Quantum theory of the real and the complexified projective line

Quantum deformations of sets of points of the real and the complexified projective line are constructed. These deformations depend on the deformation parameter q and certain further parameters \lambda_{ij}. The deformations for which the subspace of polynomials of degree three has a basis of ordered monomials are selected. We show that the corresponding algebras of three points have "polynomiality". Invariant elements which turn out to be cross ratios in the classical limit are defined. For the special case |\lambda_{ij}| = 1 a quantum cross ratio with properties similar to the classical case is presented. As an application a quantum version of the real Euclidean distance is given.Comment: 25 page

Constraining the Star Formation Histories in Dark Matter Halos: I. Central Galaxies

Using the self-consistent modeling of the conditional stellar mass functions across cosmic time by Yang et al. (2012), we make model predictions for the star formation histories (SFHs) of {\it central} galaxies in halos of different masses. The model requires the following two key ingredients: (i) mass assembly histories of central and satellite galaxies, and (ii) local observational constraints of the star formation rates of central galaxies as function of halo mass. We obtain a universal fitting formula that describes the (median) SFH of central galaxies as function of halo mass, galaxy stellar mass and redshift. We use this model to make predictions for various aspects of the star formation rates of central galaxies across cosmic time. Our main findings are the following. (1) The specific star formation rate (SSFR) at high $z$ increases rapidly with increasing redshift [$\propto (1+z)^{2.5}$] for halos of a given mass and only slowly with halo mass ($\propto M_h^{0.12}$) at a given $z$, in almost perfect agreement with the specific mass accretion rate of dark matter halos. (2) The ratio between the star formation rate (SFR) in the main-branch progenitor and the final stellar mass of a galaxy peaks roughly at a constant value, $\sim 10^{-9.3} h^2 {\rm yr}^{-1}$, independent of halo mass or the final stellar mass of the galaxy. However, the redshift at which the SFR peaks increases rapidly with halo mass. (3) More than half of the stars in the present-day Universe were formed in halos with 10^{11.1}\msunh < M_h < 10^{12.3}\msunh in the redshift range $0.4 < z < 1.9$. (4) ... [abridged]Comment: 15 figures, 22 pages, Accepted for publication in Ap

Understanding the Accrual Anomaly

Interpreting accruals as working capital investment, we hypothesize that firms rationally adjust their investment to respond to discount rate changes. Consistent with the optimal investment hypothesis, we document that (i) the predictive power of accruals for future stock returns increases with the covariations of accruals with past and current stock returns, and (ii) adding investment- based factors into standard factor regressions substantially reduces the magnitude of the accrual anomaly. High accrual firms also have similar corporate governance and entrenchment indexes as low accrual firms. This evidence suggests that the accrual anomaly is more likely to be driven by optimal investment than by investor overreaction to excessive growth or over-investment.

Corporate Governance and Market Valuation in China

This paper studies the relationship between the governance mechanisms and the market valuation of publicly listed firms in China empirically. We construct measures for corporate governance mechanisms and measures of market valuation for all publicly listed firms on the two stock markets in China by using data from the firm’s annual reports. We then investigate how the market-valuation variables are affected by the corporate governance variables while controlling for a number of factors commonly considered in market valuation analysis. A corporate governance index is also constructed to summarize the information contained in the corporate governance variables. The index is found to have statistically and economically significant effect on market valuation. The analysis indicates that investors pay a significant premium for well-governed firms in China, benefiting firms that improve their governance mechanisms.http://deepblue.lib.umich.edu/bitstream/2027.42/39949/3/wp564.pd

Medium Modification of the Proton Form Factor

I argue that the double ratio of proton-recoil polarization-transfer coefficients, P'_x and P'_z, of the quasielastic 4He(e,e'p)3H reaction with respect to the elastic 1H(e,e'p) reaction is sensitive to possible medium modifications of the proton form factor in 4He. Recent measurements at both Mainz and Jefferson Lab of this double ratio at four-momentum transfers squared between between 0.4 GeV2 and 2.6 GeV2 are discussed. I show that the data challenge state-of-the-art conventional meson-nucleon calculations, as these are unable to describe the results. The data hint at the need to include medium modifications of the proton form factor, as predicted by a quark-meson-coupling model, in the calculations. A recently approved follow-up experiment at a Q2 of 0.8 GeV2 and 1.3 GeV2 with unprecedented precision will provide one of the most stringent tests of the applicability of various calculations.Comment: presented at ``Fourth International Conference on Perspectives in Hadronic Physics'', ICTP, Trieste, 12 -- 16 May 200

Star Formation and Stellar Mass Assembly in Dark Matter Halos: From Giants to Dwarfs

The empirical model of Lu et al. 2014 is updated with recent data and used to study galaxy star formation and assembly histories. At $z > 2$, the predicted galaxy stellar mass functions are steep, and a significant amount of star formation is hosted by low-mass haloes that may be missed in current observations. Most of the stars in cluster centrals formed earlier than $z\approx 2$ but have been assembled much later. Milky Way mass galaxies have had on-going star formation without significant mergers since $z\approx 2$, and are thus free of significant (classic) bulges produced by major mergers. In massive clusters, stars bound in galaxies and scattered in the halo form a homogeneous population that is old and with solar metallicity. In contrast, in Milky Way mass systems the two components form two distinct populations, with halo stars being older and poorer in metals by a factor of $\approx 3$. Dwarf galaxies in haloes with $M_{\rm h} < 10^{11}h^{-1}M_{\odot}$ have experienced a star formation burst accompanied by major mergers at $z > 2$, followed by a nearly constant star formation rate after $z = 1$. The early burst leaves a significant old stellar population that is distributed in spheroids.Comment: 17 pages, 17 figure

Random Field Representation of Horizontal Density Distribution in Partially Oriented Strandboard Mat

A random field representation of the horizontal density distribution in partially oriented strandboard mats was investigated. The orientation of strands can be characterized by both the von Mises distribution and the uniform distribution within a range of angles. Theoretical models of the correlation coefficients of any two points simultaneously covered by one strand, characteristic area of the correlation, and the degree of orientation were developed. Results indicate that the concentration factor k = 700 is sufficiently large to represent a perfectly aligned strand arrangement in oriented strandboards based on the von Mises distribution. The correlation coefficients of two points in a mat have a lower bound (random case) and an upper bound (perfectly aligned) in both von Mises and uniform distributions. Based on the concept of characteristic area, where the minimum characteristic area is the area of the strand and the maximum characteristic area is approximate to the square of strand length, the degree of orientation in a panel can be represented as a function of characteristic area. This value is found to be very close to the percent alignment definition

Score-based generative models break the curse of dimensionality in learning a family of sub-Gaussian probability distributions

While score-based generative models (SGMs) have achieved remarkable success in enormous image generation tasks, their mathematical foundations are still limited. In this paper, we analyze the approximation and generalization of SGMs in learning a family of sub-Gaussian probability distributions. We introduce a notion of complexity for probability distributions in terms of their relative density with respect to the standard Gaussian measure. We prove that if the log-relative density can be locally approximated by a neural network whose parameters can be suitably bounded, then the distribution generated by empirical score matching approximates the target distribution in total variation with a dimension-independent rate. We illustrate our theory through examples, which include certain mixtures of Gaussians. An essential ingredient of our proof is to derive a dimension-free deep neural network approximation rate for the true score function associated with the forward process, which is interesting in its own right.Comment: 33 pages, to appear in the proceedings of 12th International Conference on Learning Representation