This paper presents a rigorous mathematical analysis of transverse electromagnetic (EM) field concentration between two adjacent obstacles within the framework of the quasi-static approximation. We investigate three degenerate conductivity models recently introduced in [22], two of these incorporating nonlocal boundary conditions to capture fundamental physical phenomena, such as surface nonlocality and thin-layer interactions. Our primary results establish sharp conditions for gradient blowup and derive the corresponding optimal blowup rates. These findings elucidate how nonlocal boundary conditions modify classical gradient estimates. Furthermore, we analyze the influence of wave frequency, demonstrating that it mitigates the severity of field concentration even in the limit of a vanishing gap distance. Consequently, this work extends the classical theory of field enhancement in plasmonic and metamaterial systems to incorporate nonlocal surface effects, yielding precise asymptotic formulas that are essential for the quantitative design of nanophotonic devices.

This is a sequel to the previous work of the author Yanagihara (2025). Let $\ell$ be an odd prime, let $N \geq 1$ be an integer, and let $δ\geq 1$ be an $\ell^N$-th-power-free integer. Let $r,s,t>0$ be integers satisfying $r+s+t=\ell^N$. In Yanagihara (2025), the author computed the root number of the Fermat quotient curve $y^{\ell^N}=x^r(δ-x)^s$ under the assumptions that $\ell\nmid rst$ and that $\operatorname{ord}_{\ell}(δ)=0$ or $\ell\nmid \operatorname{ord}_{\ell}(δ)$. In this paper, we study the case where the technical assumption $\ell\nmid rst$ is dropped. As one such case, we compute the root number when $\ell^{N-1}\| r$ and $\ell\nmid stδ$.

Advanced particle acceleration methods have produced high-peak-current ion beams with broad energy spread and complex spatial distribution. There is an urgent need to develop online spatial-resolved energy spectrometers for high-energy pulsed ions. This paper introduces a novel spectrometer based on a scintillation-fiber cube for online diagnosis of proton beams with broadband energy spread and complex spatial distribution. We present its working principles, experimental setup, and comprehensive calibration using monoenergetic and spatially uniform proton beams generated by a synchrotron accelerator. Calibration results confirm an energy measurement range of 6-93 MeV, a relative energy uncertainty of 0.6% at 80 MeV, and a pixel size of 0.5 mm for beam profile reconstruction. By exploiting a custom-designed energy degrader, we generated a complex proton beam and measured it with the scintillation-fiber cube spectrometer (SFICS). The results demonstrate the spectrometer's potential for online measurement of the energy spectrum and spatial distribution of complex proton beams.

As an emerging magnetic phase, altermagnets (AMs) with collinear compensated magnetism in real space and alternating spin splitting in the band structure have attracted widespread attention. Here, based on first-principles calculations, we demonstrate that the layer stacking imposes symmetry constraints on the spin and valley degrees of freedom (DOFs) in an AM bilayer composed of two tetragonal altermagnetic monolayers, thereby enabling the tuning of these DOFs through interlayer sliding as well as by an external electric field. Using several representative AM bilayers, we reveal that the [C2||P] and [C2||Mz] symmetries intrinsically enforce spin degeneracy, while the coupling between spin and layer DOFs establishes a general framework for achieving electric field control of spin states. Appropriate interlayer sliding breaks the [C2||Md] symmetry of AM bilayers, thereby giving rise to a spontaneous valley splitting and driving a transition to a fully compensated ferrimagnetic state. Furthermore, owing to the tunable valley splitting induced by interlayer sliding, enhanced tunneling magnetoresistance (TMR) can be realized by AM bilayers. This work highlights the intrinsic correlation among spin, valley, and layer DOFs, offering symmetry-based design principles for layer-based spintronic and valleytronic devices.

For a time dependent Schrödinger equation, the scattering map is the map sending the asymptotic profile of solution as $t\to-\infty$ to its asymptotic profile as $t\to+\infty$. In this paper we show that, for certain class of metrics, the scattering maps associated to two Schrödinger operators with two time dependent metrics only differ by a compact operator if and only if these two metrics are related by a pull-back of a diffeomorphism.

We investigate the properties of dense quark matter and strange quark stars within a nonperturbative, Poincaré-covariant framework. Employing a symmetry-preserving vector$\,\otimes\,$vector contact interaction model, we extend the quark gap equation to the regime of zero temperature and finite quark chemical potential. From the resulting momentum-independent quark propagator, we construct the equation of state (EOS) and solve the Tolman-Oppenheimer-Volkoff (TOV) equations to evaluate the mass-radius relations and tidal deformabilities of strange quark stars. We systematically analyze the sensitivity of the EOS and the macroscopic stellar properties to the model parameters, specifically the effective interaction strength and the ultraviolet cutoff. We demonstrate that reducing the coupling constant stiffens the EOS, whereas increasing the ultraviolet cutoff softens it. By confronting our predictions with multi-messenger astrophysical constraints-including pulsar mass measurements and gravitational-wave data-we identify parameter regimes that successfully describe current observations. Specifically, we find that parameter sets with $α_{ir}=0.735π$, $Λ_{uv}=0.905\,\mathrm{GeV}$ and $α_{ir}=0.588π$, $Λ_{uv}=0.9955\,\mathrm{GeV}$, alongside a vacuum bag pressure of $B \approx (0.106\,\mathrm{GeV})^4$, yield stellar properties in excellent agreement with empirical constraints.

We present a systematic study of particle production in $Ne+Ne$ collisions at $\sqrt{s_{\mathrm{NN}}} = 5.36$ TeV using the A Multi-Phase Transport (AMPT) model with string melting (SM) configuration. The analysis compares spherical and deformed configurations of ${}^{20}\mathrm{Ne}$ to investigate the influence of initial-state nuclear deformation on bulk observables. Charged-particle pseudorapidity ($\langle dN_{\mathrm{ch}}/dη\rangle$) densities, identified particle yields ($dN/dy$), transverse momentum ($p_T$) spectra, mean transverse momentum ($\langle p_{\mathrm{T}} \rangle$), and $p_{\mathrm{T}}$-differential particle ratios ($K/π$ and $p/π$) are studied as functions of multiplicity and centrality. The results show that all observables exhibit the expected dependence on event activity, including smooth multiplicity scaling, mass ordering in $\langle p_{\mathrm{T}} \rangle$, and characteristic features associated with radial flow and quark coalescence. Differences between the two configurations on bulk observables remain small across all observables, typically at the level of a 2\%--6\% percent, with slightly enhanced sensitivity observed in peripheral collisions. These findings suggest that, within the AMPT-SM framework, the collective dynamics and hadrochemical composition are primarily governed by the overall system density and interaction dynamics, while the influence of initial-state deformation is subleading. This study provides a baseline for understanding deformation effects in light-ion collision systems and highlights the limited sensitivity of bulk observables to initial nuclear geometry in transport-based approaches.

We study the surjectivity of certain maps involving local cohomology modules, which we can realize as a dual version of part of the investigation developed by Bhatt, Blickle, Lyubeznik, Singh and Zhang on the sheaf cohomology of thickenings (i.e., subschemes defined by powers of ideals), where injectivity played a central role. To this end, we introduce and investigate properties of cohomologically Mittag-Leffler (cML) rings, associated to a given flat local endomorphism (for instance the Frobenius map of a regular ring of prime characteristic), a class which we show to contain, in our setting, the so-called cohomologically full rings of Dao, De Stefani and Ma (in particular, Cohen-Macaulay, Stanley-Reisner, and Du Bois singularities) as well as rings with an ideal inducing a pure endomorphism of the quotient. Our two major specific goals rely upon the prime characteristic setting. First, we extend for the class of cML rings a classical result of Peskine and Szpiro that relates the cohomological dimension and the height of a given Cohen-Macaulay ideal. Second, we prove and illustrate a Kodaira type vanishing result on the sheaf cohomology of thickenings.

Quantum coherence is an important quantum resource which plays a pivotal role in the field of quantum information. Based on metric adjusted skew information, we define a measure of quantum uncertainty to study average coherence under conical 2-designs generalized equiangular measurements, and prove the equivalence of this measure to the scaled average coherence based on metric adjusted skew information under a set of unitary groups, operator orthonormal bases, and mutually unbiased bases. We also derive two trade-off relations by this measure and solve a conjecture. Furthermore, we give two entanglement criteria by this measure and conical 2-designs generalized equiangular measurement, respectively, and illustrate the effectiveness of them by explicit examples.

As LLM-driven autonomous agents evolve to perform complex, multi-step tasks that require integrating multiple datasets, the problem of discovering relevant data sources becomes a key bottleneck. Beyond the challenge posed by the sheer volume of available data sources, data-source selection is difficult because the semantics of data are extremely nuanced and require considering many aspects of the data. To address this, we introduce the Metadata Reasoner, an agentic approach to metadata reasoning, designed to identify a small set of data sources that are both sufficient and minimal for a given analytical task. The Metadata Reasoner leverages a table-search engine to retrieve candidate tables, and then autonomously consults various aspects of the available metadata to determine whether the candidates fit the requirements of the task. We demonstrate the effectiveness of the Metadata Reasoner through a series of empirical studies. Evaluated on the real-world KramaBench datasets for data selection, our approach achieves an average F1-score of 83.16%, outperforming state-of-the-art baselines by a substantial margin of 32 percentage points. Furthermore, evaluations on a newly-created synthetic benchmark based on the BIRD data lake reveal that the Metadata Reasoner is highly robust against redundant and low-quality tables that may be in the data lake. In this noisy environment, it maintains an average of 85.5% F1-score for selecting the right datasets and demonstrates a 99% success rate in avoiding low-quality data.

The prompt-emission spectra of gamma-ray bursts (GRBs) are commonly described by the empirical Band function. The typical low-energy spectral index is $\sim -1$, which poses a challenge to standard synchrotron radiation models. We systematically investigate a fast-cooling synchrotron model with a decaying magnetic field and test, within an observation-consistent pipeline, whether it reproduces the Band-fit parameter distributions in the GBM catalog, in a statistical sense. We solve the electron continuity equation with synchrotron, adiabatic, and synchrotron self-Compton cooling to obtain the time-dependent electron distribution and synthetic spectra; we then forward-fold through the GBM response matrices and recover $(α, β, E_p)$ with Band fits. We find that magnetic-field decay can harden the recovered $α$ relative to the fast-cooling limit in part of parameter space, but the effect is not robust and is sensitive to the location of $E_p$ within the finite band and to spectral curvature; varying key physical scales reshapes the recovered $α$ distribution, indicating that catalog $α$ often represents an effective in-band slope rather than the asymptotic index. SSC cooling provides modest additional hardening and, in our setups, does not stabilize $α$ near the observed peak. Using Monte Carlo samples designed to mimic the observations, the model yields $α$ mostly between $-1.5$ and $-0.8$, but remains centered around $α\approx -1.5$. Overall, while decaying-field fast-cooling synchrotron can partially alleviate overly soft spectra expected from standard fast-cooling synchrotron emission, it still falls short of reproducing the GBM $α$ distribution at the population level, implying that additional physical processes are required.

A graph-theoretic method is introduced for analyzing fermion mass spectra in latticized theory-space models, including chain models arising from dimensional deconstruction. Fermion mass terms are mapped to bipartite graphs, with fields as vertices and nonvanishing mass terms as edges. The number of massless modes is shown to be fixed by the cardinality of a maximum matching of the associated graph. Moreover, the wave-function support of these modes is restricted to fields reachable from exposed or unmatched vertices by even-length maximum-matching-alternating paths, as characterized by the Dulmage-Mendelsohn decomposition. These results depend only on the topology of latticized theory space and are independent of model parameters. The method enables a systematic construction of latticized models with prescribed numbers and localization properties of massless modes.

Origami structures, particularly Miura-ori patterns, offer unique capabilities for surface approximation and deployable designs. In this study, a constrained mapping optimization algorithm is designed for designing surface-aligned Miura-ori via a narrow band approximation of the input surface. The Miura-fold, embedded in the narrow band, is parameterized to a planar domain, and a mapping is computed on the parameter pattern by optimizing certain energy terms and constraints. Extensive experiments are conducted, showing the significance and flexibility of our methods.

This paper investigates the collisionless quantum hydrodynamic, or quantum Euler, system in \(\mathbb{T}^3\) with the linear pressure law \(P(ρ)=ρ\). Since this pressure is associated with the logarithmic internal energy \(f(ρ)=ρ\logρ\), the model admits a natural logarithmic Schrödinger approximation. By means of a regularized logarithmic Schrödinger equation, we rigorously construct global weak solutions to the quantum isothermal Euler system. The proof relies on the Madelung transform, the polar decomposition of the wave functions, and compactness arguments. In particular, an energy identity is used to recover the strong convergence of the hydrodynamic variables. More broadly, the analysis provides a robust Schrödinger approximation framework for QHD models whose internal energy contains an isothermal component.

Altermagnets host momentum-selective spin splitting and chiral-split magnonic excitations despite vanishing net magnetization, enabling spin transport without ferromagnetism. In rutile structures, establishing altermagnetism spectroscopically has been challenging, motivating the search for a rutile platform with a resolvable exchange-driven chiral magnon splitting. Here we combine hybrid-functional first-principles calculations with linear spin-wave theory to show that rutile CuF$_2$ exhibits a meV-scale splitting between magnon modes of opposite chirality along momentum directions dictated by its $d$-wave altermagnetic symmetry. The splitting originates from an anomalously strong long-range super-superexchange channel Cu--F$\cdots$F--Cu, which enhances the symmetry-allowed difference between seventh-neighbour exchanges, $J_{7b} - J_{7a}$, controlling the chiral-mode splitting. We identify an orbital-resonance mechanism: energetic alignment between Cu $3d_{z^2}$ and F $2p_z$ states strengthens virtual hopping along the Cu--F$\cdots$F--Cu path and amplifies the anisotropic long-range exchange. Rutile CuF$_2$ therefore provides an ideal platform to validate rutile altermagnetism and suggests an orbital-energy description for engineering large chiral magnon splittings in insulating altermagnets.

Characterizing the full 6-dimensional phase-space distribution of beams from the LCLS-II photoinjector is essential for understanding and optimizing downstream accelerator performance. Long-term monitoring of this distribution is equally important for detecting drifts in machine state and implementing timely corrective actions. Continuous phase space characterization during routine operation demands reliable tomographic diagnostic measurements and fast, efficient reconstruction methods. In this work, we demonstrate the first fully autonomous 6-dimensional beam-tomography system deployed on the DIAG0 parasitic beamline at LCLS-II. Using machine-learning-based control algorithms, the system autonomously configures DIAG0 and executes tomographic manipulations within operational constraints, adaptively re-optimizing beamline parameters and scan ranges in response to changes in the incoming beam. Tomographic measurements are streamed to the S3DF computing cluster where generative analysis methods reconstruct the phase-space distribution. We demonstrate that this framework produces detailed 6-dimensional beam reconstructions at a cadence of one reconstruction every 5 to 10 minutes, enabling real-time, multi-hour monitoring of injector beam evolution with unprecedented fidelity. These results represent a significant step toward fully autonomous operation of accelerator beamlines with real-time beam diagnostics for current and next-generation accelerator facilities.

Here we report the development of SPRAY, a massively parallel GPU accelerated, smoothed particle hydrodynamics (SPH)-based, radiation hydrodynamics (RHD) code designed specifically for simulating high intensity laser-plasma interactions. When a target is irradiated by an intense laser, highly complex fluid deformation occurs due to instabilities, which is challenging to study numerically. SPRAY is particle-based, mesh-free, and Lagrangian, which addresses numerical issues that posed difficulties to existing methods. Its SPH formulations for RHD governing equations are tailored toward accurate and reliable simulations of laser-target irradiation phenomena, and are solved via a time-dependent, flux-limited diffusion method. A new laser energy coupling module, which is based on the Wentzel-Kramers-Brillouin (WKB) approximation, is implemented with a totally mesh-free ray-tracing scheme that is applicable for arbitrary geometry and dimensions. The accuracy and reliability of the code are demonstrated with a series of benchmark problems. To the authors' knowledge, this is the first attempt to employ SPH method for simulations of laser-plasma interactions in high energy density physics research. Possible expansions to the code, such as laser beam-beam interaction modeling and more sophisticated multi-group radiation transport are left for future development.

This paper investigates several classical and novel variations of the Erdős--Szekeres problem, including multicolored point sets, convex hexagons with a given number of interior points, and polygons with constraints on edge colors. We propose a comprehensive computational framework combining combinatorial modeling within the SAT/ASP paradigms with the geometric realization of configurations. To determine point coordinates, we developed the \textbf{linear subreduction method}. The core idea consists of combining the complete logical model of the problem with a system of geometric inequalities, followed by fixing the abscissae to linearize the constraints. This approach enables a simultaneous search for a realization across the entire space of admissible abstract configurations (signotopes) rather than examining them individually, while linearization significantly accelerates the SMT solving process. Using this framework, we established new exact values for several functions; in particular, we proved $h_{nc}(4,0; 4,0)=26$: any bicolored set of 26 points in general position must contain the vertices of an empty monochromatic quadrilateral.

Objective: Reliable identification of pro-eating disorder (pro-ED) content online suffers from two pervasive problems: 1) existing methods predominantly rely on text-based signals, failing to capture the inherently multimodal nature of multimedia content; and 2) these methods struggle to keep pace with the rapid evolution of references, memes, terminology, and contextual cues that underlie this content. Together, these limitations point to a gap: the absence of an expert-annotated reference standard capable of supporting real-time research and robust multimodal detection model training for pro-ED content on short-form video platforms. Method: To address this, we propose "zeitgeist-aware" multimodal (ZAM) datasets: continuously curated collections of annotated multimodal pro-ED content with inclusion criteria that evolve alongside the memetic zeitgeist: the variable essence of what is considered pro-ED as new media and references come into the cultural zeitgeist and are absorbed and interpreted in online spaces. Results: We present a rationale for such datasets, define their core characteristics, outline approaches for their curation, and describe our progress toward that end. Discussion: This dataset and pipeline architecture may benefit researchers across several fields who are interested in how pro-ED sentiment is encoded and transmitted through short-form video content across time, including for the purpose of responsive moderation efforts.

We initiate an investigation into a notion of state complexity for discrete-variable quantum systems. Specifically, we propose an information-theoretic quantifier for the complexity of quantum states within the stabilizer formalism of quantum computation. This is achieved by leveraging the symmetric Jordan product (associated with classicality) and the skew-symmetric Lie product (linked to quantumness) between the square root of the quantum state and the Heisenberg-Weyl displacement operators. We establish the fundamental properties of this quantifier and demonstrate that state complexity is closely related to the nonstabilizerness of quantum states via the $L^4$-norm of their characteristic functions.

We prove that every maximally nodal sextic surface\,(with 65 nodes) $X \subset \mathbb{P}_{\mathbb{C}}^3$ contains a symmetric half-even set of nodes of cardinality 35. It follows that the associated half-quadratic sheaf is the cokernel of a symmetric $6 \times 6$ matrix of linear forms, yielding a linear determinantal representation of $X$. In particular, after a suitable Serre twist, the half-quadratic sheaf is an Ulrich sheaf of rank 1. As an example, we exhibit an explicit $6 \times 6$ matrix of linear forms whose determinant defines the Barth sextic surface.

We establish a fractal transference principle for continued fraction expansions over the field of Laurent series. Let $S$ be an infinite subset of the set of all polynomials over a finite field of $q$ elements of positive degree with growth density exponent $α\ge 1$, and let $U \subset S$ be a subset of positive relative upper density. We prove that there exists a subset $E_{S,U}$ of the set of points whose continued fraction digits are pairwise distinct and belong to $S$ such that \[\dim_{\rm H} E_{S,U}=\frac{1}{2α}.\] Moreover, the set of digits appearing in the continued fraction expansions of points in $E_{S,U}$ recovers the relative upper density of $U$ in $S$. We also show that the same construction preserves the relative upper density of the corresponding degree sets in $\mathbb N$. As a consequence, combinatorial statements for subsets of $\mathbb N$ of positive upper density can be transferred to degree sets arising from continued fraction expansions of Laurent series on sets of optimal Hausdorff dimension.

Optical speckle patterns have been widely used for illumination in computational imaging, optical sectioning microscopy, and super-resolution imaging. However, commonly used speckles satisfy Rayleigh statistics, which are not ideal for diverse imaging applications. Here we tailor three-dimensional speckle intensity statistics for dynamic speckle illumination microscopy based on linear fluorescence. Optical sectioning is enhanced by axially varying speckle contrast, and image reconstruction noise is minimized with in-focus speckles of binary intensities. The customized speckle statistics are shown to tolerate sample-induced aberration and scattering. We apply tailored speckle illumination to mouse brain vascular imaging and demonstrate much improved image quality than optical-sectioning structured illumination. These results establish customization of speckle intensity statistics as a promising strategy for robust, high-throughput fluorescence imaging in thick, scattering biological specimens.

In previous work, we extracted the intrinsic finite algebraic state data of a finite-node conifold degeneration in the form $A_Σ:= (V_Σ,E_Σ,c_Σ)$, where $V_Σ$ is the finite node-indexed vertex set, $E_Σ$ is the nodewise coupling space, and $c_Σ$ is the coefficient vector of the corrected global extension class. The purpose of the present paper is to construct the corresponding interaction and incidence layer. Starting from the finite-node schober package $S_Σ:= (\mathcal C_{\mathrm{bulk}},\{\mathcal C_{p_k}\}_{k=1}^r,\{Φ_k,Ψ_k\}_{k=1}^r,Sh(S_Σ))$, we define the extended vertex set $V_Σ^{\mathrm{ext}} := V_Σ\sqcup \{v_{\mathrm{bulk}}\}$, the functorial coupling relation determined by the attachment functors, the resulting functorial incidence package $\mathfrak{I}_Σ:= (V_Σ^{\mathrm{ext}},\rightsquigarrow_Σ)$, and its canonical binary decategorification $\mathcal I_Σ:= (V_Σ^{\mathrm{ext}},I_Σ)$. From these data we assemble the finite quiver-theoretic package $\mathfrak Q_Σ:= (V_Σ,E_Σ,c_Σ,\mathcal F_Σ,I_Σ)$, where $\mathcal F_Σ:= \{(Φ_k,Ψ_k)\}_{k=1}^r$ is the functorial coupling datum. We prove that this package is canonically determined by the finite-node schober datum, compatible with the corrected perverse extension and its mixed-Hodge-module refinement, and invariant under equivalence of finite-node schober realizations. This yields the interaction and incidence layer required for later graded interaction, stability, BPS, and wall-crossing structures.

Higher-order networks with multiway interactions can exhibit collective dynamical phenomena that are absent in traditional pairwise network models. However, analyzing such dynamics becomes computationally prohibitive as their state space grows combinatorially in the multiway interaction order. Here we develop quantum algorithms for two central tasks -- synchronization estimation and certification of the no-phase-locking regime -- in the simplicial Kuramoto model. This model is a higher-order generalization of the celebrated Kuramoto model for coupled oscillators on graph-based networks. Under explicit assumptions on data access and types, and simplicial structure, we derive end-to-end quantum gate complexities and identify regimes with polynomial quantum advantage for synchronization estimation and super-polynomial quantum advantage for no-phase-locking certification over classical methods. More broadly, these results extend quantum algorithms for higher-order networks from structural analysis to nonlinear dynamical diagnostics, easing a major computational bottleneck and opening a route to quantum methods for probing higher-order phenomena beyond the reach of direct classical approaches.

A benchmark of 25 nonlinear optimization problems with domain-induced discontinuity is proposed to support the performance evaluation of global optimization algorithms under feasibility-scarce and structurally discontinuous landscapes. Referred to as the CPC Benchmark (Challenging Problems for Computation), the test suiteconsists of functions that are continuous on their natural domains, while infeasible regions and undefined evaluations are implicitly embedded in the objective, creating substantial challenges for global minimization. Six representative algorithms from diverse methodological paradigms are assessed to examine the structural complexity and discriminative capability of the benchmark. Numerical results show that many functions possess extremely small feasible regions and strong precision sensitivity near feasibility boundaries, complicating initialization, feasibility discovery, and reliable objective assessment. The findings demonstrate that the CPC benchmark provides clear discriminative power across algorithmic paradigms and offers a rigorous, software-oriented testbed for advancing research in global optimization.

Recently S. Goswami proved that whenever the set $\mathbb N$ of natural numbers is finitely colored, the set $\{a, b, ab, b(a+1)\}$ is monochromatic which also established a variant of the long-standing Hindman's conjecture, which asks for a monochromatic set of the form $\{a, b, ab, a+b\}$. Actually he disproved a conjecture proposed by J. Sahasrabudhe that $\{a, b, a(b + 1)\}$ is not partition regular. In this paper we prove that $\{a, b, ab, b(a+1)\}$ is monochromatic near zero which means for every finite coloring of a dense subsemigroups of $((0, \infty), +)$, the set $\{a, b, ab, b(a+1)\}$ is monochromatic near zero or in other words, we will get $a, b$ in a dense subsemigroups of $((0, \infty), +)$ as small as we want such that the set $\{a, b, ab, b(a+1)\}$ is monochromatic for every finite coloring of that dense subsemigroups of $((0, \infty), +)$, also we show that the pattern $x, y, x+y, \frac{y}{x}$ is partition regular near zero.

As AI workloads drive increases in datacenter power consumption, accurate GPU power estimation is critical for proactive power management. However, existing power models face a scalability bottleneck not in the modeling techniques themselves, but in obtaining the hardware utilization inputs they require. Conventional approaches rely on either costly simulation or hardware profiling, which makes them impractical when rapid predictions are required. This work presents EnergAIzer, which addresses this scalability bottleneck by developing a lightweight solution to predict utilization inputs, reducing the estimation walltime from hours to seconds. Our key insight is that kernels in AI workloads commonly employ optimizations that create structured patterns, which analytically determine memory traffic and execution timeline. We construct a performance model using these patterns as an analytical scaffold for empirical data fitting, which also naturally exposes module-level utilization. This predicted utilization is then fed into our power model to estimate dynamic power consumption. EnergAIzer achieves 8% power errors on NVIDIA Ampere GPUs, competitive with traditional power models with elaborate cycle-level simulation or hardware profiling. We demonstrate EnergAIzer's exploration capabilities for frequency scaling and architectural configurations, including forecasting the power of NVIDIA H100 with just 7% error. In summary, EnergAIzer provides fast and accurate power prediction for AI workloads, paving the way for power-aware design explorations.

End-to-end learned video compression has achieved strong rate-distortion performance, but rate control remains underexplored, especially in target-bitrate-driven and budget-constrained scenarios. Existing methods mainly rely on explicit R-D-lambda modeling or feed-forward prediction, which may lack stable online adjustment when video content varies dynamically. We propose a feedback-driven rate control framework for learned video compression. First, we build a single-model multi-rate coding interface on top of a DCVC-style framework, enabling continuous bitrate control through the rate-distortion parameter lambda. Then, a log-domain PI/PID closed-loop controller updates lambda online according to the error between the target bitrate and the entropy-estimated bitrate, achieving stable target bitrate tracking. To further improve frame-level bit allocation under budget constraints, we introduce a dual-branch GRU-based adjustment controller that refines the base control signal using budget-state features and causal coding statistics. Experiments on UVG and HEVC show that the proposed PI/PID controller achieves average bitrate errors of 2.88% and 2.95% on DCVC and DCVC-TCM, respectively. With the proposed adjustment controller, the method further achieves average BD-rate reductions of 5.69% and 4.49%, while reducing the average bitrate errors to 2.13% and 2.24%. These results show that the proposed method provides a practical solution for learned video compression with both controllable bitrate and improved rate-distortion performance.

Continuous-variable (CV) quantum teleportation is usually benchmarked by average fidelity, but when the teleportation is repeatedly used within optical networks or measurement-based architectures, uniformity across the input ensemble becomes equally important. We analyze this issue using two complementary figures of merit: the average fidelity and the fidelity deviation, which quantifies the input dependence of the single-shot teleportation fidelity. We prove that any deterministic unity-gain teleportation channel that is displacement covariant has vanishing fidelity deviation for coherent-state benchmarking, irrespective of whether the shared entangled resource is Gaussian or non-Gaussian. Nonzero deviation therefore diagnoses covariance breaking rather than non-Gaussianity. We then show that when a protocol raises the average fidelity through input-selective conditioning, the deviation generically increases in tandem, giving a quantitative universality cost. As a concrete example, we study teleportation enhanced by the so-called measurement-based noiseless linear amplification, where a heralded filter acts on the Bell-measurement record. The resulting trade-off among average fidelity, fidelity deviation, and success probability shows that stronger filtering can improve the conditional fidelity only by concentrating the successful events in favored regions of phase space, thereby suppressing the success probability and reducing input uniformity. Our results provide an operational framework for distinguishing genuine channel improvement from selectivity-driven post-selected advantage and suggest that the probabilistic CV teleportation should be assessed with average quality, universality, and heralding rate treated on an equal footing.