20,226 research outputs found
Spacelike hypersurfaces in standard static spacetimes
In this work we study spacelike hypersurfaces immersed in spatially open
standard static spacetimes with complete spacelike slices. Under appropriate
lower bounds on the Ricci curvature of the spacetime in directions tangent to
the slices, we prove that every complete CMC hypersurface having either bounded
hyperbolic angle or bounded height is maximal. Our conclusions follow from
general mean curvature estimates for spacelike hypersurfaces. In case where the
spacetime is a Lorentzian product with spatial factor of nonnegative Ricci
curvature and sectional curvatures bounded below, we also show that a complete
maximal hypersurface not intersecting a spacelike slice is itself a slice. This
result is obtained from a gradient estimate for parametric maximal
hypersurfaces.Comment: 50 page
Discontinuity induced bifurcations of non-hyperbolic cycles in nonsmooth systems
We analyse three codimension-two bifurcations occurring in nonsmooth systems,
when a non-hyperbolic cycle (fold, flip, and Neimark-Sacker cases, both in
continuous- and discrete-time) interacts with one of the discontinuity
boundaries characterising the system's dynamics. Rather than aiming at a
complete unfolding of the three cases, which would require specific assumptions
on both the class of nonsmooth system and the geometry of the involved
boundary, we concentrate on the geometric features that are common to all
scenarios. We show that, at a generic intersection between the smooth and
discontinuity induced bifurcation curves, a third curve generically emanates
tangentially to the former. This is the discontinuity induced bifurcation curve
of the secondary invariant set (the other cycle, the double-period cycle, or
the torus, respectively) involved in the smooth bifurcation. The result can be
explained intuitively, but its validity is proven here rigorously under very
general conditions. Three examples from different fields of science and
engineering are also reported
Bifurcations of piecewise smooth ďŹows:perspectives, methodologies and open problems
In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study
Discrete Approximations of a Controlled Sweeping Process
The paper is devoted to the study of a new class of optimal control problems
governed by the classical Moreau sweeping process with the new feature that the polyhe-
dral moving set is not fixed while controlled by time-dependent functions. The dynamics of
such problems is described by dissipative non-Lipschitzian differential inclusions with state
constraints of equality and inequality types. It makes challenging and difficult their anal-
ysis and optimization. In this paper we establish some existence results for the sweeping
process under consideration and develop the method of discrete approximations that allows
us to strongly approximate, in the W^{1,2} topology, optimal solutions of the continuous-type
sweeping process by their discrete counterparts
Optimal control of the sweeping process over polyhedral controlled sets
The paper addresses a new class of optimal control problems governed by the
dissipative and discontinuous differential inclusion of the sweeping/Moreau
process while using controls to determine the best shape of moving convex
polyhedra in order to optimize the given Bolza-type functional, which depends
on control and state variables as well as their velocities. Besides the highly
non-Lipschitzian nature of the unbounded differential inclusion of the
controlled sweeping process, the optimal control problems under consideration
contain intrinsic state constraints of the inequality and equality types. All
of this creates serious challenges for deriving necessary optimality
conditions. We develop here the method of discrete approximations and combine
it with advanced tools of first-order and second-order variational analysis and
generalized differentiation. This approach allows us to establish constructive
necessary optimality conditions for local minimizers of the controlled sweeping
process expressed entirely in terms of the problem data under fairly
unrestrictive assumptions. As a by-product of the developed approach, we prove
the strong -convergence of optimal solutions of discrete
approximations to a given local minimizer of the continuous-time system and
derive necessary optimality conditions for the discrete counterparts. The
established necessary optimality conditions for the sweeping process are
illustrated by several examples
A comparative assessment of different deviation strategies for dangerous NEO
In this paper a number of deviation strategies for dangerous Near Earth Objects (NEO) have been compared. For each strategy (i.e. Solar Collector, Nuclear Blast, Kinetic Impactor, Low-thrust Propulsion, Mass Driver) a multi criteria optimisation method has been used to reconstruct the set of Pareto optimal solutions minimising the mass of the spacecraft and the warning time, and maximising the deviation. Then, a dominance criterion has been defined and used to compare all the Pareto sets. The achievable deviation at the MOID, either for a low-thrust or for an impulsive variation of the orbit of the NEO, has been computed through a set of analytical formulas. The variation of the orbit of the NEO has been estimated through a deviation action model that takes into account the wet mass of the spacecraft at the Earth. Finally the technology readiness level of each strategy has been used to compute a more realistic value for the required warning time
Response to âComment on âElasticity of flexible and semiflexible polymers with extensible bonds in the Gibbs and Helmholtz ensemblesââ [J. Chem. Phys. 138, 157101 (2013)]
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