Four quotient set gems

Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the Monthly, despite its intense coverage of quotient sets over the years.Comment: 9 pages, to appear in the American Mathematical Monthl

Four Quotient Set Gems

Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the MONTHLY, despite its intense coverage of quotient sets over the years

Causal particle detectors and topology

We investigate particle detector responses in some topologically non-trivial spacetimes. We extend a recently proposed regularization of the massless scalar field Wightman function in 4-dimensional Minkowski space to arbitrary dimension, to the massive scalar field, to quotients of Minkowski space under discrete isometry groups and to the massless Dirac field. We investigate in detail the transition rate of inertial and uniformly accelerated detectors on the quotient spaces under groups generated by $(t,x,y,z)\mapsto(t,x,y,z+2a)$, $(t,x,y,z)\mapsto(t,-x,y,z)$, $(t,x,y,z)\mapsto(t,-x,-y,z)$, $(t,x,y,z)\mapsto(t,-x,-y,z+a)$ and some higher dimensional generalizations. For motions in at constant $y$ and $z$ on the latter three spaces the response is time dependent. We also discuss the response of static detectors on the RP^3 geon and inertial detectors on RP^3 de Sitter space via their associated global embedding Minkowski spaces (GEMS). The response on RP^3 de Sitter space, found both directly and in its GEMS, provides support for the validity of applying the GEMS procedure to detector responses and to quotient spaces such as RP^3 de Sitter space and the RP^3 geon where the embedding spaces are Minkowski spaces with suitable identifications.Comment: 47 pages, 9 figure

India in the Global and Regional Trade - Determinants of Aggregate and Bilateral Trade Flows and Firms’ Decision to Export

This paper contributes to two strands of literature on empirical models of trade flows and trade policy. The first and the older strand is that of gravity models of bilateral trade flows going back to Hans Linneman (1966) and Tinbergen (1962) and its recent applications, particularly by Adams et al (2003) and De Rosa (2007) in analyzing the impact of Preferential Trade Agreements (PTAs). Our focus is on applying the gravity model to analyze Indias trade flows (exports and imports) with its trading partners around the world and to examine the impact of various PTAs in which India or its trading partner or both are members. Clearly this is of interest, since, from 1991 India is aggressively negotiating and concluding PTAs of which South Asian preferential trade (and later free trade) agreement is the most prominent. We find that India is not well served by its pursuit of PTAs and should instead push for multilateral trade liberalisation by contributing to conclusion of the Doha round of negotiations with an agreement beneficial to all WTO members. The second and the more recent strand is the analysis of trade flows using data on exports of individual firms. It is well known that in all countries of the world relatively few firms participate in world trade, thus suggesting that characteristics of a firm (such as its size and productivity) are relevant besides country level barriers on trade matter for participation in world trade. This strand is rapidly growing. Ours is one of the very few attempts at modeling and estimating the decision of Indian firms on their participation using firm level data. The paper reports on our preliminary results. We have also collected primary data from a sample survey of firms to explore this issue deeper. While these data are yet to be fully analyzed, nevertheless some preliminary descriptive tables summarizing them are included in an Appendix.Non-discriminatory trade liberalisation, Gravity Model, Intrabloc trade effect, trade diversion, trade creation, Firm heterogeneity, Probability of exporting, Export performance, Logit, Probit, Fixed effect, Random effect, Tobit Model

On the p-adic denseness of the quotient set of a polynomial image

The quotient set, or ratio set, of a set of integers $A$ is defined as $R(A) := \left\{a/b : a,b \in A,\; b \neq 0\right\}$. We consider the case in which $A$ is the image of $\mathbb{Z}^+$ under a polynomial $f \in \mathbb{Z}[X]$, and we give some conditions under which $R(A)$ is dense in $\mathbb{Q}_p$. Then, we apply these results to determine when $R(S_m^n)$ is dense in $\mathbb{Q}_p$, where $S_m^n$ is the set of numbers of the form $x_1^n + \cdots + x_m^n$, with $x_1, \dots, x_m \geq 0$ integers. This allows us to answer a question posed in [Garcia et al., $p$-adic quotient sets, Acta Arith. 179, 163-184]. We end leaving an open question

FERENGI: Redshifting galaxies from SDSS to GEMS, STAGES and COSMOS

We describe the creation of a set of artificially "redshifted" galaxies in the range 0.1<z<1.1 using a set of ~100 SDSS low redshift (v<7000 km/s) images as input. The intention is to generate a training set of realistic images of galaxies of diverse morphologies and a large range of redshifts for the GEMS and COSMOS galaxy evolution projects. This training set allows other studies to investigate and quantify the effects of cosmological redshift on the determination of galaxy morphologies, distortions and other galaxy properties that are potentially sensitive to resolution, surface brightness and bandpass issues. We use galaxy images from the SDSS in the u, g, r, i, z filter bands as input, and computed new galaxy images from these data, resembling the same galaxies as located at redshifts 0.1<z<1.1 and viewed with the Hubble Space Telescope Advanced Camera for Surveys (HST ACS). In this process we take into account angular size change, cosmological surface brightness dimming, and spectral change. The latter is achieved by interpolating a spectral energy distribution that is fit to the input images on a pixel-to-pixel basis. The output images are created for the specific HST ACS point spread function and the filters used for GEMS (F606W and F850LP) and COSMOS (F814W). All images are binned onto the desired pixel grids (0.03" for GEMS and 0.05" for COSMOS) and corrected to an appropriate point spread function. Noise is added corresponding to the data quality of the two projects and the images are added onto empty sky pieces of real data images. We make these datasets available from our website, as well as the code - FERENGI: "Full and Efficient Redshifting of Ensembles of Nearby Galaxy Images" - to produce datasets for other redshifts and/or instruments.Comment: 11 pages, 10 figures, 3 table