Papers

We consider localized oscillations of an Euler–Bernoulli beam on a visco-elastic foundation coupled to a damped discrete oscillator. All system parameters vary slowly and independently in time. For the conservative case, we employ three different analytical approaches: an asymptotic method, a technique based on the adiabatic invariance of the action of a trapped wave, and an analysis of the equivalent Hamiltonian system. All three approaches yield the identical formula for the oscillation amplitude. In the dissipative case, the oscillation amplitude is obtained solely via the asymptotic approach. Finally, we validate our analytical findings against numerical simulations, demonstrating excellent agreement.

We prove that the support of the canonical space-time measure for a Brakke flow is a parabolic $(k+2)$-rectifiable set. As a consequence, we obtain that at almost all points along the flow, with respect to this canonical space-time measure, there exists a unique, static, planar tangent flow, and that various notions of density for the flow agree at these points. Moreover, following on from our previous work `The space-time-Grassmann measure of the Brakke flow', we continue to develop the approach to the Brakke flow as a space-time-Grassmann measure. We prove that the standard notion of convergence for Brakke flows, coming from the compactness theorem of Ilmanen (7.1 of `Elliptic regularization and partial regularity for motion by mean curvature'), is equivalent to the convergence of these space-time-Grassmann Radon measures. This gives an alternate notion of varifold convergence to the one exhibited in 7.1(ii) of `Elliptic regularization and partial regularity for motion by mean curvature'.

A dominating set is called $k$-limited if every vertex in the set has at most $k$ neighbors outside it. The minimum cardinality among all $k$-limited dominating sets of a graph $G$ is the $k$-limited domination number, denoted by $\gamma_k^{\mathrm L}(G)$. We prove that, for every fixed integer $k\ge 2$, deciding whether a graph admits a $k$-limited dominating set of size at most $\ell$ is $\mathsf{NP}$-complete. In addition, a systematic study of $k$-limited domination in Cartesian products is initiated. In particular, we establish general lower and upper bounds for $\gamma_k^{\mathrm L}(G\square H)$, show that both are sharp, and derive exact values for several natural families of graph products.

We study the one-dimensional Schrödinger operator on a fixed interval with Gaussian white-noise potential, \[ H_\omega=-\frac{\dd^2}{\dd x^2}+\rho\dot B_x(\omega), \] under Dirichlet boundary conditions. The operator is defined pathwise through the quasi-derivative realization of Sturm–Liouville operators with distributional potentials. Let $\lambda_n$ be the Dirichlet eigenvalues, $\lambda_n^+=\max\{\lambda_n,0\}$, and $k_n=\sqrt{\lambda_n^+}$. For every finite $p$, we prove the high-energy expansion \[ k_n=\frac{n\pi}{L} +\frac{\rho}{n\pi}\int_0^L \sin^2\left(\frac{n\pi s}{L}\right)\,\dd B_s +O_{L^p(\Omega)}(n^{-2}). \] Consequently, almost surely, $\lambda_n>0$ for all sufficiently large $n$ and, for every $\varepsilon>0$, \[ k_n=\frac{n\pi}{L}+O(n^{-1+\varepsilon}). \] We also obtain first-order eigenfunction asymptotics with explicit Brownian stochastic-integral corrections. In particular, for the $L^2(0,L)$-normalized Dirichlet eigenfunction $\varphi_n$, with a fixed sign convention, \[ \sup_{0\le x\le L} \left|\varphi_n(x)-\sqrt{\frac{2}{L}}\sin(k_n x)\right| =O(n^{-1+\varepsilon}) \] almost surely. The proofs use stochastic Prüfer coordinates, stochastic Volterra expansions, the Burkholder–Davis–Gundy inequality, and a Borel–Cantelli argument. The estimates provide a first step toward KAM-type small-divisor analysis for Hamiltonian PDEs with white-noise spatial potentials.

In this work, we introduce and study the notion of $k$-limited domination in graphs, motivated by applications where dominating vertices have bounded capacity and cannot be overloaded by too many external neighbors. Formally, given an integer $k \le \Delta(G)$, a set of vertices $D \subseteq V(G)$ is called a $k$-limited dominating set if it is a dominating set and, in addition, each vertex of $D$ has at most $k$ neighbors outside $D$. The minimum cardinality of such a set is the $k$-limited domination number, denoted by $\gamma_k^{\mathrm{L}}(G)$. Since $\Delta(G)$-limited domination coincides with the classical domination, we restrict our attention to the nontrivial range $1 \le k < \Delta(G)$, where the degree limitation becomes meaningful and leads to new combinatorial phenomena. In this paper, we initiate the study of this concept by deriving sharp general bounds for $\gamma_k^{\mathrm{L}}(G)(G)$ and identifying conditions under which these bounds can be further improved. We establish a connection between $k$-limited domination and $(1,t)$-domination. In particular, for $d$-regular graphs we prove that $\gamma_k^{\mathrm{L}}(G)(G)=\gamma_{1,d-k}(G)$. In the special case $k=1$, we show that $1$-limited domination is tightly linked to graph packings, yielding the bound $\gamma^{\mathrm L}_1(G) \le n - \rho(G)$ and its characterization. The study reveals several natural open questions and indicates that limited domination provides a rich ground for further research.

We study a converse problem to the theorem of Bergman and Bieri–Groves on logarithmic limit sets of algebraic subvarieties of the complex algebraic torus $(\mathbb C^*)^n$. We introduce the notion of finite logarithmic type, a boundary condition formulated on toric compactifications in terms of coherent meromorphic extensions with uniformly bounded pole orders along toric boundary divisors. We prove that a closed reduced analytic subvariety of $(\mathbb C^*)^n$ whose logarithmic limit set is a finite rational spherical polyhedral complex of the expected dimension is algebraic whenever it is of finite logarithmic type. The proof relies on toric compactifications, coherent extension theory, Serre's GAGA theorem, and Chow's theorem. A principal result of the paper is the complete treatment of the one-dimensional case. We prove that every closed analytic curve in $(\mathbb C^*)^n$ with finite logarithmic limit set is algebraic. Consequently, the finiteness of the logarithmic limit set completely characterizes algebraicity for analytic curves, yielding a genuine converse to Bergman's theorem in dimension one. These results establish new links between logarithmic limit sets, tropical geometry, toric geometry, and complex analytic geometry.

We study a class of cyclic dominance models with seasonal forcing, extending the classical May-Leonard competition framework. By introducing time-periodic coefficients into the growth rates and the interaction terms, we explore how environmental seasonality influences the dynamics of three-species systems. Through analytical estimates and numerical simulations, we reveal the emergence of complex oscillatory behavior, including multi-year periodic cycles, transitions to heteroclinic cycles, and chaotic oscillations. Our results highlight how periodic modulation can destabilize stable periodic trajectories, generate novel attractors, and give rise to coexistence mechanisms not present in autonomous systems. These findings contribute to the understanding of biodiversity maintenance under realistic, temporally varying ecological conditions.

We develop a finite-type framework for studying the Jones polynomial and its ability to distinguish the unknot. The main difficulty is to propagate the vanishing of its finite-type coefficient layers from the natural upper degree bound down to the first potentially informative orders. To overcome this, we introduce a local clasped-twist construction whose diagrammatic reductions remain uniformly controlled even when the twist is made arbitrarily long. This separates the role of the twist length from the degree bound and permits a descending vanishing argument. A single construction transfers high-order finite-type vanishing to a long residual twist, where local smoothing relations and an anchor reproducing the original knot force vanishing at all relevant lower orders. Two constructions placed in disjoint regions then generate a controlled two-crossing family. On this family, the resulting low-order relations impose a component count after successive smoothings that contradicts the topology of a suitably chosen pair of crossings. As a consequence, every nontrivial knot has a nonzero Jones finite-type coefficient at an order bounded above by three times its crossing number. In particular, a knot whose Jones polynomial is equal to that of the unknot must itself be the unknot.

Attention mechanisms have demonstrated remarkable empirical success in identifying relevant information from large collections of tokens, yet the theoretical principles underlying this behavior remain poorly understood. We study a stylized softmax-attention model in which a query vector is learned by stochastic gradient ascent from a collection of informative and nuisance tokens. Exploiting the symmetry of the model, we derive a population objective and characterize the limiting ordinary differential equation governing the learning dynamics. Using tools from stochastic approximation and dynamical systems theory, we establish a rigorous connection between the stochastic learning algorithm and its deterministic limit. Our main result shows that, under suitable high-dimensional scaling assumptions and standard step-size conditions, the learned query converges almost surely to the one-dimensional signal subspace spanned by the latent informative direction. Equivalently, the query asymptotically recovers the latent signal up to the intrinsic sign ambiguity. These results provide a rigorous theoretical foundation for understanding attention mechanisms as signal extraction procedures in high-dimensional noisy environments and offer a dynamical-systems perspective on how attention discovers relevant information in the presence of substantial noise.

We study sums of higher even powers in the coordinate rings of singular planar curves $x^M=y^m$ for coprime positive integers $m<M$. We then show that the $2s$-Pythagoras number of real algebras of the form $\mathbb{R}[x,y,\sqrt{f_1},\sqrt{f_2},\dots, \sqrt{f_n}]$ are infinite, under some mild assumptions on the polynomials $f_1,f_2,\dots, f_n \in \mathbb{R}[x,y]$. We prove that all of the higher even Pythagoras numbers are finite for the ring of $0$-regulous functions on a $0$-regulous variety. We then show that the codimension of the bad set of order $2n$, for $n>1$, can be of codimension $2$, contrary to the quadratic case.

The monotone inclusion problem is fundamental in applied mathematics and is closely related to a wide range of practical applications. However, existing solution methods typically require the underlying operator to be Lipschitz continuous. Recently, Hölder continuity, a weaker condition than Lipschitz continuity, has proven useful in characterizing certain real-world problems. To bridge this gap, we investigate the convergence rates of the Tseng's splitting method (a fundamental algorithm for monotone inclusion problem) and its two accelerated variants, the composite extra anchored gradient method and the symplectic composite extra gradient method, under the Hölder continuity assumption. Our numerical experiments demonstrate that the numerical performance of these algorithms aligns with their respective theoretical convergence rates.

Structure-agnostic (SA) models introduced by Balakrishnan et al. (2026) aim to reflect the general lack of knowledge of structural assumptions on data-generating laws such as smoothness or sparsity in practice. Roughly speaking, SA models restrict the observed-data generating law to be in some rn-neighborhood of (black-box machine learning) estimates, treated as given and fixed, where rn encodes the convergence rates of the estimates to the truth. Under SA models, Balakrishnan et al. (2026) show that the popular Double Machine Learning (DML) estimators for three functionals, the quadratic functional in the Gaussian sequence model, the quadratic density integral functional and the expected conditional covariance, are minimax. However, minimax estimators may be inadmissible. In this paper, we show that, for the first two of the three functionals, the DML estimator is asymptotically inadmissible under the SA model. In particular, we show that these two functionals fall into a class of functionals, which we refer to as the monotone bias class. For this class, we exhibit second-order (U-statistic) estimators, which asymptotically dominate DML estimators, under the SA model. These second-order estimators are empirical higher-order influence function (HOIF) estimators introduced in Liu et al. (2017). Furthermore, the empirical HOIF estimator, like the DML estimator, is minimax for the third functional (the expected conditional covariance), although neither asymptotically dominates the other.

We prove the global well-posedness of small perturbations of two-dimensional sessile droplet equilibria for the incompressible Navier–Stokes equations with surface tension, Navier-slip boundary conditions, and a dynamic contact-point law. The main difficulty is the construction of solutions in the presence of the horizontal translational degeneracy of the equilibrium manifold. To remove this degeneracy, we work in a moving polar coordinate system determined by an orthogonality condition. We then establish local well-posedness through a Galerkin construction of pressureless weak solutions, recovery of the pressure, higher-order estimates, and a contraction argument. Combining this local theory with the global energy–dissipation estimates obtained in our previous work yields a unique global solution and the corresponding exponential decay estimate.

Deep learning has demonstrated significant success for modeling biochemical molecular systems, where inputs are commonly represented as graphs or 3D grids. A major challenge is that computational cost scales with resolution, making full graph/grid computation of molecular densities expensive and often unstable. We introduce a multigrid training strategy that leverages low-resolution optimization to accelerate learning at higher resolution through parameter transfer across discretizations. For graph molecular representations, we progressively transfer parameters learned from a coarse graph to a sequence of increasingly finer graphs via biased random walk upsampling. For 3D molecular generation, we voxelize the molecular structures at multiple resolutions, pretrain a coarse-resolution conditional Variational Autoencoder (CVAE), and initialize a fine-resolution CVAE by transferring shape compatible convolutional parameters from the coarse model. Numerical experiments on receptor-conditioned 3D Ligand generation show that multigrid training accelerates convergence and improves generalization compared to training from scratch.

A vertex coloring of a graph $G$ with nonempty color classes $V_1,V_2,\dots,V_k$ is called a \emph{FAT $k$-coloring} if there exist real numbers $\alpha,\beta\in[0,1]$ such that for every vertex $v$ and every color class $V_i \in \left\{ V_1,V_2,\dots,V_k \right\} $ we have $$ \bigl| N(v) \cap V_i \bigr|= \begin{cases} \alpha\deg(v) & \text{if } v\notin V_i,\\[4pt] \beta\deg(v) & \text{if } v\in V_i . \end{cases} $$ \noindent The FAT coloring concept was originally proposed and thoroughly studied by Beers and Mulas. The set of all FAT colorings of a graph is naturally ordered by the coarsening relation, in which finer partitions are larger in the order. The maximal elements of this poset, called irreducible FAT colorings, form a generating set: every FAT coloring of the graph can be obtained by merging color classes of some irreducible one. Beers and Mulas raised the compelling question whether, for every positive integer $s$, there exists a graph that admits exactly $s$ irreducible FAT colorings. In this paper we settle this question affirmatively by exhibiting, for any given $s$, a graph possessing precisely $s$ such colorings.

Let $k$ be an algebraically closed field of characteristic zero and let $S=k[x_1,\ldots,x_5]$ with maximal ideal $\mathfrak m=(x_1,\ldots,x_5)$. For a codimension-$r$ subspace $Q\subset S_2$, set $I_Q=(Q)+\mathfrak m^3$. Then $S/I_Q$ has Hilbert function $(1,5,r)$. We prove that the translated one-step locus defined by these ideals is contained in the smoothable component for every $r\in{6,7,\ldots,15}$. We introduce a finite field differential rank certificate proving dominance, for $6\le r\le 14$, of the Erman–Velasco map $\operatorname{GL}*5\times (\mathbb A^5)^r\dashrightarrow \operatorname{Gr}(r,\operatorname{Sym}^2 k^5)$, $(g,a^{(1)},\ldots,a^{(r)})\mapsto g\cdot\langle q(a^{(1)}),\ldots,q(a^{(r)})\rangle$, where $q(a)=\sum*{i=1}^5 a_i y_i^2-\left(\sum_{i=1}^5 a_i y_i\right)^2$. The endpoint $r=15$ is handled separately by a flat degeneration of $21$ general reduced points to the fat point defined by $\mathfrak m^3$. Combined with the known small cases and with the known elementary components for $r=3$ and $r=5$, this gives the complete one-step classification in embedding dimension five: the one-step loci with Hilbert function $(1,5,r)$ are smoothable for all $r\neq 3,5$, and the cases $r=3,5$ are precisely the generically reduced elementary component cases. In this sense the one-step Shafarevich gap in embedding dimension five is completely resolved.

Let $ H_{m/n}=\{(z_1,z_2)\in\C^2:|z_1|^m<|z_2|^n<1\}, \qquad \gcd(m,n)=1, $ be a rational Hartogs triangle. We characterize the $L^p$-boundedness of Forelli–Rudin type operators associated with its Bergman kernel. For the operators with kernel $|B_{m/n}(z,w)|^{c/2}$, the characterization holds for all $a,b\in\R$ and $c>0$; for the operators with kernel $B_{m/n}(z,w)^N$, it holds for every $N\in\Z_+$. The conditions are necessary and sufficient and recover the sharp $L^p$-ranges of the Bergman projection and the Berezin transform.

We establish a dilatation-dependent Gabriel inequality for sense-preserving harmonic $K$-quasiregular mappings in $h^2$. If $f=h+\overline g\in h^2$ satisfies $|g'|\le k|h'|$, where $K=(1+k)/(1-k)$, then every convex curve $\Gamma\subset\D$ satisfies \[ \int_\Gamma |f(z)|^2\ds(z) \le 2\frac{(1+k)^2}{1+k^2} \int_\T |f^*(\zeta)|^2\,|d\zeta| =\frac{4K^2}{K^2+1} \int_\T |f^*(\zeta)|^2\,|d\zeta|. \] The coefficient recovers the sharp analytic constant $2$ at $K=1$, is strictly smaller than the general harmonic constant $4$ for every finite $K$, and tends to $4$ as $K\to\infty$. The proof retains quantitative information on the analytic and co-analytic parts through the Hardy–Stein identity and the complex dilatation bound. To place this estimate in a broader quasiregular setting, we develop a maximal-function principle based on the Carleson-measure geometry of convex curves. We also identify a log-subharmonic modulus class that inherits the sharp analytic constant and prove Stoilow-factorization criteria under explicit boundary-weight and cone-distortion assumptions. These results distinguish distortion-dependent estimates from those arising through general boundary maximal control. The resulting formulation clarifies which conclusions follow from harmonic quasiregularity and which require separate boundary regularity assumptions.

For every $n\ge3$, we construct compact Type II ancient Yamabe flows in the round conformal class of the unit $n$-sphere $\Ss^n$. After stereographic projection and rescaling of time and conformal factor, the solutions are asymptotic backward in time to finite superpositions of Aubin–Talenti bubbles with mutually comparable scales. For integers $h\ge1$ and $k\ge2$, the limiting centers form $h$ coaxial regular $k$-gons; when $h\ge2$, their distinct layer heights may be prescribed arbitrarily in the interval $(-1,1)$. If $hk\ge3$, the resulting flows are not geometrically equivalent to any rotationally symmetric ancient flow. For each flow, there is a fixed point of $\Ss^n$ at which the Ricci tensor has both positive and negative eigenvalues on every time slice. The corresponding volume-normalized Yamabe flows are also ancient. After rescaling each time slice to a fixed volume, the metrics converge to a wedge of $hk$ isometric round $n$-spheres in the Gromov–Hausdorff topology, as time tends to $-\infty$. For fixed $h,k\ge2$, the concentration configurations admit an explicit $(h-1)$-parameter description up to Möbius transformations. Consequently, the construction yields geometrically inequivalent flows indexed by a nonempty open subset of $\R^{h-1}$, all with the same fixed-volume backward Gromov–Hausdorff limit. The proof uses inner–outer gluing for the quasilinear parabolic equation in $n$-dimensional space that exhibits Kelvin symmetry. Weighted Euclidean and spherical Hölder estimates control the nonlinear terms and, together with parabolic regularity, yield smooth extension of the conformal metrics across the north pole of $\Ss^n$.

We study Reed–Solomon codes against adversarial coordinate permutations followed by insertion-deletion (insdel) errors. It was previously shown by Con (2025) that Reed–Solomon codes can attain the exact half-Singleton bound in this setting, but only over exponentially large alphabets. We prove that, by allowing an additive $\epsilon n$ gap from this bound, the alphabet size can be reduced to polynomial. More precisely, for fixed constants $R,\epsilon\in(0,1)$ satisfying $2R+\epsilon<1$ and $k=Rn$, a random Reed–Solomon code of length $n$ and dimension $k$ over an alphabet of size $n^{O_{R,\epsilon}(1)}$ is, with high probability, robust against arbitrary coordinate permutations followed by up to $(1-\epsilon)n-2k+1$ insdel errors. We also prove a complementary alphabet-size lower bound, showing that positive-rate codes, which are robust against linearly many insdel errors in the permutation-insdel setting, require a polynomially superlinear alphabet. Finally, for the explicit two-dimensional Reed–Solomon codes constructed by Con et al. (2024) over alphabet size $O(n^3)$, we give an average $O(n)$-time decoder against arbitrary coordinate permutations followed by $n-3$ insdel errors. Previously, an $O(n)$-time decoder for this code was known only for the deletion setting.

Adam is one of the most widely implemented and influential modern optimizers. Why is it effective across different optimization problems in practice? This question arguably lies at the center of the optimization community over the last decade and has motivated a substantial body of work aimed at understanding its convergence behavior. However, existing studies have mainly focused on the convergence rate of Adam in smooth nonconvex optimization, which unfortunately does not adequately capture practical settings, since many real-world problems are nonsmooth, such as those arising in training neural networks. Thus, these studies cannot fully explain the popularity and empirical success of Adam. Recently, an insightful and powerful framework called Online-to-Nonconvex Conversion has opened a new way to analyze Adam for nonsmooth nonconvex optimization. Unfortunately, prior works along this line share two common limitations. First, all of them ignore the important bias-correction term in the original Adam algorithm. Second and more importantly, many of them require extra operations that are not used in Adam, such as a clipping step. Therefore, the convergence guarantee for the original Adam method still remains unclear. In this work, we present the first finite-time analysis for the classical form of Adam, i.e., with the bias-correction step and without further algorithmic modifications, and prove that a randomly scaled learning rate ensures a convergence rate of $1/T^{\frac{2}{13}}$ for nonsmooth nonconvex optimization. Moreover, our result provably applies to the modern heavy-tailed noise regime, which is closer to practice. Interestingly, our theory is established under the parameter choice $\beta_1=\beta_2$, aligning with the recent empirical studies.

In the present article, we consider $\mathbb{Z}_p$-extensions defined by restrictions on ramification at $p$-adic primes. We study the finiteness, the existence of non-trivial finite submodules, and the triviality of the unramified Iwasawa modules of these $\mathbb{Z}_p$-extensions. In addition, we give a new sufficient condition for Greenberg's Generalized Conjecture with some non-trivial concrete examples.

The Beauville–Voisin conjecture predicts a canonical descending filtration on the Chow group of zero-cycles of a hyperkähler variety, opposite to the conjectural Bloch–Beilinson filtration. A basic test for such filtrations is a Bloch-type principle: the action on zero-cycles should be governed by the action on the holomorphic symplectic form. While this principle has been verified in several cases of hyperkähler varieties of $\mathrm{K3}^{[n]}$-type, the $\mathrm{Kum}_n$-type case remains much less understood. In this paper, we study this problem through twisted abelian surfaces and their associated $\mathrm{Kum}_n$-type varieties. We first construct a natural action of autoequivalences of twisted abelian surfaces on the Albanese kernel and prove Bloch's conjecture for all (anti-)symplectic autoequivalences. As an application, we prove the corresponding Bloch conjecture for symplectic birational automorphisms of twisted modular $\mathrm{Kum}_n$-type varieties; in particular, this applies to those admitting a birational Lagrangian fibration. Finally, we introduce and study a Shen–Yin–Zhao type filtration on twisted modular varieties and compare it with Voisin's filtration in the sixfold case. We also establish the anti-symplectic Bloch conjecture for twisted modular $\mathrm{Kum}_3$-type varieties.

We study the competition between resonance-induced integrability breaking and centrifugal confinement in a symmetric quartic deformation of the classical $k=1$ Tremblay–Turbiner–Winternitz (TTW) system, $H=\frac{1}{2}\left(P_X^2+P_Y^2\right)+X^2+Y^2 +\frac{\gamma}{X^2}+\frac{\gamma}{Y^2}+\kappa X^2Y^2.$ The inverse-square terms are the centrifugal remnants of an $SO(2)\times SO(2)$ reduction of a four-dimensional oscillator, while the $X^2Y^2$ interaction couples the reduced radial modes and connects the Smorodinsky–Winternitz and Contopoulos limits. For $\gamma>0$, the axes are impenetrable and split configuration space into invariant sectors. For sufficiently small nonzero $\kappa$, resonant averaging constructs a phase-locked periodic orbit on each energy surface $H=h>4\sqrt{\gamma}$ whose fixed-energy reduced Poincaré map has no unit characteristic multiplier. Poincaré's criterion therefore excludes a second independent $C^1$ first integral in any invariant neighbourhood of this orbit. At finite coupling, Poincaré sections and finite-time Lyapunov maps show the breakup of invariant curves and the growth of chaotic layers. Comparison with the barrier-free limit separates two effects: the quartic interaction breaks integrability, whereas the singular barriers reduce phase-space connectivity and chaotic transport without restoring it.

We study surjective phase-isometries between the unit spheres of uniform algebras. Although such maps preserve maximal convex sets up to signs, the resulting local sign ambiguity prevents a direct application of the usual Banach–Stone type arguments for isometries. The main point of the paper is to prove that these local sign structures can be globalized on the Choquet boundary. To this end, we refine an additive Bishop-type construction and use it to propagate the sign information among the maximal convex sets associated with boundary points. As a consequence, every surjective phase-isometry admits a boundary representation by means of a global sign function, a unimodular weight, a homeomorphism between the Choquet boundaries, and a clopen decomposition into complex-linear and conjugate-linear parts. We then extend this representation to the maximal ideal spaces and obtain the corresponding real-algebraic Banach–Stone type representation.

This paper proposes modulo quantization (MQ) coding as a simple, structured, and low-complexity scheme for channels with primitive (i.e., noiseless digital) relay links and correlated Gaussian noises across terminals. The key component of MQ coding is the modulo quantization operation, which maps a real-valued symbol to its uniform-quantization index taken modulo a fixed integer. This operation allows effective exploitation of the common noise component shared across the terminals. For the Gaussian primitive relay channel with perfectly correlated noises, where a relay has a finite-capacity link to the receiver, MQ coding can be shown to achieve the capacity of this channel. For the Gaussian primitive diamond channel with perfectly correlated noises, where two relays can forward information through finite-capacity links to a receiver that has no direct observation of the transmitted signal, MQ coding yields novel achievability bounds that improve upon previously known bounds and coincide with the cut-set upper bound in certain signal-to-noise ratio (SNR) regimes. In scenarios with highly but non-perfectly correlated noises, MQ coding can approach the performance of compress-forward (CF) at significantly lower complexity, while surpassing decode-forward (DF) for the Gaussian primitive relay channel in certain SNR ranges. For the Gaussian primitive diamond channel with non-perfectly correlated noises, MQ can outperform both CF and DF at moderate SNR.

The rate regions of multi-terminal Gaussian channels (multiple-access, broadcast, interference, relay) are delimited by conditional mutual informations $I(V_A;V_B\,|\,V_C)$ among groups of input and output nodes; bringing such channels under differentiable physical-layer design therefore hinges on evaluating any such conditional MI, and its gradient, on a unified computation graph. Modeling the network as a linear Gaussian directed acyclic graph (Gaussian-DAG), we obtain $I(V_A;V_B\,|\,V_C)$ in closed form: from the node-pair covariances produced by one K-recursion forward pass, it is a log-determinant difference of two sub-block Schur complements of the support covariance. The construction is built entirely from automatic-differentiation (AD) primitives, so any differentiable function of finitely many conditional MIs is end-to-end differentiable in the design parameters; this broad class includes linear objectives (weighted sum-rate, secrecy), the rate functions of standard multi-terminal rate regions, and non-linear composites of these. A single reverse-mode AD sweep yields the Wirtinger gradient with respect to all controllable factors at once, so any such objective can be handled by projected gradient iterations without problem-specific gradient derivation. We demonstrate the framework on three experiments: rate-region maximization for a two-user MIMO multiple-access channel, secure precoding on a MIMO wiretap channel, and the same rate-region objective applied to a larger multi-hop multiple-access network.

This work investigates linear elliptic equations with multiple nonlocal nonlinearities in a bounded domain, which takes the form \begin{equation*} -\nabla\cdot(\od\nabla u) +\lambda fu= \eta \sum_{j=1}^kh_j\boldsymbol{\mathsf{N}}_j[u]+h_0. \end{equation*} Here $\lambda$ and $\eta$ are positive parameters, and $\boldsymbol{\mathsf{N}}_j[u]$ represents a nonlocal term dependent on the unknown solution~$u$. All coefficients are defined in the context of a wide range of applications. As such equations generally lack a variational structure, a new approach is developed that combines fixed-point arguments with asymptotic techniques. This method establishes the existence and multiplicity of solutions under specific conditions. Of particular interest is the role of the nonlocal nonlinearities, which lead to diverse structures of solutions. To the best of our knowledge, this property is a new finding not typically observed in related nonlocal elliptic problems.

Landau solutions, when oriented along the vertical axis, represent a one parameter family of exact, self-similar, axisymmetric, swirl-free solutions to the 3D stationary Navier-Stokes equations forced by an upward facing point-source of momentum at the origin. They have an isolated singularity at the origin. Other singular steady state solutions can be derived from a similar framework. Some of these variants have been proposed as models for physical scenarios. For example, a solution first found by Squire has been proposed as a model of a fluid entrained to a radially discharging surface layer of oil. Another, Serrin's swirling vortex, exhibits qualitative features shared with some tornadoes, like a two-cell structure consisting of a central downdraft and peripheral updraft as well as swirl. The first objective of this paper is to provide a detailed analysis of these and other examples, especially when boundaries are present. In this direction we find several new, physically motivated classes of solutions and identify new connections between the physics literature and the mathematics literature. Many of these examples are formulated on the half-space, but much of the mathematical literature on singular steady-state solutions is for $\mathbb{R}^3$. The second objective of this paper is to establish asymptotic stability for many of these solutions by formulating the problem in domains with boundaries. Our most general result in this direction uses a new approach to asymptotic stability that is based on eventual regularity. In a different direction, we prove that a class of singular steady-state solutions motivated by Serrin's swirling vortex are stable under certain axisymmetric perturbations. This requires a novel observation because these “Type III” solutions are too singular to be directly amenable to existing approaches.

A unified framework is proposed to establish the energy dissipation of implicit-explicit linear multistep methods (IMEX-LMMs) for gradient flows, based on general multipliers that are linear combinations of first-order differences of numerical solutions. A generalized Dahlquist's theory is developed to establish the energy dissipation of IMEX-LMMs. It is shown that given an IMEX-LMM, to find a multiplier ensuring the energy dissipation is relaxed to solve a linear programming that can be easily solved. Within this framework, two specific multipliers are discovered to establish the energy dissipation of the sixth-order IMEX backward differentiation formula (IMEX-BDF6) method and a seventh-order IMEX weighted and shifted BDF method, and a new eighth-order energy-dissipative IMEX-LMM is provided. To the best of our knowledge, these are the first energy-dissipation results for the IMEX-BDF6 method and the IMEX-LMMs of order higher than six. In addition, this framework can be used directly to establish the $L^2$- or $H^1$-stability of general LMMs for linear parabolic problems. Numerical experiments illustrate the temporal accuracy and energy dissipation of these methods.