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A geometric approach to radial railway network improvement is presented here. It uses what we have called ‘isochrone circle graphs’, a kind of diagrams that reflect the values of two variables, mixing ideas from isochrone maps and from polar area diagrams (unlike the former, the areas of the sectors are related to population, not to ‘geographical areas’ and, unlike the latter, they do not compute...
Nonholonomic systems are described by the Lagrange-d’Alembert principle. The presence of symmetry leads to a reduced d’Alembert principle and to the Lagrange-d’Alembert-Poincaré equations. First, we briefly recall from previous works how to obtain these reduced equations for the case of a thick disk rolling on a rough surface using a three-dimensional abelian group of symmetries. The main results...
The purpose of this paper is to investigate the local regularity of the nondivergence degenerate elliptic operator with lower order terms in generalized Morrey spaces, structured on a family of Hörmander’s vector fields without an underlying group structure. The coefficients of the second order terms of the operator are real valued, bounded and measurable functions, such that the uniform ellipticity...
In this paper we study discrete second-order vakonomic mechanics, that is, constrained variational problems for second-order lagrangian systems. One of the main applications of the presented theory will be optimal control of underactuated mechanical control systems. We derive geometric integrators which are symplectic and preserve the momentum map. Additional, we show the applicability of the proposed...
We introduce the discrete counterpart of the vakonomic method in Lagrangian mechanics with non-holonomic constraints. After defining the concepts of “admissible section” and “admissible infinitesimal variation” of a discrete vakonomic system, we aim to determinate those admissible sections that are critical for the Lagrangian of the system with respect to admissible infinitesimal variations. For sections...
In this paper we summarize the main features of vakonomic mechanics (or constrained variational calculus), both from continuous and discrete points of views. In the continuous case, we focus ourselves on Lagrangian systems defined by the following data: a Riemannian metric (kinetic term) and constraints linear on the velocities. We show that, for such kind of systems, it is possible to find an explicit...
The discrete optimal control problem with discrete Lagrangian function $${\mathcal L(t_k,x^\alpha_k,u^i_k)(t_{k+1}-t_k)}$$ and constraints $$\begin{array}{ll}\varphi^\alpha\equiv\frac{x^\alpha_{k+1}-x^\alpha_k}{t_{k+1}-t_k}-f^\alpha(t_k,x^\beta_k,u^i_k)=0,\qquad 1\le \alpha,\beta\le n,\quad 1\le i\le m\end{array}$$ where $${x^\alpha_k}$$ are the dynamical variables, $${u^i_k}$$ are...
A differential-geometric setting for the dynamics of a higher-order field theory is proposed, based on the Skinner and Rusk formalism for mechanics. This approach incorporates aspects of both, the Lagrangian and the Hamiltonian description, since the field equations are formulated using the Lagrangian on a higher-order jet bundle and the canonical multisymplectic form on its affine dual. The result...
We present a procedure leading to efficient splitting schemes for the time integration of explicitly time dependent partitioned linear differential equations arising when certain partial differential equations are previously discretized in space. In the first stage we analyze the order conditions of the corresponding autonomous problem and construct new 6th-order methods. In the second stage, by following...
The aim of this paper is to present a short historical/scientific review on nonholonomic mechanics, with special emphasis on the latest developments. Indeed, the use of differential geometric tools has permitted in the last 25 years a fast and unsuspected advance in the theory, particularly in a better understanding of symmetries and reduction, Hamilton–Jacobi theory and integrability characterizations,...
Pontryagin’s Maximum Principle is an outstanding result for solving optimal control problems by means of optimizing a specific function on some particular variables, the so called controls. However, this is not always enough for solving all these problems. A high order maximum principle Krener (SIAM J Control Optim 15(2):256–293, 1977) must be used in order to obtain more necessary conditions for...
The goal of this short note is to show the constrained nature of the set of variations and the set of admissible sections of the Euler–Poincaré reduction scheme for field theories in both the continuous formalism and in a discrete model.
We introduce a geometric setup for discrete variational problems in two independent variables based on the theory of bundles modelled on cell complexes, and characterize geometrically a first variation formula and a discrete Noether theorem for symmetries of the discrete Lagrangian. We explore the existence of discrete variational integrators, which will conserve the discrete Noether currents in the...
We present criteria for determining mean ergodicity of C0-semigroups of linear operators in a sequentially complete, locally convex Hausdorff space X. A characterization of reflexivity of certain spaces X with a basis via mean ergodicity of equicontinuous C0-semigroups acting in X is also presented. Special results become available in Grothendieck spaces with the Dunford–Pettis property.
Given a decreasing sequence of weights V on a Banach space X, we consider the weighted inductive limits of spaces of entire functions V H(X) and V H0(X). We prove that V H(X) is the strong dual of a Fréchet space F for a particular class of sequence of weights, and we study some conditions to ensure that the equality V H0(X)′′ = V H(X) holds. The existence of a predual of V H(X) leads to a linearization...
We study a special kind of homology cycles of the modular curve X0(N). For a newform of weight 2 for Γ0(N), we construct a p-adic L-function by using these cycles. If the newform is defined over $${\mathbb{Q}}$$ , this p-adic L-function gives rise to algebraic points of the attached elliptic curve.
We provide a complete solution of the abstract Cauchy problem for operator valued Laplace distributions or hyperfunctions on complete ultrabornological locally convex spaces (like spaces of smooth functions and distributions). This extends results of Komatsu for operators on Banach spaces. Concrete examples are provided. The crucial tools for our solution are a general notion of a resolvent for operators...
Let $$(\mathcal{ H} (U), \tau _{\omega })$$ denote the vector space of all complex-valued holomorphic functions on an open subset U of a Banach space E, with the Nachbin compact-ported topology. Let $$(\mathcal{ H} (K), \tau _{\omega })$$ denote the vector space of all complex-valued holomorphic germs on a compact subset K of E, with its natural inductive limit topology. Let $$\mathcal{...
In the paper, we present some Baum–Katz type results for $${\varphi}$$ -mixing random variables with different distributions. Partial results generalize the corresponding one of Shao (Acta Math Sin 31(6):736–747, 1988). In addition, the Marcinkiewicz strong law of large numbers for $${\varphi}$$ -mixing random variables with different distributions is obtained.
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