Invariant vector means and complementability of Banach spaces in their second duals

Let X be a Banach space. Fix a torsion-free commutative and cancellative semigroup S whose torsion-free rank is the same as the density of $$X^{**}$$ X ∗ ∗ . We then show that X is complemented in $$X^{**}$$ X ∗ ∗ if and only if there exists an invariant mean $$M:\ell _\infty (S,X)\rightarrow X$$ M : ℓ ∞ ( S , X ) → X . This improves upon previous results due to Bustos Domecq (J Math Anal Appl 275(2):512–520, 2002), Kania (J Math Anal Appl 445:797–802, 2017), Goucher and Kania (Studia Math 260:91–101, 2021).

A multichannel learning-based approach for sound source separation in reverberant environments

In this paper, a multichannel learning-based network is proposed for sound source separation in reverberant field. The network can be divided into two parts according to the training strategies. In the first stage, time-dilated convolutional blocks are trained to estimate the array weights for beamforming the multichannel microphone signals. Next, the output of the network is processed by a weight-and-sum operation that is reformulated to handle real-valued data in the frequency domain. In the second stage, a U-net model is concatenated to the beamforming network to serve as a non-linear mapping filter for joint separation and dereverberation. The scale invariant mean square error (SI-MSE) that is a frequency-domain modification from the scale invariant signal-to-noise ratio (SI-SNR) is used as the objective function for training. Furthermore, the combined network is also trained with the speech segments filtered by a great variety of room impulse responses. Simulations are conducted for comprehensive multisource scenarios of various subtending angles of sources and reverberation times. The proposed network is compared with several baseline approaches in terms of objective evaluation matrices. The results have demonstrated the excellent performance of the proposed network in dereverberation and separation, as compared to baseline methods.

Complementary means with respect to a nonsymmetric invariant mean

It is known that if a bivariate mean K is symmetric, continuous and strictly increasing in each variable, then for every mean M there is a unique mean $$N\,$$ N such that K is invariant with respect to the mean-type mapping $$\left( M,N\right) ,$$ M , N , which means that $$K\circ \left( M,N\right) =K$$ K ∘ M , N = K and N is called a K-complementary mean for M (Matkowski in Aequ Math 57(1):87–107, 1999). This paper extends this result for a large class of nonsymmetric means. As an application, the limits of the sequences of iterates of the related mean-type mappings are determined, which allows us to find all continuous solutions of some functional equations.

Wave drag in oscillatory mean flows

<p>In both the atmosphere and ocean, large-scale (mean) flows over topography generate internal waves. A longstanding question in both fields is what forces – often known as ‘wave drag’ – are exerted on the mean flow in this process, as such forces must be parameterized in non-wave-resolving numerical models. For a time-invariant mean flow, it is well known that lee waves are generated which extract momentum from the solid earth and deposit it where they break and dissipate at height. Here, I address the equivalent problem for a time-periodic mean flow (e.g. the ocean tide) using theory and numerical simulations. In this situation, the waves influence the amplitude and phase of the periodic mean flow near the topography regardless of where they dissipate. Dissipation plays a role in terms of controlling the magnitude of wave reflections from an upper boundary (e.g. the ocean surface) which modifies the forces acting near the topography. Our results form a framework for parameterizing tidal internal wave drag in global ocean models.</p>

Mortality Forecasting with an Age-Coherent Sparse VAR Model

This paper proposes an age-coherent sparse Vector Autoregression mortality model, which combines the appealing features of existing VAR-based mortality models, to forecast future mortality rates. In particular, the proposed model utilizes a data-driven method to determine the autoregressive coefficient matrix, and then employs a rotation algorithm in the projection phase to generate age-coherent mortality forecasts. In the estimation phase, the age-specific mortality improvement rates are fitted to a VAR model with dimension reduction algorithms such as the elastic net. In the projection phase, the projected mortality improvement rates are assumed to follow a short-term fluctuation component and a long-term force of decay, and will eventually converge to an age-invariant mean in expectation. The age-invariance of the long-term mean guarantees age-coherent mortality projections. The proposed model is generalized to multi-population context in a computationally efficient manner. Using single-age, uni-sex mortality data of the UK and France, we show that the proposed model is able to generate more reasonable long-term projections, as well as more accurate short-term out-of-sample forecasts than popular existing mortality models under various settings. Therefore, the proposed model is expected to be an appealing alternative to existing mortality models in insurance and demographic analyses.

Invariant Means, Complementary Averages of Means, and a Characterization of the Beta-Type Means

We prove that whenever the selfmapping (M1,…,Mp):Ip→Ip, (p∈N and Mi-s are p-variable means on the interval I) is invariant with respect to some continuous and strictly monotone mean K:Ip→I then for every nonempty subset S⊆{1,…,p} there exists a uniquely determined mean KS:Ip→I such that the mean-type mapping (N1,…,Np):Ip→Ip is K-invariant, where Ni:=KS for i∈S and Ni:=Mi otherwise. Moreover min(Mi:i∈S)≤KS≤max(Mi:i∈S). Later we use this result to: (1) construct a broad family of K-invariant mean-type mappings, (2) solve functional equations of invariant-type, and (3) characterize Beta-type means.

A new characterization of the Haagerup property by actions on infinite measure spaces

The aim of the article is to provide a characterization of the Haagerup property for locally compact, second countable groups in terms of actions on $\unicode[STIX]{x1D70E}$ -finite measure spaces. It is inspired by the very first definition of amenability, namely the existence of an invariant mean on the algebra of essentially bounded, measurable functions on the group.