Papers

Hypergraph $p$-Laplacian regularization is a fundamental model in data analysis with successful applications in various tasks. It aims to minimize a nonsmooth and typically large-scale objective function defined as the sum of the $p$-th powers of the Lipschitz regularization over hyperedges. In this paper, we propose an operator-splitting algorithm for the hypergraph $p$-Laplacian that allows us to handle hyperedges separately in a Gauss-Seidel fashion. Each subproblem can be viewed as a generalized graph Lipschitz learning on a hyperedge, for which we introduce an auxiliary variable to overcome the nonsmoothness and solve it with one step of the alternating direction method of multipliers (ADMM). The resulting algorithm performs proximal ADMM updates sequentially over the hyperedges, and its convergence is proven. We test the algorithm on missing data recovery problems, including image sparse inpainting and semi-supervised learning, to demonstrate that it is faster than existing methods.

Let $F$ and $E$ be finite extensions of $\mathbb Q_p$, let $\mathbb G$ be a reductive group over $F$, and put $G=\mathbb G(F)$. Let $V$ be a locally algebraic representation of the form $V=\pi_{\mathrm{sm}}\otimes_E\sigma_{\mathrm{alg}}$, where $\pi_{\mathrm{sm}}$ is smooth admissible and $\sigma_{\mathrm{alg}}$ is finite-dimensional algebraic. We study the extended Goldman–Iwahori distance on the set of non-Archimedean norms on $V$. After fixing a reference norm $\alpha_0$, its finite-distance component $\mathscr N_{\alpha_0}(V)$ is the bounded projective limit of the extended Bruhat–Tits buildings attached to $V_K=\pi_{\mathrm{sm}}^K\otimes_E\sigma_{\mathrm{alg}}$. It is complete for the resulting uniform sup metric; this metric is of $\ell^\infty$ type and is generally not CAT(0). We prove directly that a $G$-orbit in $\mathscr N_{\alpha_0}(V)$ is bounded if and only if this component contains a $G$-invariant norm. The invariant norm is the pointwise supremum of the orbit. We formulate an integral group-algebra and type-Hecke condition necessary for an invariant norm. For $G=GL_n(F)$ we specialise to $V=\operatorname{BS}(r)=\pi_{\mathrm{gen}}(r)\otimes_E \pi_{\mathrm{alg}}(r)$.

Electroencephalography (EEG) and magnetoencephalography (MEG) provide noninvasive measurements of brain activity with millisecond temporal resolution, enabling the investigation of functional and effective interactions within large-scale brain networks. This chapter presents a comprehensive overview of the methodological foundations and practical workflows for EEG/MEG-based brain network analysis. We first review the physical principles underlying EEG and MEG, emphasizing their complementary strengths and limitations. We then describe the forward and inverse problems, including subject-specific head modeling, source reconstruction techniques, and the importance of accurate anatomical modeling for reliable source localization. Strategies for mitigating volume conduction and signal leakage are discussed, together with best practices for source-space connectivity analysis. The chapter reviews widely used functional and effective connectivity measures, including coherence, phase synchronization metrics, amplitude envelope correlation, Granger causality, dynamic causal modeling, and transfer entropy, highlighting their assumptions, advantages, and limitations. Modern end-to-end analysis pipelines are presented, with particular emphasis on Brainstorm and complementary open-source software for reproducible EEG/MEG research. Finally, we discuss emerging approaches, including time-varying connectivity, cross-frequency interactions, and network-based analyses, illustrating how noninvasive electrophysiology contributes to understanding brain organization in health and disease. The chapter provides both conceptual foundations and practical guidance for researchers and advanced students seeking to map and interpret human brain networks using EEG and MEG.

Spatially embedded directed networks, such as airline networks, often exhibit simultaneous high activity at nearby nodes. Preferential attachment (PA) explains hub dominance. We extend it to spatial co-movement through a directed PA model whose out- and in-node weights follow temporally persistent Gaussian-process lognormal fields. Under sublinear PA, out-degree proportions converge to explicit normalized powered weights, whereas self-loop exclusion yields a coupled in-degree limit. We derive a strictly concave inverse that recovers the in-weights from terminal degree proportions. For ordered network histories, we develop a minorization-maximization (MM) weight estimator and profile likelihood for the PA exponent; temporal pre-whitening and a spatial quasi-likelihood estimate the latent covariance. Simulations verify transmission of distance-decaying dependence and show how random segment volume creates a distance-independent common mode in raw degrees. An analysis of U.S. domestic flights (2015-2019) separates network-wide volume variation from a short-range spatial component. An observed-volume reconstruction reproduces the raw-degree common mode, and the fitted field yields an exploratory co-exceedance transition scale of roughly 150 km. A per-carrier analysis of European air traffic also reveals the same decomposition.

We establish mean-square and concentration bounds for stochastic approximation (SA) with arbitrary norm contractive mappings, under a multiplicative noise model where the noise may scale affinely with the norm of the iterates, and the iterates are potentially unbounded. These settings arise in reinforcement learning, where operators are often contractive in the $\ell_\infty$ norm and the noise scales with the iterates. To address the arbitrary norm, earlier works replace the non-smooth squared norm with a smooth Lyapunov function constructed via the generalized Moreau envelope. For concentration analysis, these works handle multiplicative noise and unbounded iterates through a multi-stage bootstrapping argument that starts from a time-varying worst-case bound and iteratively refines it. We instead present a unified and elementary analysis that yields both bounds. Using an averaged noise sequence and corresponding auxiliary iterates, we obtain a one-step Lyapunov drift inequality for the normed error directly, without smoothing the norm or constructing an envelope. For the mean-square bound, we combine this drift inequality with an induction argument showing that the iterates remain bounded in expectation. For the concentration bound, we develop a probabilistic induction over a sequence of "good" events on which the iterates are controlled, allowing the standard Azuma-Hoeffding bound to be applied. Our approach yields the first sub-Gaussian tailed maximal (all-time) concentration bound for SA under multiplicative noise, by allowing the stepsize to depend logarithmically on the confidence level. Beyond the specific setting considered here, we discuss the generalizability of these proof techniques to other noise models and iterative algorithms.

Recently, Merino and Mütze (FOCS'23+SICOMP'24) presented an algorithm for computing a Hamilton path on the skeleton of any 0/1-polytope ${\rm conv}(X)$, where $X\subseteq\{0,1\}^n$. The algorithm uses as a black box an algorithm for solving the classical linear optimization problem $\min\{w\cdot x\mid x\in X\}$ for some weight vector $w\in\mathbb{R}^n$. The resulting delay per visited vertex on the Hamilton path is only by a $\log n$ factor larger than the time to solve one instance of the optimization algorithm. In this paper, we make the Hamilton path algorithm simpler and faster. Namely, we obtain an amortized delay that is only by a constant factor larger than the running time of the optimization algorithm, thus removing the $\log n$ factor. As concrete results, this yields improved algorithms for generating bases and independent sets in a matroid, spanning trees, forests, matchings and maximum matchings in a graph, vertex covers, minimum vertex covers, independent sets and maximum independent sets in a bipartite graph, and antichains, maximum antichains and ideals in a poset. All of these listings correspond to Hamilton paths on the corresponding polytopes. Furthermore, we obtain an $\mathcal{O}(t_{\rm LP})$ amortized delay algorithm for the vertex enumeration problem on 0/1-polytopes $\{x\in\mathbb{R}^n\mid Ax\leq b\}$, where $A\in \mathbb{R}^{m\times n}$ and $b\in\mathbb{R}^m$, and $t_{\rm LP}$ is the time needed to solve the linear program $\min\{w\cdot x\mid Ax\leq b\}$. This improves upon the $\mathcal{O}(t_{\rm LP} \log n)$ delay algorithm of Merino and Mütze, and the previous $\mathcal{O}(t_{\rm LP}\,n)$ delay algorithm of Bussieck and Lübbecke from 1998.

Let \(n\ge3\) and \(1<p<n\). We prove a quantitative stability estimate for the critical Sobolev trace inequality on the upper half-space. More precisely, the Sobolev trace deficit controls the \(\max\{2,p\}\)-th power of the gradient distance to the manifold of trace bubbles. A central part of the proof is the spectral nondegeneracy of the trace bubbles: the first two eigenspaces of the linearized weighted Steklov problem are exactly the amplitude, dilation, and tangential translation modes.

Inspired by the sharp coefficient estimates established by Cho et al.\cite{CKKLS2018} for starlike functions of order $\alpha$ in the unit disk, we investigate the corresponding problems for starlike mappings of order $\alpha$ defined on the unit ball of a complex Banach space. Employing Fréchet derivatives together with suitable auxiliary lemmas, we establish sharp upper bounds for the second-order Hankel determinant, the Fekete–Szegö functional, and the Zalcman functional associated with this class of mappings. In each case, the obtained estimates are shown to be sharp by identifying the corresponding extremal mappings. Furthermore, our results reduce to the known one-dimensional sharp estimates when the underlying Banach space is the complex plane, thereby extending several classical results of Cho et al. \cite{CKKLS2018} to the setting of complex Banach spaces.

We investigate fractional heat semigroups generated by a class of anharmonic oscillators on $\mathbb R^n$ of the form $\mathcal H_{P,Q}=Q(D)+P(x),$ where $P\in\mathcal P_{2k}$ and $Q\in\mathcal P_{2\ell}$ are real-valued polynomials with anisotropic growth. Using the Weyl–Hörmander calculus associated with the natural metric determined by $(P,Q)$, we show that the fractional powers $\mathcal H_{P,Q}^s$, $s>0$, are pseudo-differential operators with symbols in adapted classes $\Sigma_{P,Q}^{2s}$. We prove fixed-time decay estimates for the fractional anharmonic heat semigroup $e^{-t\mathcal H_{P,Q}^s}$ on both Lebesgue and modulation spaces. In the Lebesgue setting, we establish sharp $L^p$–$L^q$ estimates for the full range $1\le p,q\le\infty$. For large time, the decay is exponential and governed by the smallest eigenvalue $\lambda_0$ of $\mathcal H_{P,Q}$, namely through the factor $e^{-t\lambda_0^s}$, while for small time the estimates reveal two distinct phase-space scales associated with the coercive growth of $P$ and $Q$, leading to anisotropic $L^p$–$L^q$ smoothing. As applications, we study nonlinear fractional heat equations associated with $\mathcal H_{P,Q}^s$. We prove local well-posedness in the supercritical Lebesgue range $ p>\frac{n(\beta-1)}{2\ell s},$ derive a lower blow-up rate for finite-time blow-up solutions, and obtain critical small-data global existence. We further prove global well-posedness and exponential decay for small initial data in modulation spaces. These results extend the heat semigroup theory for harmonic and model anharmonic oscillators to a broad class of anisotropic polynomial Hamiltonians.

A cornerstone of stability theory is Voiculescu's 1983 counterexample: he constructed a sequence of pairs of unitary matrices whose commutators converge to zero in the operator norm, but whose distances from the set of commuting unitary pairs remain bounded away from zero. Namely, the group $\mathbb{Z}^2$ is not stable in the operator norm. We prove, somewhat surprisingly, that stability is restored after an asymptotically negligible enlargement of the dimension. That is, the group $\mathbb{Z}^2$ is flexibly stable in the operator norm. This provides the first example, in any context, of a flexibly stable group that is not stable. Building on a construction of Eckhardt, who produced finitely generated amenable groups that are very-flexibly stable but not flexibly stable in the normalized Hilbert-Schmidt norm, we show that the same groups exhibit the analogous separation in the operator norm: they are very-flexibly stable but not flexibly stable.

Given a random variable $X$, an independent random variable $Y$ is called a symmetrizer of $X$ if their sum $X+Y$ is symmetric about the origin. The study of symmetrization resistance asks whether every such $Y$ must contain at least as much randomness as $X$. This problem was previously investigated for binary random variables in terms of variance and Shannon entropy. In this paper, we establish sharp symmetrization resistance results for exponential and geometric distributions. We prove that every independent symmetrizer $Y$ of an exponential random variable $X$ has variance, Rényi and Tsallis entropies of every positive order at least as large as that of $X$. Parallel results are obtained for integer-valued independent symmetrizers of geometric random variables. Equality in each comparison holds precisely when $Y$ is an independent copy of $-X$. Furthermore, we show the majorization of $X$ over $Y$ via establishing sharp concentration inequalities for $Y$. The proofs crucially rely on a differential/difference inversion formula for exponential/geometric convolution, which converts the symmetrization constraint into a hazard-rate inequality and enables us to identity exponential/geometric distributions as extremal distributions of a corresponding optimization problem.

This publication consists of slides from my talk at the Simons Center for Geometry and Physics in 2024. It contains a brief review of the L-functional formalism, along with a discussion of its possible applications to QED, linearized gravity, and quenched disorder. The most interesting part is the discussion of the infrared problem in QED and a conjecture on how to construct an infrared-finite perturbation theory for QED.

We solve the optimal consumption-investment problem in incomplete markets, where the investor aims to maximise an Epstein-Zin type stochastic differential utility from consumption over an infinite time horizon. We verify that the optimal strategies can be characterised by the limit of a sequence of solutions of the HJB equation in bounded domains with carefully designed boundary conditions. We also conduct numerical experiments for three market models. Explicit solutions are constructed under some parameter regimes.

This paper proposes a novel on-off division multiple access (ODMA) transmission scheme that enables efficient joint multi-user channel estimation and iterative decoding by inserting a small number of frozen symbols into the codewords. Functionally analogous to pilots, these symbols are sparsely distributed within the codeword. Unlike conventional pilot-based methods, our approach requires only a minimal number of frozen symbols (e.g., $5\sim20$ symbols per user in a 300-user system), which serve dual purposes as both estimation references and decoding aids. By employing low-complexity single-user channel estimation and decoding, combined with simple iterative interference cancellation, the scheme achieves performance equivalent to that with perfectly known user channels, even when accounting for the additional energy and bandwidth costs of the frozen symbols. Furthermore, for the large-scale ODMA sparse multiple access system, this paper proposes a fixed-point analysis method, which can accurately estimate the iterative convergence performance over multi-user block fading channels by only leveraging the decoding functions under the single-user AWGN channel. This method is applicable to performance analysis for arbitrary code lengths, code rates, and decoding algorithms. It eliminates the need for extensive Monte Carlo simulation time, and provides an efficient tool for the design of multi-user codes.

A latin square of order $n$ is an $n\times n$ array in which each of $n$ symbols occurs exactly once in each row and column. A transversal in such a square is a selection of $n$ entries that includes one representative of each row and column, and one of each symbol. For all even orders $n\ge 28$ except $n=30$, we construct a latin square of order $n$ in which every pair of transversals share at least one entry. We conjecture that in our squares there is no single entry that is common to all transversals. We prove this conjecture for $n\le10\,000$ by finding transversals using an algorithm that is likely to be of independent interest. We say that a transversal is dominant if it intersects every other transversal of the same latin square. We show that there exist latin squares of order $n$ that have a dominant transversal for $n\in\{5,7\}$ and also for all $n\ge8$ such that $n\not\equiv3\bmod4$.

In this paper, we prove a sharp Fefferman–Stein-type estimate for the fractional Schrödinger equation, which can be regarded as a generalized Strichartz estimate for data in the Fourier Lebesgue space $\widehat{L^p}$. Then, as an application of the Fefferman–Stein inequality and its off-diagonal generalization, we prove large data local well-posedness and small data global well-posedness results for the one dimensional fractional nonlinear Schrödinger equation with pure power nonlinearities in the homonegeneous and inhomogeneous Fourier–Sobolev spaces $\widehat{\dot{H}^s_p},\widehat{H^s_p}$. Solutions are established in $L_x^r(\mathbb{R} ;L^q_t(I))$ spaces in order to overcome the difficulty of a loss of derivatives in the standard Strichartz estimates.

In 1969, Halin proved that every $k$-connected graph $G$ with minimum degree at least $k+1$ contains an edge $e$ such that $G-e$ is $k$-connected. As an edge is a matching of size one, it is natural to ask whether Halin's result extends to matchings of larger size, a question recently investigated by Li, Zhou, Fujita, and Mao. A matching $M$ of a $k$-connected graph $G$ is called \emph{$k$-removable} if $G-M$ is $k$-connected. In this paper, we study minimum degree conditions that guarantee the existence of a $k$-removable matching of prescribed size. Specifically, we prove that for all positive integers $k$ and $m$, every $k$-connected graph $G$ with at least $2m$ vertices contains a $k$-removable matching of size $m$ if \[\delta(G)\ \ge\ \begin{cases} \max\bigl\{k+\bigl\lceil\tfrac m2\bigr\rceil,\ 2m\bigr\} & \text{if } k\ge m,\\[2pt] k+m & \text{if } k<m. \end{cases}\] As a consequence, every $k$-connected graph $G$ with $\delta(G)\ge2k+1$ contains a $k$-removable matching of size $\bigl\lceil(\delta(G)+1)/2\bigr\rceil$, unless $\delta(G)$ is even and $G\cong K_{\delta(G)+1}$. This verifies a conjecture of Li, Zhou, Fujita, and Mao in the range $\delta(G)\ge2k+1$. Our main tool, of independent interest, is a strengthening of Halin's result producing a $k$-removable edge that avoids a prescribed set of vertices.

Zero-shot text-guided editing of real-world music recordings requires balancing semantic modification with faithful preservation of the original musical structure. Although recent diffusion transformers trained with rectified flow have achieved remarkable success in text-to-music generation, extending them to edit existing recordings remains challenging because editing requires accurate deterministic inversion, reliable structural preservation, and numerically stable integration throughout the inversion and generation processes. We present FlowSonic, a zero-shot music editing framework built upon a pretrained diffusion transformer trained with rectified flow. FlowSonic first deterministically inverts a real-world recording into the latent space and preserves its musical structure during editing by reusing cross-attention representations extracted during inversion. To improve the numerical reliability of inversion-based editing, we introduce a high-order ODE solver and systematically investigate how different numerical integration schemes influence trajectory stability, structural preservation, and semantic controllability. Comprehensive experiments on timbre-transfer and genre-modification tasks demonstrate that FlowSonic consistently outperforms existing music editing methods across semantic alignment, harmonic preservation, structural consistency, and perceptual audio quality. We further provide geometric and empirical analyses showing how the proposed numerical integration strategy improves latent trajectory stability and leads to more reliable music editing.

Motivated by confidence-guided parallel unmasking in masked discrete diffusion, we study a single selection step in a stylized Gaussian random-field model. A locally dependent nonnegative score field represents position wise uncertainty, and the scheduler selects the K positions with the smallest scores. Dependence among the selected positions is measured through a distance-dependent Gaussian correlation model. This separation provides a tractable framework for quantifying how the geometry of low-score locations affects the dependence cost of factorized parallel decoding. We establish two complementary results. In a conservative sub-square-root regime, the conditional Gaussian total correlation of the selected block vanishes in probability. At the square-root scale, it remains non-negligible with positive asymptotic probability and admits a strictly positive expectation lower bound. Synthetic experiments support the predicted finite-size behavior. These results provide a rigorous stochastic-geometry baseline for understanding how budget size, score dependence, and spatial correlation jointly shape one-step confidence-based selection in masked discrete diffusion.

Determining the true minimum distance of an alternant code remains a notoriously difficult problem in coding theory. In this paper, we study the minimum distances of generalized BCH codes and twisted Goppa codes through their parity-check matrices. We first give a necessary and sufficient condition for an alternant code to attain its designed distance and apply it to generalized BCH codes. As applications, we prove that broad classes of generalized BCH codes have minimum distances equal to their designed distances. These classes provide explicit infinite families rather than isolated examples. We characterize when a twisted Goppa code $\Gamma(L,g,\eta)$ with $\operatorname{deg} g=t$ satisfies $d(\Gamma(L,g,\eta))=t+1$, and derive structured classes and infinite families attaining this distance.

This paper develops a systematic theory of complex powers, logarithms, and sectorial projections of elliptic pseudodifferential operators on noncommutative tori, extending to this setting the classical constructions of Seeley and others. Building on the parametric pseudodifferential calculus of the prequel~\cite{LP:Part1}, we construct the complex powers associated with a given ray, show that they form a holomorphic family of pseudodifferential operators with the semigroup property, and compute their symbols. We further establish exponential growth bounds on vertical strips in the operator, Schatten, and trace-class topologies by means of a new holomorphic calculus for pseudodifferential families. The logarithm is identified as a pseudodifferential operator whose symbol is determined by the resolvent symbol, and the associated trace formula is derived. Sectorial projections are constructed as contour integrals and shown to be of order zero. This yields analogues for noncommutative tori of results of Wodzicki, Okikiolu, and Gaarde–Grubb, and provides the analytic foundations for the spectral-geometric applications developed in subsequent papers.

Cardinal characteristics of the continuum are the cardinalities of interesting families of reals. A well-studied example is that of maximal almost disjoint (MAD) families of sets of natural numbers. Significant work has been done investigating computability-theoretic analogues of cardinal characteristics. By considering encodings of MAD families as a single `universal' set, Lempp, Miller, Nies, and Soskova (2023) studied the class of encoded MAD families. Such a class is referred to as a mass problem; one can study the relative complexity between mass problems. In Section 2, we build on the study of mass problems as analogues of cardinal characteristics. We define mass problems of maximal eventually different (MED) families of computable functions in the same way and compare them against the mass problems defined by Lempp et al. In Section 3, we survey work by Greenberg, Kuyper, and Turetsky (2019) that provides an abstract framework for cardinal characteristics and their effective counterparts. We show that this framework is suitable for obtaining results in the setting of mass problems. In Section 4, we showcase a construction by Schrittesser (2018) of an effectively closed MED family in set theory. We show that the construction is sufficiently effective that the computable members of the constructed family are MED relative to computable functions.

It is known that a mixture of three homoscedastic multivariate Gaussian densities whose centers form an equilateral triangle can have four modes, owing to the emergence of a “ghost” mode at the center. Nevertheless, obtaining a sharp upper bound on the number of modes remains open even in this seemingly simple setting. The best previously available upper bounds applicable to the homoscedastic three-component setting were 42592 and, more recently, 72. These bounds were derived for substantially more general classes of Gaussian mixtures and are therefore not optimized for the present setting. Moreover, they are conditional on the assumption that the modal set is finite. In this paper, as a sharper bound, we prove that every mixture of three homoscedastic Gaussian densities has at most 8 modes, without imposing any finiteness or non-degeneracy assumption a priori. To the best of our knowledge, this is the first unconditional finiteness result for the modal set that holds uniformly over the full class of multivariate three-component homoscedastic Gaussian mixtures. In particular, it covers genuinely multivariate asymmetric configurations with arbitrary non-collinear centers and arbitrary positive mixture weights.

This paper establishes limitations on observability inequality and unique continuation for Schrödinger equations on fractal sets. We prove that, in contrast to the heat equation, such properties can fail in fractal settings. To achieve this, we first extend the classical Turán–Nazarov inequality, which provides lower bounds of trigonometric polynomials of the form $\sum_{k=1}^nc_ke^{2\pi im_kt}$ on sets of positive measure, to the fractal setting. Unlike in the classical case, the constant in the inequality loses uniformity in the degree $n$, and we obtain sharp bounds depending on both $n$ and the frequency difference $m_n-m_1$. These refinements then enable us to construct explicit counterexamples, showing that observability and unique continuation may fail for Schrödinger equations when the observation set is fractal.

We study the recovery of a smooth nonlinearity in semilinear wave equations on globally hyperbolic Lorentzian manifolds. Our approach combines higher-order linearization with distorted plane waves and Gaussian beams, using three-wave, four-wave, and m-wave interactions to recover different Taylor coefficients.

Let $R$ be a left and right Noetherian ring. We introduce a Serre-type condition $(G_n)$, formulated in terms of the first occurrence and the flat dimension of indecomposable injective modules, and prove that $R$ is $n$-Gorenstein if and only if it satisfies $(G_n)$. We then apply the criterion to Nakayama algebras via syzygy filtration. It is shown that syzygy filtration preserves the Auslander-Gorenstein property and reflects it within the class of $2$-Gorenstein Nakayama algebras. Combined with a result of Klász, Kleinau and Marczinzik, this shows that simple modules of odd grade over Auslander-Gorenstein Nakayama algebras are regular in their grade, thereby settling their conjecture.

Non-smooth and non-convex optimization problems are pervasive in machine learning, control, and signal processing, due to the need for sparse solutions and the inherently non-convex nature of many objective functions. In this paper, we study non-smooth and non-convex distributed optimization problems. We propose a novel bi-level Consensus Alternating Direction Method of Multipliers (ADMM) algorithm, termed CADMM-Prox. The proposed algorithm integrates classical Consensus ADMM with a proximal mechanism by introducing a sufficiently large proximal term associated with an outer-level variable. Under the mild assumption that the local objective functions are semi-convex, CADMM-Prox is guaranteed to converge globally to a neighborhood of a Clarke stationary point. Numerical experiments on a phase retrieval problem demonstrate that our proposed method exhibits more stable convergence behavior compared with baseline algorithm.

This article studies the intersection of two linear sets of rank 3 of PG(1,q^3). In particular, all possible intersection sizes are determined, the structure of the largest intersection sets is described, and existence results are given.

This study provides a quantitative framework for analysis of systemic demand uncertainty and risk propagation cascades across general supply chain networks. By leveraging properties derived from stochastic networks embedded within a Newsvendor paradigm, we model multi-echelon networks under equilibrium and transient operational regimes. We mathematically validate that the systemic volatility behavior commonly referred to as the Bullwhip effect persists entirely as an unavoidable, inherent topological property of coordinated logistics networks, independent of traditional operational noise or information visibility constraints. Extending this paradigm to transient environments, we model inventory drawdown horizons as a multi-dimensional Skorokhod reflection problem. Crucially, we endogenize market-clearing feedback loops by incorporating non-linear price elasticity mechanisms and dynamic trade relation rebalancing, demonstrating how decentralized rational actions co-evolve with physical capacity bottlenecks and can accelerate systemic network degradation. Finally, we operationalize the framework through a data-driven numerical experiment mapping global oil trade dynamics, showing how localized chokepoint disruptions, such as a capacity shock in the Strait of Hormuz, trigger non-linear cascading stockouts and systemic reallocation across sovereign buffers over time.

We prove that every smooth entire solution $ u\colon\mathbb{R}^2\to\mathbb{R}^2 $ of the Ginzburg–Landau equation $ -\Delta u=u(1-|u|^2) $ with $ |u(x)|\to1 $ as $ |x|\to\infty $ has finite potential energy, i.e., \begin{equation*} \int_{\mathbb{R}^2}(1-|u|^2)^2 \mathrm{d}x<+\infty, \end{equation*} thereby resolving Brezis' Open Problem 2.5 in [4]. The main difficulty stems from the possible presence of a curl-free mode that carries nonzero circulation and decays only like $ |x|^{-1} $; such a mode lies outside $ L^2 $ and does not admit a single-valued potential. By minimizing over $ L^2 $ gradient corrections, we construct a comparison field that solves the homogeneous equation and inherits the same circulation. The Kelvin inversion, combined with the De Giorgi–Nash–Moser theory for quasilinear elliptic equations, then produces the optimal decay $ O(|x|^{-1}) $. For a Ginzburg–Landau solution, the Bernstein estimate and the coercivity of the Jacobi form produce an $ L^2 $ forcing term in the exterior phase equation. The resulting $ L^4 $ bound on the phase field implies $1-|u|^2\in L^2(\mathbb{R}^2)$, and therefore the potential energy is finite.