bluuebunny/crossref_metadata_2025 huggingface dataset:
Wright's equation has no solutions of period four
Authors: Nussbaum Roger D.
Published: November 2011

Synopsis

LetN:ℝ→ℝ be a locally Lipschitzian map such that (y+ l)N(y)>0 for ally≠ –1 and such thatN(y)=1 +yfor – 1 ≦y≦ 3. For any positive number α the equationy'(t) αy(t–1)N(y(t)) has, aside from the constantsolutionsy(t) ≡ –1, andy(t) ≡–1 solutiony(t) such thaty(t+ 4) =y(t) for all realtIfN(y) = 1 +yfor ally, one obtains Wright's equation, which isknown to have periodic solutions of minimal periodp(depending on α) arbitrarily close to 4. Some results concerning nonexistence of periodic solutions of period 4 of other differential-delay equations are also proved. In all cases the method of proof consists in analysing an associated fourth-order system of ordinary differential equationsand showing that this system has no nonconstant periodic solutions.
On a condition for a centre of cubic non-autonomous equations
Authors: Alwash M. A. M.
Published: November 2011

Synopsis

In this note we demonstrate that a conjecture about a necessary and sufficient condition for the existence of a centre of cubic equations is not true.
Periodic solutions of a class of Hamiltonian systems with singularities
Authors: Ambrosetti Antonio , Ekeland Ivar
Published: November 2011

Synopsis

This paper deals with a class of time-periodic Hamiltonian systems obtained by a time-dependent perturbation from an autonomous system with a singularity at q = 0 in configuration space. It is shown that, T being the period of the perturbation, nondegenerate families of T-periodic orbits in the unperturbed problem branch off into a certain number of T-periodic orbits for the perturbed problem.
Darcy's law for slow viscous flow past a stationary array of bubbles
Authors: Lipton Robert , Avellaneda Marco
Published: November 2011

Synopsis

We examine slow viscous flow past a concentrated bed of small stationary viscous bubbles of a second fluid, and derive Darcy's law relating the average fluid velocity to the overall pressure gradient and body force.
Some examples of rank one convex functions in dimension two
Authors: Dacorogna B. , Douchet J. , Gangbo W. , Rappaz J.
Published: November 2011

Synopsis

We study the rank one convexity of some functionsf(ξ) where ξ is a 2 × 2 matrix. Examples such as |ξ|2α+h(detξ) and | ξ |2α(| ξ |2− γdet ξ) are investigated. Numerical computations are done on the example of Dacorogna and Marcellini,indicating that this function is quasiconvex.
A dynamical systems proof of the Krein–Rutman Theorem and an extension of the Perron Theorem
Authors: Alikakos Nicholas D. , Fusco Giorgio
Published: November 2011

Synopsis

In this paper we establish Perron and Krein–Rutman-like theorems for an operator mapping a cone into the interior of the cone, by considering the discrete dynamical system for the induced operator on the projective space (= sphere). Existence of a positive eigenvector reduces to showing that the ω-limit set of the induced operator consists of a single equilibrium. A special feature of our approach is that the convexity of the cone is needed only for establishing the non-emptiness of the w-limit set. This allows us in finite dimensions to establish an abstract Perron Theorem for non-convex cones.
The structure of solutions to the pseudo-conformally invariant nonlinear Schrödinger equation
Authors: Cazenave Thierry , Weissler Fred B.
Published: November 2011

Synopsis

We study solutions in ℝn of the nonlinear Schrödinger equation iut + Δu = λ |u|γu, where γ is the fixed power 4/n. For this particular power, these solutions satisfy the “pseudo-conformal” conservation law, and the set of solutions is invariant under a related transformation. This transformation gives a correspondence between global and non-global solutions (if λ < 0), and therefore allows us to deduce properties of global solutions from properties of non-global solutions, and vice versa. In particular, we show that a global solution is stable if and only if it decays at the same rate as a solution to the linear problem (with λ = 0). Also, we obtain an explicit formula for the inverse of the wave operator; and we give a sufficient condition (if λ < 0) that the blow up time of a non-global solution is a continuous function on the set of initial values with (for example) negative energy.
A note on the homotopy type of certain gauge groups
Authors: Kono Akira
Published: November 2011

Synopsis

Let Gk be the gauge group of Pk, the principal SU(2) bundle over S4 with c2(Pk) = k. In this paper we show that Gk ≃ Gk. if and only if (12, k) = (12, k′) where (12, k) is the GCD of 12 and k.
On Saint-Venant's problem for an elastic strip
Authors: Mielke Alexander
Published: November 2011

Synopsis

The equilibrium equations for elastic deformations of an infinite strip are considered. Under the assumption of sufficiently small strains along the whole body, it is shown that all solutions lie on a six-dimensional manifold. This is achieved by rewriting the field equations as a differential equation in a function spaceover the cross-section, the axial variable taken as time. Then the theory of centre manifolds for elliptic systems applies. Thus the local Saint-Venant's problem is solved. Moreover, the structure of the finite-dimensional solution space is analysed to reveal exactly the two-dimensional rod equations of Kirchhoff. The constitutive relations for this rod model are calculated in a mathematically rigorous way out of the constitutive law of the material forming the strip.
On binary differential equations and umbilics
Authors: Bruce J.W. , Fidal D.L.
Published: November 2011

Synopsis

In this paper we give the local classification of solution curves of bivalued direction fields determined by the equation

where a and b are smooth functions which we suppose vanish at 0 ∈ ℝ2. Such fields arise on surfaces in Euclidean space, near umbilics, as the principal direction fields, and also in applications of singularity theory to the structure of flow fields and monochromatic-electromagnetic radiation. We give a classification up to homeomorphism (there are three types) but the methods furnish much additional information concerning the fields, via a crucial blowing-up construction.
On an index law and a result of Buschman
Authors: McBride Adam C.
Published: November 2011

Synopsis

A result for the Erdélyi-Kober operators, mentioned briefly by Buschman, is discussed together with a second related result. The results are proved rigorously by means of an index law for powers of certain differential operators and are shown to be valid under conditions of great generality. Mellin multipliers are used and it is shown that, in a certain sense, the index law approach is equivalent to, but independent of, the duplication formula for the gamma function. Various statements can be made concerning fractional integrals and derivatives which produce, as special cases, simple instances of the chain rule for differentiation and changes of variables in integrals.
Interpolatory and variation-diminishing properties of generalized B-splines
Authors: Goodman T. N. T. , Lee S. L.
Published: November 2011

Synopsis

We consider classes of functions satisfying certain simple criteria of sign and smoothness and a decomposition property. It is known that these properties are possessed by Chebysheffian B-splines and it is shown here that they are also possessed by certain trigonometric B-splines. For such a class of functions, we derive a variation-diminishing property and analyse interpolation both on a finite set of nodes and on an infinite, periodically spaced set of nodes. The results are also applied to interpolation by complex polynomial splines on the circle.
Existence conditions for eigenvalue problems generated by compact multiparameter operators
Authors: Binding Paul , Browne Patrick J. , Turyn Lawrence
Published: November 2011

Synopsis

Let T, V1,…, Vk denote compact symmetric linear operators on a separable Hilbert space H, and write W(λ) = T + λ1V1 + … + λkVk, λ = (λ1, …, λk) ϵ ℝk. We study conditions on the cone

related to solubility of the multiparameter eigenvalue problem

with W(λ)−I nonpositive definite. The main result is as follows.

Theorem. If 0 ∉ V, then (*) is soluble for any T. If 0 ∈ V, then there exists T such that (*) is insoluble.

We also deduce analogous results for problems involving self-adjoint operators with compact resolvent.
Singular first order perturbations of the heat equation
Authors: Avrin Joel
Published: November 2011

Synopsis

We exhibit dimension-independent conditions under which the formal operator A = −Δ + a.∇ + V can be defined on such that its closure Ā in L2(Rs, dx) is quasi-m-accretive. Here, a is real so that Ā is nonselfadjoint. the method of proof is a generalized version of the argument employed in the portion of the author's thesis where term a.∇ was originally considered. Specifically, we construct exp (−tĀ) as a limit of approximating semigroups. Since the thesis appeared, Kato has also dealt with the term a. ∇ his conditions on a and V are similar to, but more general than, the conditions that appear here; in addition, he considers magnetic vector potentials. Of interest here is the semigroup method itself, the conciseness of the arguments thereby produced, and a relaxed condition on div a.
Full- and partial-range eigenfunction expansions for Sturm-Liouville problems with indefinite weights
Authors: Kaper Hans G. , Kwong Man Kam , Lekkerkerker C. G. , Zettl A.
Published: November 2011

Synopsis

This article is concerned with eigenvalue problems of the form Au = λTu in a Hilbert space H, where Ais a selfadjoint positive operator generated by a second-order Sturm-Liouville differential expression and T a selfadjoint indefinite multiplicative operator which is one-to-one. Emphasis is on the full-range and partial-range expansionproperties of the eigenfunctions.
Well-posedness, control and computation of a one-phase Stefan problem with Neumann condition*
Authors: Chen Goong , Sun Shunhua , Zheng Quan
Published: November 2011

Synopsis

A one-phase Stefan problem can be reduced to an equivalent variational inequality by using the Baiocchi-Duvaut transformation. In this paper, we study the variational inequality by formulating it as a set-valued partial differential equation. The existence of solutions is proved by applying a generalized Schauder fixed fixed point theorem for set-valued mappings. Uniqueness and regularity of solutions are also obtained. In §3, we regard the boundary value Neumann data as boundary controls and combine both the variational inequality and the classical approaches to study the effects of controls on the free boundary and the state (i.e. temperature). In §4, we further use the theory to study an optimal “ice-melting” problem. Our results show that if the controls have fixed total input heat flux and are constrained in magnitude, then the optimal control is “bang-bang”. If the admissible controls are not constrained in magnitude, then the optimal control is a Dirac delta type distribution which is no longer admissible. In the last section, our existence theory is combined with the finite difference method and non-linear programming techniques to obtain numerical solutions.
The global Hölder-continuous solution of isentropic gas dynamics
Authors: Lu Yun-Guang
Published: November 2011

Synopsis

This paper considers the Cauchy problem for the isentropic equations of gas dynamics in Euler coordinates ρt + (ρu)x = 0, (ρu)t + (ρu)2 + P(ρ))x=0 and gives the Hölder-continuous solution by applying the method of vanishing viscosity.
On the blow-up of solutions of a convective reaction diffusion equation
Authors: Aguirre J. , Escobedo M.
Published: November 2011

Synopsis

We study the blow-up of positive solutions of the Cauchy problem for the semilinear parabolic equation

where u is a scalar function of the spatial variable x ∈ ℝN and time t > 0, a ∈ ℝV, a ≠ 0, 1 < p and 1 ≦ q. We show that: (a) if p > 1 and 1 ≦ q ≦ p, there always exist solutions which blow up in finite time; (b) if 1 < q ≦ p ≦ min {1 + 2/N, 1 + 2q/(N + 1)} or if q = 1 and 1 < p ≦ l + 2/N, then all positive solutions blow up in finite time; (c) if q > 1 and p > min {1 + 2/N, 1 + 2q/N + 1)}, then global solutions exist; (d) if q = 1 and p > 1 + 2/N, then global solutions exist.
Hyperbolic phase change problems in heat conduction with memory
Authors: Colli Pierluigi , Grasselli Maurizio
Published: November 2011

Synopsis

The aim of this paper is to formulate and study phase transition problems in materials with memory, based on the Gurtin–Pipkin constitutive assumption on the heat flux. As different phases are involved, the internal energy is allowed to depend on the phase variable (besides the temperature) and to take its past history into account. By considering the standard equilibrium condition at the interface between two phases, we deal with a hyperbolic Stefan problem reckoning with memory effects. Then, substituting this equilibrium condition with a relaxation dynamics, we represent some dissipation phenomena including supercooling or superheating. The application of a fixed point argument helps us to show the existence and uniqueness of the solution to the latter problem (still of hyperbolic type). Hence, by introducing a suitable regularisation and taking the limit as a kinetic parameter goes to zero, we prove an existence result for the former Stefan problem. Moreover, its uniqueness is deduced by contradiction.
Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit-circle singularity, I
Authors: Atkinson F. V. , Fulton C. T.
Published: November 2011

Synopsis

Asymptotic formulae for the positive eigenvalues of a limit-circle eigenvalue problem for –y” + qy = λy on the finite interval (0, b] are obtained for potentials q which are limit circle and non-oscillatory at x = 0, under the assumption xq(x)∈L1(0,6). Potentials of the form q(x) = C/xk, 0<fc<2, are included. In the case where k = 1, an independent check based on the limit-circle theory of Fulton and an asymptotic expansion of the confluent hypergeometric function, M(a, b; z), verifies the main result.
Comparison functions for a model problem related to nonlinear elasticity
Authors: Stredulinsky E. W.
Published: November 2011

Synopsis

Comparison functions are constructed for the problem of minimizing

over maps u: D(⊆ℝ2)→ℝ2 with det≥0, subject to the constraint u= f on ∂D, D the unit disk. This is accomplished for maps / which are reparameterizations of ∂D or which are “graph-like” maps. Estimates involving half derivative boundary norms are obtained.
Level and index in the modular group
Authors: Stothers W. W.
Published: November 2011

Synopsis

It is shown that the index of a congruence subgroup of the modular group cannot be less than the level of the subgroup. This allows a number of existence theorems about non-congruence subgroups.

The level of a subgroup of the modular group can be defined in terms of the action on Q ∪ {∞}. We define a similar action to get information on congruence subgroups. In fact, we get a more powerful result, but this appears to be the most useful version.
Multipliers for vector valued functions
Authors: Torrea José Luis
Published: November 2011

Synopsis

Let G be a locally compact abelian group and let Γ be the dual of G. Let A, B be Banach spaces and Lp(G,A) the Bochner-Lebesgue spaces. We prove that the space of bounded linear translation invariant operators from L1(G, A) to LX(G, B) can be identified with the space of bounded convolution invariant (in some sense) operators and also with the space of a(A, B)-valued “weak regular” measures with the relation Tf = f *μ. (A. The existence of a function m∈ L∞ (Γ,α(A,B)), such that is also proved.
Inverse semigroups generated by nilpotent transformations
Authors: Howie John M. , Marques-Smith M. Paula O.
Published: November 2011

Synopsis

Let X be a set with infinite cardinality m and let B be the Baer-Levi semigroup, consisting of all one-one mappings a:X→X for which ∣X/Xα∣ = m. Let Km=, the inverse subsemigroup of the symmetric inverse semigroup ℐ(X) generated by all products β−γ, with β,γ∈B. Then Km = , where N2 is the subset of ℐ(X) consisting of all nilpotent elements of index 2. Moreover, Km has 2-nilpotent-depth 3, in the sense that

Let Pm be the ideal {α∈Km: ∣dom α∣<m} in Km and let Lm be the Rees quotient Km/Pm. Then Lm is a 0-bisimple, 2-nilpotent-generated inverse semigroup with 2-nilpotent-depth 3. The minimum non-trivial homomorphic image of Lm also has these properties and is congruence-free.
RP-dominated regular semigroups
Authors: Blyth T. S. , Hickey J. B.
Published: November 2011

Synopsis

We describe the structure of regular semigroups in which each element is dominated, in the Nambooripad order, by a unique maximal element that is ℋ-equivalent to a mididentity. These are precisely direct products of a rectangular band and a uniquely unit regular semigroup.
Sums and products of quasi-nilpotent operators
Authors: Fong C. K. , Sourour A. R.
Published: November 2011

Synopsis

It is proved that a bounded operator on Hilbert space is the sum of two quasi-nilpotent operators if and only if it is not a non-zero scalar plus a compact operator. Necessary conditions and sufficient conditions for an operator to be the product of two quasi-nilpotent operators are given.
A note on linear ordinary quasi-differential equations
Authors: Everitt W. N.
Published: November 2011

Synopsis

The theory of differential equations is largely concerned with properties of solutions of individual, or classes of, equations. This paper is given over to the converse problem - that of seeking properties of functions which require them to be, in some respect, solutions of a differential equation, and to determining all possible such differential equations.

From this point of view this paper discusses only linear ordinary quasi-differential equations of the second order. However, the methods can be extended to quasi-differential equations of general order.
On a discrete-time version of a problem of A. J. Pritchard and J. Zabczyk
Authors: Przyłuski K. Maciej
Published: November 2011

Synopsis

It is shown that every weakly l1-stable linear and bounded operator (which represents a linear discrete-time system) on a Hilbert space is power stable. It solves (at least partially) a discrete-time version of a problem posed by A. J. Pritchard and J. Zabczyk for strongly continuous semigroups of bounded linear operators.
On the automorphism group of the integral group ring of the infinite dihedral group
Authors: Wallace D. A. R.
Published: November 2011

Synopsis

Let ℤ(G) be the integral group ring of the infinite dihedral groupG; the aim is to calculate Autℤ(G), the group of Z-linear automorphisms of ℤ(G). It is shown that Aut ℤ(G), the subgroup of Aut *ℤ(G) consisting of those automorphisms that preserve elementwise the centre of ℤ(G), is a normal subgroup of Aut *ℤ(G) of index 2 and that Aut*ℤ(G) may be embedded monomorphically intoM2(ℤ[t]), the ring of 2 × 2 matrices over a polynomial ring ℤ[t]From this embedding and by the Noether—Skolem Theorem it is shown that Inn (G), the group of inner automorphisms of ℤ(G) induced by the units of ℤ(G), is a normal subgroup of Aut*ℤ(G)/ such that Aut*ℤ(G)/Inn (G) is isomorphic to the Klein four-group.
Radicals and one-sided ideals: an addendum
Authors: Sands A. D.
Published: November 2011

Synopsis

In the paper referred to in the title [2] an open question was raised concerning the equality of the largest left hereditary radical and the largest right hereditary radical contained in each of certain radicals. In this addendum an affirmative answer is provided to this question.
Oscillation criteria of Nehari-type for Sturm–Liouville operators and elliptic differential operators of second order and the lower spectrum
Authors: Fiedler F.
Published: November 2011

Synopsis

Sufficient oscillation criteria of Nehari-type are established for the differential equation−uʺ(t) + q(t)u(t) = 0, 0<t<∞, with and without sign restrictions onq(t), respectively. These results are extended to Sturm-Liouville equations and elliptic differential equations of second order.In Section 7 we present conclusions for the lower spectrum of elliptic differential operators and also for the discreteness of the spectrum of certain ordinary differential operators of second order.
Limit cycles of a class of polynomial systems
Authors: Carbonell Marc , Llibre Jaume
Published: November 2011

Synopsis

We study the class of polynomial vector fields of the form = αx — y + Pn(x, y), = x + αy + Qn(x, y), where Pn and Qn are homogeneous polynomials of degree n. If we define the functions f(x, y) = xPn(x, y) + yQn(x, y) and g(x, y) = xQn(x, y)−yPn(x, y), we characterise the number of limit cycles for this class when the function g(αg − f) does not change sign.
A combinatorial procedure for finding isolating neighbourhoods and index pairs
Authors: Szymczak Andrzej
Published: November 2011

Synopsis

We present a purely combinatorial procedure for finding an isolating neighbourhood and an index pair contained in a given set, being a finite union of cubes in Rs. It is applied for a computer-assisted computation of the Conley index of an isolated invariant subset of the Hénon attractor. As a corollary, it is shown that the Hénon attractor contains periodic orbits of all principal periods except for 3 and 5.
Convergence of the viscosity solutions for weakly strictly hyperbolic conservation laws
Authors: Changjiang Zhu
Published: November 2011

Synopsis

This paper is an extension of papers [14–16]. Using the theory of compensated compactness, we establish the convergence of the uniformly bounded approximate solution sequence for a class of ‘weakly strictly hyperbolic’ conservation laws.
Minimal group actions on Λ-trees
Authors: Chiswell I. M.
Published: November 2011

We consider the existence and uniqueness of minimal invariant subtrees for abelian actions of groups on Λ-trees, and whether or not a minimal action is determined up to isomorphism by the hyperbolic length function. The main emphasis is on actions of end type. For a trivial action of end type, there is no minimal invariant subtree. However, if a finitely generated group has an action of end type, the action is nontrivial and there is a unique minimal invariant subtree. There are examples of infinitely generated groups with a nontrivial action of end type for which there is no minimal invariant subtree. These results can be used to study actions of cut type.
Solution of the ℐ2-radical problem in structural matrix near-rings
Authors: Smith Kirby C. , van Wyk Leon
Published: November 2011

We answer the question, raised in 1989, whether the ℐ2-radical of a structural matrix near-ring can be written as the sum of two nontrivial ideals, one of which is nilpotent, in the affirmative.
Effective moduli of plane polycrystals with a rank-one local compliance tensor
Authors: Gibiansky L. V.
Published: November 2011

Recently, Grabovsky and Milton have studied effective properties of plane polycrystals made of a crystal with a rank-one local compliance tensor and a positive definite ‘weak’ direction. This paper continues their investigation. By using several complementary approaches, it is proved that the effective compliance tensor of such a polycrystal has exactly one weak direction. The new proofs clarify the link of the degenerate polycrystal problem with previously obtained results on the polycrystal properties. Investigation of the phase boundary conditions has allowed us to find theL-closure set of the effective properties of all polycrystals that can be constructed by sequential laminations of the original degenerate crystals. Bounds on theG-closure, i.e. a set of the effective properties of all the polycrystals that can be built from the given set of degenerate crystals, are found. They are given by a polygonal set in the plane of two linear invariants of the effective compliance tensors. The effective moduli of the polycrystals made from crystals with non-definite ‘weak’ directions are also studied.
Topological transversality and periodic solutions of neutral functional differential equations
Authors: Wu Jianhong , Xia Huaxing , Zhang Bo
Published: November 2011

In this paper, we study the existence of periodic solutions of neutral functional differential equations (NFDEs). A topological transversality theorem is used to obtain fixed points of certain nonlinear compact operators, which correspond to periodic solutions of the original differential equations. The method relies on a priori bounds on periodic solutions to a family of appropriately constructed NFDEs. A general existence theorem is proved and several illustrative examples are given where we use Liapunov-like functions in deriving such a priori bounds on periodic solutions. Due to the topological nature of the approach, the theorem applies as well to NFDEs of mixed type and NFDEs with state-dependent delay. Some comparisons between our results and the existing ones are also provided.
On the KOℤ/2-Euler class, II
Authors: Crabb M. C.
Published: November 2011

Synopsis

Let ξ be an oriented n-dimensional real vector bundle over an oriented closed m-manifold X. An r-field on ξ defined outside a finite subset of X has an index in the homotopy group πm−l(Vn,r) of the Stiefel manifold of r-frames in ℝn. The principal theorems of this paper relate the d and e-invariants of an associated ℝ/2-equivariant stable homotopy class, in certain cases, to computable cohomology characteristic numbers. Results of this type were first obtained by Atiyah and Dupont [5].
Jacobi matrices and transversality
Authors: Fusco Giorgio , Oliva Waldyr Muniz
Published: November 2011

Synopsis

The paper deals with smooth nonlinear ODE systems in ℝn, ẋ = f(x), such that the derivative f′(x) has a matrix representation of Jacobi type (not necessarily symmetric) with positive off diagonal entries. A discrete functional is introduced and is discovered to be nonincreasing along the solutions of the associated linear variational system ẏ = f′(x(t))y. Two families of transversal cones invariant under the flow of that linear system allow us to prove transversality between the stable and unstable manifolds of any two hyperbolic critical points of the given nonlinear system; it is also proved that the nonwandering points are critical points. A new class of Morse–Smale systems in ℝn is then explicitly constructed.
Piecewise continuous solutions of nonlinear pseudoparabolic equations in two space dimensions*
Authors: Dai Dao-Qing , Lin Wei
Published: November 2011

Synopsis

An initial boundary value problem of Riemann type is solved for the nonlinear pseudoparabolic equation with two space variables

The complex functionHis measurable on ℂ ×I × ℂ5, withIbeing an interval of the real line ℝ, Lipschitz continuous with respect to the last five variables, with the Lipschitz constant for the last variable being strictly less than one (ellipticity condition). No smallness assumption is needed in the argument.
On multiple solutions for Berman's problem
Authors: Hwang Tzy-Wei , Wang Ching-An
Published: November 2011

Synopsis

We study Berman's problem

subjects to the conditions f(0) =f″(0) = f′(1) = f(1)− 1 = 0, which arises from the study of laminar flows in channels with porous walls. Positive (negative) Re denotes the case of suction (injection) flows. Preliminary numerical studies indicated the existence of three different types of solutions and a small portion of mathematical evidence was provided. In this paper, we are able to show that there exist connected sets in the Re-K plane on which different types of solutions occur. In particular, our result verifies that the problem possesses all three types of suction solutions for sufficiently large Re. Moreover, the limiting injection solution is also obtained.
Uniform asymptotic expansions for oblate spheroidal functions I: positive separation parameter λ
Authors: Dunster T. M.
Published: November 2011

Synopsis

Uniform asymptotic expansions are derived for solutions of the spheroidal wave equation, in the oblate case where the parameter µ is real and nonnegative, the separation parameter λ is real and positive, and γ is purely imaginary (γ = iu). As u →∞, three types of expansions are derived for oblate spheroidal functions, which involve elementary, Airy and Bessel functions. Let δ be an arbitrary small positive constant. The expansions are uniformly valid for λ/u2 fixed and lying in the interval (0,2), and for λ / u2when 0<λ/u2 < 1, and when 1 = 1≦λ/u2 < 2. The union of the domains of validity of the various expansions cover the half- plane arg (z)≦ = π/2.
Robustness of a feedback control scheme for one-dimensional diffusion equations: perturbation to the Sturm-Liouville operator
Authors: Nambu Takao
Published: November 2011

Synopsis

We study the stabilisation of a one-dimensional diffusion equation by means of static feedback. The equation contains the so-called Sturm-Liouville operator (S-L operator). A perturbation, often interpreted as an error in modelling physical systems, enters the principal part and the boundary condition of the S-L operator. Since the perturbation is not subordinate to the operator, the classical perturbation theory is no longer available. We show, however, that the feedback stabilisation scheme for the unperturbed equation is effective also for the perturbed equation as long as the perturbation is small in an adequate topology. The key idea is to show the strong continuity of the eigenfunctions for the S-L operator relative to the coefficients of the operator.
The word problem for small cancellation quotients of groups acting on trees
Authors: Mahmud R. M. S.
Published: November 2011

Synopsis

The small cancellation theory over free products with amalgamation and HNN groups is extended to groups acting on trees in which the action with inversions is possible. This will include the case of tree products of groups and treed-HNN groups.
Symmetric-like Riemannian manifolds and geodesic symmetries
Authors: Berndt Jürgen , Prüfer Friedbert , Vanhecke Lieven
Published: November 2011

We treat several classes of Riemannian manifolds whose shape operators of geodesic spheres or Jacobi operators share some properties with the ones on symmetric spaces.
The Lavrentieff phenomenon for quadratic functionals
Authors: De Arcangelis Riccardo
Published: November 2011

The paper provides an example of an integral functional in more than two dimensions, with a symmetric and positively defined quadratic integrandq, exhibiting the Lavrentieff phenomenon on a ballBand on a linear boundary datumu0, i.e. for which

The example is also utilised to discuss nonidentity between some relaxation procedures for a quadratic integral functional and to provide a weighted Sobolev space in the Hilbert case in which smooth functions are not dense.
Quadratic systems having a parabola as an integral curve
Authors: Christopher Colin
Published: November 2011

Synopsis

The class of quadratic systems having a parabola composed of integral curves is examined. Canonical forms are found for the members of this class, and conditions are obtained, using the Bendixson's Criterion and the Poincaré–Bendixson Theorem, for the existence or non-existence of limit cycles, in the case where there is a limit cycle “inside” the parabola (that is, in the convex component of its compliment).
On the wave propagation property in finite elasticity
Authors: Maremonti Paolo , Russo Remigio
Published: November 2011

Synopsis

By using a mild hypothesis on the acoustic tensor, it is shown that the disturbances in a nonlinear elastic body travel with a finite speed. Moreover, a uniqueness theorem for the displacement problem of nonlinear elastodynamics is proved. No assumption is made on the extension of the body.
Collocation methods for second-kind Volterra integral equations with weakly singular kernels
Authors: Diogo Teresa , McKee Sean , Tang Tao
Published: November 2011

In this paper it is shown that the use of uniform meshes leads to optimal convergence rates provided that the analytic solutions of a particular class of Volterra integral equations (VIEs) are smooth. If the exact solutions are not smooth, however, suitable transformations can be made so that the new VIEs possess smooth solutions. Spline collocation methods with uniform meshes applied to these new VIEs are then shown to be able to yield optimal (global) convergence rates. The general theory is applied to a typical case, i.e. the integral kernels consisting of the singular term (t − s) −½.
Large-time behaviour of solutions of a reaction–diffusion equation
Authors: Pablo Arturo de
Published: November 2011

We investigate the large-time behaviour of the solutions u = u(x, t) to the one-dimensional nonlinear heat equation with reaction

with exponents m > 1,p < 1. The initial function u(x, 0) is assumed to be measurable and nonnegative. In the case m + p ≧ 2 where the initial value does not uniquely determine the solution, we also fix the positivity set of the solution u(x, t) if the support of u(x, 0) is not the whole line ℝ, i.e. u(x, t) > 0 if and only if −s1(t) <x<s2(t), t ≧ 0, where 0≦si(t)≦∞ for t ≧ 0, i = 1, 2 are lower semicontinuous given functions. We prove that u converges to a self-similar function which depends only on the behaviour of u(x, 0) for |x| large or si(t) for t large. We classify the set of self-similar solutions and study the equation satisfied by their interfaces.
Very weak solutions of boundary value problems for the Laplace operator and the Lamé system on polyhedral domains in ℝ3
Authors: Hua Ding
Published: November 2011

The notion of very weak solutions is introduced in this paper in order to solve the boundary value problems for the Laplace operator and for the Lamé system with nonsmooth data in polyhedral domains. A continuity theorem is given for variational solutions of the above problems. This result may be used to solve problems with concentrated loads.
A remark on the resonance set for a semilinear elliptic equation
Authors: Micheletti Anna Maria
Published: November 2011

We study the resonance set ∑ of pairs (α,β) ∊ ℝ2 for which the problem ∆u + αu+ − βu− = 0 has a nontrivial solution . We show that if λ0, is an eigenvalue of multiplicity two of −Δ, then has measure zero, where are the neighbouring eigenvalues of λ0. Moreover, we have that, if the operator Δ + αIu<0 + βIu < 0 has a kernel of dimension one for(α, β) ∊ ∑ and u ≠ 0 such that Δu + αu+ − βu− = 0, then (α, β) is an isolated point on ∑ ∩ L, where L is the straight line parallel to the diagonal of ℝ+ × ℝ+ through (α, β).
Anisotropic singular perturbations—the vectorial case
Authors: Barroso Ana Cristina , Fonseca Irene
Published: November 2011

We obtain the Γ(L1(Ώ))-limit of the sequence

where Eε is the family of anisotropic perturbations

of the nonconvex functional of vector-valued functions

The proof relies on the blow-up argument introduced by Fonseca and Müller.
On a class of semilinear elliptic systems
Authors: Liu Yaping
Published: November 2011

A class of semilinear elliptic systems of two equations is considered. Sufficient conditions are given for the existence of different types of sign-definite solutions. These conditions relate the larger eigenvalues of certain 2 × 2 real matrices associated with the system to the first eigenvalue of − ∆ under the homogeneous Dirichlet boundary condition. A special case provides a complementary result to some of the recent works.
An integro-differential equation from population genetics and perturbations of differentiable semigroups in Fréchet spaces
Authors: Bürger Reinhard
Published: November 2011

Synopsis

Existence and uniqueness of solutions of an integro-differential equation that arises in population genetics are proved. This equation describes the evolution of type densities in a population that is subject to mutation and directional selection on a quantitative trait. It turns out that a certain Fréchet space is the natural framework to show existence and uniqueness. One of the main steps in the proof is the investigation of perturbations of generators of differentiable semigroups in Fréchet spaces.
A Phragmén–Lindelöf principle for the equation of a surface of constant mean curvature
Authors: Knops R. J. , Payne L. E.
Published: November 2011

This paper studies the surface of constant mean curvature on a semi-infinite strip, and shows by means of a first-order differential inequality that the solution in a given measure either becomes asymptotically unbounded at least to polynomial order, or decays at most exponentially to the solution of an associated one-dimensional problem. A proof is also presented for uniqueness in the class of functions having bounded gradient and subject to specified growth conditions for large values of the longitudinal distance. Extensions of these results to the whole strip and to more general types of equations are also described.
Bifurcation of A-proper mappings without transversality considerations
Authors: Welsh Stewart C.
Published: November 2011

Synopsis

We consider a nonlinear eigenvalue problem in the form F(x, λ) = Ax – T(λ)x – R(x, λ) = 0 with F:X × ℝ →Y where X, Y are Banach spaces. We assume that F(0, λ) = 0 for all λ ∈ ℝ and seek bifurcation points; that is, values λ0 ∈ ℝ for which there are solutions to F(x, λ) = 0 with x ≠ 0 in any neighbourhood of (0, λ0). Corresponding to these bifurcation points we obtain global properties of the maximal connected subset of solutions to F(x, λ) = 0 containing (0, λ0).

Generalised topological degree techniques are employed in the proofs of our results without requiring a transversality condition. The operators involved belong to the general class of A-proper mappings which include compact and k-ball contractive perturbations of the identity operator, accretive mappings, and many more.
Dynamic phase transitions: Existence of “cavitation” waves
Authors: Grinfeld Michael
Published: November 2011

Synopsis

In this paper we examine the influence of capillarity on existence and uniqueness of travelling wave solutions in an isothermal system of van der Waals fluids. Existence and non-uniqueness theorems are proved using phase-space analysis and topological methods.
On the initial growth of interfaces in reaction–diffusion equations with strong absorption
Authors: Alvarez Luis , Diaz Jesus Ildefonso
Published: November 2011

We study the initial growth of the interfaces of non-negative local solutions of the equation ut = (um)xx−λuq when m ≧ 1 and 0<q <1. We show that if with C < C0, for some explicit C0 = C0(λ, m, q), then the free boundary Ϛ(t) = sup {x:u(x, t) > 0} is a ‘heating front’. More precisely Ϛ(t) ≧at(m−q)/2(1−q) for any t small enough and for some a>0. If on the contrary, with C 0. Applications to solutions of the associated Cauchy and Dirichlet problems are also given.
Growth sequences of 2-generator simple groups
Authors: Obraztsov V. N.
Published: November 2011

Synopsis

A study is made of the minimum number of generators of then-th direct power of certain 2-generator groups.
On the relation between distinct particular solutions of equation
Authors: Ke-ying Guan , Everitt W. N.
Published: November 2011

Synopsis

There exists a relation (1.5) between any n + 2 distinct particular solutions of the differential equation

In this paper, we show that when and only when n = 0, 1 and 2, this relation can be represented by the following form:

provided the form of this relation function Φn depends only on n and is independent of the coefficients of the equation. This result reveals interesting properties of these non-linear differential equations.
Solutions for the microsensor thermistor equations in the small bias case
Authors: Allegretto W. , Xie H.
Published: November 2011

Synopsis

The behaviour of a microsensor thermistor is described by a system of nonlinear coupled elliptic equations subject to mixed Dirichlet-Neumann boundary conditions, to be solved on different domains. We employ the Implicit Function Theorem in Banach space to show that the system has a solution for small applied bias. It does not appear that earlier approaches for similar thermistor problems can be employed in this physically important situation. The fact that the problem is cast in a subset of R3 is significant in our presentation.
Spectral properties of a two parameter nonlinear Sturm-Liouville problem
Authors: Shibata Tetsutaro
Published: November 2011

Synopsis

We consider the nonlinear Sturm–Liouville problem with two parameters on the general level set

We establish asymptotic formulae of the n-th variational eigenvalue λ = λn(μ, α) as α→∞ and α↓(nπ)2.
Steady states of the one-dimensional Cahn–Hilliard equation
Authors: Novick-Cohen A. , Peletier L. A.
Published: November 2011

Synopsis

The steady states of the Cahn–Hilliard equation are studied as a function of interval length, L, and average mass, m. We count the number of nontrivial monotone increasing steady state solutions and demonstrate that if m lies within the spinodal region then for a.e. there is an even number of such solutions and for a.e. there is an odd number of such solutions. If m lies within the metastable region, then for a.e. L > 0 there is an even number of solutions. Furthermore, we prove that for all values of m, there are no secondary bifurcations.
Asymptotic expansions and Stokes multipliers of the confluent hypergeometric function Φ2, I
Authors: Shimomura Shun
Published: November 2011

Synopsis

The confluent hypergeometric function Φ2(β,β′, γ, x, y) satisfies a system of partial differential equations which possesses the singular loci x = 0, y = 0, x − y = 0 of regular type and x = ∞, y = ∞ of irregular type. Near x = ∞ (|y| is bounded) and near y = ∞ (|x| is bounded), asymptotic expansions and Stokes multipliers of linearly independent solutions of the system are obtained. By a connection formula, the asymptotic behaviour of Φ2(β,β′, γ, x, y) itself is also clarified near these singular loci.
On homotopy conditions and the existence of multiple equilibria in finite elasticity
Authors: Post K. D. E. , Sivaloganathan J.
Published: November 2011

In this paper we study homotopy classes of deformations and their properties under weak convergence. As an application, we give an analytic proof (in two and three dimensions) of the existence of infinitely many local minimisers for a displacement boundary-value problem from finite elasticity, posed on a nonconvex domain, under the constitutive assumption of polyconvexity.
On rank-one convex and polyconvex conformal energy functions with slow growth
Authors: Yan Baisheng
Published: November 2011

We make some remarks about rank-one convex and polyconvex functions on the set of all realn×nmatrices that vanish on the subsetKnconsisting of all conformal matrices and grow like a power function at infinity. We prove that every non-negative rank-one convex function that vanishes onKnand grows below a power of degreen/2 must vanish identically. In odd dimensionsn≧ 3, we prove that every non-negative polyconvex function that vanishes onKnmust vanish identically if it grows below a power of degreen; while in even dimensions, such polyconvex functions can exist that also grow like a power of half-dimension degree.
Uniqueness of positive solutions of a quasilinear Dirichlet problem with exponential nonlinearity
Authors: Adimurthi
Published: November 2011

Uniqueness of positive solutions of the following equation:

has been obtained, where B ⊂ ℝn is the ball of unit radius and λ > 0. Moreover, if n = 2, then the solutions are nondegenerate.
Convergence rates to travelling waves for a nonconvex relaxation model
Authors: Mei Ming , Yang Tong
Published: November 2011

In this paper we study the asymptotic behaviour of the solution for a nonconvex relaxation model. The time decay rates in both the exponential and algebraic forms of the travelling wave solutions are shown by the weighted energy method. Our results develop and improve the stability theory in [8,9].
Bounds on the effective conductivity of two-dimensional composites made of n ≧ 3 isotropic phases in prescribed volume fraction: the weighted translation method
Authors: Nesi V.
Published: November 2011

We establish lower and upper bounds which are valid for the overall conductivity of twodimensional composites. They are based on a method which modifies the so-called translation method in a way which makes it effectively much more flexible. When specialised to composites of n > 2 isotropic phases, the new bounds are often strictly better than all the previously known ones. From the mathematical point of view, the improvement is due mainly to a new regularity result in p.d.e.s [2]. From the physical point of view the latter can be interpreted as a result bounding in a suitable sense the fluctuations of the ‘electric field’.
Solutions in the large for certain nonlinear parabolic equations and applications
Authors: Zhao Huijiang , Zhu Changjiang
Published: November 2011

We prove some results on the global existence of smooth solutions for certain nonlinear parabolic systems of the form Ut + A(U)Ux = DUxx. Here U is a vector and A(U), D are matrices with D a constant, positive matrix. We show how to use our results to study the global continuous (or generalised) solutions to the corresponding nonlinear hyperbolic conservation laws and a conjecture is given.
The transmutation operator for −d2/(w(x) dx2)
Authors: Boumenir Amin
Published: November 2011

We would like to obtain the transmutation operator V, associated with the self-adjoint operators −d2/(w(x) dx2) and (−d2/(w(x) dx2)) + h(x), where w(x) ≎ xa as x → 0. We shall show that V = 1 + K, where K is a lower triangular Volterra operator.
Singular limit of quasilinear Neumann problems
Authors: Pan Xing-Bin
Published: November 2011

This paper is devoted to the study of the singular limit of the minimal solutions, as p → 1, of quasilinear Neumann problems involving p-Laplacian operators. It is established that the limit function is of bounded variation and is locally Höolder-continuous inside the domain.
Genericity of the multibump dynamics for almost periodic Duffing-like systems
Authors: Alessio Francesca , Caldiroli Paolo , Montecchiari Piero
Published: November 2011

In this paper we consider ‘slowly’ oscillating perturbations of almost periodic Duffing-like systems, i.e. systems of the form ü = u − (a(t) + α(wt))W′(u), t ∈ ℝ, u ∈ ℝN, where W ∈ C2N(ℝN, ℝ) is superquadratic and a and α are positive and almost periodic. By variational methods, we prove that if w > 0 is small enough, then the system admits a multibump dynamics. As a consequence we get that the system ü = u − a(t)W′(u), t ∈ ℝ, u ∈ ℝN, admits multibump solutions whenever a belongs to an open dense subset of the set of positive almost periodic continuous functions.
On the strong closure of strains and stresses in linear elasticity
Authors: Miettinen Markku , Raitums Uldis
Published: November 2011

We consider the following special problem related to the optimal layout problems of materials: given two linear elastic materials, the elasticity tensors of which are C1 and C2, and a force f, find the strong closure of strains and stresses as the distribution of the materials varies, or, alternatively, find the sets of elasticity tensors which generate these strong closures. In this paper, it is shown that the local incompatibility conditions depending on C1, C2 and the local properties of strains or stresses completely characterize these sets. A connection to multiple-well problems is established.
On isotropic rank 1 convex functions
Authors: Šilhavý Miroslav
Published: November 2011

Let f be a rotationally invariant function defined on the set Lin+ of all tensors with positive determinant on a vector space of arbitrary dimension. Necessary and sufficient conditions are given for the rank 1 convexity of f in terms of its representation through the singular values. For the global rank 1 convexity on Lin+, the result is a generalization of a two-dimensional result of Aubert. Generally, the inequality on contains products of singular values of the type encountered in the definition of polyconvexity, but is weaker. It is also shown that the rank 1 convexity is equivalent to a restricted ordinary convexity when f is expressed in terms of signed invariants of the deformation.