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The relation between two branches of solutions (radiative and subradiative) of wave equations on Minkowski spacetime is investigated, for any integer spin, in flat Bondi coordinates where remarkable simplifications occur and allow for exact boundary-to-bulk formulae. Each branch carries a unitary irreducible representation of the Poincare group, though an exotic one for the subradiative sector. These two branches of solutions are related by an inversion and, together, span a single representation of the conformal group. While radiative modes are realised in the familiar holographic way (either as boundary data at null infinity or as bulk fields with radiative asymptotic behavior), the whole tower of subradiative modes forms an indecomposable representation of the usual Poincare group, which can be encoded into a single boundary field living on an interior null cone. Lorentz transformations are realised in both cases as conformal transformations of the celestial sphere. The vector space of all subradiative modes carries a unitary representation of a group isomorphic to the Poincare group, where bulk conformal boosts play the role of bulk translations., Comment: 41+2 pages, no figure; v2: refs added, version accepted for publication in JHEP

In 1987, A. Gy\'arf\'as in his paper ``Problems from the world surrounding perfect graphs'' posed the problem of determining the smallest $\chi$-binding function for $\mathcal{G}(F,\overline{F})$, when $\mathcal{G}(F)$ is $\chi$-bounded. So far the problem has been attempted for only forest $F$ with four or five vertices. In this paper, we address the case when $F=P_3\cup 2K_1$ and show that if $G$ is a $\{P_3\cup 2K_1,\overline{P_3\cup 2K_1}\}$-free graph with $\omega(G)\neq 3$, then it admits $\omega(G)+1$ as a $\chi$-binding function. Moreover, we also construct examples to show that this bound is tight for all values of $\omega\neq 3$.

Residential heating and cooling currently account for approximately 7% of electricity consumption of India. A warming climate will increase residential cooling requirements, while heating needs will decrease which is an alarming consequence for India, which has predominantly cooling requirements. Thus, to reduce the energy and carbon footprint of Indian building sector, it is essential to assess the impact of climate change on future heating and cooling needs and develop energy-efficiency solutions. This research evaluated the effect of climate change on the heating and cooling energy needs of an archetypical residential room in India, covering all climate zones. We developed a novel approach to quantify the heating and cooling load components (walls, windows, etc.) for identifying building elements to be targeted for improving energy efficiency. Our median climate model predicted that the cooling energy demand of the room archetype would increase by 23-155% by the 2090s compared to the 1990s, depending on the city and emission scenario. Walls and windows account for over 60% of the cooling needs and should be the prime targets for energy-efficiency measures. We also predict between 45-100% reductions in the heating requirements of the archetype room by the 2090s. Walls contribute over 67% to heating needs and should be targeted for heating energy reductions.

Finding families that admit a linear $\chi$-binding function is a problem that has interested researchers for a long time. Recently, the question of finding linear subfamilies of $2K_2$-free graphs has garnered much attention. In this paper, we are interested in finding a linear subfamily of a specific superclass of $2K_2$-free graphs, namely $(P_3\cup P_2)$-free graphs. We show that the class of $\{P_3\cup P_2,gem\}$-free graphs admits $f=2\omega$ as a linear $\chi$-binding function. Furthermore, we give examples to show that the optimal $\chi$-binding function $f^*\geq \left\lceil\frac{5\omega(G)}{4}\right\rceil$ for the class of $\{P_3\cup P_2, gem\}$-free graphs and that the $\chi$-binding function $f=2\omega$ is tight when $\omega=2$ and $3$.

Cryo-Electron Microscopy (cryo-EM) has emerged as a key technology to determine the structure of proteins, particularly large protein complexes and assemblies in recent years. A key challenge in cryo-EM data analysis is to automatically reconstruct accurate protein structures from cryo-EM density maps. In this review, we briefly overview various deep learning methods for building protein structures from cryo-EM density maps, analyze their impact, and discuss the challenges of preparing high-quality data sets for training deep learning models. Looking into the future, more advanced deep learning models of effectively integrating cryo-EM data with other sources of complementary data such as protein sequences and AlphaFold-predicted structures need to be developed to further advance the field.