Related papers: Towards Generalized Riesz Systems Theory
In this paper I give overviews of the polysymplectic approach to covariant Hamiltonian field theory and the simplest geometric quantization of classical particle theories. I then give a synopsis of a recently proposed toy model for applying…
This work is devoted to the study of Bessel and Riesz systems of the type $\big\{L_{\gamma}\mathsf{f}\big\}_{\gamma\in \Gamma}$ obtained from the action of the left regular representation $L_{\gamma}$ of a discrete non abelian group…
This thesis is devoted to the study of three problems on the Wess-Zumino-Witten (WZW) and Chern-Simons (CS) supergravity theories in the Hamiltonian framework: 1) The two-dimensional super WZW model coupled to supergravity is constructed.…
Rank regression offers robustness to outliers and heavy-tailed response distributions, invariance to monotonic transformations, and improved efficiency under non-Gaussian errors, making it a versatile tool for analyzing complex data. This…
As a common non-trivial generalization of the notion of a generalized co-Bassian group, recently defined by the third author, we introduce the notion of a semi-generalized co-Bassian group and initiate its comprehensive study. Specifically,…
Non-hermitian quantum graphs possessing real (i.e., in principle, observable) spectra are studied via their discretization. The discretized Hamiltonians are assigned, constructively, an elementary pseudometric and/or a more complicated…
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review…
We extend the definition of generalized parity $P$, charge-conjugation $C$ and time-reversal $T$ operators to nondiagonalizable pseudo-Hermitian Hamiltonians, and we use these generalized operators to describe the full set of symmetries of…
The discussion in this study delves into Dirac's bra-ket formalism for a quasi-Hermitian quantum composite system based on the rigged Hilbert space (RHS). We establish an RHS with a positive definite metric suitable for a quasi-Hermite…
A Hamiltonian renormalization group is presented. Such a formulation is relevant for chiralic systems and more appropriate than the Lagrangian formalism. An application to 1D system is presented.
The notion of {\it generalised structure groups} and {\it generalised holonomy groups} has been introduced in supergravity, in order to discuss the spinor rotations generated by commutators of supercovariant derivatives when non-vanishing…
We generalize the integrable Heisenberg ferromagnet model according to each Hermitian symmetric spaces and address various new aspects of the generalized model. Using the first order formalism of generalized spins which are defined on the…
The relativistic Klein-Gordon system is studied as an illustration of Quantum Mechanics using non-Hermitian operators as observables. A version of the model is considered containing a generic coordinate- and energy-dependent…
We construct large classes of exactly solvable pseudo-Hermitian 2D spin Hamiltonians. The ground states of these systems depend only on the spatial topology of the system. We identify the ground state system on a surface with the value…
We discuss a particular problem of enumerating rational curves on a Grassmannian from several perspectives, including systems theory, real enumerative geometry, and symbolic computation. We also present a new transversality result, showing…
We consider restricted Boltzmann machines with a binary visible layer and a Gaussian hidden layer trained by an unlabelled dataset composed of noisy realizations of a single ground pattern. We develop a statistical mechanics framework to…
Apart from a few remarks on lattice systems with global or gauge symmetries, most of this talk is devoted to some interesting ancient examples of symmetries and their breakdowns in elasticity theory and hydrodynamics. Since Galois Theory is…
A non-Hermitian generalisation of the Marsden--Weinstein reduction method is introduced to construct families of quantum $\mathcal{PT}$-symmetric superintegrable models over an $n$-dimensional sphere $S^n$. The mechanism is illustrated with…
Three ways of constructing a non-Hermitian matrix with possible all real eigenvalues are discussed. They are PT symmetry, pseudo-Hermiticity, and generalized PT symmetry. Parameter counting is provided for each class. All three classes of…
Random matrix theory (RMT) provides a successful model for quantum systems, whose classical counterpart has a chaotic dynamics. It is based on two assumptions: (1) matrix-element independence, and (2) base invariance. Last decade witnessed…