We present a new family of information-theoretic generalization bounds within the framework of conditional mutual information (CMI). Most of our results are established based on the leave-$m$-out (L$m$O) cross-validation error, with $m$ denoting the number of the hold-out supersamples. Under this setting, we propose a unified CMI-based bound, allowing to envelop and reproduce many known CMI-based bounds and also bridge the gap between the MI- and CMI-based bounds when $m$ tends to infinity. The proposed framework not only provides a unified description of the existing bounds but also develops new, sharper bounds. We show the benefits of the proposed bounds through several simple examples, where the existing results are either inapplicable or looser. Moreover, under the premise that the loss function is bounded, we tighten the CMI quantities involved in the proposed bounds by reducing the number of conditional terms, thereby enhancing the proposed framework. We show empirically that the resulting new bounds improve upon the previously known ones.

The rise of decentralized social networks (DSNs), and in particular the rapid uptake of the Fediverse (e.g., Pleroma, Mastodon, Lemygrad), introduces new challenges in content moderation. Independent instances host their own data, follow different moderation policies, and often observe only partial views of conversations. We present DeTox-Fed, a federated graph-learning framework for detecting toxic conversations in DSNs without requiring instances to share raw conversations or moderation labels. Each instance constructs a local conversation graph, where nodes represent conversation trees and edges capture shared user participation across conversations. A Graph Neural Network (GNN) is then trained in a federated learning setup, allowing instances to collaboratively learn a toxicity classifier while preserving data locality. Unlike text-only moderation approaches, DeTox-Fed combines conversational structure, user-interaction patterns, conversation-level statistics, and aggregate sentiment signals. We evaluate the framework on a large Pleroma conversation dataset and show that it achieves stable toxic conversation detection under limited local labels, partial client participation, and varying moderation thresholds. Our results indicate that federated graph-based moderation is a promising direction for semi-automated moderation in decentralized social networks.

These proceedings summarize recent CMS measurements at the LHC that extract the squared speed of sound, $c_s^2$, of strongly interacting matter at extreme temperatures from the multiplicity dependence of the mean transverse momentum in ultra-central lead-lead (PbPb) collisions at $\sqrt{s_{\mathrm{NN}}} = 5.02\ \mathrm{TeV}$. The analysis yields $c_s^2 = 0.241 \pm 0.002\, (\mathrm{stat}) \pm 0.016\, (\mathrm{syst})$ at an effective temperature of $T_{\mathrm{eff}} = 219 \pm 8\, (\mathrm{syst})\,\mathrm{MeV}$, in good agreement with lattice-QCD calculations. Complementary studies in proton-lead (pPb) collisions are also presented to investigate possible quark-gluon plasma signatures in smaller systems.

In this paper, we propose an end-to-end transcoding pipeline, to create 3D Gaussian splatting (3DGS) models from existing 3D plenoptic point cloud or mesh models, when the original multi-view images of the captured 3D object or scene are not available. We also propose a custom initialisation to guide the 3DGS model learning, with constraints to ensure that the final 3DGS model aligns closely with the input point cloud or mesh surface. Tests on a high-quality, standard plenoptic point cloud dataset show that our pipeline produces 3DGS models of high visual quality, with many fewer splats than points in the original dense point clouds. Additionally, our custom initialisation leads to much faster convergence and cleaner surface representation than when starting from the default SfM-based initialisation that is typically used for 3DGS model learning.

We investigate a generalized entropic cosmology obtained by applying the gravity-thermodynamics conjecture to the Universe horizon using Kaniadakis statistics, namely a relativistic extension of the standard Boltzmann--Gibbs formalism. The resulting deformation of the horizon entropy naturally modifies the Friedmann dynamics and provides a phenomenologically consistent extension of the $Λ$CDM paradigm. Within this framework, we explore the implications of the modified cosmological dynamics for the physics of the early Universe, focusing in particular on primordial gravitational waves (PGWs) and slow-roll inflation in a Starobinsky-like scenario. We show that the generalized entropic corrections simultaneously affect the evolution of tensor perturbations and the inflationary slow-roll dynamics, inducing characteristic deviations in the PGW spectrum as well as nontrivial corrections to the main inflationary observables. By confronting the theoretical predictions with the latest Planck and BICEP/Keck observations, we derive stringent constraints on the Kaniadakis parameter and assess the observational viability of the model. Our results establish a direct connection between generalized horizon thermodynamics and inflationary cosmology, showing that quantum-statistical modifications of the entropy-area law can propagate into potentially observable signatures in the physics of the early Universe.

For dynamical systems satisfying the approximate $\mathbb{Z}^{d}$ or $\mathbb{Z}_+^{d}$-product property and asymptotically entropy expansiveness, we establish a precise description of the structure of their space of invariant measures. In particular, we prove that the set of ergodic measures with any given intermediate entropy is generic in certain natural subspaces. As a consequence, this result confirms Katok's conjecture on the existence of ergodic measures with arbitrary intermediate entropy for such systems.

Stochastic Galerkin methods offer unexplored potential for the numerical simulation of parabolic problems with random variables, in particular if they are combined with variational discretizations of the space and time variables. Due to the high dimensionality, the solution of the arising algebraic systems do not become feasible without efficient solvers, preconditioners, and software architectures. A stochastic Galerkin discretization with an embedded slabwise finite element approximation of the space and time variables is proposed and analyzed numerically. For solving the linear systems, GMRES iterations are block-preconditioned by a geometric multigrid (GMG) technique using a local Vanka smoother for the space-time subsystems. Monte-Carlo methods are also used for solving random parabolic problems and studied here for the purpose of comparison. The Monte-Carlo approach is built on the space-time finite element formulation together with the GMRES-GMG solver technology. All algorithms have been implemented in a unified matrix-free framework based on the deal.II software library. Comparative numerical evaluations illustrate the performance properties of both approaches, including convergence of the discretizations and statistics of the algebraic solver. Superiority of the stochastic Galerkin approach is observed.

In this work, we propose and study in depth a universal quantum computing architecture based on a quantum construction of transistors. Our teleportation-based quantum transistors, called ``telesistors'', are ground states of systems with symmetry-protected topological order, hence suppress certain noises and provide high-fidelity Clifford gates without the need for active error correction. This physical protection, quantified by the string order parameters, serves as a low-overhead foundation upon which conventional fault-tolerant encoding (e.g., with stabilizer codes) can be built to achieve universal quantum computation. This architecture shows rich connections with current known architectures, and some desirable merits especially compared with the qubit-based circuits regarding modularity, integration, and program storage. Our study shows that it is plausible to realize it with current technology in the near future.

Data-Driven Continuum Mechanics -- the continuous counterpart of Data-Driven Computational Mechanics -- is a modern paradigm that enhances classical continuum mechanics by incorporating finite sets of experimental material data directly, avoiding any form of constitutive modeling. Despite recent progress, its analytical foundations remain at an early stage. In this work, we establish a rigorous functional-analytic framework for the data-driven stress problem under purely homogeneous normal Neumann boundary conditions. The problem is formulated as finding a stress field (satisfying the balance of linear and angular momenta and the boundary conditions) that is closest, in an $L^p$-sense, to an auxiliary stress field that is simultaneously sought and locally resembles a finite discrete set of experimental stress states. Our analysis relies on two key ingredients. First, the divergence operator induces a topological isomorphism between the space of symmetric stress fields modulo its kernel and the space of loads balanced by rigid-body motions, ensuring the existence of an equilibrated response. Second, the finiteness of the material data set guarantees proximinality in the stress space, which in turn yields a complete existence and uniqueness theory for solution equivalence classes. Together, these two properties provide a rigorous mathematical foundation for the data-driven stress problem under purely homogeneous normal Neumann boundary conditions.

For many years I have taught an advanced statistical inference course for master's students using the text of Casella and Berger (2002). The book gives a comprehensive treatment of the core topics at a level that avoids measure theory while remaining mathematically precise, but it does not cover the increasingly important concept of copulas. The present notes are intended to complement the book by adding two sections on copulas in a style that is as close as possible to that of the original text. Numbering of definitions, theorems, examples, and exercises is consistent with Casella and Berger (2002), but the material may also be read as a brief, stand-alone introduction to copula theory.

A modified gravitational model whose action is given by an arbitrary function of the Ricci scalar, the matter Lagrangian density, a scalar field, and its kinetic term is investigated as an extension of the gravitational sector including an additional dynamical degree of freedom. Within this framework, the causal structure of rotating cosmological solutions is analyzed by considering a cylindrically symmetric Pertov-type N space-times and an axially symmetric Petrov type-III with a cosmological constant as background geometries used as theoretical probes of the model consistency. In both cases, pure radiation as matter sources are examined, including a scalar-field configurations. We demonstrate that, although the considered space-times are exact solutions to the field equations of general relativity with a matter source, they are inconsistent within the modified gravity theory considered here.

We determine cuspidal character sheaves explicitly for all (GIT) stably graded exceptional Lie algebras.

In this article, we establish integration by parts formulas for the solutions of McKean-Vlasov stochastic differential equations with jumps under elliptic coefficients. The derived formulas accommodate both derivatives with respect to real-valued variables and measure-valued variables, interpreted through the Lions' derivative. As applications, we obtain estimates for the derivatives of the density functions of the McKean-Vlasov SDEs, and relying on the integration by parts formulas, we subsequently prove the existence and uniqueness of classical solutions to the associated PDEs with irregular terminal conditions.

A configuration-interaction time-dependent density functional theory (CI-TDDFT) for nuclear dynamics is developed. In this framework, the correlated nuclear many-body wave function is expanded in terms of time-dependent many-particle configurations built from a common set of orthonormal single-particle states. The equations of motion for both the expansion coefficients and the single-particle states are derived self-consistently using the Dirac-Frenkel time-dependent variational principle. This formulation extends conventional time-dependent density functional theory (TDDFT) by incorporating configuration mixing and beyond-mean-field correlations, while preserving energy and particle-number conservation. As an illustrative application, the method is implemented using the relativistic point-coupling functional PC-PK1 in the particle-hole channel and a monopole pairing interaction in the particle-particle channel, and is applied to the study of isoscalar giant monopole resonance in $^{58}$Ni and $^{60}$Ni. Numerical tests show that both the total energy and particle number are conserved, with relative deviations within $4\times 10^{-4}$ during the time evolution. Compared with conventional TDDFT, CI-TDDFT yields broader strength distributions for giant monopole resonances while keeping the main peak positions close to those from TDDFT. This broadening is associated with configuration mixing in the valence space and suggests a coupling of the monopole oscillation to additional collective degrees of freedom. These results demonstrate the potential of CI-TDDFT as a quantum, microscopic beyond-mean-field framework for nuclear dynamics.

We investigate the quantum properties of a nonlinear Kerr oscillator driven by a three-photon pump. We derive both exact and approximate analytical expressions for the ground state of this interacting model. The exact solution arises at an exact spectral degeneracy, while the approximate solution describes regimes of quasi-degeneracy of the energy spectrum. In both cases, the threefold (quasi)degenerate ground-state manifold consists of quantum superpositions of three macroscopically distinct states. These states differ qualitatively from conventional three-component Schrödinger's cat states due to the presence of squeezing with a distinctive parametric dependence. By varying the detuning between the oscillator and the three-photon pump, we show that the squeezing can be enhanced, suppressed, or even reversed, leading to a squeezing-to-anti-squeezing transition. We analyze the generation and stabilization of these superposition states, their robustness against perturbations and analytically quantify the leakage to excited states. Our analysis elucidates how the three-photon Kerr parametric oscillator can be used to encode a Kerr-cat qutrit protected against phase-flip errors.

Recent human-computer interaction (HCI) research has revealed a widespread misalignment between how developers design workplace artificial intelligence (AI) systems, and what workers actually need from them. Yet, little research has examined the effects of this gap, or how it may cause harm. We analyzed 1,524 reports of incidents in which AI systems were used to perform 171 occupational tasks across 12 industry sectors. Using an Large Language Model (LLM)-as-an-expert approach, we extracted the main traits of the AI systems involved in those incidents using an established framework of twelve traits. We then compared them with the traits that 202 workers highly familiar with those tasks would have preferred. We found that as many as 83\% of workplace incidents stem from worker-AI misalignments. In most cases, workers wanted systems that are precise, insightful, or personal, but instead received systems that are basic, simple, or general. Over the years, fast AI caused a considerable number of incidents, yet these declined, and imaginative AI, with the mass introduction of generative AI, started to cause incidents. We also compared the traits causing the incidents with the traits that 197 developers building AI systems for those tasks would have preferred. If the traits causing the incidents were the same as those designed by developers, then developers may be responsible for those incidents. We found that 74\% of task misalignments could be attributed to developers who tended to overfocus on efficiency and speed, especially for systems performing tasks in people-facing occupations such as those in the human resources sector. Our results call for design interventions that better align AI development with workers' needs, as without such corrections, workplace AI incidents are likely to persist, causing the invisible erosion of worker agency and organizational productivity.

We identify anomalous scale-free localization and the associated impurity-induced loss bursts in a non-Hermitian dissipative cross-stitch lattice. By a local basis rotation, the model is mapped onto an effective non-Hermitian Su-Schrieffer-Heeger lattice, where local impurities act as tunable effective boundaries. For the parameter choice considered here, tuning the impurity strength $η$ connects two effective open-boundary-condition-like limits, reached for $η\to0$ and $η\to\infty$, through generalized-boundary-condition regimes and the impurity-free periodic-boundary-condition point at $η=1$. For finite $η\notin\{0,1\}$, the spectral loops remain separated from the real-energy axis, while the eigenstates exhibit scale-free localization pinned by the impurity. Unlike conventional impurity-induced scale-free localization, the Lyapunov exponent depends explicitly on the eigenenergy, making the localization strength eigenstate dependent. We further show that this anomalous eigenmode structure produces an impurity-induced loss burst: the long-time integrated dissipation probability is strongly enhanced near an impurity-generated effective boundary even when the initial wave packet is far away. In the single-impurity case, the burst region consists of the impurity site and its adjacent effective-boundary site, and the effect occurs without imaginary-gap closing. For multiple impurities, local burst regions emerge around all impurities, while the dominant burst boundary is selected by the initial wave-packet position and the nonreciprocal drift direction. These results connect anomalous scale-free localization with controllable dissipation dynamics in non-Hermitian lattices.

We present a fixed-grid conservative affine-constrained modal/multiwavelet coefficient method for one-dimensional Buckley--Leverett saturation transport. The saturation is evolved directly in a local orthonormal coefficient basis with a mean/detail structure: the first mode carries the conservative cell average, whereas higher modes carry zero-mean local details. The hyperbolic inflow condition is imposed as a linear trace constraint on the coefficient vector and enforced by affine lifting. For $(p>1)$, the boundary reprojection is applied in the detail subspace of the inflow cell, so that the prescribed trace is restored without modifying the conservative cell-average update. The transport operator is discretized in conservative weak form with monotone numerical fluxes, and shock-induced oscillations are controlled by a troubled-cell limiter acting on modal details. The method is validated on a Berea-core waterflood benchmark against an independent \texttt{pywaterflood} reference solution using the same Corey fractional-flow closure, physical parameters, and pore-volume-injected scaling. The affine-constrained coefficient solver reproduces the reference breakthrough curve and saturation profiles, preserves the imposed inflow trace to roundoff accuracy, controls saturation bounds through mean-preserving detail rescaling, and gives small accumulated global mass-balance defects. Mesh-refinement, flux-comparison, and modal-order studies show that $(p=2)$, corresponding to a piecewise-linear local representation, provides the most favorable accuracy--cost compromise among the tested orders for this shock-dominated benchmark.

Let $G$ be a finite group. Let $\mathcal{N}(G)$ be the lattice of normal subgroups ordered by inclusion, regarded as an abstract lattice. Define $\operatorname{LatAut}(G) := \operatorname{Aut}(\mathcal{N}(G))$. The \emph{LatAut tower} is the sequence defined by $G_0 = G$, $G_{n+1} = \operatorname{LatAut}(G_n)$. Let $G$ be a \emph{tower group} if $G \cong \prod_{k \geq 3} S_k^{a_k}$ with finitely many $a_k \neq 0$. We establish the following for tower groups. \emph{Product Formula.} $\operatorname{LatAut}\!\bigl(\prod_{k \geq 3} S_k^{a_k}\bigr) \cong S_{a_4} \times S_B$, where $B = \sum_{k \geq 3,\, k \neq 4} a_k$. \emph{Termination Theorem.} For every tower group $G_0$, we prove that $G_3 = 1$, and that this bound is sharp. The proof applies Goursat's lemma to classify $\mathcal{N}(G)$ into three families parameterised by admissible triples $(J,\mathbf{P},H)$ as sub-products, sign-parity elements, and mixed elements, and uses the Krull--Schmidt theorem to identify the direct factors $S_k^{(k,i)}$ as precisely the nontrivial indecomposable complemented elements of $\mathcal{N}(G)$ (the complemented elements being exactly the full sub-products). These results do not extend to groups outside the tower-group family.

In this paper, we study the notion of smooth $\infty$-categories within the framework of a six-functor formalism. By leveraging the theory of condensed mathematics and analytic stacks, we apply these results to demonstrate that a rigid analytic variety is smooth if and only if its associated category of nuclear sheaves is smooth. Furthermore, we relate the compact generation of the category of nuclear sheaves to the algebraization of the rigid analytic variety; these results are then employed to obtain an example of a non atomically generated but internally smooth category.

We construct a hypersimplicial subdivision of the $r$-dilation of the $i$-th hypersimplex of dimension $d$ that provides a geometric proof of the Brenti--Welker identity.

Purely organic room-temperature phosphorescence (RTP) emitters are key components of programmable luminescent tags (PLTs), photonic devices for rewritable information storage and UV dosimetry. In this work, we systematically explore the design space of donor-acceptor and donor-acceptor-donor organic phosphorescent emitters in symmetric and asymmetric architectures. Phenoxathiine (PX) is introduced as an alternative donor to thianthrene (TA), combined with benzophenone (BP) or pyridine (Py) as acceptors. Through photophysical characterization, quantum chemical simulations, and PLT device testing, we identify structure-property relationships and, in particular, investigate the impact of the individual moieties on the emission properties and stability. The RTP emission wavelength is primarily tunable through the donor moiety: PX-based emitters emit sky-blue (λ_P = 480 nm), while TA-based emitters emit in the green (λ_P = 520 nm) due to an increased Stokes shift. The acceptor unit strongly influences the phosphorescence quantum yield, with Py-based emitters systematically outperforming BP-based ones. All newly synthesized PX-containing emitters show sufficient performance in PLT devices, though with reduced photostability compared to TA-based counterparts. Together, these results demonstrate that systematic donor-acceptor design enables predictable control over RTP emission properties, advancing the rational development of high-performance RTP-based photonic devices.

Multiple-input multiple-output (MIMO) radar has waveform diversity and large spatial degrees of freedom (DoFs), making it attractive for high-resolution sensing. Scaling MIMO radar to massive arrays can further improve sensing performance, but it also increases hardware cost, power consumption, and digital processing complexity. The microwave linear analog computer (MiLAC) can tackle these challenges by moving linear operations from the digital domain to the analog domain. MiLAC has shown promising benefits for communications in recent studies and this paper identifies its potential for radar sensing. Specifically, we consider both MiLAC-aided transmit beamforming and receiver-side two-dimensional discrete Fourier transform (2D-DFT)-based direction-of-arrival (DoA) estimation. For transmit beamforming, we formulate a weighted Cramer Rao bound (CRB) minimization problem under lossless and reciprocal MiLAC constraints and propose a penalty dual decomposition (PDD)-based iterative algorithm to address the non-convex problem. We further prove that MiLAC-aided and fully-digital beamforming achieve the same CRB. For receiver processing, we show that the 2D DFT can be implemented by a lossless reciprocal MiLAC, which enables analog-domain DoA estimation without digital optimization. Numerical results confirm the theoretical finding and show that the MiLAC-aided approach achieves the same CRB and DoA estimation performance as the fully-digital benchmark. Meanwhile, hardware cost and power consumption are reduced because only low-resolution DACs are required at the transmitter, while RF chains and ADCs are eliminated at the receiver. Moreover, performing the 2D DFT in the analog domain eliminates all digital DFT operations for DoA estimation.

Persistent inequalities in political knowledge are a central concern in political communication. We organize the mechanisms underlying the knowledge-gap literature by distinguishing between individual preconditions, structural features of the information environment, and topic characteristics. Within this framework, we note that self-directed information seeking, a prototypical form of intentional exposure, has received little attention despite its importance in navigating today's complex information environment. We conducted a field experiment in Germany combining randomized encouragements and passive browser tracking to examine how individuals with varying education levels acquire policy-specific knowledge through online search. Participants were randomly assigned to one of three conditions (verbal encouragement, financial encouragement, or control) to seek information on three salient policy topics differing in divisiveness and complexity (child support, energy transition, and cannabis legalization). We estimate both intention-to-treat (ITT) and local average treatment effects (LATE) of information seeking on post-search knowledge outcomes, with a focus on education and civic knowledge as moderators. While the interventions equalized information-seeking behavior, the results provide some support for the knowledge gap hypothesis: knowledge gains were concentrated among participants with higher education or baseline civic knowledge, who, according to our post-hoc exploratory analyses, appeared more effective at navigating search results. These findings indicate that a narrowing of knowledge inequalities goes beyond motivation: it calls for both individual-level interventions to strengthen citizens' skills and structural-level adaptations to foster more equitable learning environments.

We investigate entanglement-based quantum key distribution protocols, with particular emphasis on their efficiency under realistic conditions of satellite quantum communications, where performance is limited by the low power of a received signal and background radiation. We focus on scenarios where each photon pair is used to encode multiple qubits in order to optimally utilize the weak signal. By optimizing over the source intensity and the number of encoded qubits we study the theoretical information limit for the QKD efficiency. We show that the optimal efficiency is attained for finite entangled photons pair production probability which is in contrast to conventional communication efficiency maximized in the limit of vanishing signal strength. The multiqubit encoding can enhance the secret key rate by up to an order of magnitude compared to single-qubit schemes.

Artificial intelligence (AI) empowers innovative diagnostic tools for common diseases, yet its clinical application in skeletal health evaluation is constrained by unsatisfactory accuracy, owing to the inherent porous and poroelastic biophysical features of bone. To address such bottlenecks amid global population aging, this study targets skeletal health and develops a reliable AI framework for precise bone microstructural characterization. We proposed Biot-PINN, a physics-informed neural network embedded with Biot's poroelasticity theory to characterize mechanical responses and wave propagation in poroelastic bone tissues. By decoding photoacoustic signals encoding bone mineral and microstructural features, the framework enables automatic bone microstructural grading. Experimental results reveal that Biot-PINN reaches an accuracy of 97%, markedly surpassing traditional data-driven approaches and providing a robust solution for early skeletal health diagnosis.

There is a surge of need for fresh information with the overwhelming proliferation of the Internet of Things (IoT) applications. To characterize the information freshness perceived by the destination, the age of information (AoI) has been proposed. In this paper, we consider an IoT system with multiple devices sending status update packets to a central controller through time-correlated Markov channels and assume that the instantaneous channel states are not available to the central controller before making scheduling decisions. To ensure information freshness, we investigate a timely scheduling problem that minimizes the total expected time-average AoI under a strict communications bandwidth constraint. We formulate this problem as a partially observable restless multi-armed bandit problem. Using Lagrangian relaxation, we decouple the relaxed problem into multiple sub-problems and prove the threshold structure of their optimal policies. Armed with this property, we establish the indexability for the decoupled problem and design an algorithm to compute the Whittle's index. To reduce implementation complexity, we further derive the Whittle-like index in closed-form for low-complexity scheduling. Simulation results show that the proposed index-based policies outperform the baselines, remain close to the optimal policy or relaxed lower bound, and are especially effective when scheduling resources are limited or the network size is large.

In this paper, we show that the treewidth of the $n \times n$ toroidal grid is $2n-1$ for all $n \ge 5$. This closes the gap between the previously known upper bound of $2n-1$ (Ellis and Warren, DAM 2008) and the lower bound of $2n-2$ (Kiyomi, Okamoto, and Otachi, DAM 2016). To establish the matching lower bound, we construct a bramble of maximum order by utilizing maximum components obtained after removing $2n-1$ vertices. Our construction relies on the vertex-isoperimetric properties of the infinite grid to establish tight lower bounds on neighborhood sizes, combined with a careful analysis of balls of radius $n/2-1$ and their boundaries to overcome structural obstructions when $n$ is even.

We report an optically modulated experimental realization of a photonic time crystal (PTC) in the microwave regime, demonstrating for the first time that the PTC exponential growth can overcome losses and finite-size constraints of a practical spatio-temporal system and yield stable positive terminal gain over a continuous broadband frequency range. The developed experimental platform is a purely time-modulated capacitor (TMC) microwave circuit based on a microstrip transmission line, in which synchronized optical modulation of reverse-biased photodiodes generates strong (94.5 %) temporal modulation of the effective capacitance at 200 MHz. Broadband amplification consistent with a momentum band gap (MBG), a defining signature of photonic time-crystal physics, is observed, with a peak gain of 3.8 dB over a 65 MHz bandwidth. In addition, a narrow parametric resonance appears at the center of the band gap, reaching 4.8 dB. This sharp peak is associated with the spatial inhomogeneities of the lumped-element realization, while the corresponding homogeneous distributed system retains the Floquet-mode structure of a photonic time crystal. We show that finite microwave TMC implementations inherit the defining physics of PTCs, including phase-invariant non-resonant amplification and slow-light behavior inside the momentum band gap, while finite-size and loss mechanisms transform the ideal semicircular PTC gain profile into a continuous asymmetric non-Lorentzian gain band characterized by a Pearson type IV distribution.

We provide a first systematic treatment of so-called rectangular multispectral perturbation theory. With their paper from 2003, Hochstenbach and Plestenjak ["Backward Error, Condition Numbers, and Pseudospectra for the Multiparameter Eigenvalue Problem" in Linear Algebra and its Applications] extended perturbation theory from one-parameter eigenvalue problems to multiple spectral parameters. After two decades, we take it one step further and consider a different manifestation of the multiparameter eigenvalue problem that consists of one matrix equation with rectangular coefficient matrices. We perform a norm-wise backward error analysis, define condition numbers for both eigenvalues and eigenvectors, and introduce the pseudospectrum while also considering the computational implications of working with multiple spectral parameters. The rectangular shape hampers a direct application of the existing definitions and properties. For example, the left null space at a given eigenvalue is non-trivial and the dimensions of the left and right eigenvectors are different. Through numerical examples, we illustrate and link the different concepts from the perturbation theory. A system identification application seem to suggest that, in optimization-driven problems for which multiparameter reformulations exist, the globally optimal solutions tend to coincide with the best-conditioned eigenvalues.