Rankin–Cohen Brackets and Associativity

Don Zagier introduced and discussed in Zagier [Proc Indian Acad Sci (Math Sci) 104(1) 57–75, 1994] a particular algebraic structure of the graded ring of modular forms. In this note we interpret it in terms of an associative deformation of this graded ring.     

On Cohen–Macaulayness of Algebras Generated by Generalized Power Sums

Generalized power sums are linear combinations of ith powers of coordinates. We consider subalgebras of the polynomial algebra generated by generalized power sums, and study when such algebras are Cohen–Macaulay. It turns out that the Cohen–Macaulay property of such algebras is rare, and tends to be related to quantum integrability and representation theory of Cherednik algebras. Using representation theoretic results and deformation theory, we establish Cohen–Macaulayness of the algebra of q, t-deformed power sums defined by Sergeev and Veselov, and of some generalizations of this algebra, proving a conjecture of Brookner, Corwin, Etingof, and Sam. We also apply representation-theoretic techniques to studying m-quasi-invariants of deformed Calogero–Moser systems. In an appendix to this paper, M. Feigin uses representation theory of Cherednik algebras to compute Hilbert series for such quasi-invariants, and show that in the case of one light particle, the ring of quasi-invariants is Gorenstein.     

Generalized Froggatt–Nielsen mechanism

We propose a Generalized Froggatt–Nielsen mechanism in which the GUT gauge symmetry breaking and the generation of hierarchical flavor structure have the same origin. Assuming universality conditions for coefficients corresponding to different contractions, we can get the realistic standard model mass hierarchy and mixings in SU(5) GUT model as well as in SO(10) GUT model in several ways. With very free parameters, the Generalized Froggatt–Nielsen mechanism is very predictive and it can be very useful in GUT model building as well as in solving the flavor puzzle in standard model.     

Generalized Borcea–Voisin Construction

C. Voisin and C. Borcea have constructed mirror pairs of families of Calabi–Yau threefolds by taking the quotient of the product of an elliptic curve with a K3 surface endowed with a non-symplectic involution. In this paper, we generalize the construction of Borcea and Voisin to any prime order and build three and four dimensional Calabi–Yau orbifolds. We classify the topological types that are obtained and show that, in dimension 4, orbifolds built with an involution admit a crepant resolution and come in topological mirror pairs. We show that for odd primes, there are generically no minimal resolutions and the mirror pairing is lost.     

Gamma function to Beck–Cohen superstatistics

In this paper, we show a mathematical construction of Beck–Cohen superstatistics in the Bayesian point of view with the help of the two representations of a gamma function. Furthermore, it is shown how some results for superstatistics are related to each other.     

A generating function for Rankin–Cohen brackets

We give a closed formula for the Rankin–Cohen formal covariant quantization in terms of generating functions of Jacobi polynomials.     

Generalized Null-Ellipsometry in the Polarizer–Sample–Analyzer Scheme

Optics and Spectroscopy - The null ellipsometry technique of generalized ellipsometry based on using a compensator-free polarizer–sample–analyzer scheme for the case of incidence of an...     

On Convexity of Generalized Wigner–Yanase–Dyson Information

The convexity of the Wigner–Yanase–Dyson information, as first proved by Lieb, is a deep and fundamental result because it leads to the strong subadditivity of quantum entropy. The Wigner–Yanase–Dyson information is a particular kind of quantum Fisher information with important applications in quantum estimation theory. But unlike the quantum entropy, which is the unique natural quantum extension of the classical Shannon entropy, there are many different variants of quantum Fisher information, and it is desirable to investigate their convexity. This article is devoted to studying the convexity of a direct generalization of the Wigner–Yanase–Dyson information. Some sufficient conditions are obtained, and some necessary conditions are illustrated. In a particular case, a surprising necessary and sufficient condition is obtained. Our results reveal the intricacy and subtlety of the convexity issue for general quantum Fisher information.      

On a Generalized Frobenius–Stickelberger Addition Formula

In this Letter we obtain a generalization of the Frobenius–Stickelberger addition formula for the (hyperelliptic) -function of a genus 2 curve in the case of three vector-valued variables. The result is given explicitly in the form of a polynomial in Kleinian -functions.     

Generalized Kerr–NUT–de Sitter metrics in all dimensions

We classify all spacetimes with a closed rank-2 conformal Killing–Yano tensor. They give a generalization of Kerr–NUT–de Sitter spacetimes. The Einstein condition is explicitly solved and written as an indefinite integral. It is characterized by a polynomial in the integrand. We briefly discuss the smoothness conditions of the Einstein metrics over compact Riemannian manifolds.     

Generalized Jordanian R-Matrices of Cremmer–Gervais Type

An explicit quantization is given of certain skew-symmetric solutions of the classical Yang–Baxter equation, yielding a family of R-matrices which generalize to higher dimensions the Jordanian R-matrices. Three different approaches to their construction are given: as twists of degenerations of the Shibukawa–Ueno, Yang–Baxter operators on meromorphic functions; as boundary solutions of the quantum Yang–Baxter equation; via a vertex-IRF transformation from solutions to the dynamical Yang–Baxter equation.     

Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model

Inspired by the work of D.G.Kelly and S.Sherman on general Griffiths inequalities on correlations in Ising ferromagnets, we formulate and prove Griffith–Kelly–Sherman-type inequalities for the ferromagnetic Potts model with a general number q of local states. We take as local state space for the q-state Potts model the set F ^c  = { − l, − l + 1, ⋯ ,l − 1,l},where l=fracq-12l=frac{q-1}{2}. The important properties of F ^c for what follows are that |F ^c | = q and F ^c  = − F ^c .     

The Gallavotti–Cohen Fluctuation Theorem for a Nonchaotic Model

We test the applicability of the Gallavotti–Cohen fluctuation formula on a nonequilibrium version of the periodic Ehrenfest wind-tree model. This is an one-particle system whose dynamics is rather complex (e.g., it appears to be diffusive at equilibrium), but its Lyapunov exponents are nonpositive. For small applied field, the system exhibits a very long transient, during which the dynamics is roughly chaotic, followed by asymptotic collapse on a periodic orbit. During the transient, the dynamics is diffusive, and the fluctuations of the current are found to be in agreement with the fluctuation formula, despite the lack of real hyperbolicity. These results also constitute an example which manifests the difference between the fluctuation formula and the Evans–Searles identity.     

Generalized spheroidal spacetimes in 5-D Einstein–Maxwell–Gauss–Bonnet gravity

The field equations for static EGBM gravity are obtained and transformed to an equivalent form through a coordinate redefinition. A form for one of the metric potentials that generalizes the spheroidal ansatz of Vaidya–Tikekar superdense stars and additionally prescribing the electric field intensity yields viable solutions. Some special cases of the general solution are considered and analogous classes in the Einstein framework are studied. In particular the Finch–Skea ansatz is examined in detail and found to satisfy the elementary physical requirements. These include positivity of pressure and density, the existence of a pressure free hypersurface marking the boundary, continuity with the exterior metric, a subluminal sound speed as well as the energy conditions. Moreover, the solution possesses no coordinate singularities. It is found that the impact of the Gauss–Bonnet term is to correct undesirable features in the pressure profile and sound speed index when compared to the equivalent Einstein gravity model. Furthermore graphical analyses suggest that higher densities are achievable for the same radial values when compared to the 5-dimensional Einstein case. The case of a constant gravitational potential, isothermal distribution as well as an incompressible fluid are studied. All exact solutions derived exhibit an equation of state explicitly.     

Generalized Kinetic Criterion of the Liquid–Glass Transition

Physics of the Solid State - Justification and generalization of the glass transition criterion of Schmelzer is proposed with the involvement of the model of delocalized atoms. Unlike the...     

Generalized Chern–Simons modified gravity in first-order formalism

We propose a generalization of Chern–Simons (CS) modified gravity in first-order formalism. CS modified gravity action has a term that comes from the chiral anomaly which is Pontryagin invariant. First-order CS modified gravity is a torsional theory and in a space-time with torsion the chiral anomaly includes a torsional topological term called Nieh-Yan invariant. We generalize the CS modified gravity by adding the Nieh-Yan term to the action and find the effective theory. We compare the generalized theory with the first-order CS modified gravity and comment on the similarities and differences.     

Generalized Arago–Fresnel laws: the EME-flow-line description

We study experimentally and theoretically the influence of light polarization on the interference patterns behind a diffracting grating. Different states of polarization and configurations are considered. The experiments are analyzed in terms of electromagnetic energy (EME) flow lines, which can be eventually identified with the paths followed by photons. This gives rise to a novel trajectory interpretation of the Arago–Fresnel laws for polarized light, which we compare with interpretations based on the concept of “which-way” (or “which-slit”) information.     

Generalized Klauder–Perelomov and Gazeau–Klauder coherent states for Landau levels

Based on a pair of representations obtained for Lie algebra h₄, the Hilbert space corresponding to all quantum states of Landau levels is split into an infinite direct sum of infinite-dimensional Hilbert subspaces. For any one of the Hilbert subspaces, we get linear combinations of their bases as generalized coherent states—the so-called Klauder–Perelomov and Gazeau–Klauder.     

Periodic solutions for discrete-time Cohen–Grossberg neural networks with delays

In this paper, the existence of periodic solutions for general discrete-time Cohen–Grossberg neural networks with delays (DCGNND) is investigated. Based on graph theory, coincidence degree theory, and Lyapunov method, a sufficient criterion ensuring the existence of periodic solutions for DCGNND is established. In the end, an example and its numerical simulation are given to demonstrate the effectiveness of the theoretical result.