3,735 research outputs found
Radial and cylindrical symmetry of solutions to the Cahn-Hilliard equation
The paper is devoted to the classification of entire solutions to the
Cahn-Hilliard equation in , with particular
interest in those solutions whose nodal set is either bounded or contained in a
cylinder. The aim is to prove either radial or cylindrical symmetry, under
suitable hypothesis
Quantum memories with zero-energy Majorana modes and experimental constraints
In this work we address the problem of realizing a reliable quantum memory
based on zero-energy Majorana modes in the presence of experimental constraints
on the operations aimed at recovering the information. In particular, we
characterize the best recovery operation acting only on the zero-energy
Majorana modes and the memory fidelity that can be therewith achieved. In order
to understand the effect of such restriction, we discuss two examples of noise
models acting on the topological system and compare the amount of information
that can be recovered by accessing either the whole system, or the zero-modes
only, with particular attention to the scaling with the size of the system and
the energy gap. We explicitly discuss the case of a thermal bosonic environment
inducing a parity-preserving Markovian dynamics in which the introduced memory
fidelity decays exponentially in time, independent from system size, thus
showing the impossibility to retrieve the information by acting on the
zero-modes only. We argue, however, that even in the presence of experimental
limitations, the Hamiltonian gap is still beneficial to the storage of
information.Comment: 18 pages, 7 figures. Updated to published versio
Hybrid SAT-Based Consistency Checking Algorithms for Simple Temporal Networks with Decisions
A Simple Temporal Network (STN) consists of time points modeling temporal events and constraints modeling the minimal and maximal temporal distance between them. A Simple Temporal Network with Decisions (STND) extends an STN by adding decision time points to model temporal plans with decisions. A decision time point is a special kind of time point that once executed allows for deciding a truth value for an associated Boolean proposition. Furthermore, STNDs label time points and constraints by conjunctions of literals saying for which scenarios (i.e., complete truth value assignments to the propositions) they are relevant. Thus, an STND models a family of STNs each obtained as a projection of the initial STND onto a scenario. An STND is consistent if there exists a consistent scenario (i.e., a scenario such that the corresponding STN projection is consistent). Recently, a hybrid SAT-based consistency checking algorithm (HSCC) was proposed to check the consistency of an STND. Unfortunately, that approach lacks experimental evaluation and does not allow for the synthesis of all consistent scenarios. In this paper, we propose an incremental HSCC algorithm for STNDs that (i) is faster than the previous one and (ii) allows for the synthesis of all consistent scenarios and related early execution schedules (offline temporal planning). Then, we carry out an experimental evaluation with KAPPA, a tool that we developed for STNDs. Finally, we prove that STNDs and disjunctive temporal networks (DTNs) are equivalent
Periodic solutions to the Cahn-Hilliard equation in the plane
In this paper we construct entire solutions to the Cahn-Hilliard equation
in the Euclidean
plane, where is the standard double-well potential . Such solutions have a non-trivial profile that shadows a Willmore
planar curve, and converge uniformly to as . These
solutions give a counterexample to the counterpart of Gibbons' conjecture for
the fourth-order counterpart of the Allen-Cahn equation. We also study the
-derivative of these solutions using the special structure of Willmore's
equation
Qualitative properties and construction of solutions to some semilinear elliptic PDEs
This thesis is devoted to the study of elliptic equations. On the one hand, we study some qualitative properties, such as symmetry of solutions, on the other hand we explicitly construct some solutions vanishing near some fixed manifold. The main techniques are the moving planes method, in order to investigate the qualitative properties and the Lyapunov-Schmidt reduction
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