The Eigenvalues of Mathieu's Equation and their Branch Points

A comprehensive account is given of the behavior of the eigenvalues of Mathieu's equation as functions of the complex variable q. The convergence of their small-q expansions is limited by an infinite sequence of rings of branch points of square-root type at which adjacent eigenvalues of the same type become equal. New asymptotic formulae are derived that account for how and where the eigenvalues become equal. Known asymptotic series for the eigenvalues apply beyond the rings of branch points; we show how they can now be identified with specific eigenvalues.     

Calculation of the branch points of the eigenfunctions corresponding to wave spheroidal functions

A method for calculating eigenvalues λ_mn(c) corresponding to the wave spheroidal functions in the case of a complex parameter c is proposed, and a comprehensive numerical analysis is performed. It is shown that some points c _s are the branch points of the functions λ_mn(c) with different indexes n ₁ and n ₂ so that the value λ_{mn 1} (c _s ) is a double one: λ_{mn 1} (c _s ) = λ_{mn 2} (c _s ). The numerical analysis suggests that, for each fixed m, all the branches of the eigenvalues λ_mn(c) corresponding to the even spheroidal functions form a complete analytic function of the complex argument c. Similarly, all the branches of the eigenvalues λ_mn(c) corresponding to the odd spheroidal functions form a complete analytic function of c. To perform highly accurate calculations of the branch points c _s of the double eigenvalues λ_mn(c _s), the Padé approximants, the Hermite-Padé quadratic approximants, and the generalized Newton iterative method are used. A large number of branch points are calculated.     

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     

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     

Spin depolarization decay rates in α-symmetric stable fields on cubic lattices

We study the asymptotic, long-time behavior of the energy function where {X_s : 0 ≤ s < ∞} is the standard random walk on the d-dimensional lattice Z^d, 1 < α ≤ 2, and f:R⁺ → R⁺ is any nondecreasing concave function. In the special case f(x) = x, our setting represents a lattice model for the study of transverse magnetization of spins diffusing in a homogeneous, α-stable, i.i.d., random, longitudinal field {λV(x) : x ∈ Z^d} with common marginal distribution, the standard α-symmetric stable distribution; the parameter λ describes the intensity of the field. Using large-deviation techniques, we show that S_c(λ α f) = lim_t→∞ E(t; λ f) exists. Moreover, we obtain a variational formula for this decay rate S_c. Finally, we analyze the behavior S_c(λ α f) as λ → 0 when f(x) = x^β for all 1 ≥ β > 0. Consequently, several physical conjectures with respect to lattice models of transverse magnetization are resolved by setting β = 1 in our results. We show that S_c(λ, α, 1) ≈ λ^α for d ≥ 3, λ^agr;(ln 1/λ)^α−1 in d = 2, and in d = 1. © 1996 John Wiley & Sons, Inc.     

Solutions of the Differential Equation u‴+λ2zu+(α−1)λ2u=0

This paper deals with the solutions of the differential equation u?+λ²zu+(α?1)λ²u=0, in which λ is a complex parameter of large absolute value and α is an arbitrary constant, real or complex. After a discussion of the structure of the solutions of the differential equation, an integral representation of the solution is given, from which the series solutions and their asymptotic representations are derived. A third independent solution is needed for the special case when α?1 is a positive integer, and two derivations for this are given. Finally, a comparison is made with the results obtained by R. E. Langer.     

Scoring rule and majority agreements for large electorates with arbitrary preferences

This paper examines elections among three candidates when the electorate is large and voters can have any of the 26 nontrivial asymmetric binary relations on the candidates as their preference relations. Comparisons are made between rule-λ rankings based on rank-order ballots and simple majorities based on the preference relations. The rule-λ ranking is the decreasing point total order obtained when 1, λ and 0 points are assigned to the candidates ranked first, second and third on each voter's ballot, with 0 ? λ ? 1.Limit probabilities as the number of voters gets large are computed for events such as ‘the first-ranked rule-λ candidate has a majority over the second-ranked rule-λ candidate’ and ‘the rule-λ winner is the Condorcet candidate, given that there is a Condorcet candidate’. The probabilities are expressed as functions of λ and the distribution of voters over types of preference relations. In general, they are maximized at λ = 1/2 (Borda) and minimized at λ = 0 (plurality) and at λ = 1 for any fixed distribution of voters over preference types. The effects of more indifference and increased intransitivity in voter's preference relations are analyzed when λ is fixed.     

Calculating the branch points of the eigenvalues of the Coulomb spheroidal wave equation

A method for computing the eigenvalues λ _mn (b, c) and the eigenfunctions of the Coulomb spheroidal wave equation is proposed in the case of complex parameters b and c. The solution is represented as a combination of power series expansions that are then matched at a single point. An extensive numerical analysis shows that certain b _s and c _s are second-order branch points for λ _mn (b, c) with different indices n ₁ and n ₂, so that the eigenvalues at these points are double. Padé approximants, quadratic Hermite-Padé approximants, the finite element method, and the generalized Newton method are used to compute the branch points b _s and c _s and the double eigenvalues to high accuracy. A large number of these singular points are calculated.     

Geometrical solutions of some quadratic equations with non-real roots

This note gives geometrical/graphical methods of finding solutions of the quadratic equation ax ² + bx + c = 0, a p 0, with non-real roots. Three different cases which give rise to non-real roots of the quadratic equation have been discussed. In case I a geometrical construction and its proof for finding the solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when a,b,c ] R, the set of real numbers, are presented. Case II deals with the geometrical solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when b ] R, the set of real numbers; and a,c ] C, the set of complex numbers. Finally, the solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when a,c ] R, the set of real numbers, and b ] C, the set of complex numbers, are presented in case III.     

Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity

We study the branch of semistable and unstable solutions (i.e., those whose Morse index is at most 1) of the Dirichlet boundary value problem ? Δu = λf(x)/(1 ? u)² on a bounded domain Ω ? ?^N, which models—among other things—a simple electrostatic microelectromechanical system (MEMS) device. We extend the results of 11 relating to the minimal branch, by obtaining compactness along unstable branches for 1 ≤ N ≤ 7 on any domain Ω and for a large class of “permittivity profiles” f. We also show the remarkable fact that powerlike profiles f(x) ? |x|^α can push back the critical dimension N = 7 of this problem by establishing compactness for the semistable branch on the unit ball, also for N ≥ 8 and as long as As a byproduct, we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for λ in a natural range. In all these results, the conditions on the space dimension and on the power of the profile are essentially sharp. © 2007 Wiley Periodicals, Inc.     

Asymptotic methods for spherically symmetric MHD α2-dynamos

We consider two models of spherically-symmetric MHD α²–dynamos; one with idealized boundary conditions (BCs); and one with physically realistic BCs. As it has been shown in our previous work, the eigenvalues λ of a model with idealized BCs and constant α–profile α₀ are linear functions of α₀ and form a mesh in the (α₀, λ)–plane. The nodes of the spectral mesh correspond to double-degenerate eigenvalues of algebraic and geometric multiplicity 2 (diabolical points). It was found that perturbations of the constant α –profile lead to a resonant unfolding of the diabolical points with selection rules of the resonant unfolding defined by the Fourier coefficients of the perturbations. In the present contribution we present new exact results on the spectrum of the model with physically realistic BCs and constant α. For non-degenerate (simple) eigenvalues perturbation gradients are found at any particular α₀. We briefly discuss the spectral behavior of the α²–dynamo operator over a family of homotopic deformations of the BCs between idealized ones and physically realistic ones. Furthermore, we demonstrate that although the spectral singularities are lifted, a memory about their locations remains deeply imprinted in the homotopic family of spectral deformations due to a hidden underlying invariance. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic analysis and scaling of friction parameters

We consider an eigenvalue problem associated to the antiplane shearing on a system of collinear faults under a slip-dependent friction law. Firstly we consider a periodic system of faults in the whole plane. We prove that the first eigenvalues/eigenfunctions of different physical periodicity are all equal and that the other eigenvalues converge to this first common eigenvalue as their physical period becomes indefinitely large. Secondly we consider a large scale fault system composed on a small scale collinear faults periodically disposed. If β₀^* is the first eigenvalue of the periodic problem in the whole plane, we prove that the first eigenvalue of the microscopic problem behaves as β₀^*/∈ when ∈→ 0 regardless the geometry of the domain (here ∈ is the scale quotient). The geophysical implications of this result is that the macroscopic critical slip D_c scales with D_c^∈/∈ (here D_c^∈ is the small scale critical slip).     

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     

Polynomials in a matrix

Let A be a matrixp(x) a polynomial. Put B=p(A). It is shown that necessary and sufficient conditions for A to be a polynomial in B are (i) if λ is any eigenvalue of A, and if some elementary divisor of A corresponding to λ is nonlinear, thenp ^′(λ)≠0;and (ii) if λ,μ are distinct eigenvalues of A, then p(λ)p(μ) are also distinct. Here all computations are over some algebraically closed field.     

3-(v, 4, 1) covering designs with chromatic numbers 2 and 3

A t-(v, k, λ) covering design is a pair (X, B) where X is a v-set and B is a collection of k-sets in X, called blocks, such that every t element subset of X is contained in at least λ blocks of B. The covering number, C_λ(t, k, v), is the minimum number of blocks a t-(v, k, λ) covering design may have. The chromatic number of (X, B) is the smallest m for which there exists a map φ: X → Z_m such that ∣φ((β)∣ ≥2 for all β ∈ B, where φ(β) = {φ(x): x ∈ β}. The system (X, B) is equitably m-chromatic if there is a proper coloring φ with minimal m for which the numbers ∣φ^?1(c)∣ c ∈ Z_m differ from each other by at most 1. In this article we show that minimum, (i.e., ∣B∣ = C λ (t, k, v)) equitably 3-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0 (mod 6), v ≥ 18 for v ≥ 1, 13 (mod 36), v ≡ 13 and for all numbers v = n, n + 1, where n ≡ 4, 8, 10 (mod 12), n ≥ 16; and n = 6.5^a 13^b 17^c ?4, a + b + c > 0, and n = 14, 62. We also show that minimum, equitably 2-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0, 5, 9 (mod 12), v ≥ 0, v = 2.5^a 13^b 17^c + 1, a + b + c > 0, and v = 23. © 1993 John Wiley & Sons, Inc.     

Maximal triangle-free graphs with restrictions on the degrees

We investigate the problem that at least how many edges must a maximal triangle-free graph on n vertices have if the maximal valency is ≤D. Denote this minimum value by F(n, D). For large enough n, we determine the exact value of F(n, D) if D ≥ (n ? 2)/2 and we prove that lim F(n, cn)/n = K(c) exists for all 0 < c with the possible exception of a sequence c_k → 0. The determination of K(c) is a finite problem on all intervals [γ, ∞). For D = cn^?, 1/2 < ? < 1, we give upper and lower bounds for F(n, D) differing only in a constant factor. (Clearly, D < (n - 1)^1/2 is impossible in a maximal triangle-free graph.)     

ASYMPTOTICS OF THE NUMBER OF RESONANCES IN THE TRANSMISSION PROBLEM

We consider the resonances for the transmission problem associated with a strictly convex transparent obstacle. Under some natural assumptions we show that there is a free of resonances region in the complex upper half plane given by {C ≤ Im λ ≤ C ₁|λ|^1/3 ? C ₂}, where C, C ₁ and C ₂ are positive constants. Moreover, we obtain asymptotics for the number of resonances counted with multiplicities in the region {0 < Im λ ≤ C, 0 < Re λ ≤ r} as r → ∞, where C > 0 is the same constant as above.

     

Half-Linear Eigenvalue Problems

We consider the eigenvalue problem for t ? [0, b], where aⁿ = |a|ⁿ sgna, a ? ?, λ ? ?, the constants μ, v are real such that 0 ≤ μ < n and derive asymptotic estimates for solutions of the differential equation in the definite case q(t)> 0 which corresponds to the well-known WKB-approximation in the linear case n = 1, μ = 0. In the second part we investigate the asymptotic distribution of the eigenvalues in the general case of two -point boundary conditions and refine these results for the so called separated boundary conditions.     

Stress Singularities in Viscoelastic Media 2. Plane-Strain Stress Singularities at Corners

The stress singularities that evolve at the corner of a notchedviscoelastic angular plate subject to mode I deformation isdiscussed when prescribed, but arbitrary, displacements aresymmetrically applied to both radial edges of the sector. Thesolution procedure, based on Laplace and Mellin transforms (withtransform parameters p and s, respectively), leads to an eigenvalueproblem in the complex p-plane, which is dependent on Poisson'sratio and characterizes the singular behaviour of the stressfields. Although simple solutions to the transcendental eigenequationare not available, the real-time evolution of the stress concentrationsis obtained by monitoring a particular branch of the eigenvalueequation as it moves in the complex p-plane. A correspondingpath is traced in an appropriately cut strip in the s-plane,in which solutions to the eigenequation are single-valued. Analyticcontinuation in the p-plane thus allows the Laplace and Mellininversions to be performed and the real-time behaviour of theplane-stress components to be expressed as contour integralswithin the strips in the complex s-plane. Cast in this form,the stress components are evaluated numerically when the viscoelasticmaterial is represented as a standard linear solid. Their dependenceon the angular variation within the plate, the applied load,and the effects of the viscoelastic material properties is exhibitedfor a number of situations, and in each case contrasted withshort- and long-time asymptotic curves based on Tauberian theorems.