We propose a joint multi-epoch profile-likelihood analysis of axion-like particles (ALPs) using five sets of simultaneous Fermi-LAT and LHAASO observations of the TeV blazar Mrk 421. Photon-ALP oscillations are calculated self-consistently together with EBL absorption for two representative jet emission models: a two-zone hybrid model and a single-zone hadronic model. To account for weak correlations among different observational epochs, we introduce a Gaussian copula with a conservative correlation coefficient $ρ= 0.03$ and perform a global optimization of nuisance parameters under the no-ALP hypothesis before profiling the ALP parameters. In the low-mass regime relevant to CAST ($m_a \lesssim 1$ neV), we obtain a 95\% CL upper limit of $g_{aγ} = 7.46 \times 10^{-13}\,\mathrm{GeV}^{-1}$. Over the full mass range $0.1$--$500$ neV, the most conservative 95\% CL upper limits are $g_{aγ} < 6.50 \times 10^{-12}\,\mathrm{GeV}^{-1}$ (two-zone) and $g_{aγ} < 7.34 \times 10^{-12}\,\mathrm{GeV}^{-1}$ (single-zone). These constraints benefit from the broadband VHE coverage and long-term monitoring provided by LHAASO. The analysis framework developed here offers a statistically consistent approach for future ALP searches with multi-messenger gamma-ray data.

We present a simple computer-assisted procedure to construct $SO(d_1+1)\times SO(d_2+1)$-invariant cohomogeneity one Einstein metrics, and use it to recover known Einstein metrics on $S^{10}$ and $S^{12}$, as well as find new ones on $S^{11}$, $S^{12}$, $S^{13}$ and $S^7\times S^3$.

Reliable reconstruction of missing observations in environmental panel datasets is essential for accurate exposure assessment and policy analysis. Traditional nuclear norm matrix completion methods effectively impute missing entries in low-rank matrices, yet often overlook the spatial dependence inherent to air quality processes. This paper introduces the Eigenvector Spatial Filters Nuclear Norm Matrix Completion (ESFNNMC) method, an extension of nuclear norm fixed-effects matrix completion that replaces unit-specific intercepts with a set of Moran-type eigenvectors capturing spatial autocorrelation in the data. To estimate the model, we propose a Block-Coordinate Descent (BCD) approach for multiconvex optimization problems, with soft-thresholded singular value decomposition and cross-validated regularization. Through comprehensive simulations varying missingness patterns, the level of spatial and temporal autocorrelation, and dimension, shape, and rank structure of the matrices, ESFNNMC demonstrates substantial improvements in imputation accuracy over the standard fixed-effects approach, while keeping the computational cost approximately unchanged. The method is applied to impute missing entries in daily PM10 measurements in 64 monitoring stations in Lombardy, Italy, during the year 2021.

The classical theorems of Kolmogorov--Nagumo--de Finetti and of Aczel--Maksa characterize quasi-arithmetic means from two complementary directions: the former for compatible families of means satisfying the replacement axiom, and the latter for bisymmetric means of fixed arity. We refine both representation results by showing that the required continuity follows automatically. Our main result states that every reflexive, symmetric, bisymmetric and partially strictly increasing $n$-variable operation on a real interval is continuous and hence quasi-arithmetic. The proof is based on a recursive construction on $n$-adic rationals given by bisymmetry, and a dense-domain continuity argument. The same method also yields the regularity-free Kolmogorov--Nagumo--de Finetti theorem for compatible families of strictly increasing symmetric means.

A reduced-order Particle-in-Cell method is introduced for kinetic simulation of otherwise axisymmetric cylindrical plasmas that exhibit azimuthal instabilities. The method spatially decomposes all field quantities into a small number of (mesh-less) azimuthal Fourier modes $m=0,...,N_m,\quad N_m \ll N_θ$, reducing the costly three-dimensional field solve $\mathcal{O}(N_zN_rN_θ)$ to a family of decoupled independent two-dimensional problems $\mathcal{O}((N_m+1)N_zN_r)$ on the meridional plane - one per mode - while particles continue to move in full three-dimensional space. Fields are reconstructed at particle positions by coherent superposition of these modal contributions, preserving complete azimuthal variation at a fraction of the cost of a conventional three-dimensional simulation. The method is validated against the diocotron instability of a hollow electron annulus across three geometrically distinct configurations, recovering linear growth rates and eigenmode structures within 7% of closed-form analytic predictions, and reproducing the non-linear vortex dynamics characteristic of the instability saturation. A further benchmark against the community-standard Landmark Penning discharge problem recovers the rotating-spoke frequency, radial plasma profiles, and modal energy hierarchy in quantitative agreement with long-time reference simulations, at approximately 640 CPU-hours - a factor of 46 speed-up compared to the median benchmark cost. The approach addresses the important gap between computationally prohibitive full three-dimensional kinetic simulation and the physically limited reduced-dimensionality models on which predictive modelling of anomalous transport in magnetised plasma devices currently relies.

Multidimensional Gaussian Process (multi-GP) regression is widely used to disentangle stellar and planetary signals in radial velocities (RVs) by jointly modelling ancillary activity indicators. However, identifying the combination of indicators that best constrains the stellar signal in the RVs is non-trivial, as classical model comparison methods are not directly applicable when multi-GPs involve different time series combinations. In this work, we present an information criterion to compare multi-GP models based on their ability to explain the RV component, $\mathrm{MGIC}_{\rm rv}$. This metric combines the conditional RV likelihood with an effective parameter count that accounts for the regularisation imposed by the multi-GP model on the RV component. We demonstrate that $\mathrm{MGIC}_{\rm rv}$ provides a quantitative and robust framework for multi-GP model comparison, identifying the activity indicators that most effectively constrain the RV signal. Although developed in the context of RV analysis, the proposed criterion is general and applicable to multi-GP problems in which the inference focuses on a specific observable.

We experimentally demonstrate atom interferometry using the transverse phase profile of an optical mode. As proof-of-principle, we use the helical phase windings of Hypergeometric Gaussian beams for Ramsey interferometry with ensembles of ballistically-expanding cold Rb87 atoms, and we show that the interferometer can measure rotations induced by a motor with a sensitivity that scales linearly with orbital angular momentum and interferometer time. We characterize the thermal decoherence of the interferometer, deriving and experimentally confirming a closed-form expression for the spatially-varying interferometer visibility arising near the singularity of the helical phase winding, motivating the use of condensed atoms in ring-shaped traps.

We extend the exponential stability theorem for the three-dimensional incompressible Navier-Stokes equations from hyperbolic 3-space $\HH^3$ (established in a companion paper) to complete simply connected Riemannian 3-manifolds $(M^3, g)$ with pinched negative sectional curvature $-b^2 \leq K \leq -a^2 < 0$ and bounded geometry (including a strictly positive injectivity radius). The deformation Laplacian $Δ_\Def = Δ_B + \Ric$ remains the viscous operator, selected by Lagrangian kinematics. We prove that the {exact} system admits a unique global mild solution for small $L^3$ data, with exponential decay at a rate determined by the spectral gap of the Stokes operator. The extension overcomes three obstacles absent on $\HH^3$: (i) the semigroup factorisation $e^{tΔ_\Def} = e^{-2t}e^{tΔ_B}$ fails because $\Ric$ is not a scalar multiple of the metric; (ii) the Leray projector no longer commutes with $Δ_\Def$; (iii) the exact spectral gap is unknown. We resolve (i) unconditionally, without any curvature restriction, by observing that the Ricci perturbation $V = \Ric + 2a^2 g$ is negative semi-definite and applying a Trotter product bound with the diamagnetic inequality. We resolve (ii) by an algebraic reduction of the commutator $[\PP, Δ_\Def]$ to the complementary projector $(I-\PP)$ applied to the shifted Ricci endomorphism, giving a clean zeroth-order bound proportional to the curvature variation $b^2 - a^2$. This is the sole source of a curvature pinching constraint. We resolve (iii) via McKean's theorem, the diamagnetic inequality, and the Weitzenböck identity. The Fujita-Kato temporal singularity exponent $1/2 - 3/(2p)$ is unchanged from the $\HH^3$ case, confirming that the ultraviolet scaling obstruction is local and geometry-independent, driven fundamentally by an unresolvable temporal scaling mismatch.

Let $F$ be a $k$-uniform hypergraph, abbreviated as $k$-graph. The codegree Turán density $π_{\mathrm{co}}(F)$ is the supremum over all $γ\in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $γn$ edges. In this paper, we prove that there is a linear $k$-graph $F$ with $0<π_{co}(F) < \varepsilon$ for any $\varepsilon>0$. The special case $k=3$ solve a question proposed by Ding, Lamaison, Liu, Wang and Yang (JLMS, 2025). The main method combines an affine-plane-type incidence structure over a finite field and elementary number-theoretic arguments.

We introduce the Polymarket-v1 Database: the complete on-chain trade archive of Polymarket's first-generation CTF Exchange on Polygon, spanning 2022-11-21 to 2026-04-28 and covering the full contract lifecycle from first settlement to natural termination. The dataset comprises 1.20 billion trade records across 1.30 million markets with $61 billion in nominal volume. Its defining feature is 100% ground-truth aggressor direction derived from the blockchain settlement layer, a property unavailable in existing prediction market archives, which rely on heuristic inference. We use this truth-aligned archive to benchmark standard microstructure tools and document three findings. First, the tick rule and bulk volume classification achieve near-random aggregate accuracy (49.83% and 50.51%), but this masks a systematic, correctable price-level gradient driven by positive trade direction autocorrelation and concentrated market-making -- two structural features of prediction markets that violate the mean-reversion assumption embedded in classical classifiers. Second, these classification errors propagate into downstream metrics: inferred VPIN diverges substantially from ground-truth VPIN, and OFI estimates are directionally biased, with material consequences for Transaction Cost Analysis. Third, ground-truth microstructure quality predicts forecasting performance in ways that classification-based proxies cannot recover: True VPIN positively predicts Brier scores, while Gibbs spread negatively predicts them -- a selection effect reflecting that high-spread niche markets attract informed specialists rather than noise traders. Replacing ground-truth metrics with classified proxies attenuates both relationships, illustrating that measurement accuracy at the transaction level is a prerequisite for reliable inference about prediction market design and probability calibration.

Two-dimensional (2D) axisymmetric simulations are conducted to investigate convection in a rotating cylindrical annulus with localized heating at the outer bottom edge and uniform cooling at the inner cylindrical wall. The resulting radial and vertical temperature gradients generate buoyancy-driven motion and produce a stratification pattern relevant to atmospheric circulation. The effects of aspect ratio (\(Γ\)), Rayleigh number (\(Ra = 2.4 \times 10^{7}\)--\(1.2 \times 10^{9}\)), and Taylor number (\(Ta = 1.6 \times 10^{7}\)--\(1.2 \times 10^{9}\)), including the non-rotating limit (\(Ta=0\)), are examined. Convection is largely confined to thin boundary layers, while the fluid interior remains diffusion dominated. Without rotation, the temperature field exhibits nearly horizontal isotherms. Rotation establishes quasi-hydrostatic and geostrophic balances that redistribute heat and promote deeper penetration of isotherms into the interior. Heat transfer, quantified by the Nusselt number (\(Nu\)), depends strongly on \(Ra\), \(Ta\), and \(Γ\). For moderate and high \(Ra\), \(Nu\) follows the scaling \(Nu \sim Ra^{1/4}\) and is only weakly influenced by rotation. At low \(Ra\) and high \(Ta\), rotational suppression of buoyancy reduces \(Nu\) significantly. Increasing \(Γ\) enhances heat transfer, although the growth rate diminishes for \(Γ> 1\). The relative thermal and Ekman boundary-layer thicknesses govern the sensitivity of heat transfer to rotation.

In heterogeneous OLG economies, age alone does not tell us who counts as young or old in a local welfare comparison. For such a comparison to have normative content, it must specify the represented margin, the private unit of account, the current information that identifies the relevant cell, and the common support over which claims are made. This paper develops such a discipline, separating raw resource weights from residual weights expressed in private value units and showing that imposed local discount labels are valid precisely when they are induced by a potential on the supported comparison graph, whose vertices are current cells and whose edges are admissible age comparisons. A calibrated audit of an OLG model then maps the frontier of certified claims, with support, units, wedges, and anchors determining which welfare signs are justified.

We give a proof of the cosmological splitting conjecture of Robert Bartnik from 1988, which expresses the rigidity of the cosmological Hawking--Penrose singularity theorem. It states that a timelike geodesically complete, globally hyperbolic spacetime which has compact Cauchy surfaces and satisfies the strong energy condition must split isometrically as a Lorentzian product. Our methods combine the construction of global viscosity solutions to the Lorentzian eikonal equation by Zhu--Wu--Cui with our recently developed elliptic approach to the proof of Lorentzian splitting theorems in joint work with Braun, Gigli and Sämann, where we make use of the $p$-d'Alembertian operator for $p < 1$.

This article presents a finite-horizon linear quadratic regulator for the control of the first-order Lighthill-Whitham-Richards traffic model with a triangular fundamental diagram. The in-domain control action is realized through variable speed limits implemented as a source term in the governing hyperbolic partial differential equation. Unlike prior studies on infinite-horizon formulations, this article develops a finite-horizon LQR framework, deriving a space and time varying state feedback function for hyperbolic PDEs. The solution to the finite time optimal control problem relies on the solution of another PDE, called the Riccati PDE. The resulting nonlinear Riccati PDE is solved analytically via the parametric method of characteristics. The Riccati PDE solution is a function of both time and space, as well as the traffic regime. A sensitivity analysis demonstrates the effects of the LQR parameters for both the infinite and finite time horizon problem in different traffic situations, while siulations validate the finite-horizon LQR's ability to guarentee finite-time convergence. Comapred to the infinite-horizon LQR, the proposed approach achieves significantly improved control performance across various scenarios, making it particularly suitable for time-sensitive traffic management applications.

Diffusion-based generative models have achieved remarkable success in high-dimensional data generation; however, they fundamentally rely on isotropic diffusion processes that destroy meaningful geometric structures in the forward process. For complex, multimodal, and highly correlated distributions such as biologically constrained genetic data, isotropic noise merges distinct modes and distorts intrinsic dependencies. This forces the reverse process to recover structure from heavily degraded signals, leading to slow convergence, mode averaging, and biologically implausible samples. To address this, we introduce the Geometry-aware Ornstein-Uhlenbeck (GOU) process, a structured drift design that embeds data geometry into forward and backward dynamics. By employing a variance-aware anisotropic drift, GOU contracts low-variance directions rapidly while preserving high-variance directions longer, maintaining key multimodal structures as stable channels over time. Crucially, we show that GOU's backward initialization error is governed by local rather than global variance. This geometry-adaptive initialization improves convergence rates by reducing initial mismatch and preserving cluster-level structures. Synthetic and real-world genetic experiments demonstrate that GOU significantly improves mode separation, correlation preservation, and statistical validity over standard isotropic models.

Despite the prevalence of nonautonomous systems in applications, their statistical properties are much less understood than in the autonomous setting. Building on recent results on response theory for nonautonomous systems, we study the approximation of equivariant families and of their linear response by Ulam-type finite-dimensional reductions. First, we show that coarse-graining procedures associated with the classical Ulam method, and more generally with suitable finite-element projections, provide rigorous approximation of equivariant families for sequential systems with memory loss. Second, for systems whose transfer operators are regularizing, we prove that the linear response of the reduced finite-state Markov model converges to the projected linear response of the original system. To the best of our knowledge, a general approximation result of this type has not previously been established in this form, even in the autonomous case. We complement the analysis with numerical experiments on simple but representative time-dependent diffusive models. These results provide a rigorous foundation for the use of Markov approximations in the study of statistical properties of nonautonomous complex systems which almost invariably relies on finite-scale and finite-precision descriptions of their states and dynamics.

We prove global in time regularity of solutions of the Navier-Stokes equations defined in the complex space.

Let $f(k,s)$ denote the minimum integer $m$ such that any family $\mathcal{F}$ consisting of $k$-sized sets of cardinality at least $m$ always contain a sunflower of size $s$. The Erdős-Rado Sunflower Conjecture states that for every $s >2$, there is an constant $C=C(s)$ such that $f(k,s) \leq C^k$. In this paper, we prove the conjecture for shifted families.

The fractional Schrödinger equation is studied with a kinetic energy that slightly deviates from the usual nonrelativistic form. The harmonic oscillator and the Kepler problem are both treated in the context of small perturbations. The usual perturbation theory is used and compared with the envelope theory. The analytical results show good agreement between both methods, indicating possible future developments for many-body systems. A possible connection with experimental observations is briefly discussed.

A low-order Galerkin model is developed for a rotating fluid annulus driven by localized heating at the outer bottom periphery, with uniform cooling at the inner cylindrical wall. The model retains the full cylindrical geometry and employs Bessel-function radial eigenfunctions satisfying physically correct Dirichlet-Neumann boundary conditions. A dual-series least-squares procedure determines the conductive base state under the mixed thermal boundary condition. Galerkin projection onto the leading radial and vertical basis functions yields a 10-variable dynamical system governing the mean meridional overturning, thermal wind, baroclinic wave amplitudes, and their nonlinear interactions. Linear stability analysis yields explicit critical Rayleigh numbers for both mean and wave instabilities, showing that rotation raises Ra_c in proportion to T^2. The model reproduces the Nu ~ Ra^(1/4) scaling, rotational suppression at low Ra, and the boundary-layer-dominated flow structure observed in companion axisymmetric simulations.

We investigate the linear stability of a rotating fluid annulus subjected to localized heating at the outer periphery of the bottom surface and uniform cooling at the inner cylindrical wall through a rigorous stability analysis. The localized forcing generates a non-axisymmetric base state, invalidating the classical normal-mode decomposition. We employ Floquet-Bloch theory in the azimuthal coordinate combined with a BiGlobal eigenvalue formulation in the meridional plane. The non-axisymmetric base state is expanded in azimuthal Fourier harmonics; perturbations are expressed as quasi-periodic Bloch modes that couple all azimuthal wavenumbers through base-state harmonics. Full linearised perturbation equations, the BiGlobal block-operator structure, pressure elimination, solenoidal projection, and the modal energy budget are derived. Instability is driven by cross-modal baroclinic energy release and shear production - mechanisms absent in classical axisymmetric theory.

Half-metallic Heusler compounds like $\mathrm{Co_2MnSi}$ allow to bridge magnonic and spintronic functionality for hybrid unconventional computing approaches with sought-after properties like 100% spin polarization and associated low Gilbert damping $α\leq 10^{-3}$. However, the desirable material parameters are inherently tied to the crystal lattice with a particularly critical dependence on structural order in $\mathrm{Co_2MnSi}$. To date, the successful fabrication of nanoscale devices with robust structural integrity remains yet a challenge, and consequently the impact of the material parameters on the resulting nonlinear spin-wave dynamics remains largely unexplored. Here, we report on a study of linear and nonlinear spin-wave dynamics in transversally magnetized $\mathrm{Co_2MnSi}$ waveguides with impeccable crystalline ordering. We show that epitaxial, $\mathrm{L2}_1$-ordered $\mathrm{Co_2MnSi}$ exhibits an intrinsic cubic anisotropy with first- and second-order contributions, stabilizing a magnetization alignment along the crystal $\langle110\rangle$ directions. We confirm the implication of an unaffected crystal structure resulting in preserved magnetic properties in the patterned structures. Herein, the persistent magnetocrystalline anisotropy reshapes the spin-wave dispersion which yields a first-order nonlinear instability suppression range extending over several GHz - even for vanishing bias fields. Moreover, the intrinsic magnetocrystalline anisotropy can be exploited to counteract shape demagnetization for a stabilized low bias field operation in the favourable Damon-Eshbach geometry with high group velocities and decay lengths. Together with the proven half-metallicity and ultralow Gilbert damping, this research establishes $\mathrm{Co_2MnSi}$ as a robust, scalable platform towards bias-field-free nonlinear half-metal magnonics.

We find that commensurate resonances in superconducting films endowed with a $SISIS$ structure, where $S$ and $I$ stand for superconducting and insulating layers, respectively, enhance the gap to a value three to four times the bulk gap. Such resonances rely on spatially localized quantum states that arise due to the commensurability between the total film thickness and the distance between the two insulating barriers. Our results are obtained in the context of the Bogoliubov-de Gennes equations within the Anderson approximation, applied here to Bi films, where quantum size effects are possible due to the abnormally large mean free path.

A reduction mechanism of the theoretically predicted excessive abundance of $^7$Li via baryons evaporated by primordial black holes is suggested. It is shown that the fraction of $^7$Li with respect to the number density of baryons can be diminished down to the observed value via the process of $^7$Li transformation by neutron capture. The created in this process $^8$Li or $^8$Be quickly decay into a pair of $^4$He nuclei.

Strong-to-weak spontaneous symmetry breaking (SW-SSB) has recently emerged as a useful framework for studying phases of matter in open systems, quantum or classical. Beginning with the simple idea of extending symmetry breaking to general mixed states, and the familiar equivalence between canonical and grand-canonical ensembles in statistical mechanics, the concept has grown into a unifying perspective connecting many different ideas in physics, including topological orders, emergent hydrodynamics, and information-theoretic characterization of phases of matter. This review provides a bird's-eye view of some of these recent developments.

Matrix and noncommutative convexity constitute an important area of modern noncommutative analysis and have found significant applications in mathematical physics. In the first part of our paper we give an abstract characterization of matrix convex sets, and compact matrix convex sets. Our approach is in some part via a universal Banach space (resp.\ operator space) $X_K$ of an abstract compact convex set (resp.\ matrix convex set) $K$. This turns out to be a concrete construction of the base norm space (resp.\ nc base norm space) with base $K$, together with a natural TVS topology. Noncommutative (nc for short) base norm spaces, recently developed by the first author and Hay, are an important class of operator spaces which include duals and preduals of unital $C^*$-algebras and von Neumann algebras, and operator systems, where the `base' is exactly the noncommutative convex set of (matrix) states on these. In the later parts of the paper we give many applications, mostly to base norm spaces (classical and noncommutative). We also refine some of our recent results concerning regularity of convex sets (classical and noncommutative). We give several interesting characterizations of base norm spaces (classical and noncommutative). Any such characterization will correspond by duality to a new characterization of operator systems, or in the classical case, of function systems. For example, (complex) nc dual base norm spaces are the matrix ordered LCTVS's $V$ such that $V$ (at level 1) has a linear base which is compact.

Detecting coordination among unmanned aerial vehicle (UAV) fleets operating in shared airspace and identifying the route-lead aircraft whose navigation decisions govern fleet behavior presents a fundamental speed--accuracy trade-off: fast methods enable real-time traffic management but sacrifice detection fidelity, while accurate methods may exceed the time budget for actionable airspace deconfliction. This paper presents a game-theoretic decision framework that resolves this trade-off by formulating method selection as a two-player zero-sum game between a Monitor (selecting computational methods and parameters) and Nature (selecting the unknown traffic scenario). We construct an end-to-end pipeline from trajectory surveillance data through eight candidate detection algorithms, a Monte Carlo sensitivity analysis characterizing their stochastic performance, and finally a multi-objective optimization layer that identifies Pareto-optimal method portfolios. The minimax solution provides a robust mixed strategy with a probability distribution over methods that guarantees worst-case performance regardless of scenario uncertainty. Experimental evaluation across 200 randomized configurations spanning 5--50 aircraft demonstrates that the framework recommends distinct method portfolios depending on operational priority: Koopman Phase dominates balanced (70.6%) and speed-priority (79.7%) profiles, while CRQA emerges as primary (47.4%) when route-lead identification is prioritized. The framework achieves a guaranteed game value of 0.29--0.53 (normalized utility) across all tested preference profiles, providing the first principled, scenario-adaptive methodology for computational method selection in UTM fleet monitoring operations.

Burr and Erdős conjectured in 1976 that for all integers $k>\ell\geq 0$ such that $k\mathbb{Z}+\ell$ contains an even integer, every $n$-vertex graph without cycles of length $\ell$ modulo $k$ has at most a linear number of edges in $n$. Bollobás confirmed the conjecture in 1977, and Erdős further asked for the exact extremal number. To the best of our knowledge, this problem has been solved only for all residues when $k\leq 4$, and for $\ell\in \{0,2\}$ when $k\geq 5$ is odd. In particular, Bai {\it et al.} [arXiv:2503.03504] proved that if $G$ is an $n$-vertex graph with no cycles of length $1$ modulo $3$, then $e(G)\le \frac{5}{3}(n-1)$, and when $9\mid (n-1)$ the equality holds if and only if each block of $G$ is isomorphic to the Petersen graph. Note that for $n> 18$ every extremal graph contains a cut-vertex. In this paper, we investigate the 2-connected setting and determine the maximum number of edges in a 2-connected graph with no cycles of length $1$ modulo $3$. Our results provide a sharp extremal bound and a complete characterization of the extremal graphs, revealing structural differences from the general case. Combining this with the result of Bai {\it et al.}, we also obtain a complete characterization of all extremal graphs in the general setting, including the cases where $9\nmid (n-1)$. Finally, we determine the maximum number of edges in a $2$-connected graph with no cycles of length $2$ modulo $4$, whose extremal graphs differ substantially from those in the general setting. Consequently, the extremal numbers for $2$-connected graphs with no cycle of a fixed length modulo $k$ are now determined for all $k\leq 4$.

We propose a new voting algorithm based on the pairwise majority-comparison matrix derived from voters' preference profiles. We show that this algorithm induces exactly the winner set of the Schulze rule (Schulze, 1997). Our algorithm successively eliminates weaker candidates in terms of all-pairs comparisons, thereby reflecting a dual spirit to Condorcet's original idea of splitting preference cycles (de Condorcet, 1785). We further show that the direct sum of the survival sets obtained at each elimination round coincides with the Schwartz set (Schwartz, 1972). These two equivalence results provide a formal mathematical foundation for the ``folklore'' relationship between the Schulze winner set and the Schwartz set, as well as a new Condorcetian interpretation of the Schulze winner set.

The $k$-th power $G^k$ of a graph $G$ is the graph on the same vertex set where the edge set consists of those pairs of distinct vertices of $G$ that are at distance at most $k$ from each other. A. Abiad, G. Coutinho, and M. A. Fiol [On the $k$-independence number of graphs, Discrete Mathematics 342 (2019), 2875--2885] proposed extensions of the classical ratio (for regular graphs) and inertia bounds to the independence number of $G^k$ for $k\ge 2$. Continuing a line of work comparing these two parameters with other known bounds, we show that the $\vartheta$-function of L. Lovász and the weighted inertia bound of A. R. Calderbank and P. Frankl, when applied directly to $G^k$, perform at least as well as the ratio and inertia bounds of Abiad-Coutinho-Fiol, respectively. In particular, $\vartheta(G^k)$ provides a polynomial-time computable upper bound on the independence number of $G^k$ that is at least as strong as the ratio bound when the latter applies (i.e.,\ when the graph $G$ is regular).