The interdependence between electric power systems and transportation systems is rapidly increasing due to the high adoption of Electric Vehicles (EVs) and their charging infrastructures. Electric vehicles can represent additional load for the power system, but can also bring new opportunities for contributing to the efficient and resilient operations of the power grid. This is mainly because of their ability to provide back power to the system when it is not used for transportation, essentially serving as a moving battery source for the power grid. This so-called Vehicle-to-Grid (V2G), Vehicle-to-Building (V2B), or, generally, Vehicle-to-X (V2X) capability of EVs has been extensively studied in the literature. However, the upcoming development of autonomous driving systems and their integration within sharing mobility services can significantly add to the possibilities of interaction between the transportation system and the power grid. This paradigm is studied to a much lesser extent in the existing literature. Shared Autonomous Electric Vehicles (SAEVs) could allow for more control of the actions of the fleet allowing for large-scale coordinated responses both to mobility and energy demands. This coordinated response can be particularly useful for providing emergency power services in case of power loss while maintaining a high level of transportation service. Thus, improving the overall resilience of the system. In this work, we develop a dynamic optimization framework to evaluate the potential contribution of the SAEV fleet for improving critical buildings' energy resilience via V2B services. The model considers passengers' pick-up and transportation, relocation of vehicles, and battery charging and discharging. Power outage scenarios for critical buildings are considered and the potential of the SAEV fleet to fully or partially respond to the emergency power outage is studied. In addition, sensitivity analysis for key parameters such as the outage parameters is introduced. The results of the case study for the Ile-de-France region in France shows that the SAEV fleet has the potential to provide V2B service for critical building at an acceptable loss of passenger total waiting time on the transportation side. Furthermore, it is shown that it is capable of satisfying the emergency power load at a lower cost compared to investing in extra backup generators unless the outage occurs at significantly high frequencie

A graph $G$ is $\{F_{1}, F_{2},\dots,F_{k}\}$-free if $G$ contains no induced subgraph isomorphic to any $F_{i}$ $(1\leq i \leq k)$. A connected graph $G$ is a split graph if its vertex set can be partitioned into a clique and an independent set. Ryjáček et al. [J. Comb. Theory, Ser. B 134 (2019) 239--263] conjectured that every $4$-connected $\{K_{1,4},K_{1,4}+e\}$-free graph with minimum degree at least 6 is Hamiltonian and they confirmed the case with connectivity at least 5, where $K_{1,4}+e$ is the graph obtained from $K_{1,4}$ by adding a new edge. In this paper, we show that every 3-connected $\{K_{1,4},K_{1,4}+e\}$-free split graph of order at least $13$ is Hamilton-connected. It implies that Ryjáček et al.'s conjecture holds for split graphs of order at least $13$.

In this paper we introduce a generalization of classical risk measures in which the risk is represented by a step function taking two values, corresponding to two endogenously determined market regimes. This extends the traditional framework where risk measures map random variables to single real numbers. For the quadratic loss function, we study the optimization problem of determining the optimal regime threshold and corresponding values. In the case of log-concave distributions we give conditions for the uniqueness of the regime changing. We treat the case of one dimension and also of multi-dimensions for elliptic distributions. We demonstrate the necessity of convexity through counterexamples.

We investigate approximate null-controllability for semi-discrete heat equations on the lattice $h\mathbb{Z}^d$ with a potential. By establishing spectral inequalities for the discrete Schr{ö}dinger operator $P_h = -Δ_h + V$ on equidistributed sets, we derive observability estimates via the Lebeau-Robbiano method and the Hilbert Uniqueness Method. For bounded potentials, we obtain quantitative controllability results with explicit dependence on the potential and show near optimality of the geometric condition on the observation set. We also treat polynomial growth potentials, for which similar properties hold with weaker control cost estimates. These results extend discrete Carleman techniques to the full-space lattice setting and provide new spectral estimates for discrete Schr{ö}dinger operators.

We consider decision-making under incomplete information about an unknown state of nature. We show that a decision problem yields a higher value of information than another, uniformly across information structures, if and only if it is obtained by adding an independent, parallel decision problem.

A $q$-ary $t$-$(n,w,λ)$ design is a collection $\mathcal{A}$ of vectors of weight $w$ in $\mathbb{F}_{q}^{n}$ with the property that every vector of weight $t$ in $\mathbb{F}_{q}^{n}$ is contained in exactly $λ$ members of $\mathcal{A}$. The supports of the vectors in a $q$-ary $t$-design form an ordinary $t$-design, possibly with repeated blocks. While linear codes supporting ordinary combinatorial designs have been extensively studied, the case where codes hold $q$-ary designs remains largely unexplored. This motivates a systematic investigation into whether codewords of a fixed weight in a linear code can form a $q$-ary $t$-design. Building on previous work, we develop two new criteria for this purpose. Applying these criteria, we show that several families of linear codes hold $q$-ary $2$-designs, including one- and two-weight codes, extremal self-dual codes, as well as certain dual codes, shortened codes, and punctured codes derived from them. Moreover, for linear codes that do not satisfy these criteria, we provide an alternative approach based on the automorphism group of the code. This method enables the construction of $q$-ary $2$-designs from doubly-extended Reed-Solomon codes. Notably, for a class of linear codes previously known to support $4$-designs, we demonstrate that their codewords of certain weights give rise to $q$-ary $2$-designs whose parameters are precisely determined.

In this study, we investigate the chemical properties of the GD-1 stream using cross-matched, data-driven elemental abundances. The results reveal no clear $α$-knee in the [Mg/Fe]-[Fe/H] plane, and strong abundance consistency between the thin stream and cocoon, supporting a common origin. The absence of multiple-population signatures (e.g., C-N anti-correlation) suggests a low-mass progenitor. Using a test-particle simulation with the particle spray method and including perturbations from the Sagittarius (Sgr) dwarf galaxy, it shows that Sgr does not significantly heat the stream to form the cocoon, but modifies the intrinsic $ϕ_2$ distribution, in agreement with observations. The trailing arm narrowly distributed across the width of the stream, while the leading arm is more diffuse, indicating that major fraction of cocoon stars are present towards the leading arm. Sgr also drags more stream particles moving toward the Galactic center, producing an excess at $V_{\text{GSR}}<0$, consistent with data. Our study confirms the Sgr has a non-negligible dynamical influence on the GD-1 stream. Other heating mechanisms (e.g., dark matter sub-halo encounters and pre-stripping process inside the parent halo) remain to be considered, and higher-resolution spectroscopy is needed to further constrain chemical abundances.

In this paper, we study the effect of the modified dispersion relation (MDR) on the thermodynamics of the Kerr-Newman black hole surrounded by quintessence and a cloud of string. The thermodynamic properties of the Kerr-Newman black hole are shown to rely not only on the black hole's properties but also on the parameters associated with the modified dispersion relation, quintessence, and the cloud string. Additionally, the equation of state is impacted by these parameters. The remnant and the stability of the black hole are also discussed under the correction of MDR, quintessence, and a cloud of string. In addition, the null geodesic structure of the spacetime is studied using the Hamilton--Jacobi formalism. The effective potential for photon motion is obtained, and the radii of the prograde and retrograde circular photon orbits are determined. The instability of these circular photon orbits is further characterized by the Lyapunov exponent.

Absolute anonymization, conceived as an irreversible transformation that prevents re-identification and sensitive value disclosure, has proven to be a broken promise. Consequently, modern data protection must shift toward a privacy-utility trade-off grounded in risk mitigation. Differential Privacy (DP) offers a rigorous mathematical framework for balancing quantified disclosure risk with analytical usefulness. Nevertheless, widespread adoption remains limited, largely because effective translation of complex technical concepts, such as privacy-loss parameters, into forms meaningful to non-technical stakeholders has yet to be achieved. This difficulty arises from the inherent use of randomization: both legitimate analysts and potential adversaries must draw conclusions from uncertain observations rather than deterministic values. In this work, we propose a new interpretation of the privacy-utility trade-off based on hypothesis testing. This perspective explicitly accounts for the uncertainty introduced by randomized mechanisms in both membership inference scenarios and general data analysis. In particular, we introduce the concept of relative disclosure risk to quantify the maximum reduction in uncertainty an adversary can obtain from protected outputs, and we show that this measure is directly related to standard privacy-loss parameters. At the same time, we analyze how DP affects analytical validity by studying its impact on hypothesis tests commonly used to assess the statistical significance of empirical results. Finally, we provide practical guidance, accessible to non-experts, for navigating the privacy-utility trade-off, aiding in the selection of suitable protection mechanisms and the values for the privacy-loss parameters.

We derive bolometric correction functions for oscillation mode amplitudes observed by the different cameras of the ESA PLATO mission. Such corrections between bolometric (full light) and mission instrument-specific amplitudes enable comparisons to theoretical expectations and amplitude conversion between different photometric missions, which is essential for proper detectability yields and target selection. Bolometric correction functions were calculated assuming a Planck function approximation for the stellar spectral flux distribution. The calculations follow the procedures applied in earlier analyses for the NASA Kepler and TESS missions. We derived power-law and polynomial parametrisations of the bolometric corrections with $T_{\rm eff}$. We find that on average, oscillation mode amplitudes from PLATO's normal cameras (N-CAMs) are expected to be ~6.7% lower compared to Kepler, and ~12.5% higher compared to TESS. A significant average amplitude ratio of ~25% is expected for amplitudes measured using the blue PLATO fast camera (F-CAM) compared to TESS. We find that observations of bright solar-like oscillators, especially with PLATO's F-CAMs, would provide an important test of the predicted corrections.

We clarify the relationship between rotation and the energy condition for stationary rotating wormhole solutions of the Einstein equations coupled to a phantom field in five-dimensional spacetime with equal angular momenta, particularly with large asymmetry between the two sides. It was shown by Dzhunushaliev et al. that the violation of the null energy condition can become arbitrarily small due to rotation. We find that the degree of violation of the null energy condition is essentially determined by the angular momentum and shows little dependence on asymmetry, that is, the mass difference between the two asymptotic regions. We also discuss the relation between the wormhole spacetime and the Myers-Perry black hole. We find that the geometry asymptotes to the extremal Myers-Perry spacetime in the limit of large angular momentum, while the non-extremal black hole geometry cannot be reproduced in any limit.

We present a cold-fluid analysis of the purely transverse Weibel (current-filamentation) instability across four regimes: non-relativistic (NR) single-species, NR multi-species, relativistic single-species, and relativistic multi-species (electron--positron and electron--proton). Beginning from linearized fluid equations, we derive the dispersion relations in each regime and extract scaling laws for the maximum growth rate $γ_{\rm max}$ and characteristic unstable wavenumber $k_{\rm max} = ω_{pi}/c$. Relativistic corrections suppress $γ_{\rm max}$ by up to 40 per cent above $v_0 \approx 0.2c$, peaking near $v_0 \approx 0.9c$. Multi-species effects are significant only for $m_e/m_i \gtrsim 1/500$. For the tabletop laser experiment of Bai et al., Nat.Commun., 16, 3770 (2025), the cold-fluid prediction gives $d_i = c/ω_{pi} \approx 31.7\,μ{\rm m}$, within 2 per cent of the measured filament spacing $λ_F \approx 31\,μ{\rm m}$. The saturation field estimate $B_{\rm sat} \approx 2.3\times10^4$ T is an upper bound, consistent with the measured $\approx 5000$ T under kinetic suppression. Two MMS burst-mode bow shock crossings (October 16, 2015 and November 25, 2017) confirm $k_{\rm max} d_i = 1$ from FGM/FPI data. A multi-environment scatter plot spans 21 orders of magnitude in $n_i$, with all points within a factor of 3 of the 1:1 line.

Geometrical frustration is a powerful route to realize exotic phases such as quantum spin liquids. Despite extensive efforts, systematic searches targeting specific frustration motifs and their potential to host unconventional magnetic ground states remain rare, thus highlighting the need for a more focused and predictive materials discovery approach. Here we present a new strategy combining high-throughput first-principles calculations, magnetic force theory, and spin Hamiltonian analysis. Starting from the 150,000 material database, we catalogue candidate materials that may host competing exchange interactions and new types of magnetic states with the focus on kagome or triangular lattices. Our workflow not only reproduces the majority of known frustrated magnetic materials, validating our approach, but also predicts novel candidate compounds with targeted frustration profiles that have not yet been experimentally synthesized. Among these, we identify six promising new materials: one triangular lattice compound, KMgNiIO6, and five kagome lattice compounds; Li4Fe3WO8, Li2V3F8, Li5VP2(O4F)2, and Li2MgCo3O8 (P2/m and C2/m). For each candidate, we identify detailed magnetic properties and further propose their potential magnetic ground states, revealing that some of them may host entirely new magnetic phases driven by their distinct frustration characteristics.

We establish the inviscid limit of the Yudovich solution to the heat conductive Boussinesq equation with initial velocity and temperature/buoyancy in $L^2$ and initial vorticity in $L^\infty$ on the two-dimensional periodic domain ${\bf T}^2$. Given any finite time $T>0$ and $p \in [1,\infty[$, we show that the solution to the diffusive Boussinesq equation converges in $L^\infty(0,T; W^{1,p}({\bf T}^2))$ to the solution to the Euler--Boussinesq equation as the viscosity tends to zero, provided that the initial vorticity, velocity, and temperature/buoyancy converge strongly in $L^2$. Our proof adapts and extends the arguments in [P. Constantin, T. D. Drivas, and T. M. Elgindi, Comm. Pure Appl. Math. 75 (2022), 60--82] to forcing terms in $L^1(0,T; L^\infty({\bf T}^2))$.

The rapid advancement and impressive capabilities of large language models (LLMs) have given rise to the field of prompt engineering, the practice of crafting inputs to guide LLMs toward high-quality, task-relevant outputs. A critical challenge facing the field is the lack of standardised prompt documentation and evaluation practices. Prompts can be long, complex and difficult to evaluate on subjective tasks. To address this challenge, we propose the use of prompt cards, structured summaries of prompt engineering practices inspired by the concept of model cards. Through prompt cards, the specific goals, considerations and steps taken during prompt engineering can be systematically documented and assessed. We present the prompt card approach and illustrate it on a specific task called wordalisation, in which structured numerical data is transformed into text. We argue that a well-structured prompt card can enable better reproducibility, transparency, improve prompt methodology and give an effective alternative to benchmarking for judging the quality of generated texts. By systemically capturing underlying model details, prompt intent, contextualisation strategies, evaluation practices and ethical considerations, prompt cards make explicit the often implicit design decisions that shape system behaviour. Documenting these choices is important as prompting increasingly involves complex pipelines with multiple moving parts.

Quantum contextuality is a fundamental nonclassical property of quantum systems, regarded as a key resource that demonstrates the computational and informational advantages of quantum over classical systems. Our present work aims to construct Hardy's paradoxes, a set of possibilistic conditions witnessing contextuality, for Yu-Oh set, which is the state-independent contextual quantum system with the least number of vectors. To achieve the aim, we systematically enumerate all logically contextual pure states on Yu-Oh set, and theoretically prove that no mixed states in this scenario are logically contextual. Based on the identified logically contextual quantum states, we construct 12 Hardy's paradoxes with identical success probability SP=11.1%. Furthermore, we present corresponding observables to experimentally witness these Hardy's paradoxes.

Simulation of chemical reactions on quantum computing platforms using quantum classical hybrid algorithms such as the Variational Quantum Eigensolver (VQE) is challenged by the need for a reaction consistent treatment of electron correlation in reaction energy evaluations. In this work, we employ a previously reported symmetry guided active space selection protocol to compute ring strain energies of cyclic hydrocarbons using homodesmotic reaction schemes. The protocol enforces symmetry consistency across all reactants and products by selecting active spaces that yield identical symmetry matched fraction (SMF) values, thereby ensuring balanced correlation treatment at the reaction level. When multiple active spaces satisfy this criterion for a given molecule, larger active spaces often provide improved correlation treatment; however, smaller symmetry consistent active spaces can also yield comparable agreement due to favorable error cancellation within the homodesmotic framework. Using this framework, ring strain energies were evaluated for a series of saturated and unsaturated cyclic hydrocarbons, ranging from cyclopropane to the structurally complex adamantane. The resulting energies achieve chemical accuracy relative to density functional theory (DFT) and remain in close agreement with coupled cluster singles and doubles (CCSD) benchmarks. The systematic performance across increasing molecular complexity highlights the effectiveness of combining homodesmotic reaction design with symmetry-consistent VQE calculations. This approach, which enforces physically grounded consistency across reaction species, demonstrates clear potential for extending reaction based quantum simulations to larger molecular systems and broader classes of chemical reactions.

Purely baryonic weak decays of heavy baryons are investigated within the framework of the Skyrme model. These decays belong to a new class of unobserved decay channels, which would help us to test the standard model, particularly potential sources of CP violation in the baryonic sector. By interpreting the heavy baryon as a bound state of a heavy meson and a baryon (Skyrmion), a direct calculation of the decay process $Λ_b \to p\,\bar p\,n$ is performed. The resulting branching fraction is of $\mathcal{O}(10^{-6})$, in agreement with previous estimates.

Zero-bias current induced by injected phonons in metals and one-dimensional charge density wave (CDW) systems attached on the surface of the piezoelectric substrate is investigated microscopically based on the second order response theory. In contrast to the shift currents discovered by von Baltz and Kraut in which the zero-bias current is induced by AC electric field in systems without inversion symmetry, propagating phonons break the inversion symmetry in the presesnt case. The effects of both deformation potential and piezoelectric potential are taken into account. In the CDW system, zero-bias current appears below the transition temperature and its magnitude strongly depends on the position of the chemical potential. Possible experimental consequences are discussed.

Alternating (AC) magnetic fields can induce not only an alternating magnetization in materials, but also an alternating temperature via the magnetocaloric effect. The latter effect is typically neglected when performing AC susceptibility measurements, but consideration of both effects on an equal footing is necessary in order to reliably distinguish between internal and external causes of magnetic response and accurately extract quantitative information about relaxation processes. In order to address this, we have developed a method to measure the AC magnetocaloric effect that is compatible with AC susceptibility measurements, and also a framework to analyze these data in combination. We demonstrate the efficacy of this approach using YbVO4, a material for which strong single-ion anisotropy leads to slow spin-lattice relaxation at low temperatures via a phonon bottleneck effect. We report AC magnetic susceptibility and AC magnetocaloric effect measurements for this material as a function of field and frequency at a temperature of 3 K. We analyze the data using a discretized thermal model, and extract the field-dependence of the intrinsic spin-lattice relaxation rate. This demonstration experiment illustrates a general approach to quantitatively address multiple measured quantities in driven systems using a unified thermal circuit analysis. The thermal analysis methods presented in this report can be extended to study other magnetic, dielectric, and elastic materials exhibiting a complex response to an external driving field in the presence of internal and external relaxation, particularly when an energy dissipation process is within an accessible frequency regime.

Conventional LiDAR uses time-of-flight data from laser pulses scanned across a scene to provide accurate multi-meter-scale three-dimensional models at cm precision, limited by the tens-of-picoseconds precision of time-tagging electronics. Here, by using two-photon dual-comb ranging, we introduce an analog of LiDAR imaging using the time-of-flight of sub-picosecond laser pulses to render cm-scale point-cloud datasets with $μ$m precision. Using only free-running femtosecond lasers, the technique combines absolute accuracy with near-interferometric precision, is applicable to discontinuous surfaces with poor optical quality, and provides a stand-off range exceeding that of other optical metrologies. We demonstrate imaging of an aluminum test object and assess its accuracy by comparing our results with those from a touch-probe coordinate measurement machine. At a stand-off distance of 40 cm, we obtain ranging accuracies of 9 $μ$m - 38 $μ$m, and precisions averaging to 1.0 $μ$m after 500 ms.

We theoretically investigate the current-induced excitation of torsional vibrations in a suspended carbon nanotube (CNT) quantum dot. By considering a CNT clamped between half-metallic ferromagnetic electrodes with an antiparallel magnetization configuration, we demonstrate that the spin-rotation coupling enables the transfer of angular momentum from electron spins to the mechanical torsional mode under a constant source-drain voltage. Using a master-equation approach to analyze the coupled dynamics of the dot levels and a quantized torsional oscillator, we evaluate the steady-state current and phonon distribution. We find that when the Zeeman splitting matches the torsional phonon energy, the system exhibits a sharp resonant behavior in the current, accompanied by a significant increase in the phonon population. Our estimates for realistic device parameters indicate that this spin-driven mechanism can drive CNT torsional vibrations with detectable amplitudes. This work provides a theoretical basis for current-controlled actuation of nanoelectromechanical systems via the spin angular momentum of electrons.

Kondo lattice systems, in which localized magnetic moments coherently hybridize with itinerant electrons, exhibit a rich landscape of emergent quantum phenomena. Within this framework, the hybridization strength itself has been theoretically proposed as a spatially modulated order parameter, giving rise to a so-called hybridization wave. However, direct experimental evidence of this quantum state has remained an outstanding challenge. Here, we report the direct observation of a hybridization wave in the layered transition metal dichalcogenide 6R-TaS2, a naturally occurring heterostructure composed of alternating 1T- and 1H-TaS2 layers. Using scanning tunneling microscopy and spectroscopy (STM/STS), we identify the hybridization gap in 1T layer, demonstrating the establishment of a coherent Kondo lattice. Notably, we discover that the hybridization gap present a uniaxial unit-cell doubling modulation, which breaks the both translational and rotational symmetries of the underlying Star-of-David superlattice. Such unit-cell doubling is not caused by structural topography, and therefore, constitutes the real-space visualization of the hybridization-wave order. Furthermore, the hybridization wave correlates with an energy-dependent nematic order that shares the same periodicity and orientation, revealing intertwined electronic instabilities. Our findings not only validate a long-standing prediction but also establish layer-engineered van der Waals materials as a versatile platform for exploring and controlling hybridization-driven quantum phases.

Based on a compactness method, we establish regularity criteria for suitable weak solutions to the surface growth model with a forcing term. These criteria imply that the Hölder regularity of solutions follows from smallness conditions on several scale-invariant quantities. As a consequence, we obtain a partial regularity result stating that the one-dimensional biparabolic Hausdorff measure of the singular set is zero.

Tissues must maintain macroscopic homeostasis despite the continuous microscopic accumulation of cellular damage. Theoretical models of this process often suffer from a disconnect between microscopic biophysics and macroscopic phenomenological games. Here, we bridge this gap by deriving an exact dimensionality reduction of a physiologically structured partial differential equation (PDE) into a low-dimensional dynamical system. Under the condition of uniform mortality, we mathematically demonstrate that tissue homeostasis operates as an induced Nash equilibrium, where the per-capita net growth rates of stem and differentiated phenotypes perfectly equalize. This reduction yields closed-form algebraic rules, the Ratio and Equalization Laws, that map continuous microscopic state dynamics to measurable macroscopic observables. To demonstrate the biological utility of this framework, we present a concrete, falsifiable case study of the murine intestinal crypt. By modeling crypt regeneration following irradiation-induced stem cell depletion, our framework successfully recovers the experimentally observed reliance on progenitor dedifferentiation. Furthermore, the model generates explicit, testable predictions, enabling the in vivo estimation of hard-to-measure lineage plasticity rates directly from aggregate static cell counts. This work provides a rigorous, predictive mathematical foundation for understanding how fast-renewing tissues filter microscopic noise to sustain macroscopic regenerative capacity.

In this paper, we extend the celebrated global regularity theory of Naber-Valtorta [Ann. Math. 2017] to 1/2-harmonic mappings into manifolds. Inspired by their work, we first adapt Lin's defect measure theory [Ann. Math. 1999] to such maps building on the partial regularity established by Millot-Pegon-Schikorra [Arch. Ration. Mech. Anal. 2021]. Then apply it to show that the set of singular points of such maps can be quantitatively stratified via a new notion of boundary symmetry with the aid of {the celebrated harmonic extension method by Caffarelli-Silverstre}. As in that of Naber-Valtorta, developing the necessary quantitative regularity estimates, and then combining it with the Reifenberg type theorems and a delicate covering argument allow us to get sharp growth estimates on the volume of tubular neighborhood around singular points and establish the rectifiability of each singular stratum.

Quantum phase estimation (QPE) is a cornerstone of quantum algorithms designed to estimate the eigenvalues of a unitary operator. QPE is typically implemented through two paradigms with distinct circuit structures: quantum Fourier transform-based QPE (QFT-QPE) and Hadamard test-based QPE (HT-QPE). Existing performance assessments fail to separate the statistical information inherent in the quantum circuit from the efficiency of classical post-processing, thereby obscuring the limits intrinsic to the circuit structure itself. In this study, we employ Fisher information and the Cramer-Rao lower bound to formulate the performance limits of circuit designs independent of the efficiency of classical post-processing. Defining the circuit depth as $T$ and the total runtime as $t_{\rm total}$, our results demonstrate that the achievable scaling is constrained by a non-trivial lower bound on their product $T\,t_{\rm total}$, although previous studies have typically treated the circuit depth $T$ and the total runtime $t_{\rm total}$ as separate resources. Notably, QFT-QPE possesses a more favorable scaling with respect to the overlap between the input state and the target eigenstate corresponding to the desired eigenvalue than HT-QPE. Numerical simulations confirm these theoretical findings, demonstrating a clear performance crossover between the two paradigms depending on the overlap. Furthermore, we verify that practical algorithms, specifically the quantum multiple eigenvalue Gaussian filtered search (QMEGS) and curve-fitted QPE, achieve performance levels closely approaching our derived limits. By elucidating the performance limits inherent in quantum circuit structures, this work concludes that the optimal choice of circuit configuration depends significantly on the overlap.

Nonlinear crystals are widely used to efficiently generate second or third harmonics, and also for frequency mixing. Generation of multiple cascaded harmonics simultaneously, however, appears much less studied, although supported by the usually large nonlinearity of these crystals. Here, we demonstrate the generation of multiple harmonics up to 6th order in a \b{eta}-BBO crystal, extending into the vacuum ultraviolet spectral range. Phase matched generation of the second or the third harmonic is directly driven by a single 800 nm Ti:sapphire oscillator beam. Our study reveals that the harmonics are generated by \c{hi}(2):\c{hi}(2) and \c{hi}(2):\c{hi}(3) cascaded processes and intense harmonic signals are obtained even for the 5th and 6th harmonics, where the \b{eta}-BBO crystal is mostly absorbing. While higher order harmonics cannot be phase matched simultaneously, limiting the harmonic amplitudes, even the weakest harmonics are more than three orders of magnitude above the measurement noise limit, making them suitable for many spectroscopic applications.

Nematic phases that break rotational symmetry are widely observed in quantum materials, and clarifying their origin and relationship with other symmetry-breaking phases remains an important but challenging task. In this work, we investigate nematic fluctuations in CoTa$_3$S$_6$ using elastoresistivity experiments to resolve the nature of the proposed nematic phase intertwined with collinear and non-coplanar antiferromagnetic orders. We observe a divergence-like antisymmetric elastoresistivity that rapidly develops below the stripe antiferromagnetic transition, consistent with a distinct nematic degree of freedom coupled to the magnetic order. While nematic fluctuations are strongly modulated by an external out-of-plane magnetic field and the onset temperature of resistivity anisotropy shows pronounced strain dependence, the antiferromagnetic transition temperatures remain nearly unchanged under either magnetic field or strain. Additionally, complementary magnetoresistance measurements reveal characteristic signatures of three-state nematicity in a hexagonal system. Our findings demonstrate CoTa$_3$S$_6$ as a unique case of intertwined nematic and AFM orders with distinct origins.

We generalize well-known bijections between alternative tableaux and permutations to bijections between rhombic alternative tableaux (RAT) and assemblées of permutations. We show how these various bijections are connected. As a consequence, we find a refined enumeration formula for RAT. One of our bijections carries many statistics from RAT to assemblées; notably, it sends the number of free cells to the number of crossings, which answers a question of Mandelshtam and Viennot. We also find an $r!$-to-$1$ map from marked Laguerre histories to assemblées, answering a question of Corteel and Nunge.