We describe explicitly the Voevodsky's triangulated category of motives (and give a ‘differential graded enhancement’ of it). This enables us to able to verify that DMgm ℚ is (anti)isomorphic to Hanamura's (k).

We obtain a description of all subcategories (including those of Tate motives) and of all localizations of . We construct a conservative weight complex functor ; t gives an isomorphism . A motif is mixed Tate whenever its weight complex is. Over finite fields the Beilinson–Parshin conjecture holds if and only if tℚ is an equivalence.

For a realization D of we construct a spectral sequence S (the spectral sequence of motivic descent) converging to the cohomology of an arbitrary motif X. S is ‘motivically functorial’; it gives a canonical functorial weight filtration on the cohomology of D(X). For the ‘standard’ realizations this filtration coincides with the usual one (up to a shift of indices). For the motivic cohomology this weight filtration is non-trivial and appears to be quite new.

We define the (rational) length of a motif M; modulo certain ‘standard’ conjectures this length coincides with the maximal length of the weight filtration of the singular cohomology of M.

We apply the collapse techniques to Poizat's red differential field in order to obtain differentially closed fields of Morley rank ω·2 each equipped with an additive definable subgroup of rank ω. By means of the logarithmic derivative, we obtain a green field of rank ω·2 with a multiplicative definable divisible subgroup containing the field of constants, which is again definable in the reduct of the green field.

This paper is concerned with the existence of a global attractor for a semi-flowgoverned by the weak solutions to a nonlinear one-dimensional thermoviscoelasticsystem with clamped boundary conditions in shape memory alloys. The constitutiveassumptions for the Helmholtz free energy include the model for the study ofmartensitic phase transitions in shape memory alloys. To describe physicallyphase transitions between different configurations of crystal lattices, we workin a framework in which the strain u belongsto L∞. New approachesare introduced and more delicate estimates are derived to establish the crucialL∞ -estimate ofstrain u in the course of showing thecompactness of the orbit of the semi-flow and existence of an absorbing set.

We consider a new subgroup In(G) in any group G of finite Morley rank. This definably characteristic subgroup is the smallest normal subgroup of G from which we can hope to build a geometry over the quotient group G/ In(G). We say that G is a geometric group if In(G) is trivial.

This paper is a discussion of a conjecture which states that every geometric group G of finite Morley rank is definably linear over a ring K1 ⊕…⊕ Kn where K1,…,Kn are some interpretable fields. This linearity conjecture seems to generalize the Cherlin–Zil'ber conjecture in a very large class of groups of finite Morley rank.

We show that, if this linearity conjecture is true, then there is a Rosenlicht theorem for groups of finite Morley rank, in the sense that the quotient group of any connected group of finite Morley rank by its hypercentre is definably linear.

This paper concerns the asymptotic behaviours of pulse-like solutions for a 3 × 3 semilinear hyperbolic system in the limit of short wavelength ε. When two pulses interact with each other, we construct a pulse-like approximate solution up to Ο(ε), at which order a new pulse appears. The existence of a solution to the 3 × 3 semilinear problem with the initial data being the interaction of two pulses in a domain independent of the wavelength is proved in the space of co-normal distributions. Meanwhile, we obtain that the error between this exact solution and the approximate solution is of Ο(ε2) as ε → 0, which rigorously shows that there are three pulses propagated after the interaction of two pulses for the 3 × 3 semilinear system.

In order to construct a counterexample to Zilber's conjecture—that a strongly minimal set has a degenerate, affine or field-like geometry—Ehud Hrushovski invented an amalgamation technique which has yielded all the exotic geometries so far. We shall present a framework for this construction in the language of standard geometric stability and show how some of the recent constructions fit into this setting. We also ask some fundamental questions concerning this method.

Ion channels are proteins with a narrow hole down their middle that control a wide range of biological function by controlling the flow of spherical ions from one macroscopic region to another. Ion channels do not change their conformation on the biological time scale once they are open, so they can be described by a combination of Poisson and drift-diffusion (Nernst–Planck) equations called PNP in biophysics. We use singular perturbation techniques to analyse the steady-state PNP system for a channel with a general geometry and a piecewise constant permanent charge profile. We construct an outer solution for the case of a constant permanent charge density in three dimensions that is also a valid solution of the one-dimensional system. The asymptotical current–voltage (I–V) characteristic curve of the device (obtained by the singular perturbation analysis) is shown to be a very good approximation of the numerical I–V curve (obtained by solving the system numerically). The physical constraint of non-negative concentrations implies a unique solution, i.e., for each given applied potential there corresponds a unique electric current (relaxing this constraint yields non-physical multiple solutions for sufficiently large voltages).

For m > 0 and p ∈ (1, (N + 2)/(N − 2)), we show the uniqueness and a linearized non-degeneracy of solutions for the following problem:

In this paper, we study basic properties of the diffusive wave approximation of the shallow water equations (DSW). This equation is a doubly non-linear diffusion equation arising in shallow water flow models. It has been used as a model to simulate water flow driven mainly by gravitational forces and dominated by shear stress, that is, under uniform and fully developed turbulent flow conditions. The aim of this work is to present a survey of relevant results coming from the studies of doubly non-linear diffusion equations that can be applied to the DSW equation when topographic effects are ignored. In fact, we present proofs of the most relevant results existing in the literature using constructive techniques that directly lead to the implementation of numerical algorithms to obtain approximate solutions.

A new approach is presented for the solution of spectral problems on infinite domains with regular ends, which avoids the need to solve boundary-value problems for many trial values of the spectral parameter. We present numerical results both for eigenvalues and for resonances, comparing with results reported by Aslanyan, Parnovski and Vassiliev.

A steady-state bubble solution to the constrained mass conserving Allen–Cahn equation in a two-dimensional domain is constructed in the limit of small diffusivity. The solution is asymptotically constant inside a circle of radius rb centred at some unknown location x0 and has a sharp interface at the bubble radius that allows for a transition to a different asymptotically constant state outside the bubble. In a study by M. J. Ward (Metastable bubble solutions for the Allen–Cahn equation with mass conservation. SIAM J. Appl. Math. 56, 1996, 247–1279), the bubble centre was determined by a limiting solvability condition. The solution found by Ward suggests the existence of a corner type boundary layer where the normal derivative of the bubble solution readjusts to satisfy the no-flux condition at the boundary of the domain. This work is concerned with the details of the readjustment. A variational approach similar to the one of W. L. Kath, C. Knessl and B. J. Matkowsky (A variational approach to nonlinear singularly perturbed boundary-value problems. Stud. Appl. Math. 77, 1987, 61–88) shows the formation of a corner layer (for the derivative of the solution) which influences as a high-order correction the available determination of the bubble centre. This corner layer describes to leading order the readjustment of the level lines of the bubble to lines parallel to the boundary of the container; moreover, it provides to leading order a smooth solution across the corner layer.

The notion of an angular function has been introduced by Zilber as one possible way of connecting non-commutative geometry with two ‘counterexamples’ from model theory: the non-classical Zariski curves of Hrushovski and Zilber, and Poizat's field with green points. This article discusses some questions of Zilber relating to existentially closed structures in the class of algebraically closed fields with an angular function.

We study a scale of two-component composite structures of equal proportions with infinitely many microlevels. The structures are obtained recursively and we find that their effective conductivities are power means of the local conductivities.

Online parameter identification in time-dependent differential equations from time course observations related to the physical state can be understood as a non-linear inverse and ill-posed problem and appears in a variety of applications in science and engineering. The feature as well as the challenge of online identification is that sensor data have to be continuously processed during the operation of the real dynamic process in order to support simulation-based decision making. In this paper we present an online parameter identification method that is based on a non-linear parameter-to-output operator and, as opposed to methods available so far, works both for finite- and infinite-dimensional dynamical systems, e.g., both for ordinary differential equations and time-dependent partial differential equations. A further advantage of the method suggested is that it renders typical restrictive assumptions such as full state observability, linear parametrisation of the underlying model and data differentiation or filtering unnecessary. Assuming existence of a solution for exact data, a convergence analysis based on Lyapunov theory is presented. Numerical illustrations given are by means of online identification both of aerodynamic coefficients in a 3DoF-longitudinal aircraft model and of a (distributed) conductivity coefficient in a heat equation.

We study the geodesic interpolating spline with a biharmonic regulariser for solving the landmark image registration problem. We show existence of solutions, discuss uniqueness and show how the problem can be efficiently solved numerically. The main advantage of the geodesic interpolating spline is that it provides a diffeomorphism and we show this is preserved under our numerical approximation.