Mathematical Physics

We provide a geometric extension of the generalized Noether theorem for scaling symmetries recently presented in \cite{zhang2020generalized}. Our version of the generalized Noether theorem has several positive features: it is constructed in the most natural extension of the phase space, allowing for the symmetries to be vector fields on such manifold and for the associated invariants to be first integrals of motion; it has a direct geometrical proof, paralleling the proof of the standard phase space version of Noether's theorem; it automatically yields an inverse Noether theorem; it applies also to a large class of dissipative systems; and finally, it allows for a much larger class of symmetries than just scaling transformations which form a Lie algebra, and are thus amenable to algebraic treatments.

We provide a graphical calculus for computing averages of tensor network diagrams with respect to the distribution of random vectors containing independent uniform complex phases. Our method exploits the order structure of the partially ordered set of uniform block permutations. A similar calculus is developed for random vectors consisting of independent uniform signs, based on the combinatorics of the partially ordered set of even partitions. We employ our method to extend some of the results by Johnston and MacLean on the family of local diagonal unitary invariant matrices. Furthermore, our graphical approach applies just as well to the real (orthogonal) case, where we introduce the notion of triplewise complete positivity to study the condition for separability of the relevant bipartite matrices. Finally, we analyze the twirling of linear maps between matrix algebras by independent diagonal unitary matrices, showcasing another application of our method.

We prove that any G=SU(2)?U(1) symmetric spacetime that is Ricci flat (i.e. solves the matter-free ?=0 Einstein equations) with non-null G -orbits is locally isometric to some maximally extended generalised Taub-NUT spacetime.

In this paper, we propose a modified formal Lagrangian formulation by introducing dummy dependent variables and prove the existence of such a formulation for any system of differential equations. The corresponding Euler--Lagrange equations, consisting of the original system and its adjoint system about the dummy variables, reduce to the original system via a simple substitution for the dummy variables. The formulation is applied to study conservation laws of differential equations through Noether's Theorem and in particular, a nontrivial conservation law of the Fornberg--Whitham equation is obtained by using its Lie point symmetries. Finally, a correspondence between conservation laws of the incompressible Euler equations and variational symmetries of the relevant modified formal Lagrangian is shown.

Superoscillations are band-limited functions that can oscillate faster than their fastest Fourier component. These functions (or sequences) appear in weak values in quantum mechanics and in many fields of science and technology such as optics, signal processing and antenna theory. In this paper we introduce a new method to generate superoscillatory functions that allows us to construct explicitly a very large class of superoscillatory functions.

We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second order term, as predicted by the Lee-Huang-Yang formula. This result has been first established in J. Stat. Phys. 136(3) by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported V??L 3 ( R 3 ) , gives better rates and, in our opinion, is substantially simpler.

In this paper we study generalized time-fractional diffusion equations on the Poincarè half plane H + 2 . The time-fractional operators here considered are fractional derivatives of a function with respect to another function, that can be obtained by starting from the classical Caputo-derivatives essentially by means of a deterministic change of variable. We obtain an explicit representation of the fundamental solution of the generalized-diffusion equation on H + 2 and provide a probabilistic interpretation related to the time-changed hyperbolic Brownian motion. We finally include an explicit result regarding the non-linear case admitting a separating variable solution.

The sine-Gordon expansion method; which is a transformation of the sine-Gordon equation has been applied to the potential-YTSF equation of dimension (3+1) and the reaction-diffusion equation. We obtain new solitons of this equation in the form hyperbolic, complex and trigonometric function by using this method. We plot 2D and 3D graphics of these solutions using symbolic software.

The aim of this note is to compare work of Formanek \cite{formanek2} on a certain construction of central polynomials with that of Collins \cite{Coll} on integration on unitary groups. These two quite disjoint topics share the construction of the same function on the symmetric group, which the second author calls {\em Weingarten function}. By joining these two approaches we succeed in giving a simplified and {\em very natural} presentation of both Formanek and Collins's Theory.

We revisit the renormalization group (RG) theoretical perturbation theory on oscillator-type second-order ordinary differential equations. For a class of potentials, we show a simple functional relation among secular coefficients of the harmonics in the naive perturbation series. It leads to an inversion formula between bare and renormalized amplitudes and an elementary proof of the absence of secular terms in all orders of the RG series. The result covers nonautonomous as well as autonomous cases and refines earlier studies, including the classic examples of Van der Pol, Mathieu, Duffing, and Rayleigh equations.