Continuous Dynamic Time Warping (CDTW) measures the similarity of polygonal curves robustly to outliers and to sampling rates, but the design and analysis of CDTW algorithms face multiple challenges. We show that CDTW cannot be computed exactly under the Euclidean 2-norm using only algebraic operations, and we give an exact algorithm for CDTW under norms approximating the 2-norm. The latter result relies on technical fundamentals that we establish, and which generalise to any norm and to related measures such as the partial Fréchet similarity.

We establish an effective upper bound for the Brascamp-Lieb constant associated to a weighted family of linear maps.

This paper presents a unified framework that connects sequential quadratic programming (SQP) and the iterative linear-parameter-varying model predictive control (LPV-MPC) technique. Using the differential formulation of the LPV-MPC, we demonstrate how SQP and LPV-MPC can be unified through a specific choice of scheduling variable and the 2nd Fundamental Theorem of Calculus (FTC) embedding technique and compare their convergence properties. This enables the unification of the zero-order approach of SQP with the LPV-MPC scheduling technique to enhance the computational efficiency of robust and stochastic MPC problems. To demonstrate our findings, we compare the two schemes in a simulation example. Finally, we present real-time feasibility and performance of the zero-order LPV-MPC approach by applying it to Gaussian process (GP)-based MPC for autonomous racing with real-world experiments.

The Scaled Relative Graph (SRG) is a promising tool for stability and robustness analysis of multi-input multi-output systems. In this paper, we provide tools for exact and computable constructions of the SRG for closed linear operators, based on maximum and minimum gain computations. The results are suitable for bounded and unbounded operators, and we specify how they can be used to draw SRGs for the typical operators that are used to model linear-time-invariant dynamical systems. Furthermore, for the special case of state-space models, we show how the Bounded Real Lemma can be used to construct the SRG.

This paper tackles the challenge of performing multiple quantile regressions across different quantile levels and the associated problem of controlling the familywise error rate, an issue that is generally overlooked in practice. We propose a multivariate extension of the rank-score test and embed it within a closed-testing procedure to efficiently account for multiple testing. Then we further generalize the multivariate test to enhance statistical power against alternatives in selected directions. Theoretical foundations and simulation studies demonstrate that our method effectively controls the familywise error rate while achieving higher power than traditional corrections, such as Bonferroni.

Randomness is fundamental for secure communication and information processing. While continuous-variable optical systems offer an attractive platform for this task, certifying genuine quantum randomness in such setups remains challenging. We present a semi-device-independent scheme for randomness certification tailored to continuous-variable implementations, where the dimension assumption is operationally implemented by restricting state preparations to the two-level Fock subspace. Using standard homodyne and displacement-based measurements, we show that simple optical setups can achieve dimension-witness violations that certify positive min-entropy, even in the presence of realistic losses and misaligned reference frames. These results demonstrate that practical and scalable quantum randomness generation is achievable with minimal experimental complexity on continuous-variable platforms.

The classical one-component plasma (OCP) bounded by a spherical surface reflecting ions (BOCP) is studied using molecular dynamics (MD). Simulations performed for a series of sufficiently large BOCP's make it possible to establish the size dependencies for the investigated quantities and extrapolate them to the thermodynamic limit. In particular, the total electrostatic energy per ion is estimated in the limit of infinite BOCP in a wide range of the Coulomb coupling parameter $Γ$ from 0.03 to 1000 with the relative error of the order 0.1%. Calculated energies are by about 0.5% lower as compared to the modern Monte Carlo (MC) simulation data obtained by different authors at $Γ<30$ and almost coincide with the MC results at $Γ>175$. We introduce two more converging characteristic energies, the excess interatomic electrostatic energy and the excess ion-background electrostatic energy, which enable us to calculate the ionic compressibility factor inaccessible in conventional MC and MD simulation of the OCP with periodic boundary conditions. The derived wide-range ionic equation of state can be recommended for testing OCP simulations with various effective interaction potentials. Based on this equation, we propose an improved cutoff radius for the interionic forces implemented in LAMMPS and perform MD simulation of the OCP to demonstrate that location of the metastable region of the fluid-solid phase transition depends sensitively on this radius.

Computing the cardinality of a maximum induced acyclic vertex set in a digraph is NP-hard. Since finding an exact solution is computationally difficult, a fruitful approach is to establish high-quality lower bounds that are efficiently computable. We build on the Akbari--Ghodrati--Jabalameli--Saghafian (AGJS) bound for digraphs by adapting refinement techniques used by (a) Selkow and Harant--Mohr and (b) Angel--Campigotto--Laforest in their respective improvements of the Caro--Wei bound for undirected graphs. First, inspired by (a), we prove a neighborhood-based refinement of the AGJS bound that incorporates local degree data of each vertex. Second, inspired by (b), we compute the variance of the size of a feedback vertex set returned by a randomized algorithm. This result, combined with the Bhatia--Davis inequality, yields a tighter lower bound than the AGJS bound.

Dynamic wireless charging (DWC) is an emerging technology that has the potential to reduce charging downtime and on-board battery size, particularly in heavy-duty electric vehicles (EVs). However, its spatiotemporal, dynamic, high-power demands pose challenges for power system operations. Since DWC demand depends on traffic characteristics such as speed, density, and dwell time, effective infrastructure planning must account for the coupling between traffic behavior and EV energy consumption. In this paper, we propose a novel traffic-aware microgrid planning framework for DWC. First, we use the macroscopic cell transmission model to estimate spatio-temporal EV charging demand along DWC corridors and integrate this demand into an AC optimal power flow formulation to design a supporting microgrid. Our framework explicitly links traffic patterns with energy demand and demonstrates that traffic-aware microgrid planning yields significantly lower system costs than worst-case traffic-based approaches. We demonstrate the performance of our model on a segment of I-210W in California under a wide range of traffic conditions.

The search for life in the Solar System hinges on data from planetary missions. Detecting biosignatures based on molecular identity, isotopic composition, or chiral excess requires measurements that current and planned missions can only partially provide. We introduce a new class of biosignatures, defined by the statistical organization of molecular assemblages and quantified using diversity metrics. Using this framework, we analyze amino-acid diversity across a dataset spanning terrestrial and extraterrestrial contexts. We find that biotic samples are consistently more diverse -- and therefore distinct -- from their sparser abiotic counterparts. This distinction also holds for fatty acids, indicating that the diversity signal reflects a fundamental biosynthetic signature. It also proves persistent under modeled space-like degradation. Relying only on relative abundances, this biogenicity assessment strategy is applicable to any molecular composition data from archived, current, and planned planetary missions. By capturing a fundamental statistical property of life's chemical organization, it may also transcend biosignatures that are contingent on Earth's evolutionary history.

We propose a novel non-iterative domain decomposition time integrator for acoustic wave equations using a discontinuous Galerkin discretization in space. It is based on a local Crank-Nicolson approximation combined with a suitable local prediction step in time. In contrast to earlier work using linear continuous finite elements with mass lumping, the proposed approach enables higher-order approximations and also heterogeneous material parameters in a natural way.

Classical studies of the Fibonacci sequence focus on its periodicity modulo $m$ (the Pisano periods) with canonical initialization. We investigate instead the complete periodic structure arising from all $m^2$ possible initializations in $(\mathbb{Z}/m\mathbb{Z})^2$. We discover perfect mirror symmetry between the Fibonacci recurrence $a_n = a_{n-1} + a_{n-2}$ and its parity transform $a_n = - a_{n-1} + a_{n-2}$ and observe fractal self-similarity in the extension from prime to prime power moduli. Additionally, we classify prime moduli based on their quadratic reciprocity and demonstrate that periodic sequences exhibit weight preservation under modular extension. Furthermore, we define a minima distribution $P(n)$ governed by Lucas ratios, which satisfies the symmetric relation $P(n)=P(1-n)$. For cyclotomic recurrences, we propose explicit counting functions for the number of distinct periods with connections to necklace enumeration. These findings imply potential connections to Viswanath's random recurrence, modular forms and L-functions.

Anonymizing microdata requires balancing the reduction of disclosure risk with the preservation of data utility. Traditional evaluations often rely on single measures or two-dimensional risk-utility (R-U) maps, but real-world assessments involve multiple, often correlated, indicators of both risk and utility. Pairwise comparisons of these measures can be inefficient and incomplete. We therefore systematically compare six visualization approaches for simultaneous evaluation of multiple risk and utility measures: heatmaps, dot plots, composite scatterplots, parallel coordinate plots, radial profile charts, and PCA-based biplots. We introduce blockwise PCA for composite scatterplots and joint PCA for biplots that simultaneously reveal method performance and measure interrelationships. Through systematic identification of Pareto-optimal methods in all approaches, we demonstrate how multivariate visualization supports a more informed selection of anonymization methods.

We investigate the relativistic dynamics of a spin half particle in the presence of a Lorentz-violating background within the framework of effective field theory. A modified Dirac Hamiltonian is considered, arising from a CPT odd coupling involving the Lorentz violating gauge tensor of the Standard Model Extension (SME). The velocity and effective force operators are derived from the Heisenberg equations of motion. Using Ehrenfest s theorem and the correspondence principle, we obtain the classical limit of the dynamics and identify an effective force exhibiting a generalized Lorentz force structure. This formalism is applied to a Penning trap system, known for its high precision measurements of charged particle properties. Our analysis shows that the effective cyclotron frequency acquires a correction due to the Lorentz violating term, leading to deviations in the particle trajectory and offering a potentially observable signature of Lorentz violation in precision experiments. By comparing our results with current bounds from high precision Penning traps, we establish an upper limit on the Lorentz violating coupling, $\bar{g}k_{AF}\lesssim 2.66 \times 10^{-4}\mathrm{eV}^{-1}$ corresponds to LIV effects. This bound is compatible with the interpretation of an effective Lorentz violation, consistent with current observational constraints, and it reinforces the phenomenological nature of the term under consideration, in agreement with previous analyses based on cosmological birefringence and photon propagation in a Lorentz violating background.

We investigate the problem of identifying planted cliques in random geometric graphs, focusing on two distinct algorithmic approaches: the first based on vertex degrees (VD) and the other on common neighbors (CN). We analyze the performance of these methods under varying regimes of key parameters, namely the average degree of the graph and the size of the planted clique. We demonstrate that exact recovery is achieved with high probability as the graph size increases, in a specific set of parameters. Notably, our results reveal that the CN-algorithm significantly outperforms the VD-algorithm. In particular, in the connectivity regime, tiny planted cliques (even edges) are correctly identified by the CN-algorithm, yielding a significant impact on anomaly detection. Finally, our results are confirmed by a series of numerical experiments, showing that the devised algorithms are effective in practice.

We investigate the ferromagnetic-paramagnetic phase transition in coherently (Rabi) coupled Bose-Einstein condensates at zero and finite temperatures, exploring different routes to the transition by tuning the Rabi coupling or increasing the temperature at a fixed coupling. Using the Hartree-Fock-Bogoliubov theory within the Popov approximation, we map out the finite-temperature phase diagram of a three-dimensional homogeneous condensate and identify the critical line through the softening of the spin gap. Magnetization and the spin dispersion branch reveal the progressive suppression of the ferromagnetic order with increasing temperature. In quasi-one-dimensional harmonic traps, the transition, driven by Rabi coupling, is inferred through the softening of the spin breathing mode with its minimum shifting to lower coupling values with increasing temperature. Notably, the thermally driven transition causes monotonic hardening of all the spin modes. For both coupling and temperature-driven transition, the hybridized density modes in the ferromagnetic phase acquire more density character while approaching the critical point.

Considerable evidence now exists for partonic energy loss due to interaction with the hot, dense medium created in ultra-relativistic heavy-ion collisions. A primary signal of this energy loss is the suppression of high transverse momentum $p_{\mathrm{T}}$ hadron yields in A-A collisions relative to appropriately scaled $pp$ collisions at the same energy. Measuring the collision energy dependence of this energy loss is vital to understanding the medium, but it is difficult to disentangle the medium-driven energy loss from the natural kinematic variance of the steeply-falling $p_{\mathrm{T}}$ spectra across different $\sqrt{s_{\mathrm{NN}}}$. To decouple these effects, we utilize a phenomenologically motivated spectrum shift model to estimate the average transverse momentum loss $Δp_{\mathrm{T}}$ imparted on high $p_{\mathrm{T}}$ partons in A-A collisions, a proxy for the medium induced energy loss. We observe a striking correlation between $Δp_{\mathrm{T}}$ and Glauber-derived estimates of initial state energy density $\varepsilon_{\mathrm{Bj}}$, consistent across two orders of magnitude in collision energy for a variety of nuclear species. To access the path-length dependence of energy loss, we couple our model to geometric event shape estimates extracted from Glauber calculations to produce predictions for high-$p_{\mathrm{T}}$ hadron elliptic flow $v_2$ that agree reasonably with data.

In this work, we show that suitably designed spatially varying linearly polarized light provides a versatile route to generate and control pseudo-magnetic fields in Weyl semimetals through Floquet engineering. Within a high-frequency expansion, we derive an effective axial gauge potential $\mathbf{A}_5(\mathbf{r})$ whose curl gives the pseudo-magnetic field $\mathbf{B}_5(\mathbf{r})$. By mapping the light profile to $\mathbf{A}_5(\mathbf{r})$, we establish design principles for pseudo-magnetic field textures that mimic strain-induced gauge fields while offering key advantages like dynamic control, full reversibility, spatial selectivity, and absence of material deformation. We compare the Landau-level spectra produced by uniform real and pseudo-magnetic fields and also analyze both their linear optical conductivity and the second-order dc responses. Our results enable real-time manipulation of pseudo-magnetic fields and predict clear experimental signatures for optically engineered gauge fields in topological semimetals.

In this paper we extend several classical results on pointed torsion theories -- also known as torsion pairs -- to the setting of non-pointed torsion theories defined via kernels and cokernels relative to a fixed class of trivial objects (often referred to as pretorsion theories). Our results are developed in the recently introduced framework of (non-pointed) prenormal categories and other related contexts. Within these settings, we recover some characterisations of torsion and torsion-free subcategories, as well as the classical correspondences between torsion theories and closure operators. We also suitably extend a correspondence between torsion theories and (stable) factorisation systems on the ambient category, known in the homological case. Some of these results are then further specialised to an appropriate notion of hereditary torsion theory. Finally, we apply the developed theory to construct new examples of pretorsion theories.

We present CANOPI, a novel algorithmic framework, for solving the Contingency-Aware Nodal Power Investments problem, a large-scale nonlinear optimization problem that jointly optimizes investments in generation, storage, and transmission upgrades, including representations of unit commitment and long-duration storage. The underlying problem is nonlinear due to the impact of transmission upgrades on impedances, and the problem's large scale arises from the confluence of spatial and temporal resolutions. We propose algorithmic approaches to address these computational challenges. We pose a linear approximation of the overall nonlinear model, and develop a fixed-point algorithm to adjust for the nonlinear impedance feedback effect. We solve the large-scale linear expansion model with a specialized level-bundle method leveraging a novel interleaved approach to contingency constraint generation. We introduce a minimal cycle basis algorithm that improves the numerical sparsity of cycle-based DC power flow formulations, accelerating solve times for the operational subproblems. CANOPI is demonstrated on a 1493-bus Western Interconnection test system built from realistic-geography network data, with hourly operations spanning 52 week-long scenarios and a total possible set of 20 billion individual transmission contingency constraints. Numerical results quantify reliability and economic benefits of incorporating transmission contingencies in integrated planning models and highlight the computational advantages of the proposed methods.

We study the optimization of Steklov eigenvalues with respect to a boundary density function $ρ$ on a bounded Lipschitz domain $Ω\subset \mathbb{R}^N$. We investigate the minimization and maximization of $λ_k(ρ)$, the $k$th Steklov eigenvalue, over admissible densities satisfying pointwise bounds and a fixed integral constraint. Our analysis covers both first and higher-order eigenvalues and applies to general, not necessarily convex or simply connected, domains. We establish the existence of optimal solutions and provide structural characterizations: minimizers are bang--bang functions and may have disconnected support, while maximizers are not necessarily bang--bang. On circular domains, the minimization problem admits infinitely many minimizers generated by rotational symmetry, while the maximization problem has infinitely many distinct maximizers that are not symmetry-induced. We also show that the maps $ρ\mapsto λ_k(ρ)$ and $ρ\mapsto 1/λ_k(ρ)$ are generally neither convex nor concave, limiting the use of classical convex optimization tools. To address these challenges, we analyze the objective functional and introduce a Fréchet differentiable surrogate that enables the derivation of optimality conditions. We further design an efficient numerical algorithm, with experiments illustrating the difficulty of recovering optimal densities when they lack smoothness or exhibit oscillations.

Molecular chirality plays an important role in chemistry and biology, allows control of biological interactions, affects drugs efficacy and safety, and promotes synthesis of new materials. In general, chirality manifests itself in optical activity (circular dichroism or circular birefringence). Chiral plasmonic nanoparticles have been recently developed for molecular enantiomer separation, chiral sensing and chiral photocatalysis. Here, we show that optical chirality of plasmonic nanoparticles exhibiting strong scattering can remain completely undetected using standard characterisation techniques, such as circular dichroism measurements. This phenomenon, which we term meso-chiral in analogy to meso-compounds in chemistry, is based on mutual cancellation of absorption and scattering chiral responses. As a prominent example, the meso-optical behaviour has been numerically demonstrated in multi-wound-SiO2/Au nanoparticles over the entire visible spectral range and in other prototypical chiral nanoparticles in narrower spectral ranges. The meso-chiral property has been experimentally verified by demonstrating chiral absorption of gold helicoid nanoparticles at the wavelength where conventional circular dichroism measurements show absence of chiral response (gext=0). These findings demonstrate a valuable link between microscopic to macroscopic manifestations of chirality and can provide insights for interpreting a wide range of experimental results and designing chiral properties of plasmonic nanoparticles.

We introduce new adaptive schemes for the one- and two-dimensional hyperbolic systems of conservation laws. Our schemes are based on an adaption strategy recently introduced in [{\sc S. Chu, A. Kurganov, and I. Menshov}, Appl. Numer. Math., 209 (2025)]. As there, we use a smoothness indicator (SI) to automatically detect ``rough'' parts of the solution and employ in those areas the second-order finite-volume low-dissipation central-upwind scheme with an overcompressive limiter, which helps to sharply resolve nonlinear shock waves and linearly degenerate contact discontinuities. In smooth parts, we replace the limited second-order scheme with a quasi-linear fifth-order (in space and third-order in time) finite-difference scheme, recently proposed in [{\sc V. A. Kolotilov, V. V. Ostapenko, and N. A. Khandeeva}, Comput. Math. Math. Phys., 65 (2025)]. However, direct application of this scheme may generate spurious oscillations near ``rough'' parts, while excessive use of the overcompressive limiter may cause staircase-like nonphysical structures in smooth areas. To address these issues, we employ the same SI to distinguish contact discontinuities, treated with the overcompressive limiter, from other ``rough'' regions, where we switch to the dissipative Minmod2 limiter. Advantage of the resulting adaptive schemes are clearly demonstrated on a number of challenging numerical examples.

Intracortical brain-machine interfaces require decoders that adapt continuously to neural signal instability while operating within strict memory budgets. We introduce a dual-timescale Hebbian accumulator learning rule for spiking neural networks that enables per-timestep online supervised updates with training memory constant in sequence length, avoiding backpropagation through time. The rule combines synapse-specific fast and slow eligibility traces, error-modulated three-factor updates, and integer-friendly RMS homeostasis, operating without adaptive gradient optimizers (Adam, RMSProp) or replay buffers. On two primate intracortical datasets, the method achieves Pearson correlations of $R \geq 0.81$ on MC~Maze and $R \geq 0.63$ on Zenodo~Indy, with 63--86\% measured memory reduction versus BPTT at sequence length $T = 1000$. Closed-loop simulations demonstrate online adaptation to neural disruptions and learning from scratch without offline calibration.

The permutohedron $P_n$ of order $n$ is a polytope embedded in $\mathbb{R}^n$ whose vertex coordinates are permutations of the first $n$ natural numbers. It is obvious that $P_n$ lies on the hyperplane $H_n$ consisting of points whose coordinates sum up to $n(n+1)/2$. We prove that if the vertices of $P_n$ are contained in the union of $m$ affine hyperplanes different from $H_n$, then $m\geq n$ when $n \geq 3$ is odd, and $m \geq n-1$ when $n \geq 4$ is even. This result has been established by Pawlowski in a more general form. Our proof is shorter, rather different, and gives an algebraic criterion for a non-standard permutohedron generated by $n$ distinct real numbers to require at least $n$ non-trivial hyperplanes to cover its vertices.

Circular external difference families (CEDFs) are a recently-introduced variation of external difference families with applications to non-malleable threshold schemes: a $(v,m,\ell,1)$-CEDF is an $m$-sequence $(A_0, \ldots, A_{m-1})$ of $\ell$-subsets of an additive group $G$ of order $v$ such that $G\setminus\{0\}$ equals the multiset of all differences $a-a'$, with $(a,a')\in A_{i+1}\times A_{i}$ for some $i \in \mathbb{Z}_m$. When $G$ is the cyclic group, we speak of a cyclic CEDF. The existence of cyclic $(v,m,\ell,1)$-CEDFs is well understood when $m$ is even, while nonexistence is known when both $m$ and $\ell$ are odd. However, the case where $m$ is odd and $\ell$ is even has only been resolved in a few special cases. In this paper, we address this gap by constructing cyclic $(v,m,\ell,1)$-CEDFs for any odd $m>1$ when $\ell=2$, and for any even $\ell \ge 2$ when $m=3$. Notably, the latter result relies on the existence of a suitable tiling of the multiplicative semigroup of $\mathbb{Z}_v\setminus\{0\}$. Our approach is based on representing the blocks as arithmetic progressions and analyzing their step patterns. We present two different ways to construct cyclic $(v,m,2,1)$-CEDFs for every odd $m>1$. Their step patterns show that the resulting CEDFs are inequivalent. Many additional inequivalent CEDFs are obtained by translating suitable subsets within the CEDF.

We introduce a framework for the enhanced estimation of photometric redshifts using Self-Organising Maps (SOMs). Our method projects galaxy Spectral Energy Distributions (SEDs) onto a two-dimensional map, identifying regions that are sparsely sampled by existing spectroscopic observations. These under-sampled areas are then augmented with simulated galaxies, yielding a more representative spectroscopic training dataset. To assess the efficacy of this SOM-based data augmentation in the context of the forthcoming Legacy Survey of Space and Time (LSST), we employ mock galaxy catalogues from the OpenUniverse2024 project and generate synthetic datasets that mimic the expected photometric selections of LSST after one (Y1) and ten (Y10) years of observation. We construct 501 degraded realisations by sampling galaxy colours, magnitudes, redshifts and spectroscopic success rates, in order to emulate the compilation of a wide array of realistic spectroscopic surveys. Augmenting the degraded mock datasets with simulated galaxies from the independent CosmoDC2 catalogues has markedly improved the performance of our photometric redshift estimates compared to models lacking this augmentation, particularly for high-redshift galaxies ($z_\mathrm{true} \gtrsim 1.5$). This improvement is manifested in notably reduced systematic biases and a decrease in catastrophic failures by up to approximately a factor of 2, along with a reduction in information loss in the conditional density estimations. These results underscore the effectiveness of SOM-based augmentation in refining photometric redshift estimation, thereby enabling more robust analyses in cosmology and astrophysics for the NSF-DOE Vera C. Rubin Observatory.

We investigate the effects of ram-pressure stripping on four galaxies within the massive, strong-lensing cluster MACS-J0138.0-2155 ($z=0.336$). Of these, three are classified as jellyfish galaxies, with significant elongated tails. Two of these jellyfish galaxies, J1 and J2, are in a late-stage of stripping and show post-starburst features within their disk regions with star formation only in the tails. Using VLT/MUSE integral field spectroscopic data, we spatially resolve the stellar and gas kinematics to examine extraplanar gas associated with ram-pressure stripping. We complement this analysis with optical and near-infrared imaging from the Hubble Space Telescope to visualize the galactic structure of each member. The jellyfish galaxies are all blue-shifted with respect to the cluster and show velocity gradients of a few hundred $\mathrm{kms}^{-1}$ across their tails. From the resolved gas kinematics, we derive H$α$-based star formation rates; these are generally low reaching a maximum of approximately 0.49 $\mathrm{M_{\odot}\text{yr}^{-1}kpc^{-2}}$ in galaxy J3. We also report the kinematics for galaxy J4, which lies in the foreground of the cluster but close in projection to one of the lensed arcs.

Spectral graph theory studies how the eigenvalues of a graph relate to the structural properties of a graph. In this paper, we solve three open problems in spectral extremal graph theory which generalize the classical Turán-type supersaturation results. (a) We prove that every $m$-edge graph $G$ with the spectral radius $λ(G) > \sqrt{m}$ contains at least $\frac{1}{144} \sqrt{m}$ triangles sharing a common edge. This result confirms a conjecture of Nikiforov, and Li and Peng. Moreover, the bound is optimal up to a constant factor. (b) Next, for $m$-edge graph $G$ with $λ(G) > \sqrt{(1-\frac{1}{r})2m}$, we show that it must contain $Ω_r (\sqrt{m})$ copies of $K_{r+1}$ sharing $r$ common vertices. This confirms a conjecture of Li, Liu and Feng and unifies a series of spectral extremal results on books and cliques. Moreover, we also show that such a graph $G$ contains $Ω_r (m^{\frac{r-1}{2}})$ copies of $K_{r+1}$. This extends a result of Ning and Zhai for counting triangles. (c) We prove that every $m$-edge graph $G$ with $λ(G) > \sqrt{m}$ contains at least $(\frac{1}{8}-o(1)) m^2$ copies of 4-cycles, and we provide two constructions showing that the constant $\frac{1}{8}$ is the best possible. This result settles a problem raised by Ning and Zhai, and it gives the first asymptotics for counting degenerate bipartite graphs. The key to our proof are two structural results we obtain for graphs with large spectral radii on their maximum degree and on existence of large structured subgraphs, which we believe to be of independent interest.

Quantum simulations of metal surfaces are critical for catalytic innovation. Yet existing methods face a cost-accuracy dilemma: density functional theory is efficient but system-dependent in accuracy, while wavefunction-based theories are accurate but prohibitively costly. Here we introduce FEMION (Fragment Embedding for Metals and Insulators with Onsite and Nonlocal correlation), a systematically improvable quantum embedding framework that resolves this challenge by capturing partially filled electronic states in metals. FEMION combines auxiliary-field quantum Monte Carlo for local catalytic sites with a global random phase approximation treatment of nonlocal screening, yielding a scalable approach across diverse catalytic systems. Employing FEMION, we address two longstanding challenges: determining the preferred CO adsorption site and quantifying the H2 desorption barrier on Cu(111). Furthermore, our calculations demonstrate that the recently discovered 10-electron-count rule can also be extended to the single-atom catalysis processes on 3d metal surfaces, resolving the controversies arising from density functional theory calculations. We thus open a predictive, first-principles route to modeling complex catalytic systems.