Emergent $${{{{{{{\mathcal{PT}}}}}}}}$$-symmetry breaking of collective modes with topological critical phenomena

The spontaneous breaking of parity-time ($${{{{{{{\mathcal{PT}}}}}}}}$$ PT ) symmetry yields rich critical behavior in non-Hermitian systems, and has stimulated much interest, albeit most previous studies were performed within the single-particle or mean-field framework. Here, by studying the collective excitations of a Fermi superfluid with $${{{{{{{\mathcal{PT}}}}}}}}$$ PT -symmetric spin-orbit coupling, we uncover an emergent $${{{{{{{\mathcal{PT}}}}}}}}$$ PT -symmetry breaking in the Anderson-Bogoliubov (AB) collective modes, even as the superfluid ground state retains an unbroken $${{{{{{{\mathcal{PT}}}}}}}}$$ PT symmetry. The critical point of the transition is marked by a non-analytic kink in the speed of sound, which derives from the coalescence and annihilation of the AB mode and its hole partner, reminiscent of the particle-antiparticle annihilation. The system consequently becomes immune to low-frequency external perturbations at the critical point, a phenomenon associated with the spectral topology of the complex quasiparticle dispersion. This critical phenomenon offers a fascinating route toward perturbation-free quantum states.

EEG-based emotion recognition with deep convolutional neural networks

The emotional state of people plays a key role in physiological and behavioral human interaction. Emotional state analysis entails many fields such as neuroscience, cognitive sciences, and biomedical engineering because the parameters of interest contain the complex neuronal activities of the brain. Electroencephalogram (EEG) signals are processed to communicate brain signals with external systems and make predictions over emotional states. This paper proposes a novel method for emotion recognition based on deep convolutional neural networks (CNNs) that are used to classify Valence, Arousal, Dominance, and Liking emotional states. Hence, a novel approach is proposed for emotion recognition with time series of multi-channel EEG signals from a Database for Emotion Analysis and Using Physiological Signals (DEAP). We propose a new approach to emotional state estimation utilizing CNN-based classification of multi-spectral topology images obtained from EEG signals. In contrast to most of the EEG-based approaches that eliminate spatial information of EEG signals, converting EEG signals into a sequence of multi-spectral topology images, temporal, spectral, and spatial information of EEG signals are preserved. The deep recurrent convolutional network is trained to learn important representations from a sequence of three-channel topographical images. We have achieved test accuracy of 90.62% for negative and positive Valence, 86.13% for high and low Arousal, 88.48% for high and low Dominance, and finally 86.23% for like–unlike. The evaluations of this method on emotion recognition problem revealed significant improvements in the classification accuracy when compared with other studies using deep neural networks (DNNs) and one-dimensional CNNs.

Spectral primal spaces

Let [Formula: see text] be a mapping. Consider [Formula: see text] Then, according to Echi, [Formula: see text] is an Alexandroff topology. A topological space [Formula: see text] is called a primal space if its topology coincides with an [Formula: see text] for some mapping [Formula: see text]. We denote by [Formula: see text] the set of all fixed points of [Formula: see text], and [Formula: see text] the set of all periodic points of [Formula: see text]. The topology [Formula: see text] induces a preorder [Formula: see text] defined on [Formula: see text] by: [Formula: see text] if and only if [Formula: see text], for some integer [Formula: see text]. The main purpose of this paper is to provide necessary and sufficient algebraic conditions on the function [Formula: see text] in order to get [Formula: see text] (respectively, the one-point compactification of [Formula: see text]) a spectral topology. More precisely, we show the following results. (1) [Formula: see text] is spectral if and only if [Formula: see text] is a finite set and every chain in the ordered set [Formula: see text] is finite. (2) The one-point(Alexandroff) compactification of [Formula: see text] is a spectral topology if and only if [Formula: see text] and every nonempty chain of [Formula: see text] has a least element. (3) The poset [Formula: see text] is spectral if and only if every chain is finite. As an application the main theorem [12, Theorem 3. 5] of Echi–Naimi may be derived immediately from the general setting of the above results.

Some results in types of extensions of MV-algebras

We introduce the notions of zero divisor and extension, contraction of ideals in MV-algebras and several interesting types of extensions of MV-algebras. In particular, we show what kinds of extensions MV-algebras will lead in a homeomorphism of the spectral topology and inverse topology on minimal prime ideals. Finally, we investigate the relations among types of extensions of MV-algebras.

Spectral Topology on MV-Modules

In this paper, the spectral topology and quasi-spectral topology of proper prime A-ideals in an MV-module are introduced. We show that the spectral topology of proper ⋅-prime ideals of a PMV-algebra with unity for product, is the same as the spectral topology of proper prime ideals in an MV-algebra. Also we show that the set of all prime A-ideals in an MV-module with spectral topology is not T0 and T1 topological spaces but quasi-spectral topology is T0-space and is not T1-space. Finally, we investigate when the set of all prime A-ideals in an MV-module are Hausdorff and disconnected.