Harmonic Univalent Mappings and Linearly Connected Domains

We investigate the relationship between the univalence of $f$ and of $h$ in the decomposition $f=h+\bar{g}$ of a sense-preserving harmonic mapping defined in the unit disk $\mathbb{D}\subset\mathbb{C}$. Among other results, we determine the holomorphic univalent maps $h$ for which there exists $c>0$ such that every harmonic mapping of the form $f=h+\bar{g}$ with $|g'|< c|h'|$ is univalent. The notion of a linearly connected domain appears in our study in a relevant way

Fields, Strings and Branes

Lecture notes reviewing most recent developments in string/M/brane theory given by C. G. at the CIME Summer International Center of Mathematics at Cetraro. July 1997.Comment: Latex file. 176 pages. Six figure

Lattice models with long-range and number-non-conserving interactions with Zeeman excitations of ultracold magnetic atoms

We show that Zeeman excitations of ultracold Dy atoms trapped in an optical lattice can be used to engineer extended Hubbard models with tunable inter-site and particle number-non-conserving interactions. We show that the ratio of the hopping amplitude and inter-site interactions in these lattice models can be tuned in a wide range by transferring the atoms to different Zeeman states. We propose to use the resulting controllable models for the study of the effects of direct particle interactions and particle number-non-conserving terms on Anderson localization.Comment: 29 pages, 8 figure

Twin disc assessment of wheel/rail adhesion

Loss of adhesion between a railway wheel and the track has implications for both braking and traction. Poor adhesion in braking is a safety issue as it leads to extended stopping distances. In traction, it is a performance issue as it may lead to reduced acceleration which could cause delays. In this work, wheel/rail adhesion was assessed using a twin disc simulation. The effects of a number of contaminants, such as oil, dry and wet leaves and sand were investigated. These have been shown in the past to have significant effect on adhesion, but this has not been well quantified. The results have shown that both oil and water reduce adhesion from the dry condition. Leaves, however, gave the lowest adhesion values, even when dry. The addition of sand, commonly used as a friction enhancer, to leaves, brought adhesion levels back to the levels without leaves present. Adhesion levels recorded, particularly for the wet, dry and oil conditions are in the range seen in field measurements. Relatively severe disc surface damage and subsurface deformation was seen after the addition of sand. Leaves were also seen to cause indents in the disc surfaces. The twin disc approach has been shown to provide a good approach for comparing adhesion levels under a range of wheel/rail contact conditions, with and without contaminants

Stopping time convergence for processes associated with Dirichlet forms

Convergence is proved for solutions of Dirichlet problems in regions with many small excluded sets (holes), as the holes become smaller and more numerous. The problem is formulated in the context of Markov processes associated with general Dirichlet forms, for random and nonrandom excluded sets. Sufficient conditions are given under which the sequence of entrance times or hitting times of the excluded sets converges in the stable topology. Convergence in the stable topology is a strengthened form of convergence in distribution, introduced by Renyi. Stable convergence of the entrance times implies convergence of the solutions of the corresponding Dirichlet problems. Some additional results are given in a supplement on random center models.Comment: Fixed some typos which were introduced while "de-macroing

Evidence for dark matter in the inner Milky Way...Really?

The following is a comment on the recent letter by Iocco et al. (2015, arXiv:1502.03821) where the authors claim to have found "...convincing proof of the existence of dark matter...". The letter in question presents a compilation of recent rotation curve observations for the Milky Way, together with Newtonian rotation curve estimates based on recent baryonic matter distribution measurements. A mismatch between the former and the latter is then presented as "evidence for dark matter". Here we show that the reported discrepancy is the well known gravitational anomaly which consistently appears when dynamical accelerations approach the critical Milgrom acceleration a_0 = 1.2 \times 10^{-10} m / s^2. Further, using a simple modified gravity force law, the baryonic models presented in Iocco et al. (2015), yield dynamics consistent with the observed rotation values.Comment: 2 pages, 1 figur

Integrability, Jacobians and Calabi-Yau Threefolds

The integrable systems associated with Seiberg-Witten geometry are considered both from the Hitchin-Donagi-Witten gauge model and in terms of intermediate Jacobians of Calabi-Yau threefolds. Dual pairs and enhancement of gauge symmetries are discussed on the basis of a map from the Donagi-Witten ``moduli'' into the moduli of complex structures of the Calabi-Yau threefold.Comment: 8 pages, Latex, based on a talk given by C. Gomez at the "VIII Regional Meeting on Mathematical Physics", Oct. 1995, Ira

MONDian predictions for Newtonian M/L ratios for ultrafaint dSphs

Under Newtonian gravity total masses for dSph galaxies will scale as $M_{T} \propto R_{e} \sigma^{2}$, with $R_{e}$ the effective radius and $\sigma$ their velocity dispersion. When both of the above quantities are available, the resulting masses are compared to observed stellar luminosities to derive Newtonian mass to light ratios, given a physically motivated proportionality constant in the above expression. For local dSphs and the growing sample of ultrafaint such systems, the above results in the largest mass to light ratios of any galactic systems known, with values in the hundreds and even thousands being common. The standard interpretation is for a dominant presence of an as yet undetected dark matter component. If however, reality is closer to a MONDian theory at the extremely low accelerations relevant to such systems, $\sigma$ will scale with { stellar mass} $M_{*}^{1/4}$. This yields an expression for the mass to light ratio which will be obtained under Newtonian assumptions of $(M/L)_{N}=120 R_{e}(\Upsilon_{*}/L)^{1/2}$. Here we compare $(M/L)_{N}$ values from this expression to Newtonian inferences for this ratios for the actual $(R_{e}, \sigma, L)$ observed values for a sample of recently observed ultrafaint dSphs, obtaining good agreement. Then, for systems where no $\sigma$ values have been reported, we give predictions for the $(M/L)_{N}$ values which under a MONDian scheme are expected once kinematical observations become available. For the recently studied Dragonfly 44 { and Crater II systems}, reported $(M/L)_{N}$ values are also in good agreement with MONDian expectations.Comment: 6 pages, 3 figures, accepted for publication in MNRA

Families of continuous retractions and function spaces

In this article, we mainly study certain families of continuous retractions ($r$-skeletons) having certain rich properties. By using monotonically retractable spaces we solve a question posed by R. Z. Buzyakova in \cite{buz} concerning the Alexandroff duplicate of a space. Certainly, it is shown that if the space $X$ has a full $r$-skeleton, then its Alexandroff duplicate also has a full $r$-skeleton and, in a very similar way, it is proved that the Alexandroff duplicate of a monotonically retractable space is monotonically retractable. The notion of $q$-skeleton is introduced and it is shown that every compact subspace of $C_p(X)$ is Corson when $X$ has a full $q$-skeleton. The notion of strong $r$-skeleton is also introduced to answer a question suggested by F. Casarrubias-Segura and R. Rojas-Hern\'andez in their paper \cite{cas-rjs} by establishing that a space $X$ is monotonically Sokolov iff it is monotonically $\omega$-monolithic and has a strong $r$-skeleton. The techniques used here allow us to give a topological proof of a result of I. Bandlow \cite{ban} who used elementary submodels and uniform spaces

Connectedness like properties on the hyperspace of convergent sequences

This paper is a continuation of the work done in \cite{sal-yas} and \cite{may-pat-rob}. We deal with the Vietoris hyperspace of all nontrivial convergent sequences $\mathcal{S}_c(X)$ of a space $X$. We answer some questions in \cite{sal-yas} and generalize several results in \cite{may-pat-rob}. We prove that: The connectedness of $X$ implies the connectedness of $\mathcal{S}_c(X)$; the local connectedness of $X$ is equivalent to the local connectedness of $\mathcal{S}_c(X)$; and the path-wise connectedness of $\mathcal{S}_c(X)$ implies the path-wise connectedness of $X$. We also show that the space of nontrivial convergent sequences on the Warsaw circle has $\mathfrak{c}$-many path-wise connected components, and provide a dendroid with the same property