A novel algorithm for the computation of the quadratic numerical range is presented and exemplified yielding much better results in less time compared to the random vector sampling method. Furthermore, a bound on the probability for the random vector sampling method to produce a point exceeding a neighborhood of the expectation value in dependence on norm and size of the matrix is given.

As Part II of a three-part tutorial on holographic multiple-input multiple-output (HMIMO), this Letter focuses on the state-of-the-art in performance analysis and on holographic beamforming for HMIMO communications. We commence by discussing the spatial degrees of freedom (DoF) and ergodic capacity of a point-to-point HMIMO system, based on the channel model presented in Part I. Additionally, we also consider the sum-rate analysis of multi-user HMIMO systems. Moreover, we review the recent progress in holographic beamforming techniques developed for various HMIMO scenarios. Finally, we evaluate both the spatial DoF and the channel capacity through numerical simulations.

General rational solutions for the nonlocal resonant nonlinear Schrodinger equations are derived by using the Hirota bilinear method and the KP hierarchy reduction method. These rational solutions are presented in terms of determinants in which the elements are algebraic expressions. A weaker condition is given for KP reduction in the nonlocal case. The dynamics of first-order solutions are investigated in details. As a special case, we studied rantional solutions of the RNLS equation. Moreover, soliton solutions of the RNLS equation are given by using the Backlund transformation and nonlinear superposition formula.

This study builds on a recent paper by Lai et al [Appl. Comput. Harmon. Anal., 2018] in which a novel boundary integral formulation is presented for scalar wave scattering analysis in two-dimensional layered and half-spaces. The seminal paper proposes a hybrid integral representation that combines the Sommerfeld integral and layer potential to efficiently deal with the boundaries of infinite length. In this work, we modify the integral formulation to eliminate the fictitious eigenvalues by employing Burton-Miller's approach. We also discuss reasonable parameter settings for the hybrid integral equation to ensure efficient and accurate numerical solutions. Furthermore, we extend the modified formulation for the scattering from a cavity in a half-space whose boundary is locally perturbed. To address the cavity scattering, we introduce a virtual boundary enclosing the cavity and couple the integral equation on it with the hybrid equation. The effectiveness of the proposed method is demonstrated through numerical examples.

We characterise high-dimensional topology that arises from a random Cech complex constructed on the circle. Expected Euler characteristic curve is computed, where we observe limiting spikes. The spikes correspond to expected Betti numbers growing arbitrarily large over shrinking intervals of filtration radii. Using the fact that the homotopy type of the random Cech complex is either an odd-dimensional sphere or a bouquet of even-dimensional spheres, we give probabilistic bounds of the homotopy types. By departing from the conventional practice of scaling down filtration radii as the sample size grows large, our findings indicate that the full breadth of filtration radii leads to interesting systematic behaviour that cannot be regarded as "topological noise".

Given an action of a discrete countable group $G$ on a countable set $\mathfrak{X}$, it is studied the relationship between properties of the associated Calkin representation and the dynamics of the group action on the boundary of the Stone-\v{C}ech compactification of $\mathfrak{X}$. The first section contains results about amenability properties of actions of discrete countable groups on non-separable spaces and is of independent interest. In the second section these results are applied in order to translate regularity properties of the Calkin representation and the topological amenability on the Stone-{\v C}ech boundary within the common framework of measurable dynamics on certain extensions of the Stone-{\v C}ech boundary of $\mathfrak{X}$.

Ensembling is among the most popular tools in machine learning (ML) due to its effectiveness in minimizing variance and thus improving generalization. Most ensembling methods for black-box base learners fall under the umbrella of "stacked generalization," namely training an ML algorithm that takes the inferences from the base learners as input. While stacking has been widely applied in practice, its theoretical properties are poorly understood. In this paper, we prove a novel result, showing that choosing the best stacked generalization from a (finite or finite-dimensional) family of stacked generalizations based on cross-validated performance does not perform "much worse" than the oracle best. Our result strengthens and significantly extends the results in Van der Laan et al. (2007). Inspired by the theoretical analysis, we further propose a particular family of stacked generalizations in the context of probabilistic forecasting, each one with a different sensitivity for how much the ensemble weights are allowed to vary across items, timestamps in the forecast horizon, and quantiles. Experimental results demonstrate the performance gain of the proposed method.

A manifestation of the black hole information loss problem is that the two-point function of probe operators in a large Anti-de Sitter black hole decays in time, whereas, on the boundary CFT, it is expected to be an almost periodic function of time. We point out that the decay of the two-point function (clustering in time) holds important clues to the nature of observable algebras, states, and dynamics in quantum gravity. We call operators that cluster in time "mixing" and explore the necessary and sufficient conditions for mixing. The information loss problem is a special case of the statement that in type I algebras, there exists no mixing operators. We prove that, in a thermofield double (KMS state), if mixing operators form an algebra (close under multiplication) the resulting algebra must be a von Neumann type III$_1$ factor. In other words, the physically intuitive requirement that all non-conserved operators should diffuse is so strong that it fixes the observable algebra to be an exotic algebra called a type III$_1$ factor. More generally, for an arbitrary out-of-equilibrium state of a general quantum system (von Neumann algebra), we show that if the set of operators that mix under modular flow forms an algebra it is a type III$_1$ von Neumann factor. In a theory of Generalized Free Fields (GFF), we show that if the two-point function of GFF clusters in time all operators are mixing, and the algebra is a type III$_1$ factor. For instance, in $\mathscr{N=4}$ SYM, above the Hawking-Page phase transition, clustering of the single trace operators implies that the algebra is a type III$_1$ factor, settling a recent conjecture of Leutheusser and Liu. We explicitly construct the C$^*$-algebra and von Neumann subalgebras of GFF associated with time bands and more generally, open sets of the bulk spacetime using the HKLL reconstruction map.

This paper studies the online node classification problem under a transductive learning setting. Current methods either invert a graph kernel matrix with $\mathcal{O}(n^3)$ runtime and $\mathcal{O}(n^2)$ space complexity or sample a large volume of random spanning trees, thus are difficult to scale to large graphs. In this work, we propose an improvement based on the \textit{online relaxation} technique introduced by a series of works (Rakhlin et al.,2012; Rakhlin and Sridharan, 2015; 2017). We first prove an effective regret $\mathcal{O}(\sqrt{n^{1+\gamma}})$ when suitable parameterized graph kernels are chosen, then propose an approximate algorithm FastONL enjoying $\mathcal{O}(k\sqrt{n^{1+\gamma}})$ regret based on this relaxation. The key of FastONL is a \textit{generalized local push} method that effectively approximates inverse matrix columns and applies to a series of popular kernels. Furthermore, the per-prediction cost is $\mathcal{O}(\text{vol}({\mathcal{S}})\log 1/\epsilon)$ locally dependent on the graph with linear memory cost. Experiments show that our scalable method enjoys a better tradeoff between local and global consistency.

It is proved that the Weisfeiler-Leman dimension of the class of permutation graphs is at most 18. Previously it was only known that this dimension is finite (Gru{\ss}ien, 2017).