Search

In this work, we obtain estimates for the upper bound of gaps between consecutive eigenvalues for the eigenvalue problem of a class of second-order elliptic differential operators in divergent form, with Dirichlet boundary conditions, in a limited domain of n-dimensional Euclidean space. This class of operators includes the well-known Laplacian and the square Cheng-Yau operator. For the Laplacian case, our estimate coincides with that obtained by D. Chen, T. Zheng, and H. Yang, which is the best possible in terms of the order of the eigenvalues. For pinched Cartan-Hadamard manifolds the estimates were made in particular cases of this operator.

Smooth solutions of the forced incompressible Euler equations satisfy an energy balance, where the rate-of-change in time of the kinetic energy equals the work done by the force per unit time. Interesting phenomena such as turbulence are closely linked with rough solutions which may exhibit {\it inviscid dissipation}, or, in other words, for which energy balance does not hold. This article provides a characterization of energy balance for physically realizable weak solutions of the forced incompressible Euler equations, i.e. solutions which are obtained in the limit of vanishing viscosity. More precisely, we show that, in the two-dimensional periodic setting, strong convergence of the zero-viscosity limit is both necessary and sufficient for energy balance of the limiting solution, under suitable conditions on the external force. As a consequence, we prove energy balance for a general class of solutions with initial vorticity belonging to rearrangement-invariant spaces, and going beyond Onsager's critical regularity.

Measurements of the temperature-dependent ac magnetic susceptibility of superconducting films reveal reversible responses, i.e., irrespective of the magnetic and thermal history of the sample. This experimental fact is observed even in the presence of stochastic and certainly irreversible magnetic flux avalanches which, in principle, should randomly affect the results. In this work, we explain such an apparent contradiction by exploring the spatial resolution of magneto-optical imaging. To achieve this, we successfully compare standard frequency-independent first harmonic ac magnetic susceptibility results for a superconducting thin film with those obtained by ac-emulating magneto-optical imaging (acMOI). A quantitative analysis also provides information regarding flux avalanches, reveals the presence of a vortex-antivortex annihilation zone in the region in which a smooth flux front interacts with pre-established avalanches, and demonstrates that the major impact on the flux distribution within the superconductor happens during the first ac cycle. Our results establish acMOI as a reliable approach for studying frequency-independent ac field effects in superconducting thin films while capturing local aspects of flux dynamics, otherwise inaccessible via global magnetometry techniques., Comment: 15 pages, 10 figures

We prove existence of solutions to the Kuramoto-Sivashinsky equation with low-regularity data, in function spaces based on the Wiener algebra and in pseudomeasure spaces. In any spatial dimension, we allow the data to have its antiderivative in the Wiener algebra. In one spatial dimension, we also allow data which is in a pseudomeasure space of negative order. In two spatial dimensions, we also allow data which is in a pseudomeasure space one derivative more regular than in the one-dimensional case. In the course of carrying out the existence arguments, we show a parabolic gain of regularity of the solutions as compared to the data. Subsequently, we show that the solutions are in fact analytic at any positive time in the interval of existence., Comment: 26 pages

The execution of large deep neural networks (DNN) at mobile edge devices requires considerable consumption of critical resources, such as energy, while imposing demands on hardware capabilities. In approaches based on edge computing the execution of the models is offloaded to a compute-capable device positioned at the edge of 5G infrastructures. The main issue of the latter class of approaches is the need to transport information-rich signals over wireless links with limited and time-varying capacity. The recent split computing paradigm attempts to resolve this impasse by distributing the execution of DNN models across the layers of the systems to reduce the amount of data to be transmitted while imposing minimal computing load on mobile devices. In this context, we propose a novel split computing approach based on slimmable ensemble encoders. The key advantage of our design is the ability to adapt computational load and transmitted data size in real-time with minimal overhead and time. This is in contrast with existing approaches, where the same adaptation requires costly context switching and model loading. Moreover, our model outperforms existing solutions in terms of compression efficacy and execution time, especially in the context of weak mobile devices. We present a comprehensive comparison with the most advanced split computing solutions, as well as an experimental evaluation on GPU-less devices.

In this paper, we compute universal inequalities of eigenvalues of a large class of second-order elliptic differential operators in divergence form, that includes, e.g., the Laplace and Cheng-Yau operators, on a bounded domain in a complete Riemannian manifolds isometrically immersed in Euclidean space. A key step in order to obtain the sequence of our estimates is to get the right Yang-type first inequality. We also prove some inequalities for manifolds supporting some special functions and tensors., Comment: 15 pages. arXiv admin note: text overlap with arXiv:2206.09431

We study the three-dimensional incompressible Navier-Stokes equations in a smooth bounded domain $\Omega$ with initial velocity $u_0$ square-integrable, divergence-free and tangent to $\partial \Omega$. We supplement the equations with the Navier friction boundary conditions $u \cdot n = 0$ and $[(2Su)n + \alpha u]_{\rm tang} = 0$, where $n$ is the unit exterior normal to $\partial \Omega$, $Su = (Du + (Du)^t)/2$, $\alpha \in C^0(\partial\Omega)$ is the boundary friction coefficient and $[\cdot]_{\rm tang}$ is the projection of its argument onto the tangent space of $\partial \Omega$. We prove global existence of a weak Leray-type solution to the resulting initial-boundary value problem and exponential decay in energy norm of these solutions when friction is positive. We also prove exponential decay if friction is non-negative and the domain is not a solid of revolution. In addition, in the frictionless case $\alpha = 0$, we prove convergence of the solution to a steady rigid rotation, if the domain is a solid of revolution. We use the Galerkin method for existence, Poincar\'{e}-type inequalities, with suitable adaptations to account for the differential geometry of the boundary, and a global-in-time, integral Gronwall-type inequality.

In this work we determine the consequences of the quantum entanglement of a system of two identical particles when a generalized uncertainty principle (GUP) is considered. GUP's are usually associated with the existence of a minimal length. We focus on the main GUP's (KMM, ADV, Pedram and Nouicer) and then we determine the minimal uncertainties in position induced by those modified GUP's. Our results point out that the minimal uncertainty is reduced by half of its usual value independently of the GUP employed. This implies that the minimal length is also reduced by half. On the other hand, it is generally expected that the minimal length must not depend on physical system. We overcome this apparent paradox by realizing that the entangled system is composed by two particles so that an effective parameter related to the minimal length must be employed., Comment: Accepted for publication in the International Journal of Modern Physics D (IJMPD)

The appearance of dark sunspots over the solar photosphere is not considered to be symmetric between the northern and southern hemispheres. Among the different conclusions obtained by several authors, we can point out that the North-South asymmetry is a real and systematic phenomenon and is not due to random variability. In the present work, we selected the sunspot area data of a sample of 13 solar cycles divided by hemisphere extracted from the Marshall Space Flight Centre (MSFC) database to investigate the behavior of probability distributions using an out-of-equilibrium statistical model a.k.a non-extensive statistical mechanics. Based on this statistical framework, we obtained that the non-extensive entropic parameter $q$ has a semi-sinusoidal variation with a period of $\sim$22 year (Hale cycle). Among the most important results, we can highlight that the asymmetry index $q(A)$ revealed the dominance of the northern hemisphere against the southern one. Thus, we concluded that the parameter $q(A)$ can be considered an effective measure for diagnosing long-term variations of the solar dynamo. Finally, our study opens a new approach to investigating solar variability from the nonextensive perspective., Comment: 17 pages, 2 tables and 5 figures. Submitted to Solar Physics

Astrophysical data, in the domains of time, involve a wide range of stellar variability phenomena, among them the magnetic activity of the order of a few hours until the signature of an extra-solar planet which can cover a scale of time of a few days until tens of years. Numerous instruments are being developed to detect Earth-sized exoplanets. Exoplanets with this dimension challenge scientific instrumentation and the field of research in the data processing. In this context, our study offers a powerful framework to explain dynamical properties as a function of timescale in light curves with the planetary signal. For that, we selected the stellar target Kepler-30 to test our methods and procedures. In this sense, we investigate the multifractal behavior of the Kepler-30 system composed of a sun-like star with a rotation period of ~16 days and three planets with masses between 2 Earth and 2.5 Jupiter masses. Furthermore, this system has an orbital period varying from 29 to 143 days and orbits almost coplanar. This system is highly interesting because starspots dynamics are strongly affected by the passing of a planet in front of the star. We used about 1600 days of high-precision photometry collected by the Kepler mission to investigate the quasi-periodic variation caused by the rotation of the star and the effect of spot evolution as a function of timescale. We applied indexes extract from multifractal analysis to model the flux rotational modulation induced by active regions. Our results that stellar flux variations in Kepler-30 star caused by rotational modulation can be replicated in detail with just four recent-known multifractal indexes. These indexes will greatly simplify spot modelling of current TESS and future PLATO data., Comment: 20 pages, 1 table, 5 figures (submitted to BJD)

In this paper, we obtain eigenvalue estimates for a larger class of elliptic differential operators in divergence form on a bounded domain in a complete Riemannian manifold isometrically immersed in Euclidean space. As an application, we give eigenvalue estimates in the Gaussian shrinking soliton, and we find a domain that makes the behavior of these estimates similar to the estimates for the case of the Laplacian. Moreover, we also give an answer to the generalized conjecture of P\'olya., Comment: Final version which has been accepted for publication in Zeitschrift f\"ur Angewandte Mathematik und Physik

Lei and Lin have recently given a proof of a global mild solution of the three-dimensional Navier-Stokes equations in function spaces based on the Wiener algebra. An alternative proof of existence of these solutions was then developed by Bae, and this new proof allowed for an estimate of the radius of analyticity of the solutions at positive times. We adapt the Bae proof to prove existence of the Lei-Lin solution in the spatially periodic setting, finding an improved bound for the radius of analyticity in this case., Comment: 15 pages