We provide a new foundational approach to the generalization of terms up to equational theories. We interpret generalization problems in a universal-algebraic setting making a key use of projective and exact algebras in the variety associated to the considered equational theory. We prove that the generality poset of a problem and its type (i.e., the cardinality of a complete set of least general solutions) can be studied in this algebraic setting. Moreover, we identify a class of varieties where the study of the generality poset can be fully reduced to the study of the congruence lattice of the 1-generated free algebra. We apply our results to varieties of algebras and to (algebraizable) logics. In particular we obtain several examples of unitary type: abelian groups; commutative monoids and commutative semigroups; all varieties whose 1-generated free algebra is trivial, e.g., lattices, semilattices, varieties without constants whose operations are idempotent; Boolean algebras, Kleene algebras, and Gödel algebras, which are the equivalent algebraic semantics of, respectively, classical, 3-valued Kleene, and Gödel-Dummett logic. Finally, we prove that the variety of MV-algebras, the equivalent algebraic semantics of Lukasiewicz logic, has nullary type.

We formalize symmetry breaking as a set-covering problem. For the case of breaking symmetries on graphs, a permutation covers a graph if applying it to the graph yields a smaller graph in a given order. Canonical graphs are those that cannot be made smaller by any permutation. A complete symmetry break is then a set of permutations that covers all non-canonical graphs. A complete symmetry break with a minimal number of permutations can be obtained by solving an optimal set-covering problem. The challenge is in the sizes of the corresponding set-covering problems and in how these can be tamed. The set-covering perspective on symmetry breaking opens up a range of new opportunities deriving from decades of studies on both precise and approximate techniques for this problem. Application of our approach leads to optimal LexLeader symmetry breaks for graphs of order $n\leq 10$ as well as to partial symmetry breaks which improve on the state-of-the-art.

A novel Integrated Sensing-Communication (ISAC) system is proposed that can accommodate high mobility scenarios while making efficient use of bandwidth for both communication and sensing. The system comprises a monostatic multiple-input multiple-output (MIMO) radar that transmits orthogonal time frequency space (OTFS) waveforms. Bandwidth efficiency is achieved by making Doppler-delay (DD) domain bins available for shared use by the transmit antennas. For maximum communication rate, all DD-domain bins are used as shared, but in this case, the target resolution is limited by the aperture of the receive array. A low-complexity method is proposed for obtaining coarse estimates of the radar targets parameters in that case. A novel approach is also proposed to construct a virtual array (VA) for achieving a target resolution higher than that allowed by the receive array. The VA is formed by enforcing zeros on certain time-frequency (TF) domain bins, thereby creating private bins assigned to specific transmit antennas. The TF signals received on these private bins are orthogonal, enabling the synthesis of a VA. When combined with coarse target estimates, this approach provides high-accuracy target estimation. To preserve DD-domain information, the introduction of private bins requires reducing the number of DD-domain symbols, resulting in a trade-off between communication rate and sensing performance. However, even a small number of private bins is sufficient to achieve significant sensing gains with minimal communication rate loss. The proposed system is robust to Doppler frequency shifts that arise in high mobility scenarios.

Spin-orbit coupling in systems with broken inversion symmetry gives rise to the Edelstein effect, which is the spin polarization induced by an electric field or current, and the inverse-Edelstein effect (also known as the spin-galvanic effect), which is the electric current induced by an oscillatory magnetic field or spin polarization. At the same time, an interplay between spin-orbit coupling and electron-electron interaction leads to a special type of collective excitations -- chiral-spin modes -- which are oscillations of spin polarization in the absence of a magnetic field. As a result, both Edelstein and inverse-Edelstein effects exhibit resonances at the frequencies of chiral-spin collective modes. Here, we present a detailed study of the effect of electron correlation on the resonances in Edelstein and inverse-Edelstein effects in a single-valley two-dimensional electron gas and in a multi-valley Dirac system with proximity-induced spin-orbit coupling. While the chiral-spin modes involve both in-plane and out-of-plane oscillations of spins, we show that only the in-plane modes are responsible for the above resonances. In the multi-valley system, electron correlation splits the in-plane modes into two. We study the spectral weight distribution between the two modes over a large parameter space of intra- and inter-valley interactions. Finally, we demonstrate that using the chiral-spin modes one can get a resonant enhancement of charge-to-spin conversion and gain a directional control of the injected spins in the spin-pumping process, both of which are relevant to spintronics.

In this paper, we combine the k-reduction map, the moment method, and the classical shadow method into a practical protocol for certifying the entanglement dimensionality. Our approach is based on the observation that a state with entanglement dimensionality at most k must stay positive under the action of the k-reduction map. The core of our protocol utilizes the moment method to determine whether the k-reduced operator, i.e., the operator obtained after applying the k-reduction map on a quantum state, contains negative eigenvalues or not. Notably, we propose a systematic method for constructing reduction moment criteria, which apply to a much wider range of states than fidelity-based methods. The performance of our approach gets better and better with the moment order employed, which is corroborated by extensive numerical simulation. To apply our approach, it suffices to implement a unitary 3-design instead of a 4-design, which is more feasible in practice than the correlation matrix method. In the course of study, we show that the k-reduction negativity, the absolute sum of the negative eigenvalues of the k-reduced operator, is monotonic under local operations and classical communication for pure states.

Spin-orbit coupling in systems with broken inversion symmetry gives rise to the Edelstein effect, which is the induced spin polarization in response to an applied electric field or current, and the inverse Edelstein effect, which is the induced electric current in response to an oscillatory magnetic field or spin polarization. At the same time, an interplay between spin-orbit coupling and electron-electron interaction leads to a special type of collective excitations -- chiral-spin modes -- which are oscillations of spin polarization in the absence of a magnetic field. As a result, both Edelstein and inverse Edelstein effects exhibit resonances at the frequencies of spin-chiral collective modes. Here, we present a detailed study of the effect of electron correlation on the Edelstein and inverse Edelstein effects in a single-valley two-dimensional electron gas and a multi-valley Dirac system with proximity-induced spin-orbit coupling. While the chiral-spin modes involve both in-plane and out-of-plane oscillations of spins, we show that only the in-plane modes are responsible for the above resonances. In the multi-valley system, electron correlation splits the in-plane modes into two. We also study the spectral weight distribution between the two resonances over a large parameter space of intra- and inter-valley interactions.

We provide novel probabilistic portrayals of two multivariate models designed to handle zero-inflation in count-compositional data. We develop a new unifying framework that represents both as finite mixture distributions. One of these distributions, based on Dirichlet-multinomial components, has been studied before, but has not yet been properly characterised as a sampling distribution of the counts. The other, based on multinomial components, is a new contribution. Using our finite mixture representations enables us to derive key statistical properties, including moments, marginal distributions, and special cases for both distributions. We develop enhanced Bayesian inference schemes with efficient Gibbs sampling updates, wherever possible, for parameters and auxiliary variables, demonstrating improvements over existing methods in the literature. We conduct simulation studies to evaluate the efficiency of the Bayesian inference procedures and present applications to a human gut microbiome dataset to illustrate the practical utility of the proposed distributions.

We show that an overtwisted contact structure on a closed, oriented 3-manifold can be defined by a contact form having a Bott-integrable Reeb flow if and only if the Poincaré dual of its Euler class is represented by a graph link.

We revisit factorizations of classical characters under various specializations, some old and some new. We first show that all characters of classical families of groups twisted by odd powers of an even primitive root of unity factorize into products of characters of smaller groups. Motivated by conjectures of Wagh and Prasad (Manuscr. Math. 2020), we then observe that certain specializations of Schur polynomials factor into products of two characters of other groups. We next show, via a detour through hook Schur polynomials, that certain Schur polynomials indexed by staircase shapes factorize into linear pieces. Lastly, we consider classical and universal characters specialized at roots of unity. One of our results, in parallel with Schur polynomials, is that universal characters take values only in $\{0, \pm 1, \pm 2\}$ at roots of unity.

We study a networked economic system composed of $n$ producers supplying a single homogeneous good to a number of geographically separated markets and of a centralized authority, called the market maker. Producers compete à la Cournot, by choosing the quantities of good to supply to each market they have access to in order to maximize their profit. Every market is characterized by its inverse demand functions returning the unit price of the considered good as a function of the total available quantity. Markets are interconnected by a dispatch network through which quantities of the considered good can flow within finite capacity constraints and possibly satisfying additional linear physical constraints. Such flows are determined by the action of a system operator, who aims at maximizing a designated welfare function. We model such competition as a strategic game with $n+1$ players: the producers and the system operator. For this game, we first establish the existence of pure-strategy Nash equilibria under standard concavity assumptions. We then identify sufficient conditions for the game to be exact potential with an essentially unique Nash equilibrium. Next, we present a general result that connects the optimal action of the system operator with the capacity constraints imposed on the network. For the commonly used Walrasian welfare, our finding proves a connection between capacity bottlenecks in the market network and the emergence of price differences between markets separated by saturated lines. This phenomenon is frequently observed in real-world scenarios, for instance in power networks. Finally, we validate the model with data from the Italian day-ahead electricity market.

Nonanalytic features are interesting in physics as they carry valuable information about the physical properties of the system. These properties manifest themselves in observables containing a one- or two-particle spectral function. In this work, we use a stationary-point analysis to deduce the nonanalytic features of spectral functions that appear while computing dynamical correlation functions. We focus on the correlation functions relevant to inelastic light scattering from anisotropic superconductors and show that nodal regions of the order parameters are, quite generally, associated with linear-in-frequency scaling at low frequencies, the minima points of the order parameters are associated with step jumps, while the maxima points are associated with $\ln$ singularities. Despite this general association, we show that depending on the anisotropy of the light-scattering vertex, these features can manifest themselves as various power laws, and even not remain singular at all. We demonstrate the conditions under which these happen. We are also able to demonstrate that the association of these nonanalytic features with the extrema and nodal points of the order parameter is, in fact, derived from a more universal behaviour of functions near parabolic-like and saddle-like stationary points. We provide a general prescription that maps the universal behaviour of systems near such stationary points to different scenarios. The approach is readily extendable to other types of spectral functions and we exemplify it by also analyzing the density of states on a square lattice.

This article introduces the $L_p$-Gauss dual curvature measure and proposes its related $L_p$-Gauss dual Minkowski problem as: for $p,q\in\mathbb{R}$, under what necessary and/or sufficient condition on a non-zero finite Borel measure $μ$ on unit sphere does there exist a convex body $K$ such that $μ$ is the $L_p$ Gauss dual curvature measure? If $K$ exists, to what extent is it unique? This problem amounts to solving a class of Monge-Ampère type equations on unit sphere in smooth case: \begin{align} e^{-\frac{|\nabla h_K|^2+h_K^2}{2}}h_K^{1-p} (|\nabla h_K|^2+h_K^2)^{\frac{q-n}{2}} \det(\nabla^2h_K+h_KI)=f,\qquad (0.1) \end{align} where $f$ is a given positive smooth function on unit sphere, $h_k$ is the support function of convex body $K$, $\nabla h_K$ and $\nabla^2h_K$ are the gradient and Hessian of $h_K$ on unit sphere with respect to an orthonormal basis, and $I$ is the identity matrix. We confirm the existence of solution to the new problem with $p,q>0$ and the existence of smooth solution to the equation (0.1) with $p ,q\in\mathbb{R}$ by variational method and Gaussian curvature flow method, respectively. Furthermore, the uniqueness of solution to the equation (0.1) in the case $p,q\in\mathbb{R}$ with $q<p$ is established.

Simulating quantum many-body systems represents a fundamental challenge where classical machine learning methods are severely bottlenecked by the exponential curse of dimensionality. Variational Quantum Algorithms (VQAs) offer a native paradigm to tackle this by optimizing parameterized unitary evolutions to find the ground states of problem Hamiltonians. However, the efficacy of these VQA is deeply hindered by the challenge of balancing the preservation of critical physical symmetries with the strict constraints of hardware implementability. In this work, we address this dilemma by proposing a hardware-efficient, symmetry-preserving ansatz fortified with complete theoretical guarantees for fermionic systems, termed the Hamming Weight Preserving (HWP) ansatz. We establish the necessary and sufficient conditions for 2-local HWP operators to achieve subspace universality, formally debunking the prevailing assumption that truncation-free simulation requires complex high-order interactions. Empirical validations corroborate our theoretical guarantees, showcasing the exact approximation of arbitrary unitary matrices within the HWP subspace. Crucially, we demonstrate the exceptional versatility of the proposed approach by deploying the exact same ansatz across distinct fermionic models, including diverse molecular electronic structures and the Fermi-Hubbard model. Our proposed HWP ansatz consistently suppresses ground-state energy errors below $1 \times 10^{-10}$ Ha, achieving a level of precision that surpasses the stringent threshold of chemical accuracy by multiple orders of magnitude. This work establishes a complete, theoretically fortified 2-local framework for symmetry-preserving computation, offering a highly universal and hardware-efficient building block for advancing quantum machine learning and fermionic many-body simulations.

Pressure-solution is a chemo-mechanical process, involving dissolution at grain/asperity contacts and precipitation away from them. It induces a compaction in time of rocks and sediments. The present study investigates numerically the impact of precipitation on the slowdown of creep behavior induced by pressure-solution. A recently published framework, called the Phase-Field Discrete Element Model, is carefully calibrated against existing indentation experiments and validated for other rate-limiting scenarios. It is shown that when precipitation is relatively slow, the slowdown of pressure-solution is due to a chemical mechanism (accumulation of solute concentration within the pore space), whereas, at fast precipitation, the slowdown is due to a mechanical mechanism (stress reduction at the contact).

While grey-body factors for a test scalar field in stringy black holes described by the renowned Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) solution have been analyzed in the literature, no such analysis exists for gravitons, likely due to the complexity of the perturbation equations. In this study, we utilize known data on quasinormal modes and the relationship between quasinormal modes and grey-body factors to derive these factors for gravitational and electromagnetic perturbations of the dilaton black hole. Our findings indicate that grey-body factors are significantly suppressed by the dilaton parameter as the black hole's charge approaches its extreme value. The iso-spectrality between axial and polar channels of perturbations is broken in the presence of the dilaton field, which leads to different grey-body factors for different types of perturbations.

This paper addresses the secure state estimation problem for continuous linear time-invariant systems with non-periodic and asynchronous sampled measurements, where the sensors need to transmit not only measurements but also sampling time-stamps to the fusion center. This measurement and communication setup is well-suited for operating large-scale control systems and, at the same time, introduces new vulnerabilities that can be exploited by adversaries through (i) manipulation of measurements, (ii) manipulation of time-stamps, (iii) elimination of measurements, (iv) generation of completely new false measurements, or a combination of these attacks. To mitigate these attacks, we propose a decentralized estimation algorithm in which each sensor maintains its local state estimate asynchronously based on its measurements. The local states are synchronized through time prediction and fused after time-stamp alignment. In the absence of attacks, state estimates are proven to recover the optimal Kalman estimates by solving a weighted least square problem. In the presence of attacks, solving this weighted least square problem with the aid of $\ell_1$ regularization provides secure state estimates with uniformly bounded error under an observability redundancy assumption. The effectiveness of the proposed algorithm is demonstrated using a benchmark example of the IEEE 14-bus system.

Dark axion portal connects an axion-like-particle (ALP), which can be the QCD axion, with the SM by the coupling to a photon and a dark photon, leading to a rich and distinct phenomenology related to dark matter, astrophysics, and cosmology. We note that due to the gauge invariance, the $Z$ boson-dark photon-ALP coupling should also be generated with sizable strength, and we study its phenomenological consequences at $Z$ boson factories: LEP, FCC-ee, and forward physics detectors at the LHC and FPF@FCC - FASER and MATHUSLA. Due to the large number of $Z$ bosons produced, DAP particles can be efficiently produced and detected by semi-visible displaced decays of the heavier DS species to $γ$+inv. or $f^+ f^-$+inv., and via missing energy signature with zero or one photon. Because of the complementarity of the two approaches, we find great prospects in exploring both short and long-lived lifetime regimes of DAP, especially when the heavier dark species has mass above $0.1\,$GeV.

Homeomorphism types of compression bodies form the vertices of a graph where two vertices are joined by an edge if one compression body is obtained by gluing a $2$-handle onto the other. Motivated by earlier work of Lackenby and Purcell on geodesicity of unknotting tunnels for hyperbolic links, we show that it is possible to realise all of the edges in the graph of compression bodies by paths of cone manifold holonomy groups such that the handle that is glued in is obtained as a limit of singular arcs of cone angle increasing from $0$ to $ 2π$. We apply standard techniques from the theory of $ \mathrm{CAT}(0) $ spaces, and do not rely on the harmonic deformation theory of Hodgson and Kerckhoff. Along the way we prove a generalisation of a classic theorem of Koebe and Maskit on existence of function groups which implies existence results for reflex angled hyperbolic cone structures on a wide range of compression bodies.

This paper explores the behavior of the torsional rigidity of a precompact domain as the ambient manifold evolves under a geometric flow. Specifically, we derive bounds on torsional rigidity under the Ricci Flow for Heisenberg spaces and homogeneous spheres. Additionally, we establish bounds under the Inverse Mean Curvature Flow for strictly convex, free-boundary, disk-type hypersurfaces within a ball. In this latter case, by extending the analysis to the maximal existence time of the flow, we obtain inequalities of comparison with the flat disk for both volume and torsional rigidity.

The Oja depth (simplicial volume depth) is one of the classical statistical techniques for measuring the central tendency of data in multivariate space. Despite the widespread emergence of object data like images, texts, matrices or graphs, a well-developed and suitable version of Oja depth for object data is lacking. To address this shortcoming, a novel measure of statistical depth, the metric Oja depth applicable to any object data, is proposed. Two competing strategies are used for optimizing metric depth functions, i.e., finding the deepest objects with respect to them. The performance of the metric Oja depth is compared with three other depth functions (half-space, lens, and spatial) in diverse data scenarios. Keywords: Object Data, Metric Oja depth, Statistical depth, Optimization, Metric statistics

We establish the exponential clustering of correlation functions for the high-temperature Gibbs states of Bose-Hubbard type models. To overcome the technical difficulties arising from the unboundedness of bosonic operators, we develop the interaction-picture cluster-expansion technique. This method also allows us to systematically bound the moments of the local particle number. This result provides an analytical justification for the low-boson-density condition frequently assumed in the study of bosonic many-body systems. As direct mathematical consequences of the clustering property, we derive a uniform upper bound on the specific heat density and establish a bosonic thermal area law with improved temperature dependence.

A MIMO dual-function radar communication (DFRC) system transmitting orthogonal time frequency space (OTFS) waveforms is considered. A key advantage of MIMO radar is its ability to create a virtual array, achieving higher sensing resolution than the physical receive array. In this paper, we propose a novel approach to construct a virtual array for the system under consideration. The transmit antennas can use the Doppler-delay (DD) domain bins in a shared fashion. A number of Time-Frequency (TF) bins, referred to as private bins, are exclusively assigned to specific transmit antennas. The TF signals received on the private bins are orthogonal and thus can be used to synthesize a virtual array, which, combined with coarse knowledge of radar parameters (i.e., angle, range, and velocity), enables high-resolution estimation of those parameters. The introduction of $N_p$ private bins necessitates a reduction in DD domain symbols, thereby reducing the data rate of each transmit antenna by $N_p-1$. However, even a small number of private bins is sufficient to achieve significant sensing gains with minimal communication rate loss.

Van der Waals layered ferromagnetic compounds with high two-dimensional electronic conductivity holds strong potential for quantum computing, future unconventional superconductors, catalysts, batteries, and fuel cells. We suggest a minimal theoretical model to understand the magnetic properties of the metal-organic framework CrCl$_2$(pyz)$_2$ (pyz=pyrazine). Using a Hubbard model we show that the groundstate is dominated by a specific configuration of delocalized electrons on the pyz sites with a ferrimagnetic coupling to the localized spins on the Cr sites. This model suggests a magnetic moment of $2μ_B$ which is remarkably close to the experimental value of $1.8 μ_B$ [K. S. Pedersen et al., Nat. Chem. 10, 1056-1061 (2018)]. From Weiss mean-field theory we predict a weak ferromagnetic Cr-Cr coupling of $\approx 0.9$ meV. This is consolidated by second order perturbation theory of the RKKY interaction yielding a Cr-Cr coupling of $\approx 5$ meV. Understanding the interactions in these types of compounds can facilitate designs of metal-organic compounds with tailored magnetic properties.

Our main result is a combinatorial characterization of when a horospherical variety has (at worst) quotient singularities. Using this characterization, we show that every quasiprojective horospherical variety with quotient singularities is globally the quotient of a smooth variety by a finite abelian group.

Squeezed states of light belong to the most prominent nonclassical resources. They have compelling applications in metrology, which has been demonstrated by their routine exploitation for improving the sensitivity of a gravitational-wave detector since 2010. Here, we report on the direct measurement of 15 dB squeezed vacuum states of light and their application to calibrate the quantum efficiency of photoelectric detection. The object of calibration is a customized InGaAs positive intrinsic negative (p-i-n) photodiode optimized for high external quantum efficiency. The calibration yields a value of 99.5% with a 0.5% (k = 2) uncertainty for a photon flux of the order 10^17/s at a wavelength of 1064 nm. The calibration neither requires any standard nor knowledge of the incident light power and thus represents a valuable application of squeezed states of light in quantum metrology.

The study of the structure and mechanical properties of complex fluid interfaces has gained increasing interest in recent decades as a result of its significant scientific relevance to the understanding of biological systems, drug development, and industrial applications. The in situ combination of molecular-level structural measurements with the assessment of dynamical (rheological) properties is particularly valuable, as comparing measurements conducted on separate samples under challenging-to-reproduce experimental conditions can be problematic. In this work, we present a new sample environment at the FIGARO instrument, the horizontal neutron reflectometer at the Institut Laue-Langevin, which includes an interfacial shear rheometer operating in the Double Wall-Ring (DWR) geometry, compatible with commercial rotational rheometers. This innovative setup enables simultaneous structural (neutron reflectometry) and dynamical (shear interfacial rheology) measurements on the same sample.

We construct several models with multiple quantum many-body scars (QMBS) using integrable boundary states (IBS). Specifically, we focus on the tilted Néel states, which are parametrized IBS for the spin-1/2 Heisenberg chain, and show that these states can be used to construct a tower of scar states. Our models exhibit periodic revival dynamics, showcasing a characteristic behavior of superpositions of QMBS. Furthermore, the tower of QMBS found in this study possesses a restricted spectrum generating algebra (RSGA) structure, indicating that QMBS are equally spaced in energy. This approach can be extended to two-dimensional models, which can be decomposed into an array of one-dimensional models. In this case, the tilted Néel states again serve as parent states for multiple scar states. These states demonstrate low entanglement entropy, marking them as exact scar states. Notably, their entanglement entropy adheres to the sub-volume law, further solidifying the nonthermal properties of QMBS. Our results provide novel insights into constructing QMBS using IBS, thereby illuminating the connection between QMBS and integrable models.

In MPLS, packets are encapsulated with labels that add domain-specific forwarding information. Special purpose labels were introduced to trigger special behavior in MPLS nodes but their number is limited. Therefore, the IETF proposed the MPLS Network Actions (MNA) framework. It extends MPLS with new features, some of which have already been defined to support relevant use cases. This paper provides a comprehensive technological overview of MNA concepts and use cases. It compares MNA to IPv6 extension headers (EHs) that serve a similar purpose, and argues that MNA can be better deployed than EHs. It then presents P4-MNA, a first hardware implementation running at 400 Gb/s per port. Scalability and performance of P4-MNA are evaluated, showing negligible impact on processing delay caused by network actions. Moreover, the applicability of MNA is demonstrated by implementing the use cases of link-specific packet loss measurement using the alternate-marking-method (AMM) and bandwidth reservation using network slicing. We identify header stacking constraints resulting from hardware resources and from the number of network actions that must be supported according to the MNA encoding. They make an implementation for hardware that can only parse a few MPLS headers infeasible. We propose to make the number of supported network actions a node parameter and signal this in the network. Then, an upgrade to MNA is also feasible for hardware with fewer available resources. We explain that for MNA with in-stack data (ISD), some header bits must remain unchanged during forwarding, and give an outlook on post-stack data (PSD).

For any given submersion $π:X\to B$ with closed, oriented and spin$^c$ fibers of even dimension, equipped with a Riemannian and differential spin$^c$ structure, we apply the Atiyah-Singer-Gorokhovsky-Lott approach to the local family index theorem without the kernel bundle assumption to construct an analytic index $\textrm{ind}^a_k$ in odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory at the cocycle level. This is achieved by associating to every cocycle $(\mathbf{E}, \mathbf{F}, α)$ of the odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory group of $X$ a cocycle $\textrm{ind}^a_k(\mathbf{E}, \mathbf{F}, α)$ of the odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory group of $B$. We also prove a Riemann-Roch-Grothendieck-type formula in odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory, which expresses the Cheeger-Chern-Simons form of $\textrm{ind}^a_k(\mathbf{E}, \mathbf{F}, α)$ in terms of that of $(\mathbf{E}, \mathbf{F}, α)$. Furthermore, we show that the analytic index $\textrm{ind}^a_k$ and the Riemann-Roch-Grothendieck-type formula in odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory refine the underlying geometric bundle of the analytic index and the Riemann-Roch-Grothendieck theorem in $\mathbb{R}/\mathbb{Z}$ $K$-theory, respectively.

The rapid growth of virtual reality (VR) has led to increased use of social VR platforms for interaction. However, these platforms lack adequate features to support blind and low vision (BLV) users, posing significant challenges in navigation, visual interpretation, and social interaction. One promising approach to these challenges is employing human guides in VR. However, this approach faces limitations with a lack of availability of humans to serve as guides, or the inability to customize the guidance a user receives from the human guide. We introduce an AI-powered guide to address these limitations. The AI guide features six personas, each offering unique behaviors and appearances to meet diverse user needs, along with visual interpretation and navigation assistance. We aim to use this AI guide in the future to help us understand BLV users' preferences for guide forms and functionalities.