Papers

Stochastic gradient descent (SGD) optimization schemes are the methods of choice for the optimization of deep neural networks (DNNs) in artificial intelligence (AI) systems. Often not the standard SGD method is used but instead suitable accelerated, adaptive, and/or normalized variants of standard SGD such as Adam, AdamW, and MUON are employed to train large scale AI systems in practically relevant settings. The acceleration (higher order convergence speed) in all these popular optimizers relies on the momentum SGD optimizer. In this work we provide a rigorous error analysis for the momentum SGD optimizer. In particular, we establish convergence rates for the momentum optimizer in terms of the size of the learning rate (step size), the size of the mini-batch, and the size of the one-point convexity constant.

Let $S_k$ be the space of cusp forms of weight $k$ for the full modular group ${\rm SL}_2(\mathbb{Z})$. In this paper, we will exhibit an explicit rational spanning subset of $S_k$ constructed from Eisenstein series at prime levels. The main ingredient of our proof is to show a result on determining cusp forms by noncentral quadratic-twist $L$-values.

Spectral submanifold (SSM) reduction has emerged as a mathematically principled route to reliable nonlinear reduced-order models, capturing dynamics beyond the reach of linear techniques such as Dynamic Mode Decomposition (DMD). The computation of SSMs, however, remains computationally expensive, particularly for high-dimensional systems. In this work, we introduce equivariant spectral submanifold (eSSM) reduction, a novel extension of the SSM framework that explicitly incorporates symmetries of the full-order model into the reduction process. We establish the mathematical foundations of this approach by showing that SSMs are naturally equivariant submanifolds and that the associated charts and reduced dynamics inherit the appropriate induced group actions. Building on this framework, we develop a novel equivariant SSM reduction algorithm that exploits these symmetries to achieve substantially faster computations while also improving model robustness. We demonstrate the advantages of this approach on several benchmark problems including a test from the Common Task Framework for Science.

We show that the dual approach to Bourgain–Brezis estimates for Hodge systems is substantially more flexible than previously understood. For $1\leq l\leq n-1$, we introduce the trace-free Beurling–Ahlfors transform $S=\frac{n-l}{n}P-\frac lnP^\perp$, a canonical normalization of the generalized Beurling–Ahlfors transform on $l$-forms in $\mathbb{R}^n$. Its matrix symbol decomposes into scalar multipliers that are odd under suitable orthogonal reflections, yielding an endpoint cancellation estimate from finite measures to $L^\infty$ for $|D|^{-n}S$. This cancellation allows us to complete the Hilbertian case of the Bourgain–Brezis conjecture in every dimension and for every form degree. We then develop multilinear reflection estimates and obtain new critical Triebel–Lizorkin and Besov Bourgain–Brezis estimates. In particular, for every dimension and form degree, the Sobolev Bourgain–Brezis conjecture in $\dot W^{\frac np,p}$ holds for $p=\frac{2k}{2k-1}$, $k\geq1$, and hence for exponents arbitrarily close to $1$. We also derive endpoint Hodge decompositions and Hodge–Sobolev inequalities. Finally, except in the endpoint Besov case where the critical space already embeds into $L^\infty$, we prove that the associated bounded selections cannot be linear.

We study the problem of aligning data from multiple modalities into a shared representation space, focusing on settings where strong pretrained unimodal encoders are available but cross-modal paired data are scarce. We propose a structure-preserving alignment framework, joint kernel entropic Gromov–Wasserstein Optimal Transport (JK-EGW), which maps multiple modalities into a common latent space by minimizing a quadratic optimal transport objective. JK-EGW leverages fine-grained similarity relationships within and across modalities to construct a global affinity kernel instead of relying on raw feature-space distances. Our framework naturally provides explicit control over the geometry and distribution of the latent embedding. On the theory side, we establish parametric sample complexity rate of $n^{-1/2}$, matching the corresponding rates for standard, entropic and Gromov–Wasserstein optimal transport. On the algorithmic side, we derive a scalable alternating procedure to solve JK-EGW with entropic optimal transport (EOT) updates through a low-rank kernel approximation and a variational lifting. This lifting scheme effectively relieves the burden of a quadratic objective, and allowing us to take the advantage of existing EOT solvers. Empirically, we focus on post-hoc alignment of embeddings from pretrained encoders in data-scarce regimes, and show that our proposed method achieves improved multimodal retrieval performance compared to existing alignment baselines.

For an arbitrary Artin algebra $A$, we construct a minimal and consistent scattering diagram by approximating its module category $\mathrm{mod}\,A$ using the subcategories $(\mathrm{mod}\,A)_\ell$ of modules of length at most $\ell \in \mathbb{N}$. We prove that each subcategory $(\operatorname{mod}A)_\ell$ possesses a well-behaved lattice of torsion classes, a finite wall-and-chamber structure $\mathfrak{D}_\ell(A)$ and an associated picture group $G_\ell(A)$ with a natural categorical interpretation. Using these properties, we build a finite, minimal, consistent scattering diagram for each $\ell \in \mathbb{N}$. Passing to the inverse limit, we establish the existence of a minimal consistent scattering diagram for $A$. In particular, when $A$ is a finite-dimensional algebra over $\mathbb{C}$, our inverse limit construction is canonically isomorphic to Bridgeland's stability scattering diagram.

A computational path records the steps used to establish an equality. Topologies on computational equality witnesses can depend on the chosen generators and execution schedule, even when their algebraic and homotopical meanings agree. We study continuous presentations of finite computational paths whose declared rewrites are realized by endpoint-fixed homotopies. We compare a topology retaining the full signed trace with an observable topology recording endpoints, length, and exact path realization. Continuous finite-word interpretations that are inverse up to scoped rewriting preserve trace-sensitive quotient arrows and their final composable-pair spaces. Observable quotients behave differently. Over a point with a nonempty primitive alphabet, we classify their topology: it records length parity when all named relations preserve parity, and is otherwise indiscrete. Adjoining a conservative abbreviation can therefore turn a discrete two-element quotient into an indiscrete space, even though ordinary multiplication remains continuous. A circle presentation also gives nonhomeomorphic quotients under binary and equal-slot timing. Geometric completeness alone does not identify the trace-sensitive and observable topologies. Multiplication is continuous on the quotient of composable representatives. We characterize when this topology agrees with the ordinary pullback topology and give an open-projection criterion. Universal presentations recover classical fundamental-groupoid topology. Normal forms, primitive interchange, and a Cayley covering prove completeness for circle, torus, product, and rose presentations. A literal-word correspondence relates the semantics to the reflexivity, symmetry, and transitivity fragment of equality derivations. Selected results are formalized in Lean, with the realization models and proof coverage stated explicitly.

Evacuation planning for disaster preparedness requires making critical decisions under uncertainty before the number and spatial distribution of evacuees are known, including shelter location, evacuation route assignment, and relief supply prepositioning. Because these decisions are highly interdependent, planners must balance the competing objectives of maximizing relief demand coverage and minimizing evacuation time. We propose, to our knowledge, the first adaptive robust evacuation planning model to jointly optimize shelter locations, evacuation route assignments, relief supply prepositioning, and post-disaster relief item distribution. The model minimizes the worst-case weighted sum of unmet demand for relief items across shelters and the congestion-dependent evacuation time. We characterize theoretical complexity drivers of the resulting problem with mixed-integer recourse and develop a partition-and-bound algorithm that maintains tractability by selectively partitioning only the most critical subpartition of the uncertainty set while producing strong upper and lower bounds. To quantify the value of centralized route planning, we also formulate a user route choice alternative in which evacuees choose among acceptable routes. Computational experiments quantify the value of centralized route planning, which reduces worst-case unmet demand and evacuation time by up to 90.6% and 79.3%, respectively, relative to decentralized user route choice. Adaptive post-disaster supply redistribution further improves relief demand coverage. Coordination between evacuation routing and relief distribution creates substantial operational value under uncertainty. Centralized route planning primarily mitigates congestion by coordinating evacuee flows across shelters, whereas adaptive redistribution primarily improves relief demand coverage when relief supplies are scarce or inflexibly prepositioned.

The demand matching problem generalizes both the knapsack problem and the $b$-matching problem. In this problem, each edge of a graph has a demand and a weight, each vertex has a capacity, and the goal is to find a maximum weight subset of edges whose total incident demand at every vertex does not exceed its capacity. We study $(\alpha, \beta)$-bicriteria approximation algorithms, which return a solution of weight at least $1/\alpha$ times the optimum while allowing an additive capacity violation of at most $\beta$ times the maximum edge demand. We give an iterative relaxation algorithm for the demand matching problem that exploits a structural characterization of strictly fractional extreme points of the natural LP relaxation, which reduces the residual rounding problem to odd-cycle instances. Combined with a better-of-two rounding strategy, this yields $(7/6, 1)$- and $(1, 1)$-bicriteria approximation algorithms for general and bipartite graphs, respectively. We further generalize this approach to obtain a parametric family of algorithms, including a $(1, 4/3)$-bicriteria approximation. Separately, for the more general $k$-hypergraph demand matching problem, we give a greedy, combinatorial $(k, 1)$-bicriteria approximation algorithm. We complement these algorithmic results with matching lower bounds relative to the natural LP relaxation for $\beta = 0$ and all $\beta \geq 1$, completely characterizing the trade-off between weight approximation and additive capacity violation in this range.

We classify smooth complex affine surfaces admitting a finite surjective morphism from $\mathbf{C}^*\times\mathbf{C}^*$. Using the classification theory of smooth affine surfaces according to logarithmic Kodaira dimension, we show that every such surface has logarithmic Kodaira dimension either $-\infty$ or $0$, and determine all possibilities. More precisely, the only surfaces of logarithmic Kodaira dimension $-\infty$ are $\mathbf{C}^2$ and $\mathbf{C}\times\mathbf{C}^*$, while those of logarithmic Kodaira dimension $0$ are precisely $\mathbf{C}^*\times\mathbf{C}^*$ and Fujita's surface $H[-1,0,-1]$. This establishes the classification of smooth affine surfaces properly dominated by $\mathbf{C}^*\times\mathbf{C}^*$ anticipated by M.~Furushima in 1989.

Analytical solutions to Maxwell's equations are essential for understanding the causal and instantaneous behavior of electromagnetic fields, yet they are challenging to obtain in open-space settings with general initial values and source terms. This work uses an infinite-order estimation scheme that yields closed-form analytical solutions to Maxwell's equations. The scheme is expressed as general function-to-function transformations that directly map initial values and source terms to the fields. Several case studies demonstrate the method, producing exact solutions for different initial and source configurations. The results provide both theoretical insight and practical benchmarks for computational electromagnetics.

We construct three tamely ramified local equivalences of factorization module categories. The first is a factorization version of the Arkhipov-Bezrukavnikov equivalence at a point. The second is a factorization version of the Bezrukavnikov equivalence at a point. The third is an Iwahori-ramified version of the factorizable Fundamental Local Equivalence.

Nonlinear state estimation requires sequentially fusing model-based predictions with noisy measurements. Under imperfect dynamics and unknown, time-varying noise statistics, this fusion can degrade in both accuracy and statistical consistency. Existing learning-aided filters largely treat accuracy and uncertainty estimation separately, limiting their ability to correct model-mismatch-induced bias while retaining an explicit, calibrated posterior covariance. This paper introduces Unscented KalmanNet (UKN), a model-based deep learning architecture that extends the Unscented Kalman Filter (UKF) with learned mechanisms for these two sources of filtering error while preserving explicit posterior covariance propagation. NoiseNet learns time-varying process and measurement covariances as bounded multiplicative corrections to baseline covariances, guaranteeing positive definiteness, while GainNet learns a bounded residual correction to the analytical UKF gain to compensate for model-mismatch-induced bias. A calibration-aware training objective couples state error with posterior covariance and innovation consistency terms through adaptive weighting, jointly optimizing accuracy and calibration. UKN is benchmarked against UKF, KalmanNet, and Bayesian KalmanNet on three synthetic systems and real-flight UZH-FPV data. It achieves the lowest state-estimation error in all four examples and reduces RMSE by 26.4-49.7% compared with UKF in the synthetic cases. Leave-one-sequence-out cross-validation over 11 flights shows 22.4% and 34.3% reductions in mean position and velocity RMSE, respectively. UKN also yields the lowest fold-to-fold variability, with dimension-normalized NEES and empirical coverage closest to nominal values among covariance-reporting filters. These results show that structured learned adaptation improves estimation accuracy while retaining calibrated uncertainty.

We study a physics-informed neural network (PINN) for the unsteady, two-dimensional incompressible Navier–Stokes equations in which the stiff divergence-free constraint is replaced by an artificial-compressibility (AC) relaxation governed by a single scalar parameter $\eps$. The relaxation reintroduces a pressure time derivative, converting a differential-algebraic constraint into an ordinary residual that a PINN can minimise directly. On the Taylor–Green vortex, which admits a closed-form unsteady solution, we quantify the effect of $\eps$: the residual divergence scales as $\eps\,|\partial_t p|$, so larger $\eps$ raises both the divergence and the velocity error, and both decrease monotonically and saturate as $\eps$ is reduced. On the $Re=100$ cylinder wake the plain forward AC-PINN collapses to the steady symmetric branch and does not reproduce von Kármán shedding; assimilating a few hundred sparse velocity sensors from a boundary-layer-resolved finite-element reference (whose Strouhal number, $0.176$, we bring close to the $0.164$–$0.172$ literature band by resolving the separating shear layer, though it remains just above it) recovers the unsteady vortex street to $7\%$ over the wake and its shedding frequency to within $3\%$ of that same reference — a bound set by the reference's own fidelity rather than an independent validation against the true flow.

The double extension method allows the construction of quadratic Lie algebras from smaller ones. Using this approach, in this work we describe all solvable indecomposable quadratic Lie algebras of dimension at most $7$ over arbitrary fields of characteristic not $2$. In particular, we get complete classifications up to isometric isomorphism over algebraically and real closed fields.

We develop a nonlinear continuum transport model for localized traveling structures in a coupled two-phase medium. The model is derived from an Einstein-type material-balance formulation combining diffusion with directed transport induced by the relative gradient of an evolving background phase. The resulting system couples diffusion, nonlinear gradient-driven transport, and irreversible background depletion. The framework provides a mathematical model for chemotactic transport and a possible modeling perspective for processes associated with the formation of oil and gas deposits. We analyze traveling-wave solutions and identify beta = chi/D = 1 as the critical threshold for globally positive localized traveling bands under prescribed asymptotic conditions as the traveling coordinate s tends to plus or minus infinity. Such bands exist for beta greater than or equal to 1, whereas for 0 < beta < 1 the positive branch reaches zero background concentration at a finite traveling coordinate. For beta greater than or equal to 1, explicit profiles are obtained and, for fixed wave speed c > 0 and right-end state V_plus > 0, are unique up to spatial translation in the traveling coordinate s. We further derive the variable-coefficient linearized perturbation operator and establish a conditional finite-time a priori estimate for the mobile-phase perturbation under bounded background forcing using a maximum-principle argument.

Control theory and dynamical systems are closely intertwined fields. Pontryagin's maximum principal offers a strong connection by providing a constructive way to synthesize feedback control laws via lifting the control system to a (Hamiltonian) dynamical system. Feedback laws can then be encoded as invariant manifolds of this induced dynamical system - under the condition that these manifolds project diffeomorphically back to the base control system. While feedback stabilization is impossible for many control systems, the above procedure can still be carried out. In this setting, the invariant manifold no longer projects diffeomorphically which results in the emergence of caustics. This paper offers an overview of the above connection by translating target sets from controls to isotropic submanifolds in symplectic geometry and associates feedback controllability to singularities of the induced invariant manifolds. Multiple low-dimensional examples are included to elucidate the theory.

For a natural number $n$ let $[n] = \{1,\ldots,n\}$. We say that a family ${\cal{S}}\subseteq 2^{[n]}$ is representing if every singleton set of $[n]$ is an intersection of some sets from ${\cal{S}}$. We show that the smallest possible cardinality of a representing set for $[n]$ is the discrete inverse $s(n)$ of the Sperner's function $n\mapsto \binom{n}{\lfloor n/2\rfloor}$, which by Sperner's Theorem is the maximum number of elements in an antichain in $2^{[n]}$ when viewed as subset (or boolean) lattice. Specifically, $s(n)$ is then the smallest positive integer such that $2^{[n]}$ contains an $n$-element antichain. Some generalization, further applications and asymptotics in terms of the second real branch of the Lambert $W$ function are presented.

The set of $n\times n$ correlation matrices, known as the elliptope, has volume decaying at the super-exponential rate $\exp\{-\tfrac14 n^2\log n\}$. We characterize where this vanishing volume concentrates. A uniform draw is entrywise close to the identity yet globally far from it and nearly singular. Its maximum absolute correlation is of order $\sqrt{\log n/n}$, its Frobenius distance is asymptotic to $\sqrt n$, its empirical spectral distribution converges to the Marchenko-Pastur law with ratio one, and its smallest eigenvalue has the exact $\operatorname{Beta}(1,d)$ distribution, where $d=n(n-1)/2$, and is therefore of order $n^{-2}$. Distinct off-diagonal entries are exactly pairwise independent under every $\operatorname{LKJ}(\eta)$ law, which yields a Chen-Stein proof of the extreme-correlation point-process limit for the whole LKJ family and an $O(n^{-1})$ total-variation bound for finite-dimensional exceedance counts relative to Poisson laws with their exact finite-$n$ means. Finally, for a bounded, centered i.i.d. off-diagonal perturbation of any correlation matrix, the nearest-correlation projection removes a fraction of the squared perturbation that tends to one and recovers the original matrix in average squared error per entry.

We study subhomogeneity of the minimal operator system over a polyhedral cone, or equivalently the size of irreducible Arveson boundary points of free polyhedra. We obtain a complete classification in dimension three for cones (equivalently, dimension two for polytopes): cones with three extreme rays are 1-subhomogeneous, cones with four extreme rays are 2-subhomogeneous, and cones with at least five extreme rays are not subhomogeneous. In the last case, we construct irreducible Arveson boundary points at every even matrix level. We prove that subhomogeneity passes to faces and face quotients. We also determine the size of irreducible Arveson boundary points of free polyhedra over products of two simplices, obtaining a dichotomy between the product of two segments and all remaining cases. As applications, we classify n-dimensional cones with n+1 extreme rays. More generally, when every facet omits at most two extreme rays, we show that subhomogeneity occurs exactly for direct sums of simplicial cones and three-dimensional four-ray cones. We also construct, for a range of fixed dimensions and ray counts, cones exhibiting both subhomogeneous and non-subhomogeneous behavior.

In 2010, Levine found that the activity functions of parallel chip-firing games on the complete graph $K_n$ converge to a devil's staircase pattern: adding chips causes activity to progress through open intervals in which it is locally constant. In 2022, Bu, Choi, and Xu improved on an earlier bound by Kominers and Kominers to show that there exists a strict lower bound below which all games have activity $0$, and a strict upper bound above which all games have activity $1$. They thereby generalized the bottom and topmost rungs of the devil's staircase to all graphs. In 2024, Ji, Li, and Wang conjectured that a similarly general bound exists for games with activity $\tfrac12$. We use GPT-5.6-Sol to prove this conjecture, unifying existing results for trees, cycles, complete graphs, and complete bipartite graphs. This generalizes the middle rung of the devil's staircase.

Kemeny's constant is used as a measure of the average travel time on a graph. Braess' paradox for graphs is the observation that in some graphs, when an edge is added, Kemeny's constant increases. We introduce the notion of a Braess clique $K_\ell$, a clique that when inserted into a graph on an independent set of $\ell$ vertices, will create an increase in Kemeny's constant. In this context, a Braess edge is a Braess $K_2$. We provide examples of graphs that have a Braess $K_\ell$ for $\ell\geq 3$. We observe that almost every connected planar labelled graph has a Braess $K_\ell$ for each $\ell\geq 3$. We also explore the relationship between Braess edges and Braess cliques in graphs.

The direct sum problem in computational complexity asks whether solving $n$ independent instances of a computational task inherently requires $n$ times the resources needed to solve a single instance. In this paper, we resolve a central form of the direct sum conjecture in randomized communication complexity. Specifically, we prove that the "amortized expected randomized communication complexity" of any function is exactly equal to its "zero-error information complexity"—a measure of the precise amount of information the communicating parties must reveal about their inputs to compute the function without error. This result also provides a tight characterization of the amortized "worst-case" randomized communication complexity up to a constant factor. To achieve our exact characterization, we introduce a new single-instance protocol embedding equipped with a prefix-verification mechanism to accurately localize global errors. Furthermore, we apply our new structural theorems to the fundamental Set-Disjointness problem. Our resulting exact asymptotic bounds for Set-Disjointness successfully refute a conjecture in DFHL18 regarding its scaling behavior.

Let a smooth, unforced, three-dimensional Navier–Stokes flow on the flat torus approach a finite terminal time. Its kinetic-energy densities converge, along the full time variable, to a unique endpoint measure. We prove that each point atom forces a same-parent, full-tail dynamical rigidity. A preassigned level-crossing catalogue and nested local Hodge projections produce orthogonal packets. From the entire packet tail, we extract a single backward adjoint whose terminal energy concentrates at the atom. Cauchy saturation then locks this adjoint to every sufficiently late packet and yields uniform two-parameter saturation of both the constrained Oseen propagator and its adjoint, together with vanishing first-order dissipation. Consequently, every sufficiently late fixed-root descendant has infinite delayed second-order action and non-integrable positive enstrophy production. Equivalently, a delayed second-order operator budget determined solely by the Navier–Stokes parent must fail arbitrarily close to the endpoint.

Given probability distributions $\mu$ and $\nu$ on measure spaces $X$ and $Y$, and a closed set $S \subseteq X \times Y$, when is there a probability distribution on $X \times Y$ whose marginals are $\mu$ and $\nu$, and whose support is precisely $S$? We answer the question when the marginals are discrete, and when the marginals are continuous distributions on the real line. Of special interest is the case where $S \subseteq [0,1]^2$ and $\mu$ and $\nu$ are Lebesgue measure; then the above question is tantamount to “when is $S$ the support of a doubly stochastic measure?". The discrete case is generalized to determine when a (possibly infinite) edge-capacitated, node-weighted graph supports a full, nowhere-zero flow; for the continuous case we provide a particularly straightforward characterization when the set in question is regular (i.e., is the closure of its interior).

Practical Reconfigurable Intelligent Surfaces (RISs) often support only a small number of phase states, making their configuration inherently discrete. For a 1-bit RIS with $N$ elements, direct optimization requires searching among $2^N$ binary configurations. This work shows that this exponential configuration space is not unstructured. By reformulating 1-bit RIS optimization as the maximization of the norm of a signed sum of channel-dependent vectors, we prove that every globally optimal configuration must be induced by the signs of their projections onto a common direction. This geometric characterization restricts the class of configurations that can contain global optima and leads to different algorithmic consequences depending on the signal-space dimension. For general Multiple-Input-Multiple-Output (MIMO) systems, we develop a geometry-informed sampling method that evaluates only structurally admissible configurations. For Single-Input-Single-Output (SISO) systems, the same principle reduces to a two-dimensional angular partition, allowing the complete candidate set to be characterized and the global optimum to be recovered through polynomial-time enumeration, by evaluating at most $N+1$ out of the $2^N$ configurations. Finally, we apply the same binary optimization principle to an Integrated Sensing and Communication (ISAC) scenario, where communication enhancement and target localization reduce to the same underlying geometric problem. The proposed framework therefore provides a unified approach for exploiting the structure of practical 1-bit RIS configurations across communication and sensing functionalities.

Homeostasis is a biological phenomenon in which a function of the state of the system remains approximately constant as an input parameter varies. Recent mathematical work has developed a framework for studying homeostasis using methods from singularity theory. In this framework, the requirement that the function of the state remain approximately constant is replaced by the condition that the derivative of the function with respect to the input parameter vanishes at an isolated point. This zero-derivative condition is called infinitesimal homeostasis. Much of the existing theory concerns steady states. In this paper, we develop an analogous theory for oscillatory systems in which the homeostatic quantity is the period of a stable periodic solution. We refer to this behavior as period homeostasis and to the corresponding zero-derivative condition as infinitesimal period homeostasis. Experimental studies in cyanobacteria and cultured human cells suggest that circadian rhythms can disappear through Hopf bifurcation as temperature decreases to a critical value, with the oscillation amplitude tending to zero. Motivated by this evidence, we focus on stable periodic solutions arising from nondegenerate Hopf bifurcation. We show that bifurcation theory and singularity theory can be used to find period homeostasis in parameterized ODE models. Our results connect the onset of oscillations with homeostasis of the period and yield a geometric description of parameter space that locally organizes the qualitative behavior of solutions of the ODE.

As a popular optimization scheme, distributionally robust optimization (DRO) protects decisions against ambiguity in probability distributions. For (single-stage) DRO, prevailing dual reformulations can become difficult when model or ambiguity-set structures are complex. We study DRO from a primal perspective, working directly with distributions in ambiguity sets on closed, potentially unbounded sample spaces. This perspective leads to an algorithmic framework, referred to as BiCS, that constructs and leverages distribution cuts to achieve strong performance. We show that BiCS is applicable to standard DRO, almost-sure DRO, DRO with various chance constraints, and DRO with ambiguity sets strengthened by local information. Numerical experiments with moment and Wasserstein ambiguity sets show that this framework demonstrates superior performance, including solving cases where the examined compact reformulations are unavailable or computationally difficult. The local-information study also makes changes in worst-case distributions directly visible.

A strong majority edge colouring of a graph $G$ is an edge colouring in which, for every edge $e$ and every colour $\alpha$, at most half the edges adjacent to $e$ receive colour $\alpha$. Like many related colouring notions, it admits a natural interpretation as a colouring problem for an associated hypergraph. Somewhat surprisingly, although the corresponding hypergraph may have arbitrarily large vertex degrees, a universal finite upper bound on the sufficient number of colours in a strong majority edge colouring exists under a natural modest minimum degree assumption, unlike in several other closely related majority concepts. We in particular prove that every graph $G$ with minimum degree $\delta\ge5$ admits a strong majority edge colouring with three colours, improving both the previously known bound $\delta\ge9$ for three colours and the result showing that four colours suffice whenever $\delta\ge5$. Our result is best possible with respect to the number of colours and the minimum degree assumption. We also introduce a more general framework of strong $1/k$-majority edge colourings and establish corresponding bounds for this setting.

Phase reduction provides a principled relationship between high-dimensional oscillator dynamics and low-dimensional phase models such as the Kuramoto model. However, connecting analytical results derived from the Kuramoto framework to experimental data remains an open challenge, as many physical platforms violate the weak-coupling assumptions underlying standard phase reduction. Here, we present an experimental platform based on electronic quadrature oscillators whose phase dynamics are, under certain assumptions, equivalent to the Kuramoto model. Crucially, this equivalence is achieved for both weak and strong coupling through a non-diffusive coupling circuit, and the design approach supports arbitrary (weighted and directed) network topologies while preserving scalability and oscillation regularity. Across different coupling schemes, we demonstrate that the platform reliably reproduces phase transitions predicted by the Kuramoto model for both global and cluster synchronization. This cost-effective platform enables systematic experimental validation of analytical and numerical results for synchronization in complex networks.