Thor Catteau, Benjamin Glancy, Allen Holder
et al.
The central path revolutionized the study of optimization in the 1980s and 1990s due to its favorable convergence properties, and as such, it has been investigated analytically, algorithmically, and computationally. Past pursuits have primarily focused on linking iterative approximation algorithms to the central path in the design of efficient algorithms to solve large, and sometimes novel, optimization problems. This algorithmic intent has meant that the central path has rarely been celebrated as an aesthetic entity in low dimensions, with the only meager exceptions being illustrative examples in textbooks. We undertake this low dimensional investigation and illustrate the artistic use of the central path to create aesthetic tilings and flower-like constructs in two and three dimensions, an endeavor that combines mathematical rigor and artistic sensibilities. The result is a fanciful and enticing collection of patterns that, beyond computer generated images, supports math-aesthetic designs for novelties and museum-quality pieces of art.
Will Dumm, Mary Barker, William Howard-Snyder
et al.
In many situations, it would be useful to know not just the best phylogenetic tree for a given data set, but the collection of high-quality trees. This goal is typically addressed using Bayesian techniques, however, current Bayesian methods do not scale to large data sets. Furthermore, for large data sets with relatively low signal one cannot even store every good tree individually, especially when the trees are required to be bifurcating. In this paper, we develop a novel object called the "history subpartition directed acyclic graph" (or "history sDAG" for short) that compactly represents an ensemble of trees with labels (e.g. ancestral sequences) mapped onto the internal nodes. The history sDAG can be built efficiently and can also be efficiently trimmed to only represent maximally parsimonious trees. We show that the history sDAG allows us to find many additional equally parsimonious trees, extending combinatorially beyond the ensemble used to construct it. We argue that this object could be useful as the "skeleton" of a more complete uncertainty quantification.
Diffusion models have recently been recognised as efficient inverse problem solvers due to their ability to produce high-quality reconstruction results without relying on pairwise data training. Existing diffusion-based solvers utilize Gradient Descent strategy to get a optimal sample solution. However, these solvers only calculate the current gradient and have not utilized any history information of sampling process, thus resulting in unstable optimization progresses and suboptimal solutions. To address this issue, we propose to utilize the history information of the diffusion-based inverse solvers. In this paper, we first prove that, in previous work, using the gradient descent method to optimize the data fidelity term is convergent. Building on this, we introduce the incorporation of historical gradients into this optimization process, termed History Gradient Update (HGU). We also provide theoretical evidence that HGU ensures the convergence of the entire algorithm. It's worth noting that HGU is applicable to both pixel-based and latent-based diffusion model solvers. Experimental results demonstrate that, compared to previous sampling algorithms, sampling algorithms with HGU achieves state-of-the-art results in medical image reconstruction, surpassing even supervised learning methods. Additionally, it achieves competitive results on natural images.
The star formation and quenching of central galaxies are regulated by the assembly histories of their host halos. In this work, we use the central stellar mass to halo mass ratio as a proxy of halo formation time, and we devise three different models, from the physical hydrodynamical simulation to the empirical statistical model, to demonstrate its robustness. With this proxy, we inferred the dependence of the central galaxy properties on the formation time of their host halos using the SDSS main galaxy sample, where central galaxies are identified with the halo-based group finder. We found that central galaxies living in late-formed halos have higher quiescent fractions and lower spiral fractions than their early-formed counterparts by $\lesssim$ 8%. Finally, we demonstrate that the group finding algorithm has a negligible impact on our results.
Dark matter (DM) is predicted to be the dominant mass component in galaxies. In the central region of Early-type galaxies it is expected to account for a large amount of the total mass, although the stellar mass should still represent the majority of the mass budget, depending on the stellar Initial Mass Function (IMF). We discuss latest results on the DM fraction and mean DM density for local galaxies and explore their evolution with redshifts in the last 8 Gyr of the cosmic history. We compare these results with expectations from the $Λ$CDM model, and discuss the the role of the IMF and galaxy model, through the central total mass density slope. We finally present future perspectives offered by next generation instruments/surveys (Rubin/LSST, Euclid, CSST, WEAVE, 4MOST, DESI), that will provide the unique chance to measure the DM evolution with time for an unprecedented number of galaxies and constrain their evolutionary scenario.
The $γ$-ray deposition history in an expanding supernova (SN) ejecta has been mostly used to constrain models for Type Ia SN. Here we expand this methodology to core-collapse SNe, including stripped envelope (SE; Type Ib/Ic/IIb) and Type IIP SNe. We construct bolometric light curves using photometry from the literature and we use the Katz integral to extract the $γ$-ray deposition history. We recover the tight range of $γ$-ray escape times, $t_0\approx30-45\,\textrm{d}$, for Type Ia SNe, and we find a new tight range $t_0\approx80-140\,\textrm{d}$, for SE SNe. Type IIP SNe are clearly separated from other SNe types with $t_0\gtrsim400\,\textrm{d}$, and there is a possible negative correlation between $t_0$ and the synthesized $^{56}$Ni mass. We find that the typical masses of the synthesized $^{56}$Ni in SE SNe are larger than those in Type IIP SNe, in agreement with the results of Kushnir. This disfavours progenitors with the same initial mass range for these explosions. We recover the observed values of $ET$, the time-weighted integrated luminosity from cooling emission, for Type IIP, and we find hints of non-zero $ET$ values in some SE SNe. We apply a simple $ γ$-ray radiation transfer code to calculate the $γ$-ray deposition histories of models from the literature, and we show that the observed histories are a powerful tool for constraining models.
We have investigated the toroidal analog of ellipsoidal shells of matter, which are of great significance in Astrophysics. The exact formula for the gravitational potential $Ψ(R,Z)$ of a shell with a circular section at the pole of toroidal coordinates is first established. It depends on the mass of the shell, its main radius and axis-ratio $e$ (i.e. core-to-main radius ratio), and involves the product of the complete elliptic integrals of the first and second kinds. Next, we show that successive partial derivatives $\partial^{n +m} Ψ/\partial_{R^n} \partial_{Z^m}$ are also accessible by analytical means at that singular point, thereby enabling the expansion of the interior potential as a bivariate series. Then, we have generated approximations at orders $0$, $1$, $2$ and $3$, corresponding to increasing accuracy. Numerical experiments confirm the great reliability of the approach, in particular for small-to-moderate axis ratios ($e^2 \lesssim 0.1$ typically). In contrast with the ellipsoidal case (Newton's theorem), the potential is not uniform inside the shell cavity as a consequence of the curvature. We explain how to construct the interior potential of toroidal shells with a thick edge (i.e. tubes), and how a core stratification can be accounted for. This is a new step towards the full description of the gravitating potential and forces of tori and rings. Applications also concern electrically-charged systems, and thus go beyond the context of gravitation.
Galaxy assembly bias, the correlation between galaxy properties and halo properties at fixed halo mass, could be an important ingredient in halo-based modelling of galaxy clustering. We investigate the central galaxy assembly bias by studying the relation between various galaxy and halo properties in the Illustris hydrodynamic galaxy formation simulation. Galaxy stellar mass $\Mstar$ is found to have a tighter correlation with peak maximum halo circular velocity $\Vp$ than with halo mass $\Mh$. Once the correlation with $\Vp$ is accounted for, $\Mstar$ has nearly no dependence on any other halo assembly variables. The correlations between galaxy properties related to star formation history and halo assembly properties also show a cleaner form as a function of $\Vp$ than as a function of $\Mh$, with the main correlation being with halo formation time and to a less extent halo concentration. Based on the galaxy-halo relation, we present a simple model to relate the bias factors of a central galaxy sample and the corresponding halo sample, both selected based on assembly-related properties. It is found that they are connected by the correlation coefficient of the galaxy and halo properties used to define the two samples, which provides a reasonable description for the samples in the simulation and suggests a simple prescription to incorporate galaxy assembly bias into the halo model. By applying the model to the local galaxy clustering measurements in Lin et al. (2016), we infer that the correlation between star formation history or specific star formation rate and halo formation time is consistent with being weak.
Recent work has studied the interplay between a galaxy's history and its observable properties using "genetically modified" cosmological zoom simulations. The approach systematically generates alternative histories for a halo, while keeping its cosmological environment fixed. Applications to date altered linear properties of the initial conditions such as the mean overdensity of specified regions; we extend the formulation to include quadratic features such as local variance, which determines the overall importance of smooth accretion relative to mergers in a galaxy's history. We introduce an efficient algorithm for this new class of modification and demonstrate its ability to control the variance of a region in a one-dimensional toy model. Outcomes of this work are two-fold: (i) a clarification of the formulation of genetic modifications and (ii) a proof of concept for quadratic modifications leading the way to a forthcoming implementation in cosmological simulations.
We consider the problem of sparse signal recovery from 1-bit measurements. Due to the noise present in the acquisition and transmission process, some quantized bits may be flipped to their opposite states. These sign flips may result in severe performance degradation. In this study, a novel algorithm, termed HISTORY, is proposed. It consists of Hamming support detection and coefficients recovery. The HISTORY algorithm has high recovery accuracy and is robust to strong measurement noise. Numerical results are provided to demonstrate the effectiveness and superiority of the proposed algorithm.
We show that the simplicity constraints, which define the dynamics of spin foam models, imply, and are implied by, the first law of thermodynamics, when the latter is applied to causal diamonds in the quantum spacetime. This result reveals an intimate connection between the holographic nature of gravity, as reflected by the Bekenstein entropy, and the fact that general relativity and other gravitational theories can be understood as constrained topological field theories. To state and derive this correspondence we describe causal diamonds in the causal structure of spin foam histories and generalize arguments given for the near horizon region of black holes by Frodden, Gosh and Perez and Bianchi. This allows us to apply a recent argument of Jacobson to show that if a spin foam history has a semiclassical limit described in terms of a smooth metric geometry, that geometry satisfies the Einstein equations. These results suggest also a proposal for a quantum equivalence principle.
We explore the basic parameters that drive the evolution of the fundamental properties of star forming galaxies within the gas regulator model, or bathtub-model. We derive the general analytic form of the evolution of the key galaxy properties, i.e. gas mass, star formation rate (SFR), stellar mass, specific SFR, gas fraction, gas phase metallicity and stellar metallicity, without assuming that galaxies live in the equilibrium state. We find that the timescale required to reach equilibrium, tau_eq, which is determined by the product of star-formation efficiency and mass-loading factor, is the central parameter that is essentially in control of the evolution of all key galaxy properties. The scatters in most of the key scaling relations, such as the mass-SFR relation and mass-metallicity relation, are primarily governed by tau_eq. Most strikingly, the predicted sSFR evolution is controlled solely by tau_eq (apart from the cosmic time). Although the precise evolution of the sSFR depends on tau_eq, the sSFR history is largely insensitive to different values of tau_eq. The difference between the minimum and maximum sSFR at any epoch is less than a factor of four. The shape of the predicted sSFR history simply mimics that of the specific mass increase rate of the dark matter halos (sMIR_DM) with the typical value of the sSFR around 2*sMIR_DM, in good agreement with the predictions from typical Semi-Analytic Models (SAMs), but both are fundamentally different from the observed sSFR history. This clearly implies that some key process is missing in both typical SAMs and gas regulator model, and we hint at some possible culprit. We emphasize the critical role of tau_eq in controlling the evolution of the galaxy population, especially for gas rich low mass galaxies that are very unlikely to live around the equilibrium state at any epoch and this has been largely ignored in many similar studies.
Daily records of sunspot group areas compiled by the Royal Observatory, Greenwich, from May of 1874 through 1976 indicate a curious history for the penumbral areas of the smaller sunspot groups. On average, the ratio of penumbral area to umbral area in a sunspot group increases from 5 to 6 as the total sunspot group area increases from 100 to 2000 microHem (a microHem is a millionth the area of a solar hemisphere). This relationship does not vary substantially with sunspot group latitude or with the phase of the sunspot cycle. However, for the sunspot groups with total areas <100 microHem, this ratio changes dramatically and systematically through this historical record. The ratio for these smallest sunspots is near 5.5 from 1874 to 1900. After a rapid rise to more than 7 in 1905 it drops smoothly to less than 3 by 1930 and then rises smoothly back to more than 7 in 1961. It then returns to near 5.5 from 1965 to 1976. The smooth variation from 1905 to 1961 shows no indication of any step-like changes that might be attributed to changes in equipment or personnel. The overall level of solar activity was increasing monotonically during this time period when the penumbra-to-umbra area ratio dropped to less than half its peak value and then returned. If this history can be confirmed by other observations (e.g., Mt. Wilson or Kodaikanal) it may impact our understanding of penumbra formation, our dynamo models, and our estimates of historical changes in the solar irradiance.
If we are to develop a comprehensive and predictive theory of galaxy formation and evolution, it is essential that we obtain an accurate assessment of how and when galaxies assemble their stellar populations, and how this assembly varies with environment. There is strong observational support for the hierarchical assembly of galaxies, but our insight into this assembly comes from sifting through the resolved field populations of the surviving galaxies we see today, in order to reconstruct their star formation histories, chemical evolution, and kinematics. To obtain the detailed distribution of stellar ages and metallicities over the entire life of a galaxy, one needs multi-band photometry reaching solar-luminosity main sequence stars. The Hubble Space Telescope can obtain such data in the low-density regions of Local Group galaxies. To perform these essential studies for a fair sample of the Local Universe, we will require observational capabilities that allow us to extend the study of resolved stellar populations to much larger galaxy samples that span the full range of galaxy morphologies, while also enabling the study of the more crowded regions of relatively nearby galaxies. With such capabilities in hand, we will reveal the detailed history of star formation and chemical evolution in the universe.
We develop a theory of universal central extensions of Hom-Lie algebras. Classical results of universal central extensions of Lie algebras cannot be completely extended to Hom-Lie algebras setting, because of the composition of two central extensions is not central. This fact leads to introduce the notion of universal $α$-central extension. Classical results as the existence of a universal central extension of a perfect Hom-Lie algebra remains true, but others as the central extensions of the middle term of a universal central extension is split only holds for $α$-central extensions. A homological characterization of universal ($α$)-central extensions is given.
A general model of decentralized stochastic control called partial history sharing information structure is presented. In this model, at each step the controllers share part of their observation and control history with each other. This general model subsumes several existing models of information sharing as special cases. Based on the information commonly known to all the controllers, the decentralized problem is reformulated as an equivalent centralized problem from the perspective of a coordinator. The coordinator knows the common information and select prescriptions that map each controller's local information to its control actions. The optimal control problem at the coordinator is shown to be a partially observable Markov decision process (POMDP) which is solved using techniques from Markov decision theory. This approach provides (a) structural results for optimal strategies, and (b) a dynamic program for obtaining optimal strategies for all controllers in the original decentralized problem. Thus, this approach unifies the various ad-hoc approaches taken in the literature. In addition, the structural results on optimal control strategies obtained by the proposed approach cannot be obtained by the existing generic approach (the person-by-person approach) for obtaining structural results in decentralized problems; and the dynamic program obtained by the proposed approach is simpler than that obtained by the existing generic approach (the designer's approach) for obtaining dynamic programs in decentralized problems.