Anomalous hydrodynamics with dyonic charge

In this paper, we study anomalous hydrodynamics with a dyonic charge. We show that the local second law of thermodynamics constrains the structure of the anomaly in addition to the structure of the hydrodynamic constitutive equations. In particular, we show that not only the usual [Formula: see text] term but also [Formula: see text] term should be present in the anomaly with a specific coefficient for the local entropy production to be positive definite.

Variance of sums in arithmetic progressions of divisor functions associated with higher degree L-functions in 𝔽q[t]

We compute the variances of sums in arithmetic progressions of generalized [Formula: see text]-divisor functions related to certain [Formula: see text]-functions in [Formula: see text], in the limit as [Formula: see text]. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when [Formula: see text], in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual [Formula: see text]-divisor function, when the [Formula: see text]-function in question has degree one. They illustrate the role played by the degree of the [Formula: see text]-functions; in particular, we find qualitatively new behavior when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over [Formula: see text], and we illustrate them by examining in some detail the generalized [Formula: see text]-divisor functions associated with the Legendre curve.

The classical-quantum duality of nature including gravity

The classical-quantum duality at the basis of quantum theory is here extended to the Planck scale domain. The classical/semiclassical gravity (G) domain is dual (in the precise sense of the classical-quantum duality) to the quantum (Q) elementary particle domain: [Formula: see text], [Formula: see text] being the Planck scale. This duality is universal. From the gravity (G) and quantum (Q) variables [Formula: see text], we define new quantum gravity (QG) variables [Formula: see text] which include all (classical, semiclassical and QG) domains passing through the Planck scale and the elementary particle domain. The QG variables are more complete than the usual ([Formula: see text], [Formula: see text]) ones which cover only one domain (Q or G). Two [Formula: see text] or [Formula: see text] values [Formula: see text] are needed for each value of [Formula: see text] (reflecting the two dual ways of reaching the Planck scale). We perform the complete analytic extension of the QG variables through analytic (holomorphic) mappings which preserve the light-cone structure. This allows us to reveal the classical-quantum duality of the Schwarzschild–Kruskal spacetime: exterior regions are classical or semiclassical while the interior is totally quantum: its boundaries being the Planck scale. Exterior and interior lose their difference near the horizon: four Planck scale hyperbolae border the horizons as a quantum dressing or width: “l’horizon habillé”. QG variables are naturally invariant under [Formula: see text]. Spacetime reflections, antipodal symmetry and PT or CPT symmetry are contained in the QG symmetry, which also shed insight on the global properties of the Kruskal manifold and its present renewed interest.

Difference Hierarchies for nT τ-functions

We introduce hierarchies of difference equations (referred to as [Formula: see text]-systems) associated to the action of a (centrally extended, completed) infinite matrix group [Formula: see text] on [Formula: see text]-component fermionic Fock space. The solutions are given by matrix elements ([Formula: see text]-functions) for this action. We show that the [Formula: see text]-functions of type [Formula: see text] satisfy bilinear equations of length [Formula: see text]. The [Formula: see text]-system is, after a change of variables, the usual [Formula: see text] term [Formula: see text]-system of type [Formula: see text]. Restriction from [Formula: see text] to a subgroup isomorphic to the loop group [Formula: see text], defines [Formula: see text]-systems, studied earlier in [1] by the present authors for [Formula: see text].

Nambu–Poisson bracket on superspace

We propose an extension of [Formula: see text]-ary Nambu–Poisson bracket to superspace [Formula: see text] and construct by means of superdeterminant a family of Nambu–Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace [Formula: see text]. We prove in the case of the superspaces [Formula: see text] and [Formula: see text] that our [Formula: see text]-ary bracket, defined with the help of superdeterminant, satisfies the conditions for [Formula: see text]-ary Nambu–Poisson bracket, i.e. it is totally skew-symmetric and it satisfies the Leibniz rule and the Filippov–Jacobi identity (fundamental identity). We study the structure of [Formula: see text]-ary bracket defined with the help of superdeterminant in the case of superspace [Formula: see text] and show that it is the sum of usual [Formula: see text]-ary Nambu–Poisson bracket and a new [Formula: see text]-ary bracket, which we call [Formula: see text]-bracket, where [Formula: see text] is the product of two odd degree smooth functions.

Bisectors determining unique pairs of points in the bidisk

Bisectors are equidistant hypersurfaces between two points and are basic objects in a metric geometry. They play an important part in understanding the action of subgroups of isometries on a metric space. In many metric geometries (spherical, Euclidean, hyperbolic, complex hyperbolic, to name a few) bisectors do not uniquely determine a pair of points, in the following sense: completely different sets of points share a common bisector. The above examples of this non-uniqueness are all rank [Formula: see text] symmetric spaces. However, generically, bisectors in the usual [Formula: see text] metric are such for a unique pair of points in the rank [Formula: see text] geometry [Formula: see text]. This result indicates the striking assertion that non-uniqueness of bisectors holds for “most” geometries.

Simultaneous description of low-lying positive and negative parity states in spd, sdf and spdf interacting boson model

In order to investigate negative parity states, it is necessary to consider negative parity-bosons additionally to the usual [Formula: see text]- and [Formula: see text]-bosons. The dipole and octupole degrees of freedom are essential to describe the observed low-lying collective states with negative parity. An extended interacting boson model (IBM) that describes pairing interactions among s, p, d and f-boson based on affine [Formula: see text] Lie algebra in the quantum phase transition (QPT) field, such as spd-IBM, sdf-IBM and spdf-IBM, is composed based on algebraic structure. In this paper, a solvable extended transitional Hamiltonian based on affine [Formula: see text] Lie algebra is proposed to describe low-lying positive and negative parity states between the spherical and deformed gamma-unstable shape. Three model of new algebraic solution for even–even nuclei are introduced. Numerical extraction to low-lying energy levels and transition rates within the control parameters of this evaluated Hamiltonian are presented for various [Formula: see text] values. We reproduced the positive and negative parity states and our calculations suggest that the results of spdf-IBM are better than spd-IBM and sdf-IBM in this literature. By reproducing the experimental results, the method based on signature of the phase transition such as level crossing in the lowest excited states is used to provide a better description of Ru isotopes in this transitional region.

Unusual situations that arise with the Dirac delta function and its derivative

There is a situation such that, when a function ƒ(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">) is combined with the Dirac delta function δ(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">), the usual formula <img src="/img/revistas/rbef/v31n4/a04form01.gif" align="absmiddle">does not hold. A similar situation may also be encountered with the derivative of the delta function δ'(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">), regarding the validity of <img src="/img/revistas/rbef/v31n4/a04form02.gif" align="absmiddle">. We present an overview of such unusual situations and elucidate their underlying mechanisms. We discuss implications of the situations regarding the transmission-reflection problem of one-dimensional quantum mechanics.

ORIGIN OF MATTER OUT OF PURE CURVATURE

We propose a mechanism for the origin of matter in the universe in the framework of Einstein–Gauss–Bonnet gravity in higher dimensions. The new static black hole solution recently discovered by the authors,1 with the Kaluza–Klein split of space–time as a product of the usual [Formula: see text] with a space of negative constant curvature, is indeed a pure gravitational creation of a black hole which is also endowed with a Maxwell-like gravitational charge in four-dimensional vacuum space–time. This solution has been further generalized to include radially flowing radiation, which means that extra-dimensional curvature also produces matter distribution asymptotically, resembling charged null dust. The static black hole could thus be envisioned as being formed from anti–de Sitter space–time by the collapse of radially inflowing charged null dust. It thus establishes the remarkable reciprocity between matter and gravity — as matter produces gravity (curvature), gravity produces matter. After the Kaluza–Klein generation of the Maxwell field, this is the first instance of realization of matter without matter in the classical framework.

The Real Presence of Mary: Eucharistic Disbelief and the Limits of Orthodoxy in Fourteenth-Century France

On July 15, 1318, a twenty-six-year-old laywoman named Aude Fauré was called before the Inquisition tribunal at the diocesan seat of Pamiers in southern France and immediately confessed to having temporarily doubted both the real presence of Christ in the Eucharist and the transubstantiation of bread and wine into Christ's body and blood; her doubt, she explained, had been cured by intervention from the Blessed Virgin. Less than a month later, Aude abjured her errors by the usual formula and was sentenced to a series of pilgrimages and fasts stretching over the next three years. Aude's multiple confessions, along with depositions from her family, friends, and neighbors, take up a mere six folio pages in the famously detailed Register kept by Bishop Jacques Fournier, head of the Pamiers tribunal, and preserved in the Vatican Library after Fournier became Pope Benedict XII. This relatively quick-moving and insignificant case seems unrelated to the best-known activity of Fournier's tribunal, namely, the extinction of the last vestiges of Occitan Catharism. Yet Aude's case has gleaned several mentions in recent historiographic works, and these mentions are striking for their focus on the protagonist's psyche: she has been variously diagnosed as hypersensitive, neurotic, masochistic, morbid, hysterical, obsessive, afflicted with atheism, prone to fantasy, tormented by guilt, suffering from postpartum depression, and simply deviant.