We explicitly compute the perverse Leray filtration on the top cohomology of the Hilbert scheme of points on $Σ\times\mathbb{C}$, for any connected smooth projective curve $Σ$. The computation is carried out in the natural basis given by the $\mathbb{C}^*$-upward-flow cycles. The result is described by a simple symmetric-function dictionary: upward-flow classes correspond to products of complete homogeneous symmetric functions, while the perverse-homogeneous basis corresponds to products of power-sum symmetric functions. This gives an explicit triangular change-of-basis between the two bases.

Warning messages have been used to disrupt individuals seeking online child sexual abuse material (CSAM) and promote engagement with support services, yet large-scale field evidence on message content remains limited, particularly in high anonymity environments. This study reports a field experiment on Ahmia.fi, a Tor search engine, examining how warning message content influences behavior. Across a 140-day period, almost 20 million searches were observed, with over 3 million searches containing known CSAM-related terms that triggered a warning linking to an anonymous self-help program. Users were exposed to warning messages varying in thematic content and framing, or a neutral message. Across a randomized comparison, a campaign-wide analysis, and interrupted time series models, message content consistently influenced engagement with help resources. All active messages increased click-through rates to help resources relative to the neutral condition, with a harm-focused message producing the strongest effects. At the platform level, click-through rates increased from 8.73% before the intervention to 15.67% during the campaign. These findings highlight the importance of message content in shaping responses to warning interventions, supporting an approach in which messaging is refined and adapted to increase engagement with support resources.

We give a complete classification of doubly totally-umbilical submanifolds in the probability simplex. The probability simplex is one of the most standard statistical manifolds, and information geometry initiated by S. Amari and H. Nagaoka studies the statistical submanifold theory of the probability simplex. On the other hand, H. Furuhata defined doubly totally-umbilical submanifolds in the geometry of statistical manifolds, inspired by the surface theory of Euclidean space.

Triangles capture higher-order structures in graphs and are fundamental to applications such as clustering and network analysis. To enable efficient use of such structures at scale, we study the problem of triangle cut sparsification, which aims to reduce the graph size while approximately preserving triangle counts across every cut. We investigate quantum algorithms for this problem, using triangle listing as our main technical ingredient. In particular, we present a quantum algorithm for triangle listing that, for a graph with $n$ vertices, $m$ edges, and $t$ triangles, runs in time $T_{\mathrm{q\text{-}list}} =$ $\widetilde{O}\bigl(\min(n^{5/4}t^{7/12} + n^{7/6}t^{7/9}, m + m^{3/4}t^{1/2},$ $n^{3/2}t^{1/2})\bigr)$, improving upon the best known classical bounds over a broad range of parameters. Our algorithm is based on a heavy-light vertex partition and an extension of triangle detection via quantum walks and Grover search. Leveraging this result, we design a quantum algorithm for constructing $\varepsilon$-triangle cut sparsifiers of size $\widetilde{O}(n/\varepsilon^2)$ in time $\widetilde{O}(T_{\mathrm{q\text{-}list}} + \sqrt{mn}/\varepsilon)$. Finally, we demonstrate applications to clustering algorithms based on triangle-related measures and prove a lower bound of $Ω(n/\varepsilon^2)$ on the size of any $\varepsilon$-triangle cut sparsifiers.

Animals capable of powered flight range in wingspan from a few hundred microns to a few meters. The inertial turbulence to which these animals are exposed features vortices ranging from a few hundred micrometers to hundreds of kilometers in size. Yet, the impact of ambient turbulence on animal flight is virtually uncharted and most studies on animal flight are conducted in still air or under laminar conditions. Here, we propose a novel parameterization that links animal flight with turbulence, through a proxy for the energy injected into the atmosphere, $E_{sp}=b^3 f^2$, with $f$ the animal's flapping frequency and $b$ the wingspan. We model this parameter using a scaling relation in the shape of a power law $E_{sp} \propto k^α$, with $k=1/b$ the wavenumber corresponding to the animal inverse wingspan. Literature provides four theoretical predictions on the exponent $α$: two connected to aerodynamic and energetic aspects of flight, $α_{aero}=-2$ and $α_{power}=-5/3$, and two linked to physiological limits. Drawing from experimental data of over 400 species spanning 13 insect orders and two vertebrate classes, we recover $α_{power}=-5/3$ as the best scaling relation across the animal kingdom. Grouping per animal clade however reveals a secondary power law with $α=-5/2$ exponent for invertebrate orders, with a family-dependent coefficient. This new scaling relation suggests a yet-unknown universal physical mechanism in insect flight, likely depending on wing morphology and mechanical properties.

An estimated 5.3 billion mobile phones became electronic waste in 2022. Many of these devices can be repurposed and used in different contexts to extend their lifetime and to reduce ecological impacts. An often overlooked aspect of smartphone reuse is cybersecurity: these devices embed hardware-backed security mechanisms that rely on vendor-controlled provisioning and are designed for a fixed device lifecycle. In this paper, we investigate whether security mechanisms and guarantees remain effective when devices are repurposed outside their original ecosystem. We explore security features in a PinePhone, an open-hardware smartphone, and focus on three core security aspects: boot chain integrity, isolation provided by the Trusted Execution Environment, and the protection of hardware-bound secrets. Our experiments simulate realistic repurposing scenarios and highlight the complexity of reconstructing trust anchors. We generalize our observations to infer requirements for secure repurposing and illustrate how vendor locked mechanisms hinder the repurposing of a majority of discarded devices.

We study quantum algorithms for nonlinear and stochastic homogenization via a Young-measure based linear programming (LP) formulation, which lifts the nonlinear problem to a linear one in higher dimensions by treating the microscale, the gradient, and possible random variables as independent variables, thereby capturing effective macroscopic quantities without directly resolving fine-scale oscillations. The resulting LP is large but structured, and its high-dimensional nature creates regimes in which quantum LP solvers outperform direct classical solvers: in the deterministic setting, polynomial quantum speedup arises when moderate homogenized accuracy suffices; in the stochastic setting, encoding all random realizations simultaneously in a single LP yields a quantum square-root reduction in stochastic sampling cost that grows with the number of random variables. Regularity or sparsity of the Young measure may further extend these advantages to fine-scale accuracy. Numerical experiments on one- and two-dimensional benchmarks confirm the correctness of the Young-measure LP formulation.

Google Search deploys a "Onebox" feature at the top of the results page when users conduct searches for Child Sexual Abuse Material. This study evaluates the impact of a strategic shift in this feature, comparing a revised intervention, focused on repercussions and therapeutic resources, to a previous iteration that focused on reporting. Using a difference-in-differences analysis of internal Google Search logs data, we found the new messaging resulted in a 3.8 percentage point reduction as compared to the status quo in subsequent CSAM-related queries within the same Search session. We found an average click through rate of 0.73% on any of the hyperlinked buttons to help-providing resources. Together, this research presents convergent evidence that a subset of individuals can be deterred from ongoing CSAM-seeking and redirected to therapeutic services.

Spin supersolids emerge as a central topic in frustrated magnetism, motivating the search for realization in quantum materials. To this end, we study the origin of exchange anisotropy, $Δ$, in triangular-lattice cobaltate families $X_2$$Y$Co(PO$_4$)$_2$ and $X_2$Co(SeO$_3$)$_2$ ($X$ = Na, K, Rb, Cs; $Y$ = Mg, Ca, Sr, Ba) by tailoring realistic spin models. We show that $Δ$ is determined by the ratio of trigonal crystal field to spin-orbit coupling strength. This framework explains contrasting anisotropies in these families, predicts systematic trends in $Δ$ across $X/Y$-substitutions, and identifies candidate materials for spin supersolids. Our results establish trigonal field engineering as a microscopic route toward the design of spin supersolids.

We prove strong existence and pathwise uniqueness for two stochastic models of a seeker steering toward a target, written in polar coordinates. In both, the angular drift carries a $\frac1{r}$-type singularity which belongs to the supercritical regime in $\mathbb{R}^2$. Standard results for SDEs with singular drift therefore do not apply, and we give a new proof of strong well-posedness based on a pathwise argument. The two models arise from sailboat navigation and proportional navigation. We study the limiting regime in which the stopping radius around the target tends to zero and prove that, despite the singularity at the origin, each system admits a unique strong solution up to the hitting time of the target. These results provide an example of strong well-posedness in a regime where the general theory does not apply.

The paper deals with Ulrich wildness of decomposable threefold scrolls $X$ over Hirzebruch surfaces $\mathbb{F}_a$, for any $a \geqslant 0$. Our Main Theorem enstablishes that for $a=0$, the moduli space of rank-$r$ Ulrich bundles, for any $r \geqslant 2$ and of given Chern classes, contains a generically smooth, unirational component $\mathcal{M}(r)$ of computed dimension whose general point corresponds to a slope-stable Ulrich bundle; in particular $X$ turns out to be Ulrich wild. When $a \geqslant 1$ and in presence of modular obstructions, $X$ is nevertheless shown to be Ulrich wild too.

We verify the 'extendedGCD' implementation in Go's standard library ('crypto/internal/fips140/bigmod'), which plays a crucial role in the generation of RSA key pairs. Even though the Go implementation is supposedly a direct port from BoringSSL's implementation, we uncovered two deviations that each break the algorithm's invariants: (1) the Go implementation deviates in the way coefficients are updated, and (2) it permits a larger input domain. We address both deviations; the first by fixing the Go implementation, which results in an on average 24% speedup, and the second deviation by porting an existing proof for BoringSSL and extending it to cover the larger input domain. We prove correctness and termination of the fixed Go implementation using Gobra, a deductive program verifier for Go. Where necessary, we used Lean to prove key lemmata on non-linear arithmetic, which we import into Gobra. Our verification effort reveals three key insights: subtle bugs can slip into even well-reviewed code with surprising ease; formal verification is a powerful tool for uncovering them; and AI agents can facilitate the verification process by iteratively refining invariants and lemmata based on Gobra's error messages.

Accurate prediction and parameter identification of multiphase flow in porous media remain central challenges in geological carbon dioxide storage due to strong nonlinearities, high-dimensional parameter spaces, and limited observational data. We present a machine learning framework that integrates surrogate modeling and Bayesian inference to enable efficient forward prediction and inverse parameter estimation for CO2-brine flows in geological media. The approach is demonstrated using the "FluidFlower" experimental rig, a controlled laboratory system that provides high-resolution, time-resolved observations of CO2 migration in heterogeneous porous media. A convolutional neural network surrogate is trained on high-fidelity numerical simulations to learn the evolution of CO2 saturation and dissolved CO2 concentration fields over a wide range of multiphase flow properties. The trained surrogate is embedded within a Markov chain Monte Carlo framework for parameter inference conditioned on experimental observations. Results show that the surrogate accurately captures large-scale CO2 plume migration, dissolution dynamics, and multiphase flow behavior while providing orders-of-magnitude acceleration compared to traditional simulations. Embedding the surrogate within a Bayesian framework enables computationally tractable exploration of the parameter space and reveals both identifiable and non-identifiable parameter combinations that produce similar plume behavior. By leveraging spatially and temporally resolved full-field observations, the framework substantially improves agreement between simulations and experiments compared to previous manual calibrations based on limited plume-scale metrics. Analysis using progressively increasing observation horizons further shows that observations become more informative once the plume interacts with geological features such as faults and sealing layers.

We show in the $(0+1)$-dimensional Berenstein-Maldacena-Nastase matrix model, dual to M-theory on a pp-wave background, that the coupling can be changed between any two finite, non-zero values using a special class of deformations, known as conjugation deformations. Importantly, we prove that they preserve normalizability of the states. This implies that BPS states in the model cannot lift as the couplings are varied, and hence their (unsigned) number cannot change, except at the free point and Banks-Fischler-Shenker-Susskind point.

Optical photon Monte Carlo simulation is a computational bottleneck for noble liquid Time Projection Chambers. Design studies require repeated, geometry dependent simulations of timing, wavelength shifting, and optical response, while reconstruction and particle identification workflows need labeled optical datasets. We present Simphony, a GPU optical simulation tool, formerly EIC-Opticks, built on Opticks with CUDA and NVIDIA OptiX. Simphony implements a GPU version of the Geant4 G4OpWLS wavelength-shifting model and returns Monte Carlo truth for detected hits with low per-photon overhead. We validate Simphony against Geant4 11.3.2 in a simplified 14.7 kt liquid argon Time Projection Chamber benchmark with a two-stage wavelength-shifting shell and idealized photon counting detector. For three paired 2.5 GeV electron simulations, each producing about 61 M optical photons, the integrated detected-photon ratio agrees with Geant at the subpercent level. The detected arrival time and wavelength spectra give $χ^2/\mathrm{ndf}$ values of 0.98 and 1.08. Contained muon and near-Cerenkov-threshold proton samples give $R_N=1.0017\pm0.0008$ and $R_N=1.0005\pm0.0014$, confirming agreement for distinct source topologies. On an NVIDIA RTX 4090, a stacked launch of four 2.5 GeV electron events transports 243 M optical photons in $3.03\pm0.06$ s, giving $80.2\pm1.6$ M photons s$^{-1}$. Relative to a single-thread Geant reference and including GPU overheads and host-device transfers, the optical transport speedup is $1053\pm55$; the end-to-end wall time acceleration is $89\pm5$. These results show that Simphony can make explicit optical photon Monte Carlo practical for detector development studies and for generating machine learning optical response datasets.

In this series of papers, we present dynamical models of cluster members in strong lensing (SL) galaxy clusters to independently probe the persistent discrepancy reported between SL models and cosmological simulations, in terms of total mass properties for the cluster subhalos. In this work, we focused our study on early-type galaxies within Abell 2744 ($z=0.309$) and MACS J0416.1-2403 ($z=0.397$). We took advantage of deep MUSE spectroscopic data, complemented with HFF photometry. We used a pipeline based on spectral fitting to perform kinematic measurements of the LOS velocity dispersion profiles of 109 cluster members. We modeled the galaxies assuming a dPIE total mass density distribution and a Jaffe stellar mass density distribution. From the models, we inferred the values of the central stellar velocity dispersion, $σ_0$, and the truncation radius, $r_t$, for the galaxies in our sample. We found that $σ_0$ is accurately recovered for all of the cluster members, while $r_t$ is reliably measured for a fraction of galaxies in our sample, with sufficiently extended radial kinematic coverage. Our dynamical models predicted LOS velocity dispersion profiles that fit the measured ones better than those inferred from SL models. We then exploited the $σ_0$ measurements obtained from the dynamical models to calibrate the Faber-Jackson scaling relations for the cluster members in both galaxy clusters. When comparing our relations to those obtained in previous kinematics and SL works, we found systematically higher normalization and compatible slope and scatter values. We conclude that our dynamical measurements of $σ_0$ and $r_t$, along with calibrated scaling relations, are more robust than previous kinematic estimates which are biased by not taking into account the effects of the PSF, and should therefore be adopted as improved initial prescriptions in future SL models.

The laser-accessible nuclear transition in the thorium-229 isotope has been identified as a promising candidate for the realization of an optical nuclear clock. Such a nuclear clock might rival or outperform current optical clocks based on electron-shell transitions in atoms or ions, is expected to be more robust against external perturbations, and provides enhanced sensitivity in clock-based tests of fundamental principles of physics. Here, we implement a thorium-229 nuclear clock by stabilizing a continuous-wave laser to the 148 nm nuclear transition with rapid feedback based on continuous absorption spectroscopy. The thorium-229 nuclei are embedded into a millimeter-sized, room temperature calcium fluoride crystal. A subharmonic of the 148 nm radiation is continuously compared to a Yb+ single-ion clock. The nuclear clock shows a simple shot-noise limited scaling of the fractional frequency instability of $3\cdot 10^{-12} \sqrt{τ/\text{s}}$ where $τ$ is the averaging time, approaching $10^{-15}$ instabilities over 1 day of continuous operation. Improvements of the instability by several orders of magnitude can be projected for future solid-state nuclear clocks. We use the nuclear clock to constrain models of ultralight dark matter by searching for periodic fluctuations and slow drifts in the nuclear transition energy, on time scales between 20 s and 1 day. Drawing benefit from the enhanced sensitivity of the thorium-229 transition, these constraints compete with the best atomic clocks concerning dark matter coupling to photons and go beyond previous measurements regarding coupling to the strong force and quarks.

We extend the correlated B-spline basis function (C-BSBF) method to high-precision calculations of the ionization energies of helium Rydberg $n^{1,3}P$ states ($n=24$--$35$). Using a unified basis set, we evaluate nonrelativistic energies, relativistic corrections of order $mα^4$ (including finite-mass recoil), QED contributions of order $mα^5$, and partial $mα^6$ terms (singlet-triplet mixing, one- and two-loop radiative corrections). The remaining higher-order contributions are estimated via $1/n^3$ scaling. The resulting ionization energies achieve kHz-level accuracy and are in excellent agreement with independent Hylleraas calculations, thereby providing cross-validation between two distinct theoretical approaches. From these data, the quantum-defect parameters are determined and used to extrapolate the ionization energies up to $n=102$. Combining our Rydberg ionization energies with high-precision experimental $2S \rightarrow nP$ transition frequencies yields the ionization energies for the metastable $2^{1}S$ and $2^{3}S$ states as \num{960332040.533(10)}$_\mathrm{stat}(20)_ \mathrm{sys}$ MHz and \num{1152842742.7274(53)}$_\mathrm{stat}(25)_ \mathrm{sys}$ MHz, respectively. The C-BSBF result for the $2 \, ^1 S$ state is consistent with the experimental ionization energy obtained from Rydberg-series extrapolation, while for the $2 \, ^3 S$ state the difference is 0.019(10) MHz.

We provide a pedagogical review of the hydrodynamics of the FPUT chain. There are three hydrodynamic fields corresponding to the conservation of mass, momentum and energy. We provide physically motivated derivations of the hydrodynamic equations at the levels of Euler and then Navier-Stokes-Fourier. Next we consider examples to test as to how successful the hydrodynamic description is in predicting the observed time evolution of nonequilibrium initial conditions such as domain walls and blasts. We find that in some cases there is good agreement of microscopic simulations with predictions from the Euler equations while, in several other cases, there is significant departure from the Euler predictions suggesting that the role of dissipation and noise is important in general.

This study examines sequential algorithms with the Zero Jacobian Property (ZJP) for estimating structural models subject to equilibrium constraints. For the Maximum Likelihood Estimation (MLE) and the Generalized Method of Moments (GMM), the current study shows that these algorithms attains fast (near-quadratic) local convergence in large samples to the solution of the constrained optimization problem. If consistent initial estimates of the parameters are available, the algorithms yield an asymptotically efficient estimator even after one iteration. It then proposes a novel algorithm called Sequential Linearly Constrained (SLC) algorithm, which is applicable to a broader class of structural models than existing methods. A key advantage of the SLC algorithm is that it can be implemented without explicitly computing the Jacobian of the equilibrium constraints and can be multiple times faster than the Nested Fixed Point (NFXP) approach. The current study illustrates its performance through two numerical experiments: a dynamic discrete game with time-varying unobserved heterogeneity and a dynamic demand model.

Global infrasound monitoring provides a persistent means of detecting energetic bolide atmospheric entries, complementing optical observations and extending coverage over remote regions. We present a global assessment of the physical factors governing bolide infrasound detectability by correlating 623 bolide events reported by the Center for Near-Earth Object Studies between 2007 and 2025 with waveform data from the International Monitoring System. We identify 311 events with confirmed infrasound detections, corresponding to a detection rate of approximately 50%, substantially higher than inferred from earlier surveys, reflecting both the maturation of the global infrasound network and advances in automated, multi-frequency array processing. Analysis of flight parameters shows that infrasound detectability is selective rather than uniform across the bolide population. Detected events are preferentially associated with steeper entry angles and lower-altitude energy deposition, while shallow, high-altitude trajectories are less consistently observed. Very high-energy events remain detectable regardless of geometry, but for the more common lower-energy regime, observability depends on specific combinations of entry parameters and propagation conditions. This geometric dependence persists across comparable energy ranges and atmospheric conditions, indicating that entry angle exerts a primary control on detectability, with energy and propagation acting as secondary modulating factors. These results provide new physical constraints on bolide-atmosphere interactions and improve interpretation of global infrasound observations for planetary defense and atmospheric-entry studies.

We give a sharp lower bound for the total mean curvature of a convex hypersurface in Euclidean space in terms of the length of a shortest nontrivial closed geodesic, generalizing a result of Álvarez Paiva for convex surfaces. This result is based on a sharp lower bound for the mean width of a convex hypersurface in terms of its Birkhoff invariant, which gives sharp lower bounds for a broader array of total curvature functionals. We also characterize spheres as the unique convex hypersurfaces whose planar sections containing chords of maximal length are all as long as possible.

Root-MUSIC is a spectral estimation algorithm that approximates the unknown signal frequencies by constructing a high-degree polynomial and finding a subset of roots which are closest to the complex unit circle. Previous works found asymptotic expectation formulas for the performance of Root-MUSIC under the implicit assumption that the aforementioned root selection criterion does not select extraneous roots -- those which are unrelated to the correct parameters. This paper removes the need for this assumption by showing all extraneous roots lie outside an annulus of a certain thickness and therefore are not selected by the algorithm. This paper also provides sharp, non-asymptotic, and explicit error bounds for the correct roots in terms of fundamental model parameters. All results hold under a natural separation condition on the correct signal frequencies and are applicable in both the single- and multi-snapshot models. More specifically, in the multi-snapshot model, we prove that Root-MUSIC estimates the frequencies with error at most $O(σ/(m \sqrt n))$, where $σ^2$ is the noise variance, $m$ is the number of sensors, and $n$ is the number of snapshots. A novelty of this non-asymptotic bound is the explicit $1/m$ decay, which indicates that there is a significant advantage in utilizing additional sensors. Numerical simulations confirm our theory. The main mathematical insight of this paper is a geometric property of the Root-MUSIC polynomial: its correct roots are highly stable to noise while its extraneous roots must lie outside of an annulus.

Integrated sensing and communication (ISAC) enables simultaneous sensing and data transmission but exposes a critical vulnerability: probing signals may be intercepted, revealing both the transmitted information and the act of sensing itself. Existing physical layer security approaches mitigate interception yet operate with detectable signals, leaving sensing activity observable to a passive warden. This paper introduces sub-noise-floor pseudo-random probing (SNF-PRP), a covert sensing framework for OFDM-based ISAC systems under an energy-detection adversary model. SNF-PRP establishes an $ε$-covertness guarantee via Kullback-Leibler (KL) divergence, exploits an $N_{\mathrm{sc}}$-fold spreading gain absent from prior wideband analyses, and derives in closed form the minimum integration length required to achieve a target Cramér-Rao bound (CRB). Simulations under 5G~NR n78 numerology confirm sub-0.5\,m range and sub-0.5\,m/s velocity accuracy with KL divergence $5.8\times$ below the covertness threshold, validating joint feasibility at $-12$\,dB and $-15$\,dB probing powers.

Let $\mathcal{A}$ be a (not necessarily rational) conformal net with an action of a discrete group $G$. We show that the category $\text{Rep}^G(\mathcal{A})$ of $G$-twisted representations of $\mathcal{A}$ is canonically a $G$-crossed balanced $\mathrm{W}^*$-tensor category. This extends the results of Müger arXiv:math/0403322, in the language of localized endomorphisms, that $\text{Rep}^G(\mathcal{A})$ is a $G$-crossed braided tensor category.

A classical result establishes that the area of a minimal graph intersected with the unit ball is at most $2π$. A natural question is whether this upper bound is sharp. In this note, we resolve this by constructing a sequence of minimal graphs via solutions to a Dirichlet problem. We show that the areas of these graphs tend to $2π$, demonstrating that the bound is sharp.

Let $T(n)=n/2$ if $n$ is even and $T(n)=(3n+1)/2$ if $n$ is odd. We prove that for each $m\ge1$, exactly $F(m+1)$ odd integers in $\{1,\ldots,2^m\}$ have the property that their orbit under $T$ avoids the residue class $4\pmod6$ during steps $2,\ldots,m$, where $F(m+1)$ is the $(m+1)$-th Fibonacci number; the proportion decays at rate $(φ/2)^m$, $φ=(1+\sqrt{5})/2$. The proof uses the directed graph $G$ of Collatz transitions modulo $6$ and its unique absorbing strongly connected component $G'=G[\{1,2,4,5\}]$. Removing vertex $4$ from $G'$ yields a subgraph of spectral radius $φ$, against $ρ(G')=2$; the Fibonacci count follows from this spectral gap. We construct an explicit bijection $Ψ_m:\{1,\ldots,6\cdot2^m\}\to\mathcal{P}_m(G)$ onto the directed paths of length $m$ in $G$. We further show that no vertex of $G'$ is dispensable: removing any single vertex reduces the spectral radius strictly below $2$, with hierarchy $1<\sqrt{2}<φ<2$. In particular, every positive cycle of $T$ must visit residue class $2\pmod6$, and a flow conservation identity forces this class to account for more than $18\%$ of the steps in any such cycle.

Metamaterials can achieve exceptional functionality through careful engineering of their mesoscale structure. Although appropriately introduced irregularities can be advantageous, current approaches largely conform to regular structures to preserve tractability. Here, we contend that network theory, enriched with geometry and physics, provides a natural framework for designing metamaterials with controlled irregularities at relevant scales, thereby enabling the discovery of new property-enhancing structures. We examine how this augmented network theory can facilitate the creation of irregular metamaterials with enhanced or novel properties and how metamaterial research, in turn, is opening new directions in network science. Supported by machine learning and advanced self-assembly, the emerging field of irregular metamaterial networks is poised to transform inverse design and scalable manufacturing of novel materials.

This paper presents new results for fast matrix multiplication in small formats obtained by combining the meta flip graph framework with the serendipitous product construction. The framework has been extended to support all 680 rectangular formats with dimensions up to $16 \times 16 \times 16$. Compared to the previous state of the art, ranks are improved for 207 formats. For 84 formats, ternary schemes are found where previously only integer or rational coefficients were known. Additionally, 23 new schemes with asymptotic exponent $ω< \log_2 7$ are discovered, bringing the total number of such schemes to 52. The overall distribution of coefficient types across all investigated formats is 375 ternary, 18 integer, and 287 rational. All code and discovered schemes are available as open source.

Computational epitope prediction is a critical tool for exploring and understanding CD4+ T cell-mediated immune responses, a key aspect of adaptive immunity. While existing computational methods primarily focus on supervised learning approaches, they often overlook the essential role of antigen processing in determining binding specificity. To address this limitation, our group developed Antigen Processing Likelihood (APL), an algorithm that integrates crystallographic B-factor, solvent accessible surface area (SASA), hydrogen exchange protection factors (COREX), and sequence entropy. In this paper we introduce APLSuite, a comprehensive and lightweight software suite designed to streamline APL-based epitope prediction. APLSuite integrates distributed RESTful API services, a Python client for data aggregation and processing, a data science tool for efficient epitope computation, and a user-friendly graphical user interface for non-coding users. It provides a seamless and efficient pipeline for APL calculation and epitope prediction that can be finished in minutes with GPU-acceleration, which has not been implemented by existed tools. This flexible and extensible software suite is deployable on desktop and cloud environments, offering both guided and customizable workflows to meet diverse research needs in immunology research and immunotherapy development. (The project page for this work is available at: https://tulane-mettu-landry-lab.github.io/blogs/APLSuite/)