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We construct counterexamples to classical calculus facts such as the inverse and implicit function theorems in scale calculus—a generalization of multivariable calculus to infinite-dimensional vector spaces, in which the reparameterization maps relevant to symplectic geometry are smooth. Scale calculus is a corner stone of polyfold theory, which was introduced by Hofer, Wysocki, and Zehnder as a broadly applicable tool for regularizing moduli spaces of pseudoholomorphic curves.We show how the novel nonlinear scale-Fredholm notion in polyfold theory overcomes the lack of implicit function theorems, by formally establishing an often implicitly used fact: The differentials of basic germs—the local models for scale-Fredholm maps—vary continuously in the space of bounded operators when the base point changes. We moreover demonstrate that this continuity holds only in specific coordinates, by constructing an example of a scale-diffeomorphism and scale-Fredholm map with discontinuous differentials. This justifies the high technical complexity in the foundations of polyfold theory.

In 1976 Simmons conjectured that every coloring of a 2-dimensional sphere of radius strictly greater than 1/2 in three colors has a pair of monochromatic points at distance 1 apart. We prove this conjecture.

A new adaptive neural control method, with the actuators multiple constraints of amplitude and rate into consideration, is proposed in this paper for the flexible air-breathing hypersonic vehicle (AHV). In order to better reflect the characteristics of the actual AHV model, we regard the AHV as a completely unknown non-affine system in the control law design process, which is different from the existing AHV control methods, thus ensuring the reliability of the designed control law. On the basis of the implicit function theorem, the radial basis function neural network (RBFNN) is introduced to approximate the model. Meanwhile, the minimum learning parameter algorithm is adopted to adaptively adjust the weight vector of RBFNN, then the design of the ideal control law is completed. When the amplitude and rate of the actuator are saturated, the designed novel auxiliary error compensation system is used to effectively compensate for the ideal control law, and the stability of the closed-loop control system is proved via the Lyapunov stability theory. In addition, to avoid the “explosion of terms” problem in the control law design process, the finite-time-convergence-differentiator is introduced to accurately estimate the differential signal. Finally, the effectiveness of the control method designed in this paper is verified by simulation.

In this paper we consider a general control system involving a spatially heterogeneous dynamics . This means that the state space is partitioned into several disjoint regions and that each region has its own (smooth) control system. As a result, the dynamics discontinuously changes whenever the trajectory crosses an interface between two regions. In that spatially heterogeneous setting (and in contrast with the usual smooth case), a needle-like perturbation of the control may generate a perturbed trajectory that does not uniformly converge towards the nominal one, and may lead to the absence of a corresponding first-order variation vector. The first contribution of this paper is to illustrate this issue by means of a simple counterexample. Our second and main contribution is to provide a modified needle-like perturbation of the control (adapted to the spatially heterogeneous setting) which generates a perturbed trajectory that uniformly converges towards the nominal one, and leads to a corresponding first-order variation vector (which has the particularity of admitting a discontinuity jump at each interface crossing). This is made possible under several assumptions (including transverse crossing conditions), by introducing new tools such as auxiliary trajectories and auxiliary controls and by using a conic version of the implicit function theorem.

We consider an overdetermined problem for a two-phase elliptic operator in divergence formwith piecewise constant coefficients. We look for domains such that the solution u of a Dirichlet boundary value problem also satisfies the additional property that its normal derivative ∂n u is a multiple of the radius of curvature at each point on the boundary. When the coefficients satisfy some “non-criticality” condition, we construct nontrivial solutions to this overdetermined problem employing a perturbation argument relying on shape derivatives and the implicit function theorem. Moreover, in the critical case, we employ the use of the Crandall-Rabinowitz theorem to show the existence of a branch of symmetry breaking solutions bifurcating from trivial ones. Finally, some remarks on the one-phase case and a similar overdetermined problem of Serrin type are given.

Previous Mizar articles [7, 6, 5] formalized the implicit and inverse function theorems for Frechet continuously differentiable maps on Banach spaces. In this paper, using the Mizar system [1], [2], we formalize these theorems on Euclidean spaces by specializing them. We referred to [4], [12], [10], [11] in this formalization.

We prove a version of the implicit function theorem for Lipschitz mappings ƒ : ℝ n+m ⊃ A → X into arbitrary metric spaces. As long as the pull-back of the Hausdorff content H ∞ n by ƒ has positive upper n-density on a set of positive Lebesgue measure, there is then a local diffeomorphism G in ℝ n+m and a Lipschitz map π : X → ℝ n such that π ◦ ƒ ◦ G −1, when restricted to a certain subset of A of positive measure, is the orthogonal projection of ℝ n+m onto the first n-coordinates. This may be seen as a qualitative version of a simlar result of Azzam and Schul [2]. The main tool in our proof is the metric change of variables introduced in [6].

Background Chronic diseases, such as type 2 diabetes, are responsible for a substantial proportion of global deaths and are marked by an increasing number of people that suffer from them. Our objective is to systematically investigate the analytical determination of the drift in prevalence peaks over calendar-time and age, with an emphasis on understanding the intrinsic attributes of temporal displacement. This aims to enhance the understanding of disease dynamics that may contribute to refining medical strategies and to plan future healthcare activities. Methods We present two distinct yet complementary approaches for identifying and estimating drifts in prevalence peaks. First, assuming incidence and mortality rates are known, we employ a partial differential equation that relates prevalence, incidence, and mortality. From this, we derive an ordinary differential equation to mathematically describe the prevalence peak drift. Second, assuming prevalence data (rather than incidence and mortality data) are available, we establish a logistic function approach to estimate the prevalence peak drift. We applied this method to data on the prevalence of type 2 diabetes in Germany. Results The first approach provides an exact mathematical prescription of the trajectory of the prevalence peak drift over time and age, assuming incidence and mortality rates are known. In contrast, the second approach, a practical application based on available prevalence data, demonstrates the prevalence peak dynamics in a real-world scenario. The theoretical model, together with the practical application, effectively substantiates the understanding of prevalence peak dynamics in two different scenarios. Conclusion Our study shows the theoretical derivation and determination of prevalence peak drifts. Our findings underpin the dynamic nature of chronic disease prevalence, highlighting the importance of considering the related age-dependent trends for planning future healthcare activities.

We consider differential delay equations of the form _tx(t) = X_t(x(t - )) in R^n , where (X_t)_tın S^1 is a time-dependent family of smooth vector fields on R^n and is a delay parameter. If there is a (suitably non-degenerate) periodic solution x_0 of this equation for =0 , that is without delay, there are good reasons to expect existence of a family of periodic solutions for all sufficiently small delays, smoothly parametrized by . However, it seems difficult to prove this using the classical implicit function theorem, since the equation above, considered as an operator, is not smooth in the delay parameter. In this paper, we show how to use the M-polyfold implicit function theorem by Hofer–Wysocki–Zehnder (2009, 2021) to overcome this problem in a natural setup.