409 research outputs found
Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential
Many existence and nonexistence results are known for nonnegative radial
solutions to the equation with nonlinearites satisfying for some . Existence of
nonradial solutions, by contrast, is known only for , ,
and large enough. Here we show that the
equation has multiple nonradial solutions as for , , , and nonlinearities satisfying
suitable assumptions. Our argument essentially relies on the compact embeddings
between some suitable functional spaces of symmetric functions, which yields
the existence of nonnegative solutions of mountain-pass type, and the
separation of the corresponding mountain-pass levels from the energy levels
associated to radial solutions
Fast Quantum Methods for Optimization
Discrete combinatorial optimization consists in finding the optimal
configuration that minimizes a given discrete objective function. An
interpretation of such a function as the energy of a classical system allows us
to reduce the optimization problem into the preparation of a low-temperature
thermal state of the system. Motivated by the quantum annealing method, we
present three strategies to prepare the low-temperature state that exploit
quantum mechanics in remarkable ways. We focus on implementations without
uncontrolled errors induced by the environment. This allows us to rigorously
prove a quantum advantage. The first strategy uses a classical-to-quantum
mapping, where the equilibrium properties of a classical system in spatial
dimensions can be determined from the ground state properties of a quantum
system also in spatial dimensions. We show how such a ground state can be
prepared by means of quantum annealing, including quantum adiabatic evolutions.
This mapping also allows us to unveil some fundamental relations between
simulated and quantum annealing. The second strategy builds upon the first one
and introduces a technique called spectral gap amplification to reduce the time
required to prepare the same quantum state adiabatically. If implemented on a
quantum device that exploits quantum coherence, this strategy leads to a
quadratic improvement in complexity over the well-known bound of the classical
simulated annealing method. The third strategy is not purely adiabatic;
instead, it exploits diabatic processes between the low-energy states of the
corresponding quantum system. For some problems it results in an exponential
speedup (in the oracle model) over the best classical algorithms.Comment: 15 pages (2 figures
Designing experiments using digital fabrication in structural dynamics
In engineering, traditional approaches aimed at teaching concepts of dynamics to engineering students include the study of a dense yet sequential theoretical development of proofs and exercises. Structural dynamics are seldom taught experimentally in laboratories since these facilities should be provided with expensive equipment such as wave generators, data-acquisition systems, and heavily wired deployments with sensors. In this paper, the design of an experimental experience in the classroom based upon digital fabrication and modeling tools related to structural dynamics is presented. In particular, all experimental deployments are conceived with low-cost, open-source equipment. The hardware includes Arduino-based open-source electronics whereas the software is based upon object-oriented open-source codes for the development of physical simulations. The set of experiments and the physical simulations are reproducible and scalable in classroom-based environments.Peer ReviewedPostprint (author's final draft
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