In this article we study the n‐existential closure property of the block intersection graphs of infinite t‐(v, k, λ) designs for which the block size k and the index λ are both finite. We show that such block intersection graphs are 2‐e.c. when 2?t?k ? 1. When λ = 1 and 2?t?k, then a necessary and sufficient condition on n for the block intersection graph to be n‐e.c. is that n?min{t, ?(k ? 1)/(t ? 1)? + 1}. If λ?2 then we show that the block intersection graph is not n‐e.c. for any n?min{t + 1, ?k/t? + 1}, and that for 3?n?min{t, ?k/t?} the block intersection graph is potentially but not necessarily n‐e.c. The cases t = 1 and t = k are also discussed. © 2011 Wiley Periodicals, Inc. J Combin Designs 19: 85–94, 2011