We study a Ginzburg–Landau functional for a vector-valued order parameter that carries a spin and couples directly to the magnetic field. We verify that spin coupling reduces the lower critical field Hc1, the smallest value of the applied field strength at which vortices penetrate the superconductor, and discuss the related phenomena of a ‘spontaneous vortex phase’ for sufficiently strong spin coupling, predicted by physicists. The method uses energy methods and a priori bounds to confine the vortices and spin transitions to very small balls

We consider a generalized Ginzburg–Landau energy functional modelling a superconductor surrounded by a material in the normal state. In this model, the order parameter is defined in the whole space. We derive existence of a global minimizer in weighted Sobolev spaces for both square-integrable and constant-applied magnetic fields. We then prove boundedness and classical elliptic estimates for the order parameter, in order to study the loss of superconductivity for high applied magnetic fields. In two dimensions for the general case and in three dimensions for the case of constant permeability, we show the existence of an upper critical field above which the only finite-energy weak solutions are the normal states. For the three-dimensional case, we show that as the applied field tends to infinity, finite-energy weak solutions tend to the normal state

Let V be any vector space and P(V) the set of all partial linear transformations defined on V, that is, all linear α: A → B, where A, B are subspaces of V. Then P(V) is a semigroup under composition, which is partially ordered by ⊆ (that is, α ⊆ β if and only if dom α ⊆ dom β and α = β | dom α). We compare this order with the so-called 'natural partial order' ≤ on P(V) and we determine their meet and join. We also describe all elements of P(V) that are minimal (or maximal) with respect to each of these four orders, and we characterize all elements that are 'compatible' with them. In addition, we answer similar questions for the semigroup T(V) consisting of all α ∈ P(V) whose domain equals V. Other orders have been defined by Petrich on any regular semigroup: three of them form a chain below ≤, and we show that two of these are equal on the semigroup P(V) and on the ring T(V). We also consider questions for these orders that are similar to those already mentioned

We prove uniqueness of solutions to scalar conservation laws with space discontinuous fluxes. To do so, we introduce a partial adaptation of Kružkov's entropies which naturally takes into account the space dependency of the flux. The advantage of this approach is that the proof turns out to be a simple variant of the original method of Kružkov. In particular, we do not need traces, interface conditions, bounded variation assumptions (neither on the solution nor on the flux), or convex fluxes. However, we use a special ‘local uniform invertibility’ structure of the flux, which applies to cases where different interface conditions are known to yield different solutions

In this paper, we study a symmetry-breaking phenomenon of a least-energy solution to a nonlinear Schrödinger equationunder suitable assumptions on V(x), where λ > 1, p > 2 and χA is the characteristic function of the set A = [−(l + 2), −l] ∪ [l,l + 2] with l > 0. We also study asymptotic profiles of least-energy solutions for the singularly perturbed problemfor small ε > 0.

For n canonical systems of differential equations, the corresponding n copies of their domain (0, ∞) are thought of as a graph with vertex 0. An interface condition at 0 is given by a so-called Nevanlinna pair. Explicit formulae are deduced for the spectral representation of the corresponding underlying self-adjoint relation and the generalized Fourier transformation. Furthermore, results on compressions of the Fourier transformation to closed linear subspaces and the multiplicity of the eigenvalues if the spectrum is discrete are presented

We consider the homogeneous Dirichlet problem δu = −f(u) ≤ 0 in Ω with u = 0 on ∂Ω. We are interested in the inverse problem of determining the nonlinear source f from knowledge of the normal derivative of u, ∂u/δn, on an open arc Γ of ∂Ω. It is well known that this fails if Ω is a ball. On the other hand, Beretta and Vogelius proved that an analytic source f is uniquely determined from knowledge of (∂u/∂n)|Γ if Γ has at least a true corner. In this paper we try to bridge the gap finding a class of smooth domains for which the determination of analytic f is possible

Let S be a 0-bisimple inverse semigroup and let denote the contracted l1-algebra of S; that is, the Banach algebra l1 (S)/Z, where Z is the ideal spanned by the zero of S. It is shown that, if l1 (G) is primitive for some non-zero maximal subgroup G of S, then is primitive. The same result also holds with primitivity replaced by a stronger condition, here called *-primitivity

In this paper, we propose a mean curvature flow equation with nonlocal nonlinearities for an image processing model on noisy surfaces, which corresponds to the negative $L^2$-gradient flow of surface energy functional in (1.4). We investigate the initial value problem mathematically. Furthermore, numerical simulations are shown, which exhibit that the model performs smoothing operation on a complicated level surface of a given image data while preserving its geometric shape feature under suitable parameters chosen.

The aim of this paper is to give a detailed introduction to the generalized hyperbolic option pricing model. We discuss European and discrete Asian options, especially arithmetic average options, and compare the results with the classical Black–Scholes model.

We study two coupled reaction-diffusion equations of the $\lambda$–$\omega$ type [11] in $d\,{\le}\,3$ space dimensions, on a convex bounded domain with a $C^2$ boundary. The equations are close to a supercritical Hopf bifurcation in the reaction kinetics and are model equations for oscillatory reaction-diffusion equations. Global existence, uniqueness and continuous dependence on initial data of strong and weak solutions are proved using the classical Faedo-Galerkin method of Lions [15] and compactness arguments. We also present a complete case study for the application of this method to systems of nonlinear reaction-diffusion equations.

In a recent paper, a new three-parameter class of Abel type equations, so-called AIR, all of whose members can be mapped into Riccati equations, is shown. Most of the Abel equations with solution presented in the literature belong to the AIR class. Three canonical forms were shown to generate this class, according to the roots of a cubic. In this paper, a connection between those canonical forms and the differential equations for the hypergeometric functions $_2{\rm F}_1$, $_1{\rm F}_1$ and $_0{\rm F}_1$ is unveiled. This connection provides a closed form $_p{\rm F}_q$ solution for all Abel equations of the AIR class.

For a given scalar partial differential equation (PDE), a potential variable can be introduced through a conservation law. Such a conservation law yields an equivalent system (potential system) of PDEs with the given dependent variable and the potential variable as its dependent variables. Often there is also another equivalent scalar PDE (potential equation) with the potential variable as its dependent variable. The Nonclassical Method for obtaining solutions of PDEs is a generalization of the Classical Method for obtaining invariant solutions from point symmetries admitted by a given PDE. As a prototypical example, the nonlinear heat conduction equation is used to demonstrate that the Nonclassical Method applied to a potential equation can yield new solutions (nonclassical potential solutions) of a given PDE that are unobtainable as invariant solutions from admitted point symmetries of the given PDE, a related potential system or the potential equation, or from nonclassical solutions generated by applying the Nonclassical Method ($\tau{\equiv} 1$) to the given scalar PDE.

The wave equation $(u_{t} + u u_{x})_{x} - u = 0$ is a model for shallow water waves with Coriolis force, sound waves in a bubbly liquid and more generally “is the canonical asymptotic equation for genuinely nonlinear waves that are nondispersive as their wavelength tends to zero” in the words of Hunter [13]. This Ostrovsky–Hunter equation has steadily-translating, spatially periodic solutions which exist only when $c \leq c_{limit}$. The limiting wave (‘parabolic wave’) is exactly given by a piecewise quadratic polynomial in $x$ with a discontinuous slope at the crest. We show that near the limit, the travelling waves (‘paraboloidal waves’) can be approximated by matched asymptotic expansions: the inner solution rounds off the point while the outer solution, valid over most of the spatial domain, is to lowest order just the parabolic wave. In the opposite limit of small amplitude, we derive a Fourier-and-powers-of-amplitude expansion (‘Stokes' series’). We show that this is remarkably accurate even very close to the limiting wave and converges to the limiting wave for unit amplitude. We demonstrate also that the Fourier pseudospectral method gives first order convergence even for the slope-discontinuous parabolic wave.

We propose a delay differential equation model for a single species with stage-structure in which the maturation delay is modelled as a distribution, to allow for the possibility that individuals may take different amounts of time to mature. General birth and death rate functions are used. We find that the dynamics of the model depends largely on the qualitative form of the birth function, which depends on the total number of adults. If it is monotonic increasing and a non-zero equilibrium exists, then the equilibrium is globally stable for all maturation delay distributions with compact support. For the case of a finite spatial domain with impermeable boundaries, a reaction-diffusion extension of the model is rigorously derived using an approach based on the von Foerster diffusion equation. The resulting reaction-diffusion system is nonlocal. The dynamics of the reaction-diffusion system again depends largely on the qualitative form of the birth function. If the latter is non-monotone with a single hump, then the dynamics depends largely on whether the equilibrium is to the left or right of the hump, with oscillatory dynamics a possibility if it is sufficiently far to the right.

This paper deals with a dynamical inverse problem for a composite beam formed by two connected beams. The vibrations of the composite beam are governed by a differential system where a coupling takes place between longitudinal and bending motions. In this paper, we neglect bending motions and we only deal with the longitudinal motions. These motions are governed by a two-by-two second order system coupled in the lower order terms by the shearing stiffness coefficient, which models the connection between the two beams and which contains direct information on the integrity of the system. We prove that the shearing stiffness coefficient can be reconstructed from the frequency response function of the system evaluated at one end of the beam.

In this paper, we prove a certain maximality property of the Shimura subgroup amongst the multiplicative-type subgroups of $J_0(N)$, and apply this to verify conjectures of Stevens on the existence of certain canonical parametrizations of rational elliptic curves by modular curves. We are also able to verify some of Stevens’s conjectures on the characterization of the elliptic curve in an isogeny class with minimal Faltings–Parshin height.

AMS 2000 Mathematics subject classification: Primary 11G05; 11G18

Conformally invariant functionals on the space of knots are introduced via extrinsic conformal geometry of knots and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form, which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot. We show that our functionals detect the unknot as the total curvature does, and that their values explode if a knot degenerates to a singular knot with double points.

AMS 2000 Mathematics subject classification: Primary 57M25; 53A30. Secondary 57R17

Following a recent paper by Faltings, we study the integral points on $\bm{P}_2\setminus\mathcal{D}$, where $\mathcal{D}$ is the branch locus of a projection from a surface $\mathcal{X}$; a crucial point in the analysis is that the pull-back of $\mathcal{D}$ in the Galois closure of the projection often splits into several components. As in the paper by Faltings, under certain assumptions we obtain finiteness of the integral points (Theorem 3.1); for instance, we shall find that it suffices if the projection is sufficiently general and if $\mathcal{X}$ has Kodaira number $\ge0$ (Corollary 4.1). We have borrowed freely from Faltings’s paper, for the whole geometrical setting. As to the arithmetic, our method is in part different, relying on the recent paper by Corvaja and Zannier and leading to apparently new conditions. We shall also use a more elementary approach to study a similar situation in arbitrary dimension, where the projection is taken from a hypersurface (Theorem 2.1).

In concrete terms, these results deal with certain diophantine equations $F(x_0,\dots,x_n)=c$.

AMS 2000 Mathematics subject classification: Primary 11D72; 11G35; 11G99

We consider a real-analytic compact surface diffeomorphism $f$, for which the tangent space over the non-wandering set $\varOmega$ admits a dominated splitting. We study the dynamical determinant

$$d_f(z)=\exp-\sum_{n\ge1}\frac{z^n}{n}\sum_{x\in\textrm{Fix}^*f^n}|\textrm{Det}(Df^n(x)-\textrm{Id})|^{-1},$$

where $\textrm{Fix}^*f^n$ denotes the set of fixed points of $f^n$ with no zero Lyapunov exponents. By combining previous work of Pujals and Sambarino on $C^2$ surface diffeomorphisms with, on the one hand, results of Rugh on hyperbolic analytic maps and, on the other, our two-dimensional version of the same author’s analysis of one-dimensional analytic dynamics with neutral fixed points, we prove that $d_f(z)$ is either an entire function or a holomorphic function in a (possibly multiply) slit plane.

AMS 2000 Mathematics subject classification: Primary 37C30; 37D30; 37E30