House mice (Mus musculus), like other rodents, communicate using sonic and ultrasonic vocalizations (USVs), but their functions are still poorly understood. One of the main challenges for studying any acoustic communication is processing and analyzing audio files. Our aims here are to provide a critical and comprehensive review of the new bioacoustic tools available for processing and analyzing recordings of mouse vocalizations. We consider each method as used in a serial data processing pipeline and how to minimize errors at each step to prevent error propagation (or error cascades). First, we review methods for processing audio files of recordings of mice. We compare conventional approaches for visualizing vocalizations (time-frequency representations) with an alternative method adapted to the mouse auditory system. We compare machine learning (ML) and signal processing methods for automating USV detection and emphasize the need for better methods for denoising audio files and reliable frequency contour (ridge) tracking and feature extraction. Second, we review methods for analyzing detected USVs, focusing on classification and sequencing approaches. Classifying USVs is a challenging task because, while some calls are discrete, others show graded variation within and between call classes. We compare supervised classifications and unsupervised labeling, and we emphasize the importance of reliable manual (researcher-based) methods as a gold standard for automated ML approaches. We review classifications of mouse vocalizations in the literature, and we propose a new hierarchical framework for the classification of USVs. We examine methods for sequencing USVs and consider their relative advantages. Finally, we address the unresolved technological challenges for these methods to study rodent vocalizations and propose potential solutions for the future.

In this paper, we investigate whether the tensor product of two frames, each individually localised with respect to a spectral matrix algebra, is also localised with respect to a suitably chosen tensor product algebra. We provide a partial answer by constructing an involutive Banach algebra of rank-four tensors that is built from two solid spectral matrix algebras. We show that this algebra is inverse-closed, given that the original algebras satisfy a specific property related to operator-valued versions of these algebras. This condition is satisfied by all commonly used solid spectral matrix algebras. We then prove that the tensor product of two self-localised frames remains self-localised with respect to our newly constructed tensor algebra. Additionally, we discuss generalisations to localised frames of Hilbert-Schmidt operators, which may not necessarily consist of rank-one operators.

We introduce a localization concept for operator-valued frames, where the quality of localization is measured by the associated operator-valued Gram matrix belonging to some suitable Banach algebra. We prove that intrinsic localization of an operator-valued frame is preserved by its canonical dual. Moreover, we show that the series associated to the perfect reconstruction of an operator-valued frame converges not only in the underlying Hilbert space, but also in a whole class of associated (quasi-)Banach spaces which are carefully introduced and analyzed. Finally, we apply our results to irregular Gabor g-frames.

In this paper, we ask when it is possible to transform a given sequence into a frame or a lower semi frame by multiplying the elements by numbers. In other words, we ask when a given sequence is a weighted frame or a weighted lower semi frame and for each case we formulate a conjecture. We determine several conditions under which these conjectures are true. Finally, we prove an equivalence between two older conjectures. The one being that any unconditionally convergent multiplier can be written as a multiplier of Bessel sequences by shifting weights between the generating sequences of the multiplier. The second one that every unconditionally convergent multiplier which is invertible can be written as a multiplier of frames by a similar shift of weights. We also show that these conjectures are also related to one of the newly posed conjectures.