The goal of this paper is to explicitly compute the Kodaira-Spencer maps over Hilbert-Siegel modular varieties and twisted Hilbert modular varieties and their effects on the metrics of the Hodge bundle. Our result is a generalization of the result in \cite{ arXiv:2205.11334 }.

We establish and advocate for a novel branch of category theory, centered around strong dinatural transformations (herein known as "paranatural transformations"). Paranatural transformations generalize natural transformations to mixed-variant difunctors, but, unlike other such generalizations, are composable and exceptionally well-behaved. We define the category of difunctors and paranatural transformations, prove a novel "diYoneda Lemma" for this category, and explore some of the category-theoretic implications. We also develop three compelling uses for paranatural category theory: parametric polymorphism, impredicative encodings of (co)inductive types, and difunctor models of type theory. Paranatural transformations capture the essence of parametricity, with their "paranaturality condition" coinciding exactly with the "free theorem" of the corresponding polymorphic type; the paranatural analogue of the (co)end calculus provides an elegant and general framework for reasoning about initial algebras, terminal coalgebras, bisimulations, and representation independence; and "diYoneda reasoning" facilitates the lifting of Grothendieck universes into difunctor models of type theory. We develop these topics and propose further avenues of research.

Stability of power electronic converters connected to power grids is commonly assessed by using the impedance criterion while the stability of power grids is typically analysed by using the network state-space representation. It is known that the impedance criterion may lead to erroneous results if the grid frequency dynamics are not considered while eigenvalue analysis is considered as a reliable method for system stability assessment. The equivalence between these two methods has been recently explored, without considering the effect of network frequency variations. Additionally, the link of the impedance criterion with the power-frequency dynamics of power systems also remains largely unexplored. In this paper, the equivalency between the impedance method considering the grid frequency dynamics and the conventional eigenvalue analysis is demonstrated. In addition, the dynamic interaction between the apparent power flow and the network fundamental frequency is formulated and its link with the impedance representation is shown. It is demonstrated that, by using the impedance representation with the network frequency as an additional input port, the network frequency perturbation plot (NFP) can be intuitively expressed by using quantities consistent with the impedance analysis framework. The main findings are verified using detailed numerical simulations of two representative systems.

We introduce a non-parametric density estimator deemed Radial Voronoi Density Estimator (RVDE). RVDE is grounded in the geometry of Voronoi tessellations and as such benefits from local geometric adaptiveness and broad convergence properties. Due to its radial definition RVDE is continuous and computable in linear time with respect to the dataset size. This amends for the main shortcomings of previously studied VDEs, which are highly discontinuous and computationally expensive. We provide a theoretical study of the modes of RVDE as well as an empirical investigation of its performance on high-dimensional data. Results show that RVDE outperforms other non-parametric density estimators, including recently introduced VDEs.

This paper investigates the limit behavior of the Minkowski sums for sequences of set-valued random variables. When the underlying space is finite dimensional, by using the support function, we establish the weak and strong laws of large numbers for non-independent random variables in the hyperspace with respect to the Hausdorff metric $d_H$.

We use N-body cosmological simulations and empirical galaxy models to study the merger history of dwarf-mass galaxies (with M_halo~10^10 M_Sun). Our input galaxy models describe the stellar mass-halo mass relation, and the galaxy occupation fraction. The number of major and minor mergers depends on the type of dark matter; in particular, minor mergers are greatly suppressed in warm dark matter models. In addition, the number of mergers that bring in stars is strongly dependent on the galaxy occupation model. For example, minor mergers are negligible for stellar halo growth in models with a high mass threshold for galaxy formation (i.e. 10^9.3 M_Sun at z=0). Moreover, this threshold for galaxy formation can also determine the relative difference (if any) between the stellar haloes of satellite and field dwarfs. Using isolated simulations of dwarf-dwarf mergers, we show that the relative frequency of major and minor mergers predict very different stellar haloes: Typically, "intermediate" dark matter merger ratios (~1:5) maximise the growth of distant stellar haloes. We discuss the observability of dwarf stellar haloes and find that the surface brightness of these features are incredibly faint. However, when several dwarfs are stacked together models that form particularly rich stellar haloes could be detectable. Finally, we show that stellar streams in the Galactic halo overlapping in phase-space with known dwarf satellites are likely remnants of their stripped stellar haloes. The mere existence of dwarf stellar haloes can already put constraints on some small-scale models, and thus observational probes should be a high priority.

In this article, we consider a random access scheme for an indoor Internet of Things (IoT) framework that uses optical wireless communication (OWC). We focus on a Slotted ALOHA (SA)-based solution where a number of OWC IoT users contend to send data to a central OWC receiver. In any given slot, and for a randomly selected active user, we consider the reliability of decoding the user's data packet at the receiver. This is done by deriving the signal-to-noise-and-interference-ratio (SINR) statistics from a randomly chosen user and evaluating the probability that the user's SINR is below a given threshold. By placing our analysis in the context of an indoor OWC IoT uplink setup, and employing the standard OWC channel model, we investigate the trade-offs between the reliability and the OWC system parameters such as the cell area or the transmitter's semi-angle. We obtain valuable insights into the design of an SA-based random access solution for a typical indoor OWC cell.

Following the trends of index modulated (IM) techniques for optical communications, in the last few years several new waveform proposals have been made, aiming at conveying a higher density of information by driving different signal properties. One of these proposals mixes multi-pulse pulse-position modulation (MPPM) and quadrature amplitude modulation (QAM) in a system called QAM-MPPM. We present here a new way to demodulate its compound waveform, and, for the non-turbulent free space optical (FSO) channel, we provide accurate analytical expressions for its error probabilities, both in the case of the traditional and the new detector. Based on these analytical derivations, we also provide simplified expressions for the estimation of the error probabilities. We show that the new detector offers a gain of some tenths of dB in signal-to-noise ratio with respect to the previously defined one without an increase in complexity, and that our error probability estimations are more accurate than previously published results. To the best of our knowledge, this work is the first to provide simulation results validating the study of the error probability performance of QAM-MPPM.

We count the number of regular supersymmetric solutions in supergravity, called superstrata, that represent non-linear completion of linear fluctuations around empty AdS_3 x S^3. These solutions carry the same charges as the D1-D5-P black hole and represent its microstates. We estimate the entropy using thermodynamic approximation and find that it is parametrically smaller than the area-entropy of the D1-D5-P black hole. Therefore, these superstrata based on AdS_3 x S^3 are not typical microstates of the black hole. What are missing in the superstrata based on AdS_3 x S^3 are higher and fractional modes in the dual CFT language. We speculate on what kind of other configurations to look at as possible realization of those modes in gravity picture, such as superstrata based on other geometries, as well as other brane configurations.

Bound states in the continuum (BICs), exceptional points (EPs), and unidirectional guided resonances (UGRs) are key concepts in non-Hermitian and radiative photonics, enabling high-$Q$ resonances, topological singularities, and direction-selective emission. Here we show that vertical-symmetry breaking provides a unified route to control all three in photonic crystal slabs. Using a minimal two-mode non-Hermitian Hamiltonian, we show that symmetry breaking simultaneously induces near-field coherent coupling and far-field radiative-channel mixing between leaky modes of opposite vertical parity. The coherent coupling governs the weak-to-strong coupling transition and EP formation, whereas the radiation-overlap parameter controls Friedrich--Wintgen loss exchange and quasi-BIC linewidth suppression. Resolving the model into top and bottom radiation channels gives a direct criterion for UGRs as hybrid states that are dark on one side only, placing BICs, quasi-BICs, and UGRs on a common footing. We validate this framework numerically and experimentally in square-lattice SiN$_x$-on-SiO$_2$ photonic crystal slabs, where tuning the superstrate index drives an EP and partial etching produces a broad quasi-BIC plateau with strongly direction-resolved radiation. Finally, we extend the same radiation-vector picture to a laterally shifted bilayer grating, where a symmetry-protected BIC, four UGRs, and quasi-BICs emerge along a continuous pathway. Our results establish vertical-symmetry breaking as a general strategy for engineering EPs, quasi-BICs, and directional UGRs in single-layer and multilayer photonic crystal platforms.

We study an extension of first-order logic that allows to express cardinality conditions in a similar way as SQL's COUNT operator. The corresponding logic FOC(P) was introduced by Kuske and Schweikardt (LICS 2017), who showed that query evaluation for this logic is fixed-parameter tractable on classes of databases of bounded degree. In the present paper, we first show that the fixed-parameter tractability of FOC(P) cannot even be generalised to very simple classes of databases of unbounded degree, such as unranked trees or strings with a linear order relation. Then we identify a fragment FOC_1(P) of FOC(P) which is still sufficiently strong to express standard applications of SQL's COUNT operator. Our main result shows that query evaluation for FOC_1(P) is fixed-parameter tractable on nowhere-dense classes of databases.

We investigate two-parameter solutions of sigma-models on two dimensional symmetric spaces contained in E11. The sigma-models considered here are not proposed as fundamental two-dimensional quantum field theories, but as auxiliary, solution-generating constructions extending the one-parameter brane sigma-model framework. Embedding such sigma-model solutions in space-time gives classical bosonic solutions of M*- and M'-theory whose warp factors are travelling wavefunctions solving a transverse wave equation, rather than the harmonic functions appearing in the standard one-parameter brane solutions. Weyl reflection allows such solutions to be mapped to M-theory solutions where the wave functions depend explicitly on extra coordinates contained in the fundamental representation of E11.

We prove the Breuil-Mézard conjecture for 2-dimensional potentially Barsotti-Tate representations of the absolute Galois group G_K, K a finite extension of Q_p, for any p>2 (up to the question of determining precise values for the multiplicities that occur). In the case that K/Q_p is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serre's conjecture, proving a variety of results including the Buzzard-Diamond-Jarvis conjecture.