The stable Picard group of finite Adams Hopf algebroids with an application to the R\mathbb{R}-motivic Steenrod subalgebra A(1)R\mathcal{A}(1)^{\mathbb{R}}

In this paper, we investigate the rigidity of the stable comodule category of a specific class of Hopf algebroids known as finite Adams, shedding light on its Picard group. Then we establish a reduction process through base changes, enabling us to effectively compute the Picard group of the $\mathbb{R}$-motivic mod $2$ Steenrod subalgebra $\mathcal{A}(1)^{\mathbb{R}}$. Our computation shows that $\operatorname{Pic}(\mathcal{A}(1)^{\mathbb{R}})$ is isomorphic to $\mathbb{Z}^4$, where two ranks come from the motivic grading, one from the algebraic loop functor, and the last is generated by the $\mathbb{R}$-motivic joker $J$.Comment: 18 pages, 4 figure

The Allowed Parameter Space of a Long-lived Neutron Star as the Merger Remnant of GW170817

Due to the limited sensitivity of the current gravitational wave (GW) detectors, the central remnant of the binary neutron star (NS) merger associated with GW170817 remains an open question. In view of the relatively large total mass, it is generally proposed that the merger of GW170817 would lead to a short-lived hypermassive NS or directly produce a black hole (BH). There is no clear evidence to support or rule out a long-lived NS as the merger remnant. Here, we utilize the GW and electromagnetic (EM) signals to comprehensively investigate the parameter space that allows a long-lived NS to survive as the merger remnant of GW170817. We find that for some stiff equations of state, the merger of GW170817 could, in principle, lead to a massive NS, which has a millisecond spin period. The post-merger GW signal could hardly constrain the ellipticity of the NS. If the ellipticity reaches 10−3, in order to be compatible with the multi-band EM observations, the dipole magnetic field of the NS (B p ) is constrained to the magnetar level of ~1014 G. If the ellipticity is smaller than 10−4, B p is constrained to the level of ~109–1011 G. These conclusions weakly depend on the adoption of the NS equation of state

Modular average case analysis: Language implementation and extension

Motivated by accurate average-case analysis, MOdular Quantitative Analysis (MOQA) is developed at the Centre for Efficiency Oriented Languages (CEOL). In essence, MOQA allows the programmer to determine the average running time of a broad class of programmes directly from the code in a (semi-)automated way. The MOQA approach has the property of randomness preservation which means that applying any operation to a random structure, results in an output isomorphic to one or more random structures, which is key to systematic timing. Based on original MOQA research, we discuss the design and implementation of a new domain specific scripting language based on randomness preserving operations and random structures. It is designed to facilitate compositional timing by systematically tracking the distributions of inputs and outputs. The notion of a labelled partial order (LPO) is the basic data type in the language. The programmer uses built-in MOQA operations together with restricted control flow statements to design MOQA programs. This MOQA language is formally specified both syntactically and semantically in this thesis. A practical language interpreter implementation is provided and discussed. By analysing new algorithms and data restructuring operations, we demonstrate the wide applicability of the MOQA approach. Also we extend MOQA theory to a number of other domains besides average-case analysis. We show the strong connection between MOQA and parallel computing, reversible computing and data entropy analysis

MANIPULATING AND SIMPLIFYING THE INTERMOLECULAR INTERACTIONS IN LIQUID MIXTURES

Long ranged intermolecular interactions have significant influence on the structure of the liquid and present serious challenges for computer simulations. In particular, the long ranged tail of Coulomb interaction usually needs to be calculated using Ewald summation or related techniques in computer simulation, which can be too time consuming to be carried out for large systems. Local Molecular Field(LMF) theory has been developed to simplify long-ranged Coulomb and Van der Waals interactions for nonuniform liquids by approximating these long ranged interactions as effective static single-particle fields. Despite the success LMF theory made in describing the structure of nonuniform liquids, it is not appropriate to use LMF theory to describe the structure of uniform liquid mixtures, since the dynamically moving unbalanced forces produced in mixture can not be captured by the framework of LMF theory. In this thesis, we propose a new framework which approximates the unbalanced forces produced in a mixture as effective intermolecular interactions. This new framework can simplify the long ranged intermolecular interactions and produce a mimic system with short ranged solvent-solvent interactions, which is much easier to simulate or analyze. Based on this framework and other techniques introduced in this thesis, we have constructed a "Short Solvent Model", which has noticeable advantages compared to the explicit solvent model and implicit solvent model. This framework has also been used to simplify the interactions of phase-separating mixtures. The impact of using this framework on the diffusion dynamics of the solutes has also been discussed. Possible application of this framework and the Short Solvent Model to biopolymers folding problems is briefly discussed