Fundamental to the analysis of nonlinear relativistic motion is the precise characterization of the induced dynamic Doppler effects. In this work, we analyze the electromagnetic signals observed by non-inertial receivers using two frameworks to describe the relativistic motion. We first consider observer paths described by higher-order kinematic 4 vectors: relativistic acceleration and jolt. The dynamic Doppler effects of relativistic acceleration and jolt are exponential spectral broadening and exponential amplitude growth or decay. We derive compact expressions for the spectrum transformation resulting from relativistic acceleration and jolt. The jolt induces nonlinear skewed chirps in observed signals. Next we consider observer paths described by the 4D Frenet-Serret frame and the curvature and torsion of the observer path. We obtain descriptions of the amplitude and phase fluctuations of the signal in terms of the geometric parameters of curvature and torsion. Concise, interpretable descriptions of non-inertial dynamic Doppler effects provide a useful diagnostic and predictive tool for engineering applications including radar, sensing, and communications systems.

We consider closed hypersurfaces smoothly immersed in hyperbolic manifolds up to homotopy and commensurability. We prove that if a closed hyperbolic manifold $M$ contains a sequence of asymptotically geodesic hypersurfaces, then $π_1(M)$ is virtually special and hence linear over integers. If $M$ (dimension at least 3) is, in addition, arithmetic of type I, we constructs a sequence of hypersurfaces which are asymptotically geodesic (but not totally geodesic), strongly filling, and equidistributing in the Grassmann bundle over $M$. This partially answers a question of Al Assal--Lowe. As a corollary, for each cocompact arithmetic lattice $Γ$ of $SO(n+1,1)$ of type I, there exist infinitely many arithmetic and infinitely many non-arithmetic cocompact lattices $H$ of $SO(n,1)$ that admit monomorphisms into $Γ$ which do not extend to a Lie group homomorphism from $SO(n,1)$ into $SO(n+1,1)$.

We introduce Quantum Spectral Authentication (QSA), a primitive for verifying that a remote quantum endpoint still possesses a previously installed secret quantum resource, such as a hidden state or state-preparation capability, without revealing that secret. QSA uses fresh public unitary challenges and spectral features of the planted state to derive transcript-bound session material for explicit authentication. We analyse attack strategies including eigenstate propagation across challenges, repeated-session leakage, and direct online forgery. For practical implementation, we develop a symmetric verifier-driven unitary compiler compatible with low-depth quantum phase estimation. Simulations indicate that this symmetric fast-power construction is substantially more noise tolerant than an asymmetric alternative, and small-instance experiments on IBM ibm_fez provide a hardware sanity check. QSA therefore offers a plausible near-term authentication layer for quantum networks and control-plane applications.

Understanding how weather and climate influence fire risk is important for many purposes, including climate adaptation planning and decision-making in sectors such as emergency management, finance, health and infrastructure (e.g., for energy and water availability). In this study, bias-corrected 20CRv2c reanalysis data are used to investigate the climatology and long-term trends of weather conditions associated with landscape fires in Australia. The McArthur Forest Fire Danger Index (FFDI) is used here as a broad-scale representation of weather conditions known to influence fire behaviour based on wind speed, humidity, temperature and rainfall measures. In particular, using this reanalysis dataset allows analysis over a longer time period than previous studies, from 1876 to 2011. Another novel aspect is that trends are examined using several different approaches, including a method to help account for the influence of interannual drivers of climate variability not previously used for fire weather analysis. Results show increases in mean and extreme seasonal FFDI values throughout Australia in general, with all statistically significant trends being positive in sign for individual climate zones. Humidity and temperature trends, attributable to human-caused climate change, are shown to be the main cause of the increase in dangerous weather conditions for fires. These findings build on previous studies, with the novel data and methods used adding confidence to the overall understanding of fire risk factors in a changing climate.

We introduce a dual-threshold probabilistic knowing value logic for uncertain multi-agent settings. The framework captures within a single formalism both probabilistic-threshold attitudes toward propositions and high-confidence attitudes toward term values, thereby connecting probabilistic epistemic logic with classical knowing value logic. It is especially motivated by privacy-sensitive scenarios in which an attacker assigns high posterior probability to a candidate sensitive value without guaranteeing that it is the true one. The main idea is to separate the threshold domains of propositional and value-oriented operators. While $K_i^θ$ ranges over the full rational threshold interval, the knowing-value operator $Kv_i^η(t)$ is restricted to $(\frac{1}{2},1]$. This high-threshold restriction has a structural effect: once $η>\frac{1}{2}$, two distinct values cannot both satisfy the threshold, so uniqueness becomes automatic. Over probabilistic models with countably additive measures, $Kv_i^η(t)$ is interpreted as non-factive high-confidence value locking. We establish sound axiomatic systems for the framework and develop a two-layer construction based on type-space distributions and assignment-configuration mappings. This resolves the joint realization problem arising from probabilistic mass allocation and value-sensitive constraints, and yields a structured weak-completeness theorem for the high-threshold fragment.

An electron in quantum confinement takes on a discrete energy spectrum which is defined based on the solution to the Schrodinger Equation for a given potential. Well defined closed-form energy spectra are known for the particle in a box, circular potential, quarter circle potential, and an equilateral triangle. A closed-form solution for more complex shapes may not be known, but numerical methods can be used to find an approximate solution. In this research, an application of the Finite Element Method (FEM) in Wolfram Mathematica is presented and applied to Quantum Billiards with a variety of geometries. To assess the accuracy of the method, the computed energy states are analyzed in the limit of a polygon with an increasing number of sides, the numerical results are validated against analytical solutions for geometries with known exact forms, and a standard convergence test is conducted. The FEM results closely match analytical solutions for known potentials, demonstrating its high accuracy. For high energy index n, quantum scarring may emerge for certain geometries. The nature of quantum scarring and its presence in the computed models is also investigated qualitatively.

We investigate bound states in the continuum (BICs) in a hybrid normal--superconducting triple quantum dot system, where the central dot is coupled to two normal leads and the lateral dots are proximity-coupled to superconducting electrodes. Local electron--electron interactions are treated within the Hubbard approximation. Finite bias, together with lateral-dot detuning and superconducting proximity, induces interference between elastic electron tunneling (ET) and Andreev reflection (AR) channels, mediated by BIC-related modes and proximity-induced Andreev bound states. As the bias is swept through the subgap resonances, ET exhibits sharp antiresonances that evolve into exact transport zeros, signaling the emergence of (quasi-)BICs. We further find a continuous crossover from a Fano--Andreev BIC-supported regime to a Fano--Andreev quasi-BIC regime as the detuning asymmetry increases. The formation of BICs and quasi-BICs is accompanied by a pronounced change in the occupation of the side quantum dot, providing an internal diagnostic directly correlated with the transport signatures of the bound states.

While primordial black holes (PBHs) have long been a benchmark target for microlensing searches, the modern landscape of dark matter models suggests other, distinct, formation channels for compact objects made of dark matter. In the large class of self-interacting, dissipative models, dark matter has cooling channels that can enable fragmentation and gravitational collapse of some dark matter into compact objects, including black holes. The resulting populations have mass distributions, bias parameters, and abundance, spatial profile and velocity dispersion within the Milky Way that all differ from those of PBHs. We demonstrate that these population-level differences can leave imprints in the space of microlensing observables, with the differences in how the populations trace the dark matter giving the primary distinguishing lever. We discuss the possible overlap of microlensing signals from dark and baryonic lenses, and the complementarity of microlensing detection or constraints with other gravitational probes of novel populations of dark matter origin.

Secure multi-party computation (MPC) offers a practical foundation for privacy-preserving machine learning at the edge. However, current MPC systems rely heavily on communication and computation-intensive primitives-such as secure comparison for nonlinear inference, which are often impractical on resource-constrained platforms. To enable real-time inference under a resource-constrained platform, we introduce a Trusted Acceleration of Minimal-Interaction MPC framework, TAMI-MPC, for nonlinear evaluation. Specifically, we reduce communication cost by redesigning the core primitives, leaf comparison, and tree merge, reducing the interactive round from log(n) to just 1 per operation. Furthermore, unlike prior work that heavily relies on oblivious transfer (OT), a well-known computational bottleneck, we leverage synchronized seeds inside the TEE to eliminate OT for the vast majority of our designs, along with a correlated-randomness reuse technique that keeps new designs computationally lightweight. To fully realize the potential, we design a specialized accelerator that restructures the dataflow across stages to enable continuous, fine-grained streaming and high parallelism, reducing memory overhead. Our design achieves up to 4.86x speedup on ResNet-50 inference, compared with state-of-the-art CNN frameworks, and achieves up to 7.44x speedup on BERT-base inference, compared with state-of-the-art LLM frameworks.

We describe an all-electron implementation of the Bethe-Salpeter equation (BSE) for the calculation of optical absorption spectra in the full-potential linearized augmented-plane-wave (FLAPW) method. So far, FLAPW implementations have resorted to a simple plane-wave basis for the bare and screened Coulomb potentials, thereby forgoing the all-electron description to some extent. In contrast, we expand the interaction potentials in the all-electron mixed basis. As in most implementations, the BSE is solved by the diagonalization of a two-particle Hamiltonian matrix, whose dimension is proportional to the number of $\mathbf{k}$ points. Due to the large number of $\mathbf{k}$ points required to converge the BSE, the resulting matrix becomes large even for small unit cells. We describe a method that exploits the crystal symmetries to accelerate the construction and diagonalization of the two-particle Hamiltonian. In particular, we employ group theoretical tools to bring the Hamiltonian into block-diagonal form. Furthermore, it is shown that often only one of the blocks needs to be taken into account for the optical absorption spectrum leading to a considerable speedup of the diagonalization step. The code allows for the inclusion of spin-orbit coupling and is parallelized with the possibility of storing the Hamiltonian in distributed memory over many nodes, keeping the memory demands low. To validate our implementation, we show optical absorption spectra and report exciton binding energies for bulk Si, LiF, and MoS$_2$. By exploiting the crystal symmetries, we can reduce the dimension of the Hamiltonian matrix of Si by a factor of five, resulting in a 125-fold speedup in its diagonalization. The calculated exciton binding energies of 22~meV and 76~meV for Si and MoS$_2$ are closer to experimental values than in previous BSE studies.

In applications, quantities of interest are often modelled in equilibrium or an equilibrium solution is sought. The presence of confounding makes causal inference in this setting challenging. We provide interpretable graphical models for equilibrium systems with confounding using anterial graphs (Lauritzen and Sadeghi, 2018), a class of graphs containing directed acyclic graphs, ancestral graphs, and chain graphs. In this setting, we provide valid graphical representations of both counterfactual variables and observational variables, which we relate to counterfactual graphs (Shpitser and Pearl, 2007) and single-world intervention graphs (Richardson and Robins,2013). As an application of this graphical representation, we provide an element-wise procedure of selecting adjustment sets that flexibly include and exclude given covariates.

Compact-group representations on Banach spaces are known to be norm-continuous precisely when they have finite spectra. For a quantum group with continuous-function algebra $\mathcal{C}(\mathbb{G})$ norm continuity can be cast analogously as the bounded weak$^*$-norm continuity of the representation's attached map $\mathcal{C}(\mathbb{G})^*\to \mathrm{End}(E)$. While the uniformity/isotypic finiteness equivalence no longer holds generally, it does for compact quantum groups either coamenable or having dimension-bounded irreducible representations. This generalizes the aforementioned classical variant, providing two independent quantum-specific mechanisms of recovering it.

Neuromorphic VLSI systems take inspiration from biology to enable efficient emulation of large-scale spiking neural networks and to explore new computational paradigms. To establish large neuromorphic systems, a sophisticated routing infrastructure is needed to communicate spikes between chips and to/from the host computer. For the BrainScaleS wafer-scale neuromorphic system considered in this work, especially the stimulation with input spikes and the recording of spikes is demanding, requiring high bandwidth and temporal resolution due to the accelerated emulation of neural dynamics 10.000 faster than biological real time. Here, we present a systematic characterization of the BrainScaleS off-wafer communication infrastructure implemented around Kintex7 FPGAs. The communication flow is characterized in terms of throughput, transmission delay, jitter and pulse loss. Further, we analyze the effect of the communication distortions (like pulse loss and jitter) on a neural benchmark model with highly varying spike activity. The presented methods and techniques for communication evaluation are general applicable and provide useful insights for the mapping of network models to the hardware such as the distribution of input spikes across communication channels.

Peculiar features of the Josephson effect in graphene were described theoretically by Titov and Beenakker [Phys. Rev. B 74, 041401(R) (2006)], who solved the Dirac-Bogoliubov-de-Gennes equation for a superconductor-graphene-superconductor junction with rectangular geometry. Here, we adopt the analysis for graphene Corbino disks, finding out that -- for the outer to inner radii ratio $r_2/r_1\gtrsim{}5$ -- such systems may demonstrate, when varying the electrochemical potential and the spatial profile of the electrostatic barrier, crossover from standard Josephson tunneling (SJT), via graphene-specific multimode Dirac-Josephson tunneling (MDJT), towards the ballistic Josephson effect (BJE). Signatures of SJT appear only near the Dirac point when the barrier shape is close to rectangular, MDJT appears in the tripolar range and is very robust against varying the barrier shape, and BJE is restored in the unipolar range when smoothing the barrier shape. A comparison with the results of a numerical simulation of quantum transport on the honeycomb lattice is also given.

We establish nonlinear stability of fronts that describe the creation of a periodic pattern through the invasion of an unstable state. Our results concern pushed fronts, that is, fronts whose propagation is driven by a localized mode at the front interface. We prove that these pushed pattern-forming fronts attract initial data supported on a half-line, and therefore determine both propagation speeds and selected wave numbers for invasion from localized initial conditions. This provides to our knowledge the first proof of the marginal stability conjecture for pattern-forming fronts, thereby confirming universal wave number selection laws widely used in the physics literature. We present our analysis in the specific setting of the FitzHugh-Nagumo system, but our methods can be applied to general dissipative PDE models which exhibit pattern formation. The main technical challenge is to control the interaction between the localized mode driving the propagation and outgoing diffusive modes in the wake of the front. Through a subtle far-field/core decomposition of the linearized evolution, we resolve this interaction and describe the nonlinear response of the front to perturbations as a dynamically driven phase mixing problem for the pattern in the wake. The methods we develop are generally useful in any setting involving the interaction of localized modes and outward diffusive transport, such as in the nonlinear stability of undercompressive viscous shock waves or source defects.

The increasing detection of new molecules in the interstellar medium (ISM) shows that stereoisomerism is a fundamental contributor to interstellar molecular complexity. This work presents the first comprehensive overview of interstellar stereoisomerism. A total of 16 stereoisomeric pairs have been identified (13 conformational and 3 geometric), spanning molecules with 5-12 atoms and energy separations from 10 K to 2667 K. They were observed across diverse astrophysical environments with kinetic temperatures ranging from low to high values (7.5-300 K). The observed stereoisomeric ratios (OSR) - defined as the column density ratio of the higher-energy isomer divided by that of the lower-energy isomer - vary widely (0.009-4). While systems with small energy differences (1.2 kcal/mol) in hot environments (> 100 K) generally follow thermodynamic expectations (often assisted by tunneling-driven interconversion), many stereoisomers - particularly those in cold clouds or with larger energy separations - exhibit abundances far exceeding equilibrium values. This demonstrates that thermodynamics alone cannot explain interstellar stereoisomerism. Instead, stereoselective formation/destruction pathways (in the gas phase and/or in the surface of dust grains), photoisomerization, and chemical rearrangement during desorption must play a dominant role. Stereoisomeric ratios thus provide powerful constraints on interstellar chemical pathways, and about the physico/chemical conditions of the ISM. This review highlights the need for stereochemistry-sensitive astrochemical models. Progress in this field requires expanded laboratory spectroscopy of higher-energy stereoisomers, dedicated quantum chemical studies of isomerization processes, and the explicit inclusion of stereoselective chemistry in chemical networks.

Recently, the second author [Ramanujan J. 2026] introduced and proved a $q$-series identity that appears to provide the first known $q$-analogue of an evaluation for a ${}_{2}F_{1}$-series known as \emph{Gosper's strange series}. Yamaguchi's derivation of this $q$-analogue relies on three-term relations for ${}_{2}ϕ_{1}$-series along with Heine's transformation of ${}_{2}ϕ_{1}$-series. In this note, we introduce and prove, using a $q$-analogue of a series evaluation technique relying on an Abel-type summation lemma, a further $q$-analogue of Gosper's ${}_{2}F_{1}$-identity that is inequivalent to Yamaguchi's $q$-analogue, and we also apply this technique to construct an alternative and simplified proof of Yamaguchi's $q$-analogue, together with a ${}_{3}ϕ_{2}$-series variant of Heine's $q$-analogue of Gauss's hypergeometric formula, a ${}_{6}ϕ_{5}$-series variant and two ${}_{4}ϕ_{3}$-series variants of the $q$-analogue of Kummer's identity due to Bailey and Daum, along with a $q$-analogue of a result obtained by Cantarini [Ramanujan J. 2022] via Fourier-Legendre theory and related to Ramanujan's first series for $\frac{1}π$.

Non-Hermitian skin effect, which is a unique feature of non-Hermitian systems, exhibits the formation of an extensive number of boundary modes under open boundary conditions. However, its manifestation in higher dimensions remains elusive. In our work, we demonstrate a hybrid skin-topological effect arising from the interplay between first-order band topology and non-reciprocal hopping in an engineered two-dimensional brick-wall geometry. The non-Hermitian brick-wall lattice under open boundary conditions in both directions exhibits several unconventional spectral features. Notably, the eigenvalues associated with the corner skin modes do not exhibit non-trivial windings in the complex energy plane; instead, they exhibit dynamically stable exceptional point-like features that do not originate from eigenvector coalescence. In contrast, the remaining modes accumulate at the opposite pair. Of all the corner skin modes, only the two that originate from the topological corner states of the Hermitian brick-wall lattice remain localized at individual corners, while the rest accumulate at the pair of opposite corners. This spatial distribution contrasts sharply with the second-order skin effect, where corner skin modes are more uniformly distributed. Finally, for the non-Hermitian Brick-wall lattice, we design and implement the corresponding topolectrical circuit (circuit for a square lattice is included for comparison) to directly visualize the hybrid skin-topological modes.

Spacecraft development costs remain high despite falling launch costs, in part because Model-Based Systems Engineering (MBSE) tools carry the complexity of the object-oriented programming paradigm: tightly coupled data and logic, mutable state, and rigid class hierarchies that resist integration with discipline-specific analysis tools. This paper presents a data-oriented approach to MBSE that adapts the Entity-Component-System (ECS) architecture from the video game industry. Design data is stored as immutable, format-agnostic components in a generic data system; stateless analysis functions operate on this data through templates and containerized tools within a continuous integration pipeline. A prototype implementation, VVERDAD (https://github.com/VisVivaSpace/vverdad-prototype), demonstrates the approach on an example interplanetary mission concept, showing how data-oriented principles can reduce deployment complexity, simplify testing, and preserve the traceability benefits of document-based systems engineering.

We consider the application of a ubercalibration-like relative flux calibration to the grism observations of the Roman High Latitude Wide Area Survey (HLWAS). We propose a simplified model of the calibration with an independent flat field for each detector in each exposure of the focal plane. In addition, we include two wavelength dependent components: a single wavelength throughput curve, modulated by a simple parabolic model for the throughput as a function of a source's focal plane position. We consider the impact of the dither scale, as well as the calibrator magnitude cuts. We show that the width of the calibration residuals can be reduced to less than 1.5 mmag, or 0.15% in flux, within the optimal dither range 50-240". This wide range allows for significant flexibility in optimising other parts of the observing program without diminishing the effectiveness of the relative flux calibration. We also discuss some improvements to the methodology that must be strongly considered before the calibration can be applied to real data. Finally, although we focused on spectroscopic component of the HLWAS here, our formalism and results should carry over to the imaging surveys as well.

In this work we investigate the weak lensing convergence using an end-to-end nonlinear general relativistic framework. Combining numerical relativity simulations of large-scale structure formation with general relativistic ray-tracing, we compare our nonlinear calculation to the expectation based on perturbation theory for a set of 20 synthetic observers. We focus on large angular scales $\ell < 100$ across a broad range of redshifts with $0.05<z<3$. We confirm the importance of Doppler lensing for redshifts below $z\sim$0.6, as predicted by previous works. On average across our observers, linear perturbation theory predicts the nonlinear convergence to within 3-30% across all redshifts and angular scales we study. In general, we find smaller angular scales are better matched by linear theory than larger angular scales. While we cannot definitively identify the source of the discrepancy, for our particular study of redshift slices on observers' light cones the differences are mostly below the level of cosmic variance.

Large Language Models (LLMs) face critical challenges when analyzing security vulnerabilities in real world codebases: token limits prevent loading entire repositories, code embeddings fail to capture inter procedural data flows, and LLMs struggle to generate complex static analysis queries. These limitations force existing approaches to operate on isolated code snippets, missing vulnerabilities that span multiple functions and files. We introduce codebadger, an open source Model Context Protocol (MCP) server that integrates Joern's Code Property Graph (CPG) engine with LLMs. Rather than requiring LLMs to generate complex CPG queries, codebadger provides high level tools for program slicing, taint tracking, data flow analysis, and semantic code navigation, enabling targeted exploration of large codebases without exhaustive file reading. We demonstrate its effectiveness through three use cases: (1) navigating an 8,000 method codebase to audit memory safety patterns, (2) discovering and exploiting a previously unreported buffer overflow in libtiff, and (3) generating a correct patch for an integer overflow vulnerability (CVE-2025-6021) in libxml2 on the first attempt. codebadger enables LLMs to reason about code semantically across entire repositories, supporting vulnerability discovery, patching, and program comprehension at scale.

Paucity of naturally occurring noncentrosymmetric materials is stimulating growing interest in engineered two-dimensional systems for nonlinear optical applications. Here, we show that breaking inversion symmetry in centrosymmetric bilayer Bi$_2$Se$_3$ through twisting, point-defect insertion, or the application of an external electric field unlocks rich nonlinear optical responses. In twisted bilayer Bi$_2$Se$_3$ at the first commensurate angle of 21.78$^\circ$, we find peak shift and injection current conductivities of -14 $ nm.μAV^{-2}$ and 104 $\times 10^8$ $nm.A V^{-2}s^{-1}$, respectively, which lie in the visible spectrum and enable efficient THz applications. The external electric field and point-defect insertion both transform the bilayer into C$_ {3v}$ symmetry, with the selenium vacancy (V$_{Se}$) achieving peak shift and injection current conductivities of -190 nm.$μAV^{-2}$ and -170 $\times 10^8$ $nm.A V^{-2}s^{-1}$. In all three cases, the peak nonlinear optical responses are found to be comparable to those of benchmark 2D materials such as GeS, and the broadband responses, including helicity-dependent current generation, make these engineered bilayers viable candidates for next-generation 2D photovoltaics.

We introduce a flexible framework for high-dimensional matrix estimation to incorporate side information for both rows and columns. Existing approaches, such as inductive matrix completion, often impose restrictive structure-for example, an exact low-rank covariate interaction term, linear covariate effects, and limited ability to exploit components explained only by one side (row or column) or by neither-and frequently omit an explicit noise component. To address these limitations, we propose to decompose the underlying matrix as the sum of four complementary components: (possibly nonlinear) interaction between row and column characteristics; row characteristic-driven component, column characteristic-driven component, and residual low-rank structure unexplained by observed characteristics. By combining sieve-based projection with nuclear-norm penalization, each component can be estimated separately and these estimated components can then be aggregated to yield a final estimate. We derive convergence rates that highlight robustness across a range of model configurations depending on the informativeness of the side information. We further extend the method to partially observed matrices under both missing-at-random and missing-not-at-random mechanisms, including block-missing patterns motivated by causal panel data. Simulations and a real-data application to tobacco sales show that leveraging side information improves imputation accuracy and can enhance treatment-effect estimation relative to standard low-rank and spectral-based alternatives.

We study the out-of-equilibrium dynamics of a weakly interacting one-dimensional Bose gas in a box trap, subjected to a drive realized by a periodically oscillating linear potential. After a transient regime, the gas reaches a quasi-steady state, characterized by the presence of several solitons. At weak driving amplitude, the solitons are only weakly perturbed by one another, while at strong driving amplitude a regime analogous to turbulence is reached, where the solitons are strongly intertwined with each other. We show that a hallmark of both regimes can be found in the momentum distribution, which displays a power-law decay $n(k) \sim k^{-2}$ at weak driving amplitude and $n(k) \sim k^{-α}$ with a power-law exponent $α\in [7,9]$ at large amplitude. We further characterize each of the two regimes by following the space-time maps and characterizing the solitons using the inverse scattering transform. The protocol analyzed in this study is amenable to experimental realization in current experimental setups.

We study topology in gapless phases of an interacting spinful model with spin-charge separation. We focus on the gapless boundaries between $\mathbb{Z}_2$ symmetry-breaking phases. We find two topologically non-trivial gapless states that occur at the boundary between a non-trivial and a trivial insulator. They correspond to topological Luther-Emery liquid and topological Mott insulator. The Luther-Emery liquid is characterized by gapless charge excitations and features topological edge modes that carry fractional spin, while the topological Mott insulator has gapless spin sector and features edge states that carry fractional charge. Surprisingly, even though there is no mean-field description of the interacting gapless phases, as there is no local order parameter, we show that they can be adiabatically connected to a non-interacting topological metal. This non-interacting state is a phase boundary between decoupled Su-Schrieffer-Heeger chains with the winding number $ν=2$ and chains with $ν=1$.

Tracking moving objects is a critical skill for many everyday tasks, such as crossing a busy street, driving a car or catching a ball. Attention is a key cognitive function that supports object tracking; however, our understanding of the brain mechanisms that support attention is almost exclusively based on evidence from tasks that present stable objects at fixed locations. Accounts of multiple object tracking are also limited because they are largely based on behavioral data alone and involve tracking objects in a 2D plane. Consequently, the neural mechanisms that enable moment-by-moment tracking of goal-relevant objects remain poorly understood. To address this knowledge gap, we developed SABER (Spatial Attention, Brain, Extended Reality), a new framework for studying the behavioral and neural dynamics of attention to objects moving in 3D. Participants (n=32) completed variants of a task inspired by the popular virtual reality (VR) game, Beat Saber, where they used virtual sabers to strike stationary and moving color-defined target spheres while we recorded electroencephalography (EEG). We first established that standard univariate EEG metrics which are typically used to study spatial attention to static objects presented on 2D screens, can generalize effectively to an immersive VR context involving both static and dynamic 3D stimuli. We then used a computational modeling approach to reconstruct moment-by-moment attention to the locations of stationary and moving objects from oscillatory brain activity, demonstrating the feasibility of precisely tracking attention in a 3D space. These results validate SABER, and provide a foundation for future research that is critical not only for understanding how attention works in the physical world, but is also directly relevant to the development of better VR applications.

We formalize Hilbert's Seventh Problem and its solution, the Gelfond-Schneider theorem, in the Lean 4 proof assistant. The theorem states that if $α$ and $β$ are algebraic numbers with $α\neq 0,1$ and $β$ irrational, then $α^β$ is transcendental. Originally proven independently by Gelfond and Schneider in 1934, this result is a cornerstone of transcendental number theory, bridging algebraic number theory and complex analysis.

We introduce a coefficient-decoupled matrix product operator (MPO) representation for Pauli-sum operators that separates reusable symbolic operator support from a tunable coefficient bridge across a fixed bipartition. This representation provides a direct interface to linear-combination-of-unitaries (LCU) circuits: the symbolic left/right dictionaries define a static \textsc{Select} oracle that is compiled once, while coefficient updates modify only a dynamic \textsc{Prep} oracle. As a proof of concept, we construct compact state-adapted Pauli pools by sampling Pauli strings from a pretrained matrix product state (MPS), pruning them to a fixed symbolic pool, optimizing only their coefficients, and transferring the resulting weights directly to the LCU interface. The resulting workflow provides a reusable classical-to-quantum compilation strategy in which the symbolic operator structure is compiled once, and subsequent updates are confined to a low-dimensional coefficient object.

This paper introduces robust twoblock (RTB) simultaneous dimension reduction, which is the first statistically robust method to perform simultaneous dimension reduction in two blocks of variables and allows to fine-tune the model complexity in each block individually. The paper proposes both a dense and a sparse version of the new method. Sparse RTB is the first robust estimator that allows to select both model complexity and the degree of sparsity for each block individually. RTB thereby allows to optimally extract and summarize the relevant portion of information in each block of data, also in the presence of outliers. As a corollary, the estimators can be recombined into a single estimate of regression coefficients for multivariate regression that is operable when the number of variables exceeds the number of cases in each block. An extensive simulation study illustrates that the new methods are resistant to different types of outliers, while maintaining estimation efficiency. across a range of dimensionality settings. These findings both hold true for the dense and the sparse method. The methods' performance is further illustrated on two example data sets and a straightforward algorithm is presented and made accessible in an open source repository.