pseudo-Riemannian space, conformal Fedosov structure, recurrent space, symmetric space, quantum key distribution, single-photon regime, special relativity, fuzzy space, additive quantum number, conservation laws, electroweak theory, symmetry, unified field theories, Lie-Poisson algebra, quantization, matrix regularization, noncommutative geometry

From the standpoint of the History of Mathematics, beginning with pioneering work of Hurwitz on invariant integrals (or invariant measures) on Lie groups, we pick up epoch-making works successively and draw the main stream among so many contributions to the study of invariant integrals on groups, due to Hurwitz, Schur, Weyl, Haar, Neumann, Kakutani, Weil, and Kakutani-Kodaira, and explain their contents and give the relationships among them.

In this paper, we study the (1+1)-dimensional Schrödinger-Maxwell-Bloch equation (NLS-MBE) which is integrable by the Inverse Scattering Method. Its Lax pair is presented. We apply methods of the theory of integrable systems to the geometry of surfaces immersed in Euclidean spaces. Using Sym-Tafel formula we construct the first and second fundamental forms, the Gaussian (total) curvature, mean curvature and Christoffel symbols for the NLS-MBE.

The present paper aims to introduce a new algebra based on the knots and their ability to change dramatically the physical state, when we act on them with the braiding operator. The frame of all this ideas is the topological quantum computing. Nowadays, the quest for a reliable quantum computer technology refers, among others quantum phenomena, to the Quantum Hall Effect. Setting suitable quantum experiments (for instance a super lattice - two dimensional electron system that appears in the hetero structures of two semiconductors) involving exceptionally strong magnetic fields and very low temperatures, the QHE puts into evidence new quasi particles, evolving in 2+1 dimensions space. Such a topological quantum computer will successfully fight against the undesirable interaction between the quantum system and the environment (de-coherence). The knotted quantum world-lines of quasi-particles (anyons) provide the quantum computing physical support. The necessary algebra (which will deal with) is the algebra of the creation and annihilation of the knot's braids when the corresponding braiding operators act.

The article addresses the specific characteristic features of mutually intersecting regular pyramids with a common base. Advanced is a new theorem that proves to be highly effective when the objects are placed in a particular position.

In the present work, we investigate the behavior of the Locally Recurrent Probabilistic Neural Network (LRPNN) with different activation functions in the recurrent layer neurons. Specifically, we evaluate the performance of the modified activation function proposed here, which belongs to the family of Rectified Linear Units (ReLU), and compare it with other ReLU-based functions, the traditional sigmoid activation function, as well as with the Swish and E-Swish activation functions. Furthermore, we investigate the efficiency of a training procedure which simultaneously adjusts the spread factor sigma and the weights in the recurrent layer of the LRPNN. This training helps for coping with practical tasks, such as the recognition of Parkinson condition from speech signals, which operate under limited amount of training data.