An efficient method of solving the time-dependent Schrödinger equation for pure states is described: At each timestep, a restricted subspace of the total Hilbert space is systematically and naturally constructed via the image of repeated applications of the Hamiltonian operator, and the time evolution is computed exactly within said subspace. The subspace is dynamically recomputed such that it co-evolves with the state vector. The method is built from the ground up as a parallel algorithm for graphics processing units and suited to Hamiltonians that are sparse in a given basis. We benchmark the method by comparing its results for a 1D Holstein model to previously published multiset-MPS results, and then apply the method to compute optical spectra and non-equilibrium dynamics of one-, two- and three-dimensional model chromophore nanoaggregates.

One of the most interesting questions that astronomy can hope to answer is: are we alone in our Milky Way galaxy? A detection of an electromagnetic (EM) signal generated by an extraterrestrial technological intelligence, or the presence in our solar system of an alien probe, would answer this question in the negative. Purposeful interstellar communication is a 2-way street - the transmitting and receiving technological intelligences (TIs) both need to do their parts. As the receiving TI, our EM search programs should incorporate a model of what a transmitting TI is likely to be doing. Published searches for extraterrestrial technological intelligence (SETI) have generally not done so and thus have often been suboptimally designed. We propose an improved search technique that more closely corresponds to astronomical surveys that have been undertaken for reasons that have nothing to do with SETI. Published non-SETI radio and optical surveys are sufficiently extensive that they already supply meaningful constraints on the prevalence of nearby purposely communicative alien civilizations. Purposeful communication can also include the sending of spaceships (probes). The absence of evidence for alien probes in the solar system suggests that no alien civilization has passed within ~100 lt-yr of Earth during the past few billion years.

We experimentally validate a unified input--output model that incorporates internal and external coupling phases across multiple cavity modes in a room-temperature cavity magnonic system. By explicitly accounting for both phase contributions, the model provides a clear interpretation of the transition from level repulsion to level attraction at an interference-induced antiresonance, and accurately reproduces nonreciprocal transmission arising from the internal phases of the contributing modes. Quantitative agreement between experiments and simulations is obtained across all coupling regimes, establishing a predictive framework for phase-controlled cavity--magnon devices including isolators, circulators, and quantum transducers.

Delay-Doppler Map (DDM) noise-floor observations from the Cyclone Global Navigation Satellite System (CYGNSS) constellation provide a practical means for spaceborne detection of GNSS radio frequency interference (RFI). Existing CYGNSS analyses use NASA's kurtosis-based flag product or mean aggregation of the four simultaneous DDM noise-floor values at each epoch. However, these DDMs are formed from different reflected GNSS signals received through two nadir antennas with different orientations. Thus, ground-based RFI may raise only some channel noise floors, depending on antenna gain and viewing geometry. Mean aggregation can dilute the strongest anomaly with unaffected channels, causing missed detections. This paper replaces the mean with the maximum of four co-temporal DDM noise-floor values. This statistic preserves channel-level anomalies and accounts for channel-dependent exposure. A practical 41 dB threshold is established using low-RFI reference regions and documented or persistent interference environments, enabling simple detection without image-level classification or raw intermediate-frequency processing. To reduce isolated false alarms, a verification stage uses multi-satellite concurrence and temporal persistence over a 10 s window. The method is evaluated using CYGNSS Level 1 data from May 2025 over the White Sands Missile Range, where NOTAM-announced GPS jamming tests provide documented interference conditions, and the Middle East, where persistent RFI has been reported. In the White Sands case, the proposed method detected RFI on three dates where the mean-based method produced negligible detections. In the Middle East, it flagged 62% of observed epochs, compared with 46% for the mean-based method and 33% for the kurtosis-based method. These results show that maximum-based aggregation offers a simple, lightweight improvement over existing CYGNSS DDM noise-floor methods.

Stellar photospheric heterogeneity is a dominant astrophysical systematic impacting exoplanet transmission spectroscopy. NASA's Pandora SmallSat Mission is designed to address this challenge through contemporaneous visible photometry and NIR spectroscopy of exoplanet host stars. Here we present an end-to-end simulation study quantifying Pandora's ability to infer stellar photospheric properties and correct stellar contamination using out-of-transit observations. We construct eight representative stellar activity scenarios and generate 160 simulated Pandora datasets, incorporating time-dependent stellar spectra, instrument response, and noise. Given accurate models, Bayesian retrievals of Pandora spectrophotometry recover photospheric temperatures with typical uncertainties of ${\approx}30$ K, with no significant bias. Models with two spectral components (i.e., quiescent photosphere and spots) are strongly favored in 95% of cases; one-component models are preferred when true spot filling factors fall below a detection threshold of ${\approx}0.3$%. We propagate the true and inferred stellar parameters to compute true, inferred, and residual contamination signals under physically motivated spot geometries. For simple spot distributions, contamination signals of $10^2{-}10^3$ ppm are reduced to ${\lesssim}10$ ppm, well below Pandora's expected transmission spectroscopy precision (30$-$100 ppm). For more complex spot distributions, geometric degeneracies limit deterministic corrections, leaving residual contamination at the $10^3$ ppm level that must be mitigated using additional constraints, such as spot-crossing events and joint stellar-planetary retrievals of transmission spectra. These results define regimes in which stellar contamination can be corrected from stellar observations alone and show how Pandora stellar observations can identify cases where additional information is required.

We study the orbits of the solutions to the Markoff-type equation $$X^2 + Y^2 + Z^2 = XYZ +AX + BY + CZ + D$$ in $\mathbb{F}_p$ for fixed integers $A, B, C,$ and $D$ under the group of symmetries $Γ$ generated by \[\begin{split}&V_1: (x, y, z)\mapsto (A + yz - x, y, z),\\ &V_2: (x, y, z)\mapsto (x, B + xz - y, z),\text{ and}\\ &V_3: (x, y, z)\mapsto (x, y, C + xy - z).\end{split}\] This equation arises as the Relative Character Variety of the Four-Times Punctured Sphere, and $Γ$ arises from the Pure Mapping Class Group. For most parameters we show that there is a density one set of primes $p$ such that $Γ$ acts transitively on the bulk of the solutions mod $p$, with the remainder breaking up into a few small orbits arising from finite orbits within the solutions over $\mathbb{C}$. We classify those ``degenerate'' parameters to which this result does not apply, and show there are either 2 (for most degenerate parameters) or 4 (for the remaining degenerate parameters other than $(0, 0, 0, 4)$) large orbits modulo density one of primes. Our results become especially interesting when applied to two special subfamilies. The first is $$X^2 + Y^2 + Z^2 = XYZ + k$$ for $k \neq 4$, which arises in the study of the combinatorial group theory of $\text{SL}_2(\mathbb{F}_p)$. Our results very nearly prove the $Q$-classification conjecture of McCullough and Wanderley for density 1 of primes. The second subfamily is $$x_1^2 + x_2^2 + x_3^2 + a_1x_2x_3 + a_2x_1x_3 + a_3x_1x_2 = (3+a_1+a_2+a_3)x_1x_2x_3,$$ which arises from certain generalized cluster algebras. Here, our notion of degenerate parameters specializes to the degeneracy condition of de Courcy-Ireland, Litman, and Mizuno. For all nondegenerate and some degenerate surfaces in this family, their results imply that our count of large orbits (1, 2, or 4) applies to all sufficiently large primes $p$.

Exploiting the special features of four-dimensional Riemannian geometry, we derive topological and rigidity results for hypersurfaces immersed in space forms of dimension 5. First, we provide a complete description of the Weyl tensor for four-dimensional hypersurfaces, by means of which we derive a new characterization result for isoparametric hypersurfaces; then, we prove sharp topological bounds on the Weyl functional for closed, minimal hypersurfaces, involving the Euler characteristic in the case of an ambient space with constant non-negative sectional curvature. Then, inspired by a famous conjecture by Chern and the so-called second pinching problem, we find estimates for the norm of the second fundamental form in terms of the Euler characteristic in the minimal, constant scalar curvature case, under a cross-sectional area assumption. Finally, we prove some rigidity results by means of integral inequalities on the derivatives of the second fundamental form, also dealing with special curvature conditions, such as half harmonic Weyl curvature and Bach-flatness. We also extend some of the local results to the case of a locally conformally flat 5-dimensional ambient space.

Duflo--Serganova functors play an important role in the representation theory of Lie superalgebras. While it is desirable to understand the images of modules under DS, little is known beyond finite-dimensional representations. For general linear Lie superalgebras, Brundan--Goodwin study the Whittaker coinvariants functor $H_{0}$ and the associated principal $W$-superalgebra. In this paper we investigate rank-one DS functors attached to odd roots, characterized by the condition that $\operatorname{DS}_{x}(\mathfrak g)\subset \mathfrak g$ is a graded subsuperalgebra with respect to the principal good grading, and the induced functors $\overline{\operatorname{DS}}$ on $W$-superalgebra module categories via the Skryabin equivalence. In particular, we explicitly compute the DS images of $\mathfrak b$-Verma supermodules (for a suitable class of Borel subalgebras $\mathfrak b$). We also observe that, via the parabolic Miura transform, the pullbacks of tensor products of (dual) Verma modules for the $W$-superalgebra can be identified with the $H_{0}$-images of $\mathfrak b$-Verma supermodules for an appropriate choice of $\mathfrak b$.

Misfit layered compounds are natural van der Waals heterostructures in which electronically active transition-metal dichalcogenide layers are decoupled by incommensurate blocking layers, enabling bulk realization of quasi-two-dimensional quantum states. Here we investigate the superconducting, transport,and structural properties of the misfit compound $\mathrm{(SnS)_{1.15}(TaS_2)}$ under pressures up to 150 GPa. The low-pressure superconducting phase is gradually suppressed and disappears near 14.7 GPa,accompanied by increasing residual resistance. Remarkably, a distinct superconducting phase reemerges above 80 GPa and persists to the highest pressures achieved. This reentrant superconductivity follows a pressure-induced sign reversal of the Hall coefficient near 60 GPa and a nonmonotonic evolution of the normal-state resistance, indicating an electronic reconstruction. No structural phase transition is detected over the entire pressure range. Our results demonstrate a pressure-driven electronic reconstruction leading to reentrant superconductivity in a misfit layered compound, establishing pressure as an effective route to engineer superconductivity and electronic states in natural van der Waals heterostructures.

We study warm Higgs portal dark matter (DM) in the framework of freeze-in at stronger coupling. This scenario assumes that the Standard Model thermal bath temperature has always been relatively low, which suppresses dark matter production. As a result, a significant DM-Higgs coupling is allowed, enabling warm dark matter detection via Higgs decay at colliders. We find that the Lyman-α bound on the DM mass is particularly strong, excluding masses below 50-100 keV, depending on further details. The shape of the DM momentum distribution is highly non-thermal, with low momenta being effectively cut off, and not captured by the common αβγ-parametrization.

We consider the motion of a charged spinning test/probe particle -- governed by the Mathisson-Papapetrou-Dixon equations with generic, adiabatic, and conservative spin- and field-induced multipole moments -- in a background $\sqrt{\text{Kerr}}$ field on flat spacetime: the electromagnetic field of a charged spinning ring-disk singularity obtained from the $G\to 0$ limit of the Kerr-Newman solution for a charged spinning black hole. We investigate the existence of two extra hidden constants of motion, analogous to the Carter constant (for geodesic motion in a Kerr spacetime, or for its spinning-probe generalization) and Rüdiger's linear-in-spin constant for a spinning probe in a Kerr background. We find that these two constants exist only when the Wilson coefficients parameterizing the probe's multipole structure take the particular values corresponding to spin-exponentiation of the effective Compton amplitudes through second order in spin.

Full-wave numerical methods based on quasinormal modes (QNMs) offer valuable physical insights and computational efficiency for analyzing electromagnetic resonators. However, despite their advantages, many researchers in electromagnetism continue to favor real-frequency domain or time-domain approaches, often using finite element or finite-difference time-domain methods. This preference stems from various factors, including the perception that QNM theory is still developing or requires advanced mathematical tools from complex analysis. In this work, we combine numerical techniques with accurate ap-proximations to simplify the computation of QNMs and enable ultrafast reconstructions us-ing QNM expansions. The result is a new approach that is straightforwardly accessible to users familiar with real-frequency methods. We demonstrate the practicality of our ap-proach through an open-source package [Doi: 10.5281/zenodo.18708748] implemented within a widely-used commercial photonics software.

We analyze planar $n$-body Hamiltonian systems with quadratic $D_n$-invariant interactions and identify the symmetry obstruction to choreographic motion. Choreographies are taken throughout to be collision-free solutions of the equations of motion in which all bodies traverse one closed curve with uniform time shifts. By diagonalizing the dynamics into discrete Fourier sectors, we show that superintegrability, periodicity, and choreography are governed by distinct conditions: commensurability of the active frequencies closes bounded motions, whereas a sectorwise $C_n$ phase-matching condition is required for full equivariance. At the configuration level this equivariance is already equivalent to a genuine simple choreography. Thus generic resonant multi-sector motions are periodic but multi-trace, while true choreographies occur only on phase-matched loci, in single irreducible sectors, or through effective one-sector reductions produced by exact degeneracy. The cases $n=4,5,6$ exhibit this mechanism explicitly, with $n=6$ marking the first distinction between nondegenerate commensurability and additional exact degeneracy.

The classical Mpemba effect is the counterintuitive phenomenon where hotter water freezes faster than colder water due to the breakdown of Newton's law of cooling after a sudden temperature quench. The genuine nonequilibrium post-quench dynamics allows the system to evolve along effective shortcuts absent in the quasi-static regime. When the time needed for preparing the (classical or quantum) system in the hotter initial state is included, we encounter so-called Pontus-Mpemba effects. We here investigate multi-step Pontus-Mpemba protocols for open quantum systems whose dynamics is governed by non-autonomous (aka time-inhomogeneous) Lindblad master equations. In the limit of infinitely many steps, one arrives at continuous Pontus-Mpemba protocols. We study the crossover between the quasi-static and the sudden-quench regime, showing the presence of dynamically generated shortcuts achieved for time-dependent dissipation rates. Considering a two-parameter family of time-dependent rates, the parameters allowing for optimal speed-up conditions are determined. Time-dependent rates can also cause non-Markovian behavior, highlighting the existence of rich dynamical regimes accessible beyond the Markovian framework.

This paper develops a geometric diagnostic framework for classical inequality decomposability. Representing the simplest nontrivial setting of three-person income distributions as points on the two-dimensional income-share simplex, we translate population-share-weighted and income-share-weighted decomposability into concrete geometric restrictions on within- and between-group residuals, making it possible to localise and characterise violations across measures. Applied to the Mean Log Deviation, the Gini coefficient, the coefficient of variation, and the Theil index, the analysis shows that decomposability is not a binary property as measures fail in qualitatively distinct ways, and the between-group residual is consistently the primary locus of failure. Negative between-group residuals render the decomposition uninterpretable and arise for the coefficient of variation and the Theil index under population-share weighting, and for the Mean Log Deviation under income-share weighting. Stylised numerical examples quantify the resulting misinterpretation scenarios for applied researchers.

We study builder-driven MEV arbitrage on BNB Smart Chain (BSC). BSC's Proposer-Builder Separation (PBS) adopts a leaner design: only whitelisted builders can participate, blocks are produced at shorter intervals, and private order flow bypasses the public mempool. These features have long raised community concerns over centralization, which we empirically confirm by tracing the arbitrage activities of the two dominant builders from Apr. 1, 2025 to Feb. 28, 2026 (full observable activity cycle). Within months, the two leading builders, \bd{48Club} and \bd{Blockrazor}, produced over 87\% of blocks and captured about 90\%+ of MEV profits. We find that profits concentrate in short, low-hop arbitrage routes over wrapped tokens and stablecoins, and that block construction rapidly converges toward monopoly. Beyond concentration alone, our analysis reveals a structural source of inequality: BSC's short block interval and whitelisted PBS collapse the contestable window for MEV competition, amplifying latency advantages and excluding slower builders and searchers. MEV extraction on BSC is not only more centralized than on Ethereum, but also structurally more vulnerable to censorship and fairness erosion.

Quantum error correction offers a promising path to suppress errors in quantum processors, but the resources required to protect logical operations from noise, especially non-Clifford operations, pose a substantial challenge to achieve practical quantum advantage in the early fault-tolerant quantum computing (EFTQC) era. In this work, we develop a systematic scheme to encode exponential maps of the form $\exp(-iθP)$ into stabilizer codes with simple circuit structures and low qubit overhead. We provide encoded circuits with small first-order logical error rate after postselection for the [[n, n-2, 2]] quantum error-detecting codes and the [[5, 1, 3]], [[7, 1, 3]], and [[15, 7, 3]] quantum error-correcting codes. Detailed analysis shows that under the level of physical noise of current devices, our encoding scheme is 4--7 times less noisy than the unencoded operation, while at most 3% of runs need to be discarded.

We study the group of birational selfmaps of a variety birational to a conic bundle. We prove that it admits a surjective morphism to the direct sum of an uncountable number of copies of $\mathbb Z/2\IZ$.

Insider threat detection is difficult because malicious behavior is rare, irregular, and buried in long periods of inactivity. In enterprise audit data, most windows contain little activity, while attacks appear intermittently and range from brief events to sustained campaigns. Standard reconstruction-based models are therefore dominated by inactive regions and tend to learn baseline behavior rather than meaningful deviations. We separate activity presence from magnitude. Each window is decomposed into a binary mask indicating whether activity occurs and a value matrix capturing its intensity. A dual-channel autoencoder reconstructs both, with value loss applied only where activity is present, directing learning toward sparse structure. Using the CERT r5.2 dataset as a controlled setting, we examine how anomaly signal changes with temporal configuration. Short attacks are detected mainly through presence; longer attacks introduce a magnitude component; noise degrades magnitude reliability and shifts detection back toward presence. The balance between channels is not fixed and follows the data. At the campaign level, signal concentrates in a small number of anomalous windows. Simple aggregation that emphasizes extreme scores is sufficient to recover extended activity without explicit sequence modeling. Effective detection depends less on model complexity and more on aligning representation and objective with sparse temporal structure.

Network segmentation is a popular security practice for limiting lateral movement, yet practitioners lack a metric to measure how segmented a network actually is. We define segmentedness as the fraction of potential node-pair communications disallowed by policy -- equivalently, the complement of graph edge density -- and show it to be the first statistically principled scalar metric for this purpose. Then, we derive a normalized estimator for segmentedness and evaluate its uncertainty using confidence intervals. For a 95\% confidence interval with a margin-of-error of $\pm 0.1$, we show that a minimum of $M=97$ sampled node pairs is sufficient. This result is independent of the total number of nodes in the network, provided that node pairs are sampled uniformly at random. We evaluate the estimator through Monte Carlo simulations on Erdős--Rényi, stochastic block models, and real-world enterprise network datasets, demonstrating accurate estimation. Finally, we discuss applications of the estimator, such as baseline tracking, zero trust assessment, and merger integration.

We investigate homological and depth-theoretic properties of finitely generated modules of finite projective dimension over Noetherian local rings. A central theme is the study of criteria for freeness and reflexivity derived from the torsion-freeness or reflexivity of tensor products of the form \( M \otimes_R M \) and \( M \otimes_R M^* \). Under mild homological assumptions, we prove that such properties of these tensor products impose strong structural constraints on \( M \), often forcing it to be free. These results generalize classical theorems of Auslander beyond the regular case. The second part of the paper is devoted to the dimension and support of Ext-modules, particularly \( \operatorname{Ext}^i_R(M, R) \) for critical values of \( i \), when \( M \) has finite projective dimension. We establish sharp bounds on their Krull dimensions, analyze their behavior for prime and equidimensional modules, and relate these findings to the grade conjecture and other homological conjectures, i.e., whenever $\operatorname{grade}(M) = \operatorname{ht}(\operatorname{Ann}(M))$ where Gdim$(M)<\infty$. We consider the problem that asks whenever is \( \pd_R(M \otimes_R N) = 1 \)? Applications include new cases of a question of Jorgensen, which asks whether \( \operatorname{pd}(M) < i \) whenever \( \operatorname{Ext}^i_R(M, M) = 0 \) and \( M \) has finite projective dimension over a complete intersection ring. Finally, we examine the projective dimension of prime ideals in rings that fail chain conditions.

The complexity quasi-metric of Schellekens is a topological framework in which the asymmetry of computational comparisons -- ``$A$ is at most as fast as $B$'' carrying different information than ``$B$ is at most as slow as $A$'' -- is built into the distance itself. This paper develops the theory of expansive homeomorphisms on the resulting space. The central result is that the scaling transformation $ψ_α(f)(n)=αf(n)$ is expansive on the complexity space $(\C,d_\C)$ if and only if $α\neq 1$. The $δ$-stable sets of this dynamics turn out to coincide with asymptotic complexity classes, giving a dynamical characterisation of objects familiar from complexity theory. We then show that the canonical coordinates of $ψ_α$ are hyperbolic with contraction rate $λ=1/α$, and we connect orbit separation in the dynamical system to the classical time hierarchy theorem of Hartmanis and Stearns. Unstable sets, conjugate dynamics, and topological entropy estimates for the scaling map are also worked out. Concrete algorithms and Python implementations accompany every proof, so each result can be checked computationally; SageMath snippets sit alongside the examples, and the full code is in the \href{https://github.com/gabayae/expansive-homeomorphisms-complexity-qmetric}{companion repository}.

Asynchronous effects of Ahman and Pretnar complement the conventional synchronous treatment of algebraic effects with asynchrony based on decoupling the execution of algebraic operation calls into signalling that an operation's implementation needs to be executed, and into interrupting a running computation with the operation's result, to which the computation can react by installing matching interrupt handlers. Beyond providing asynchrony for algebraic effects, the resulting core calculus also naturally models examples such as pre-emptive multi-threading, (cancellable) remote function calls, and multi-party applications. In this paper, we study the normalisation properties of this calculus. We prove that if one removes general recursion from it, then the remaining calculus is strongly normalising, including both its sequential and parallel parts. To cover more interesting programs, we also prove that the sequential part of the calculus remains strongly normalising when a controlled amount of interrupt-driven recursive behaviour is reintroduced. Our normalisation proofs are structured compositionally as an extension of Lindley and Stark's $\top\top$-lifting-based approach for proving strong normalisation of effectful languages. All our results are also formalised in Agda.

Consider a population of $N$ individuals, each having $d\geq 1$ different traits, and an additive measure, called dispersion, which rewards large pairwise separations between traits. The goal is to select $M\leq N$ individuals such that their traits are as dispersed as possible. We compute analytically the full statistics (including large deviation tails) of the maximally achievable dispersion among sub-populations of size $M$ when the traits are independent and identically distributed. Two complementary approaches are developed, one based on a mean-field theory for order statistics, and the other on the replica method from the field of disordered systems. In all dimensions $d$, and for rotationally symmetric distributions, the optimal subset for large populations consists of all points lying outside a $d$-dimensional ball whose radius is determined self-consistently. For a single trait ($d=1$), the statistics of the maximal dispersion can be tackled for finite $N,M$ as well. The formulae we obtained are corroborated by numerical simulations on small instances and by heuristic algorithms that find near-optimal solutions.

Every stable 4-sphere is identified with the double branched covering space of a trivial surface-knot space. As a result of Wall, it is known that any two orthogonal bases of every stable 4-sphere are transformed into each other by an orientation-preserving diffeomorphism of the stable 4-sphere. In this paper another proof of Wall's result is presented, strengthened in the sense that the lift of an equivalence of the trivial surface-knot space can be taken as the diffeomorphism. Two applications are made. The first shows that every orientation-preserving diffeomorphism of every stable 4-sphere is nothing but the double branched covering lift of an equivalence of a trivial surface-knot space up to a smooth isotopy and a composition with an identity-shift. The second gives a similar result for TOP stable 4-spheres. Here, even if it is a smooth 4-manifold, unless it is diffeomorphic to the stable 4-sphere, the TOP trivial surface-knot space cannot be smooth.

Understanding entanglement cost in non-local quantum computation (NLQC) is relevant to complexity, cryptography, gravity, and other areas. This entanglement cost is largely uncharacterized; previous lower bound techniques apply to narrowly defined cases, and proving lower bounds on most simple unitaries has remained open. Here, we give two new lower bound techniques that can be evaluated for any unitary, based on their controllable correlation and controllable entanglement. For Haar random two qubit unitaries, our techniques typically lead to non-trivial lower bounds. Further, we obtain lower bounds on most of the commonly studied two qubit quantum gates, including CNOT, DCNOT, $\sqrt{\text{SWAP}}$, and the XX interaction, none of which previously had known lower bounds. For the CNOT gate, one of our techniques gives a tight lower bound, fully resolving its entanglement cost. The resulting lower bounds have parallel repetition properties, and apply in the noisy setting.

Given an elliptic curve $E$ over $\Q$ and a nontrivial element $σ$ of its Shafarevich--Tate group $\Sha(E)$, we introduce the \textbf{Visualization category} $\V(E; σ)$ of abelian varieties that ``visualize'' $σ$ in the sense of Mazur, and we study minimal objects in this category. In particular, we show that there can be several minimal visualizing abelian varieties of different dimensions, answering a question of Mazur. We revisit two constructions of visualizing abelian varieties: restriction of scalars (as in the work of Agashe and Stein), and a construction due to de Jong (as in the work of Cremona and Mazur). We show that restriction of scalars typically produces minimal visualizations. When $σ$ has order $2$ or $3$, we build upon the de Jong construction and make it totally explicit. While the de Jong construction can produce non-minimal objects, an appropriate choice in the construction for order $2$ elements $σ$ yields an explicit genus $2$ curve whose Jacobian is a minimal visualization. For order $3$ elements we apply our algorithmic construction to Fisher's database of such elements, and obtain computational evidence that, in the absence of a $3$-isogeny, the de Jong construction yields a minimal visualization.

In this note, we show that the weighted mass of Baldauf and Ozuch (2022) can be derived as a natural geometric mass invariant following Michel (2011), for a certain weighted curvature map. An associated weighted centre of mass definition is also derived from this. The adjoint of the linearisation of this curvature map leads to a notion of weighted static metrics, which are natural candidates for weighted mass minimisers. This weighted curvature quantity is essentially the scalar curvature of a conformally related metric that Law, Lopez and Santiago (2025) used to considerably simplify the proof of the weighted positive mass theorem. We show an equivalence between static metrics and weighted static metrics via the conformal relationship, from which we show that a uniqueness theorem holds for weighted static manifolds with weighted minimal surface boundaries. Furthermore, we show that weighted manifolds satisfy a Riemannian Penrose inequality whose equality case holds precisely for these unique weighted static metrics.

Large language models have recently shown promise for multimodal recommendation, particularly with text and image inputs. Yet real-world recommendation signals extend far beyond these modalities. To reflect this, we formalize recommendation features into four modalities: text, images, categorical features, and numerical attributes, and highlight the unique challenges this heterogeneity poses for LLMs in understanding multimodal information. In particular, these challenges arise not only across modalities but also within them, as attributes such as price, rating, and time may all be numeric yet carry distinct semantic meanings. Beyond this intra-modality ambiguity, another major challenge is the nested structure of recommendation signals, where user histories are sequences of items, each associated with multiple attributes. To address these challenges, we propose UniRec, a unified multimodal encoder for LLM-based recommendation. UniRec first employs modality-specific encoders to produce consistent embeddings across heterogeneous signals. It then adopts a triplet representation, comprising attribute name, type, and value, to separate schema from raw inputs and preserve semantic distinctions. Finally, a hierarchical Q-Former models the nested structure of user interactions while maintaining their layered organization. Across multiple real-world benchmarks, UniRec outperforms state-of-the-art multimodal and LLM-based recommenders by up to 15%, and extensive ablation studies further validate the contributions of each component.

We investigate the possibility of constraining primordial black holes (PBHs) with masses $M_\mathrm{PBH}\gtrsim 10^{15}\,\mathrm{g}$ through Galactic diffuse synchrotron emissions. Due to Hawking radiation, these types of PBHs are expected to be stable sources of cosmic-ray (CR) electrons and positrons with energies below $\mathcal{O}(10\,\mathrm{MeV})$. In many CR propagation models with diffusive re-acceleration characterized by a significant Alfvén velocity $V_a\sim \mathcal{O}(10)\,\mathrm{km/s}$, the energies of the evaporated electrons/positrons can be further enhanced to $\mathcal{O}(100)\,\mathrm{MeV}$ through their scattering with the Galactic random magnetic fields. Consequently, the observation of Galactic synchrotron emissions at frequencies above $\sim 20\,\mathrm{MHz}$ can provide useful constraints on the abundance of PBHs. Using the AMS-02 and Voyager-1 data on the boron-to-carbon nuclei flux ratio, we confirm that a significant Alfvén velocity $V_a \sim 20\,\mathrm{km/s}$ is favored in several benchmark diffusive re-acceleration models. We show that, in this scenario, the observed low-frequency synchrotron emissions (from 22 MHz to 1.4 GHz) can provide stringent constraints on PBH abundance. The obtained conservative constraints are stronger than those derived from the Voyager-1 all-electron (electron plus positron) data by more than one order of magnitude for $M_\mathrm{PBH}\gtrsim 1\times 10^{16}\,\mathrm{g}$, and also stronger than our previous constraints derived from the AMS-02 positron data for $M_\mathrm{PBH}\gtrsim 2\times 10^{16}\,\mathrm{g}$.