1,675 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Mixed formulation for the computation of Miura surfaces with Dirichlet boundary conditions

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    Miura surfaces are the solutions of a constrained nonlinear elliptic system of equations. This system is derived by homogenization from the Miura fold, which is a type of origami fold with multiple applications in engineering. A previous inquiry, gave suboptimal conditions for existence of solutions and proposed an H2H^2-conformal finite element method to approximate them. In this paper, the existence of Miura surfaces is studied using a mixed formulation. It is also proved that the constraints propagate from the boundary to the interior of the domain for well-chosen boundary conditions. Then, a numerical method based on a least-squares formulation, Taylor--Hood finite elements and a Newton method is introduced to approximate Miura surfaces. The numerical method is proved to converge at order one in space and numerical tests are performed to demonstrate its robustness

    Multiple solutions for the fractional p-Laplacian with jumping reactions

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    We study a nonlinear elliptic equation driven by the degenerate fractional p-Laplacian, with Dirichlet-type condition and a jumping reaction, i.e., (p − 1)-linear both at infinity and at zero but with different slopes crossing the principal eigenvalue. Under two different sets of hypotheses, entailing different types of asymmetry, we prove the existence of at least two nontrivial solutions. Our method is based on degree theory for monotone operators and nonlinear fractional spectral theory

    Věty o pevných bodech a jejich aplikace

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    Suppose that we have some set XX and a mapping T:XXT:X\to X, we call xXx\in X a fixed point of TT if x=T(x)x=T(x). In the first part of this thesis we will be discussing conditions under which such a fixed point exists and in the second part we will introduce several applications for these fixed point theorems.Předpokládejme že máme nějakou množinu XX a zobrazení T:XXT:X\to X, bod xXx\in X nazveme pevným bodem zobrazení TT pokud x=T(x)x=T(x). V první částí této práce se budeme zabývat podmínkami, které zaručují, že tento pevný bod existuje. V druhé části předvedeme některé aplikace těchto vět o pevných bodech.470 - Katedra aplikované matematikyvýborn

    Knots and Chaos in the R\"ossler System

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    The R\"ossler System is one of the best known chaotic dynamical systems, exhibiting a plethora of complex phenomena - and yet, only a few studies tackled its complexity analytically. In this paper we find sufficient conditions for the existence of chaotic dynamics for the R\"ossler System. This will allow us to prove the existence of infinitely many periodic trajectories for the flow, and determine their knot types.Comment: 58 pages, 56 figure

    Solvability of nonlinear elliptic boundary value problems

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    This dissertation focuses on the study of steady states of reaction diffusion problems that are motivated by applications. In particular, we focus on elliptic boundary value problems where the nonlinear reaction may appear in the interior or on the boundary of a domain in the Euclidean space. First, we study linear elliptic problems with nonlinear reaction on the boundary. In this case, we establish the existence of maximal and minimal solutions for both monotone and non monotone cases. We then extend these results to the systems case. Next, we prove the existence, nonexistence, multiplicity and global bifurcation results of positive solutions of superlinear problems. To support our analytical results we numerically approximate solutions using finite difference methods including existence and stability analysis. Second, we study problems that are nonlinear inside the domain and linear on the boundary in the context of a model arising in mathematical ecology. To begin with we perform computational simulations for the problem in the one dimensional setting. Then, motivated by the bifurcation diagrams that are obtained, we prove several analytical results such as existence, uniqueness and nonexistence

    Efficient Symbolic Reasoning for Neural-Network Verification

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    The neural network has become an integral part of modern software systems. However, they still suffer from various problems, in particular, vulnerability to adversarial attacks. In this work, we present a novel program reasoning framework for neural-network verification, which we refer to as symbolic reasoning. The key components of our framework are the use of the symbolic domain and the quadratic relation. The symbolic domain has very flexible semantics, and the quadratic relation is quite expressive. They allow us to encode many verification problems for neural networks as quadratic programs. Our scheme then relaxes the quadratic programs to semidefinite programs, which can be efficiently solved. This framework allows us to verify various neural-network properties under different scenarios, especially those that appear challenging for non-symbolic domains. Moreover, it introduces new representations and perspectives for the verification tasks. We believe that our framework can bring new theoretical insights and practical tools to verification problems for neural networks

    Observation of Josephson Harmonics in Tunnel Junctions

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    Superconducting quantum processors have a long road ahead to reach fault-tolerant quantum computing. One of the most daunting challenges is taming the numerous microscopic degrees of freedom ubiquitous in solid-state devices. State-of-the-art technologies, including the world's largest quantum processors, employ aluminum oxide (AlOx_x) tunnel Josephson junctions (JJs) as sources of nonlinearity, assuming an idealized pure sinφ\sin\varphi current-phase relation (Cφ\varphiR). However, this celebrated sinφ\sin\varphi Cφ\varphiR is only expected to occur in the limit of vanishingly low-transparency channels in the AlOx_x barrier. Here we show that the standard Cφ\varphiR fails to accurately describe the energy spectra of transmon artificial atoms across various samples and laboratories. Instead, a mesoscopic model of tunneling through an inhomogeneous AlOx_x barrier predicts %-level contributions from higher Josephson harmonics. By including these in the transmon Hamiltonian, we obtain orders of magnitude better agreement between the computed and measured energy spectra. The reality of Josephson harmonics transforms qubit design and prompts a reevaluation of models for quantum gates and readout, parametric amplification and mixing, Floquet qubits, protected Josephson qubits, etc. As an example, we show that engineered Josephson harmonics can reduce the charge dispersion and the associated errors in transmon qubits by an order of magnitude, while preserving anharmonicity
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