On the origins of the scalar and vectorial product definitions

The operations of two vector multiplication (the scalar and vector products) are introduced in physics and mathematics textbooks just as a definition, without any reference or discussion on the formal reasons and/or motivations that have led to these structures. In this work, a short pedagogical review on the origins of these definitions is presented. We discuss the formal results obtained by Hamilton in the context of quaternionic algebra and some changes performed by Gibbs and Heaviside, leading to what is now usually known as vector algebra. We present comments on some disadvantages of these changes, referring to more practical and formal systems (Grasmmann and Clifford algebras). Some basic and recent works on the subject are also mentioned and commented.Nos livros-texto de física e de matemática utilizados em cursos básicos universitários, as operações de multiplicação de dois vetores (produtos escalar e vetorial) são introduzidas apenas como definições, sem nenhuma referência ou discussão a respeito das razões formais e/ou motivações que levaram ao estabelecimento de tais estruturas. Neste trabalho, apresentamos uma breve revisão didática sobre as origens dessas definições, discutindo os resultados pertinentes, formais, estabelecidos por Hamilton no contexto da álgebra de quatérnions e certas adaptações feitas por Gibbs e Heaviside, as quais deram origem ao ramo da matemática que hoje é popularmente conhecido como álgebra vetorial. Comentamos algumas desvantagens decorrentes dessas adaptações, fazendo referência a outros sistemas algébricos práticos e formalmente bem fundamentados (álgebras de Grassmann e Clifford). Indicamos e comentamos alguns artigos e trabalhos, básicos e também recentes, nos quais o assunto pode ser aprofundado.2305.12305.1

High-energy elementary amplitudes from quenched and full QCD

Making use of the Stochastic Vacuum Model and the gluon gauge-invariant two-point correlation function, determined by numerical simulation on the lattice in both quenched approximation and full QCD, we calculate the elementary (quark-quark) scattering amplitudes in the momentum transfer space and at asymptotic energies. Our main conclusions are the following: (1) the amplitudes decrease monotonically as the momentum transfer increases; (2) the decreasing is faster when going from quenched approximation to full QCD (with decreasing quark masses) and this effect is associated with the increase of the correlation lengths; (3) dynamical fermions generate two components in the amplitude at small momentum transfer and the transition between them occurs at momentum transfer near 1 GeV$^2$. We also obtain analytical parametrizations for the elementary amplitudes, that are suitable for phenomenological uses, and discuss the effects of extrapolations from the physical regions investigated in the lattice.Comment: 6 pages, 4 figures, aps-revtex4, to be published in Physics Letters

On model-independent analyses of elastic hadron scattering

By means of an almost model-independent parametrization for the elastic hadron-hadron amplitude, as a function of the energy and the momentum transfer, we obtain good descriptions of the physical quantities that characterize elastic proton-proton and antiproton-proton scattering (total cross section, r parameter and differential cross section). The parametrization is inferred on empirical grounds and selected according to high energy theorems and limits from axiomatic quantum field theory. Based on the predictive character of the approach we present predictions for the above physical quantities at the Brookhaven RHIC, Fermilab Tevatron and CERN LHC energies.67567

Extrema bounds for the soft Pomeron intercept

By using an extended Regge parametrization and taking into account the discrepancies in the high-energy pp and p(bar)p total cross section data, in both accelerator and cosmic-ray regions, we estimate extrema bounds for the soft Pomeron intercept. First we consider two ensembles of data with either the CDF or the E710 and E811 results for the p(bar)p total cross section at 1.8 TeV, from which we obtain the bounds 1.102 and 1.081, respectively. These ensembles are then combined with the highest and lowest estimations for the pp total cross section from cosmic-ray experiments (6-40 TeV), leading to the upper and lower bounds 1.109 and 1.082, respectively. The effects of simultaneous fits to total cross sections and the rho parameter, individual fits to total cross sections, and the influence of the subtraction constant in the dispersion relations are also presented. Our global results favor the E710 and E811 data.Comment: 7 pages, 7 figures, aps-revtex, to be published in Physics Letters

Total Hadronic Cross Section and the Elastic Slope: An Almost Model-Independent Connection

An almost model-independent parametrization for the ratio of the total cross section to the elastic slope, as function of the center of mass energy, is introduced. The analytical result is based on the approximate relation of this quantity with the ratio $R$ of the elastic to total cross section and empirical fits to the $R$ data from proton-proton scattering above 10 GeV, under the conditions of asymptotic unitarity and the black-disk limit. This parametrization may be useful in studies of extensive air showers and the determination of the proton-proton total cross section from proton-air production cross section in cosmic-ray experiments.Comment: 15 pages, 4 figures, v4: few typos corrected, final version to be published in Nucl. Phys.

Scaling violations: Connections between elastic and inelastic hadron scattering in a geometrical approach

Starting from a short range expansion of the inelastic overlap function, capable of describing quite well the elastic pp and $\bar{p}p$ scattering data, we obtain extensions to the inelastic channel, through unitarity and an impact parameter approach. Based on geometrical arguments we infer some characteristics of the elementary hadronic process and this allows an excellent description of the inclusive multiplicity distributions in $pp$ and $\bar{p}p$ collisions. With this approach we quantitatively correlate the violations of both geometrical and KNO scaling in an analytical way. The physical picture from both channels is that the geometrical evolution of the hadronic constituents is principally reponsible for the energy dependence of the physical quantities rather than the dynamical (elementary) interaction itself.Comment: 16 pages, aps-revtex, 11 figure

Disordered Type-II Superconductors: A Universal Phase Diagram for Low-Tc_c Systems

A universal phase diagram for weakly pinned low-T$_c$ type-II superconductors is revisited and extended with new proposals. The low-temperature ``Bragg glass'' phase is argued to transform first into a disordered, glassy phase upon heating. This glassy phase, a continuation of the high-field equilibrium vortex glass phase, then melts at higher temperatures into a liquid. This proposal provides an explanation for the anomalies observed in the peak effect regime of 2H-NbSe$_2$ and several other low-T$_c$ materials which is independent of the microscopic mechanisms of superconductivity in these systems.Comment: 23 pages, 9 figure

Phenomenological analysis connecting proton-proton and antiproton-proton elastic scattering

Based on the behavior of the elastic scattering data, we introduce an almost model-independent parametrization for the imaginary part of the scattering amplitude, with the energy and momentum transfer dependences inferred on empirical basis and selected by rigorous theorems and bounds from axiomatic quantum field theory. The corresponding real part is analytically evaluated by means of dispersion relations, allowing connections between particle-particle and particle-antiparticle scattering. Simultaneous fits to proton-proton and antiproton-proton experimental data in the forward direction and also including data beyond the forward direction, lead to a predictive formalism in both energy and momentum transfer. We compare our extrapolations with predictions from some popular models and discuss the applicability of the results in the normalization of elastic rates that can be extracted from present and future accelerator experiments (Tevatron, RHIC and LHC).Comment: 17 pages, 17 figures, to appear in Eur. Phys. J.

Fungal community composition at the last remaining wild site of yellow early marsh orchid (Dactylorhiza incarnata ssp. ochroleuca)

The yellow early marsh orchid (Dactylorhiza incarnata ssp. ochroleuca) is a critically endangered terrestrial orchid in Britain. Previous attempts to translocate symbiotic seedlings to a site near the last remaining wild site demonstrated some success, with a 10% survival rate despite adverse weather conditions over a two-year period. However, to facilitate future reintroduction efforts or conservation translocations, a more comprehensive understanding of the fungal microbiome and abiotic soil characteristics at the final remaining wild site is required. Obtaining comprehensive information on both the fungal community and soil nutrient composition from wild sites has significant benefits and may prove critical for the success of future conservation translocations involving threatened orchids. This preliminary study, conducted at the last remaining wild site, revealed a significant correlation between the relative abundance of the orchid mycorrhizal fungal order Cantharellales and the concentrations of nitrate and phosphate in the soil. Another orchid mycorrhizal fungal group, Sebacinales, was found to be distributed extensively throughout the site. The composition of fungal communities across the entire site, orchid-hosting and non-orchid-hosting soils is discussed in relation to reinforcing the current population and preventing the extinction of this orchid

Temporal fluctuations of waves in weakly nonlinear disordered media

We consider the multiple scattering of a scalar wave in a disordered medium with a weak nonlinearity of Kerr type. The perturbation theory, developed to calculate the temporal autocorrelation function of scattered wave, fails at short correlation times. A self-consistent calculation shows that for nonlinearities exceeding a certain threshold value, the multiple-scattering speckle pattern becomes unstable and exhibits spontaneous fluctuations even in the absence of scatterer motion. The instability is due to a distributed feedback in the system "coherent wave + nonlinear disordered medium". The feedback is provided by the multiple scattering. The development of instability is independent of the sign of nonlinearity.Comment: RevTeX, 15 pages (including 5 figures), accepted for publication in Phys. Rev.