We build a new spectrum of recursive models ( $ \operatorname {\mathrm {SRM}}(T)$ ) of a strongly minimal theory. This theory is non-disintegrated, flat, model complete, and in a language with a finite signature.

Let $A\rightarrow B$ be a morphism of Artin local rings with the same embedding dimension. We prove that any $A$ -flat $B$ -module is $B$ -flat. This freeness criterion was conjectured by de Smit in 1997 and improves Diamond’s criterion [The Taylor–Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379–391, Theorem 2.1]. We also prove that if there is a nonzero $A$ -flat $B$ -module, then $A\rightarrow B$ is flat and is a relative complete intersection. Then we explain how this result allows one to simplify Wiles’s proof of Fermat’s last theorem: we do not need the so-called ‘Taylor–Wiles systems’ any more.

It is proved that flatness of an analytic mapping germ from a complete intersection is determined by its sufficiently high jet. As a consequence, one obtains finite determinacy of complete intersections. It is also shown that flatness and openness are stable under deformations.

The Blumlein pulse forming line (BPFL) consisting of an inner coaxial pulse forming line (PFL) and an outer coaxial PFL is widely used in the field of pulsed power, especially for intense electron-beam accelerators (IEBA). The output voltage waveform determines the quality and characteristics of the output beam current of the IEBA. Comparing with the conventional BPFL, an IEBA based on a strip spiral type BPFL can increase the duration of the output voltage in the same geometrical volume. However, for the spiral type BPFL, the voltage waveform on a matched load may be distorted, which influences the electron-beam quality. In this paper, the output waveform of an IEBA based on strip spiral BPFL is analyzed. It is found that there is fluctuation on the flattop of the main pulse, and the flatness is increased with the increment of the output voltage. According to the time integrated pictures of the cathode holder during the operation of the IEBA, the electron emission of the cathode holder is one of the reasons to cause the variance of the flatness. Furthermore, the distribution of the current density of spiral middle cylinder of the BPFL is calculated by using electromagnetic simulation software, and it is obtained that the current density is not uniform, and which leads to the nonuniformity of the impedance of BPFL. Meanwhile, when the nonuniformity of the BPFL is taken into account, the operation of the whole accelerator is simulated using a circuit-simulation code called PSpice. It is obtained that the nonuniformity of the BPFL influences the flatness of the output voltage waveform. In order to get an ideal square pulse voltage waveform and to improve the electron beam quality of such an accelerator, the uniformity of the spiral middle cylinder should be improved and the electron emission of the cathode holder should be avoided. The theoretical analysis and simulated output voltage waveform shows reasonable agreement with that of the experimental results.

We study the problem of flatness of two-input driftless control systems. Although acharacterization of flat systems of that class is known, the problems of describing allflat outputs and of calculating them is open and we solve it in the paper. We show thatall x-flat outputs are parameterized by an arbitrary function of threecanonically defined variables. We also construct a system of 1st order PDE’s whosesolutions give all x-flat outputs of two-input driftless systems. Weillustrate our results by describing all x-flat outputs of models of anonholonomic car and the n-trailer system.

For a locally compact group G, let LUC(G) denote the space of all left uniformly continuous functions on G. Here, we investigate projectivity, injectivity and flatness of LUC(G) and its dual space LUC(G)* as Banach left modules over the group algebra as well as the measure algebra of G.

Homological properties of several Banach left L1(G)-modules have been studied by Dales and Polyakov and recently by Ramsden. In this paper, we characterize some homological properties of and as Banach left L1(G)-modules, such as flatness, injectivity and projectivity.

Over the past few years several technical efforts have been focused on continuous improvement of quench and tempered product at ArcelorMittal Burns Harbor plate mill facility with particular attention being given to optimization of the flatness of quenched product. Incremental improvement in quench line equipment and practices is being aided by computer-based modelling tools and specialized instrumentation. The paper describes the development of different tools used to control flatness after quench and tempering units. The detailed description of a 3D model designed to predict distortion such as edge waves, center buckles, longbows and crossbows during quenching is presented. The model developed with Abaqus is able to predict the evolution of plate temperature, thermal stresses, strains, and distortions during cooling. The water cooling system is represented by top and bottom movable surfaces where a heat flux depending on width, length and temperature is applied. The effect of phase transformation is approximated by a variation of the plate thermal expansion coefficient.

Motion planning and boundary control for a class of linear PDEs with constant coefficients is presented. With the proposed method transitions from rest to rest can be achieved in a prescribed finite time. When parameterizing the system by a flat output, the system trajectories can be calculated from the flat output trajectory by evaluating definite convolution integrals. The compact kernels of the integrals can be calculated using infinite series. Explicit formulae are derived employing Mikusiński's operational calculus. The method is illustrated through an application to a model of a Timoshenko beam, which is clamped on a rotating disk and carries a load at its free end.

In this paper we consider a free boundary problem for a nonlinear parabolic partial differential equation. In particular, we are concerned with the inverse problem, which means we know the behavior of the free boundary a priori and would like a solution, e.g. a convergent series, in order to determine what the trajectories of the system should be for steady-state to steady-state boundary control. In this paper we combine two issues: the free boundary (Stefan) problem with a quadratic nonlinearity. We prove convergence of a series solution and give a detailed parametric study on the series radius of convergence. Moreover, we prove that the parametrization can indeed can be used for motion planning purposes; computation of the open loop motion planning is straightforward. Simulation results are given and we prove some important properties about the solution. Namely, a weak maximum principle is derived for the dynamics, stating that the maximum is on the boundary. Also, we prove asymptotic positiveness of the solution, a physical requirement over the entire domain, as the transient time from one steady-state to another gets large.

Let K be an algebraically closed field endowed with a complete non-archimedean norm with valuation ring R. Let $f\colon Y \to X$ be a map of K-affinoid varieties. In this paper we study the analytic structure of the image $f(Y) \subset X$; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are precisely the $\mathbf D$-semianalytic sets, where $\mathbf D$ is the truncated division function first introduced by Denef and van den Dries. This result is most conveniently stated as a Quantifier Elimination result for the valuation ring R in an analytic expansion of the language of valued rings. To prove this we establish a Flattening Theorem for affinoid varieties in the style of Hironaka, which allows a reduction to the study of subanalytic sets arising from flat maps, that is, we show that a map of affinoid varieties can be rendered flat by using only finitely many local blowing ups. The case of a flat map is then dealt with by a small extension of a result of Raynaud and Gruson showing that the image of a flat map of affinoid varieties is open in the Grothendieck topology. Using Embedded Resolution of Singularities, we derive in the zero characteristic case, a Uniformization Theorem for subanalytic sets: a subanalytic set can be rendered semianalytic using only finitely many local blowing ups with smooth centres. As a corollary we obtain the fact that any subanalytic set in the plane R2 is semianalytic. 2000 Mathematical Subject Classification: 32P05, 32B20, 13C11, 12J25, 03C10.