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The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p -Laplace operator, namely: λ 1 (β , Ω) = min ψ ∈ W 1 , p (Ω) ∖ { 0 } ⁡ ∫ Ω F (∇ ψ) p d x + β ∫ ∂ Ω | ψ | p F (ν Ω) d H N − 1 ∫ Ω | ψ | p d x , where p ∈ ] 1 , + ∞ [ , Ω is a bounded, anisotropic mean convex domain in R N , ν Ω is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on R N. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on geometrical quantities associated to Ω. More precisely, we prove a lower bound of λ 1 in the case β > 0 , and a upper bound in the case β < 0. As a consequence, we prove, for β > 0 , a lower bound for λ 1 (β , Ω) in terms of the anisotropic inradius of Ω and, for β < 0 , an upper bound of λ 1 (β , Ω) in terms of β. [ABSTRACT FROM AUTHOR]

Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator between metric spaces X and Y. We study the problem of determining the degree of approximation of such operators on a compact subset K X ⊂ X using a finite amount of information. If F : K X → K Y , a well established strategy to approximate F (F) for some F ∈ K X is to encode F (respectively, F (F)) in terms of a finite number d (respectively m) of real numbers. Together with appropriate reconstruction algorithms (decoders), the problem reduces to the approximation of m functions on a compact subset of a high dimensional Euclidean space R d , equivalently, the unit sphere S d embedded in R d + 1. The problem is challenging because d , m , as well as the complexity of the approximation on S d are all large, and it is necessary to estimate the accuracy keeping track of the inter-dependence of all the approximations involved. In this paper, we establish constructive methods to do this efficiently; i.e., with the constants involved in the estimates on the approximation on S d being O (d 1 / 6). We study different smoothness classes for the operators, and also propose a method for approximation of F (F) using only information in a small neighborhood of F , resulting in an effective reduction in the number of parameters involved. To further mitigate the problem of large number of parameters, we propose prefabricated networks, resulting in a substantially smaller number of effective parameters. The problem is studied in both deterministic and probabilistic settings. [ABSTRACT FROM AUTHOR]

The fractional heat operator (∂ t − Δ x) s and Continuous Time Random Walks (CTRWs) are interesting and sophisticated mathematical models that can describe complex anomalous systems. In this paper, we prove asymptotic mean value representation formulas for functions with respect to (∂ t − Δ x) s and we introduce new nonlocal, nonlinear parabolic operators related to a tug-of-war which accounts for waiting times and space-time couplings. These nonlocal, nonlinear parabolic operators and equations can be seen as nonlocal versions of the evolutionary infinity Laplace operator. [ABSTRACT FROM AUTHOR]

This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate solutions. A delicate issue for constructing local-energy solutions resides in the identification of weak limits of nonlinear terms for approximate solutions in a limiting procedure. Indeed, such an identification process often needs the maximal monotonicity of nonlinear elliptic operators (involved in the doubly-nonlinear equations) as well as uniform estimates for approximate solutions; however, even the monotonicity is violated due to a localization of the equations, which is also necessary to derive local-energy estimates for approximate solutions. In the present paper, such an inconsistency is systematically overcome by reducing the original equation to a localized one, where a (no longer monotone) localized elliptic operator is decomposed into the sum of a maximal monotone operator and a perturbation, and by integrating all the other relevant processes. Furthermore, the general framework developed in the present paper is also applied to the Finsler porous medium and fast diffusion equations , which are variants of the classical PME and FDE and also classified as a doubly-nonlinear equation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel projected steepest descent iterative method with frozen derivative. The classical projected steepest descent iterative method involves the computation of derivative of the nonlinear operator at each iterate. The method of this paper requires the computation of derivative of the nonlinear operator only at an initial point. We exhibit the convergence analysis of our method by assuming the conditional stability of the inverse problem on a convex and compact set. Further, by assuming the conditional stability on a nested family of convex and compact subsets, we develop a multi-level method. In order to enhance the accuracy of approximation between neighboring levels, we couple it with the growth of stability constants. This along with a suitable discrepancy criterion ensures that the algorithm proceeds from level to level and terminates within finite steps. Finally, we discuss an inverse problem on which our methods are applicable. [ABSTRACT FROM AUTHOR]

We establish theorems on the existence and compactness of solutions to the σ 2 -Nirenberg problem on the standard sphere S 2. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Möbius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Möbius invariant degenerate elliptic equations, a priori estimates of first and second order derivatives of solutions to the σ 2 -Nirenberg problem, and a Bôcher type theorem for the associated fully nonlinear Möbius invariant elliptic equations. Given these results, we are able to complete a fine analysis of a sequence of blow-up solutions to the σ 2 -Nirenberg problem. In particular, we prove that there can be at most one blow-up point for such a blow-up sequence of solutions. This, together with a Kazdan-Warner type identity, allows us to prove L ∞ a priori estimates for solutions of the σ 2 -Nirenberg problem under some simple generic hypothesis. The higher derivative estimates then follow from classical estimates of Nirenberg and Schauder. In turn, the existence of solutions to the σ 2 -Nirenberg problem is obtained by an application of the by now standard degree theory for second order fully nonlinear elliptic operators. [ABSTRACT FROM AUTHOR]

• We propose a 5G NR- multiband for Radio over Fiber link outfitted with an optimized magnitude selective affine (OMSA) method to cover enhanced mobile broadband (eMBB) situations for 2.14 GHz and 10 GHz, correspondingly. • We show experimentally that OMSA further reduces the complexity with better or at least similar performance to that of MSA by using a weighted function as a nonlinear operator to accurately capture the nonlinear behaviour of the system being modelled, which can lead to a reduction in the number of segments and improvement in the accuracy of the model. This means that it will also lead to a reduction in the number of coefficients needed. This helps to make the model more efficient and simpler to use. • The experimental results demonstrate that the OMSA model has lower complexity than the MSA model while achieving a slightly better performance that is assessed in terms of computations, error vector magnitude (EVM) or adjacent channel power ratio (ACPR). The application of analog radio over fiber (A-RoF) systems for 5G new radio (NR) multiband waveforms presents challenges, including nonlinearity introduced by the Mach Zehnder Modulator and the dispersion and attenuation of the optical signal over long fiber lengths. To address these challenges, this paper proposes a new version of the magnitude-selective affine (MSA) model, named the optimized magnitude-selective affine (OMSA) model, which incorporates a power-reliant weighting function to improve performance while reducing complexity via Digital Predistortion (DPD). The OMSA model is tested using 5G NR signals at 10 GHz with 50 MHz and flexible-waveform signals at 2.14 GHz with a 20 MHz bandwidth, transmitted over a 10 km fiber length using a Mach Zehnder Modulator and a 1550 nm optical carrier. The performance of the OMSA model is compared to the MSA and generalized memory polynomial (GMP) models in terms of adjacent channel power ratio, error vector magnitude, and complexity. The results show that the OMSA model outperforms the MSA and GMP models in terms of performance reducing the error vector magnitude to 1.9% as compared to 4% and 3% in case of GMP and MSA respectively. Additionally, in terms of complexity measured via coefficients, OMSA is 168 times more efficient as compared to GMP and offers 1.93 times higher efficiency as compared to MSA while maintaining lower complexity. The experimental demonstration of 5G NR multiband waveforms over A-RoF links using the OMSA model represents a significant step towards the practical implementation of high-performance, low-complexity 5G NR systems over fiber-optic links and provides a promising solution to the challenges associated with DPD linearization for A-RoF systems. [ABSTRACT FROM AUTHOR]

In this paper, we develop a systematical approach in applying the method of moving planes to study qualitative properties of solutions for nonlocal parabolic equations. We first establish a series of key ingredients in the proofs, such as maximum principles for antisymmetric functions, narrow region principles, maximum principles near infinity, and the Hopf lemma, then we demonstrate how these new tools can be employed to derive the radial symmetry and monotonicity of ancient positive solutions to fractional parabolic equations in the whole space. Our results are in particular applicable to entire solutions, by which we mean solutions defined for all t ∈ R. We believe that the new ideas and methods introduced here can be adapted to study many other nonlocal parabolic problems with more general operators and nonlinear terms. [ABSTRACT FROM AUTHOR]

• Derived a new class of nonlinearly stable flux reconstruction schemes in split form. • Nonlinearly stable flux reconstruction correction functions must be applied on the nonlinear volume terms. • Flux reconstruction in split form with the correction functions only reconstructing the surface is divergent. • The split form must be embedded within the discrete stiffness operator for nonlinear stability. The flux reconstruction (FR) method has gained popularity in the research community as it recovers promising high-order methods through modally filtered correction fields, such as the discontinuous Galerkin method, amongst others, on unstructured grids over complex geometries. Moreover, FR schemes, specifically energy stable FR (ESFR) schemes also known as Vincent-Castonguay-Jameson-Huynh schemes, have proven attractive as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, split forms have recently seen a resurgence in research activity due to their resultant nonlinear (entropy) stability proofs. This paper derives for the first time nonlinearly stable ESFR schemes in split form that enable nonlinear stability proofs for, uncollocated, modal, ESFR split forms with different volume and surface cubature nodes. The critical enabling technology is applying the splitting to the discrete stiffness operator. This naturally leads to appropriate surface and numerical fluxes, enabling both entropy stability and conservation proofs. When these schemes are recast in strong form, they differ from schemes found in the ESFR literature as the ESFR correction functions are incorporated on the volume integral. Furthermore, numerical experiments are conducted for Burgers' equation verifying that the new class of proposed ESFR split forms is nonlinearly stable in contrast to the standard split form ESFR approach. Lastly, the new ESFR split form is shown to obtain the correct orders of accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following problem { (− Δ) p s u (x) = f (u) , u (x) > 0 in Ω , u (x) = 0 on R n ∖ Ω , where (− Δ) p s is the fractional p -Laplacian with p ≥ 2 and Ω : = { x = (x ′ , x n) | x n > φ (x ′) } is an unbounded domain. In the case φ ≡ 0 , it reduces to the upper half space. Without assuming any asymptotic behavior of u near infinity, we first develop narrow region principles in unbounded domains, then using the method of moving planes, we establish the monotonicity of the positive solutions. In most previous literature, to apply the method of moving planes on unbounded domains, one usually needed to assume that the solution tends to zero in certain rate near infinity; or make a Kelvin transform, or divide the solution by a function, so that the new solution possesses such an asymptotic decay. Here we introduce a new idea, estimating the singular integral defining (− Δ) p s along a sequence of auxiliary functions at their maximum points. This way, we only require the solutions be bounded. We believe that this new method will become a useful tool in investigating qualitative properties of solutions for equations involving nonlinear nonlocal operators. [ABSTRACT FROM AUTHOR]