Papers

The problems of designing sparse networks arise frequently in resource allocation and operations research. In production systems, for example, sparse process flexibility designs are used to handle uncertain demand effectively: the goal is to construct the sparsest bipartite graph between supply and demand that still achieves an expected fulfilled demand comparable to that of a fully flexible system. In middle-mile transportation, sparse delivery-route subgraphs that sustain large matchings after random node deletions help reduce delivery costs; here, the goal is to design the sparsest graph whose maximum matching size remains comparable to that of the fully connected graph under node deletions. The design of sparse networks has been studied extensively, with state-of-the-art results providing order-wise optimal designs for both bipartite and unipartite networks (Chen et al., 2015; Feng et al., 2024). However, identifying designs that achieve the sharp theoretical limit – where the average degree asymptotically matches the lower bound of any graph to achieve a given loss level, has remained open. In this paper, we prove that the random regular graph achieves this sharp optimal condition in both bipartite and unipartite settings. Numerical experiments further validate this optimality. Our results highlight a practical guideline for sparse flexibility networks: designs that combine degree regularity with dispersed edge placement can achieve optimal performance under uncertainty.

This report provides an introduction to Sobolev spaces, a foundational concept in modern analysis and the theory of partial differential equations (PDEs). These spaces are useful to study, among other things, the well-posedness of partial differential equations and their approximation using finite elements. We begin with a historical overview, tracing the development of weak derivatives and the shift from classical to variational formulations of PDEs. After establishing the basic definitions and presenting key examples, we survey central theorems such as the Sobolev Embedding Theorem and Rellich's Theorem, emphasizing their significance in ensuring existence, uniqueness, and regularity of solutions. Finally, we discuss a classic application to PDEs, the Elliptic Regularity Theorem. We aim to provide a self-contained and accessible introduction for students with a background in real analysis and the theory of PDEs.

A finite element discretization of an optimal insulation problem with convective heat transfer is considered. The model is formulated as a non-smooth, two-variable convex minimization problem. It accounts for the temperature distribution in a thermally conducting body $\Omega\subseteq\mathbb{R}^d$, with $d\in \{2,3\}$, and the distribution of a given amount of insulation material on an insulated boundary part $\Gamma_I\subseteq \partial\Omega$. The surface integral over the insulated boundary $\Gamma_I$ is approximated by a mass-lumping quadrature that preserves the structure of the continuous setting and, in particular, yields discrete optimality conditions mirroring their continuous counterparts. Well-posedness, stability, and weak convergence of discrete solutions to the continuous ones are established. Furthermore, a block coordinate descent algorithm for the computation of the discrete solutions is formulated and its linear convergence is derived. Under suitable regularity assumptions, uniform $L^\infty(\Gamma_I)$-bounds and $\textit{a priori}$ error estimates for both the temperature distribution and the distribution of a given amount of insulation material are obtained. Numerical experiments are carried out that confirm the predicted error decay rates and demonstrate the method in a qualitative three-dimensional test on a realistic spacecraft crew module capsule geometry with idealized reentry-heating Robin data.

Controlling batch polymerization is challenging because the absence of a steady operating point prevents standard linearization; the dynamics are intrinsically nonlinear; and multi-phase operation induces state-triggered switching. This study systematically combines four established real-time NMPC ingredients, smooth mode blending, advanced-step warm starts, variable scaling, and a capped iteration budget, to attain real-time feasibility without ad hoc switching heuristics. We provide practice-oriented guidance for selecting smoothing gains and locating switching surfaces, and we make explicit the approximations introduced by smoothing such that, with appropriate tuning, the smoothed and original switching logic are numerically indistinguishable at solver-tolerance levels. All results are obtained in closed-loop simulation using an industrial gas-liquid polymerization benchmark with estimator-in-the-loop, compared against PID and conventional NMPC baselines. Results show improved constraint satisfaction and shorter batch duration under bounded computation, while an ablation study quantifies the specific contributions of each component individually

It is a classical theorem of Robertson and Seymour (1986) that the treewidth of a graph is linearly related to its separation number: the smallest integer $k$ such that, for every weight function on the vertices, the graph admits a balanced separator of size at most $k$. Motivated by recent progress on coarse treewidth, Abrishami, Czyżewska, Kluk, Pilipczuk, Pilipczuk, and Rzażewski (2025) conjectured the following coarse analogue: for every $r\in \mathbb{N}$ there exists an $r'\in \mathbb{N}$ such that every graph that admits balanced separators that can be covered by a bounded number of balls of bounded radius $r$ admits a tree decomposition where every bag can be covered by a bounded number of balls of radius $r'$. We verify a stronger variant of this conjecture for all $r \in \mathbb{N}$ for the hereditary class of $K_{t,t}$-induced-minor-free graphs of bounded clique number. A key step in the proof is the following result, which we expect to be of independent interest. In $K_{t,t}$-induced-minor-free graphs with clique number bounded by $s$, given a large subset of vertices $Y \subseteq V(G)$, there is a set $Z$ whose size is bounded by a function polynomial in $s$, such that no ball of radius $r$ in $G- Z$ covers a large proportion of $Y$.

Let X be a smooth, projective, geometrically integral variety over a global function field k of characteristic p. We show that if the unipotent Brauer group of X is zero, and the Picard scheme of X is torsion free, then only finitely many places of k can be potentially relevant to the Brauer–Manin obstruction for X. In the opposite case, almost all places of k are potentially relevant. If X is base changed from a finite field, then in both cases the finite set of exceptional places is empty. We give an example of a supersingular K3 surface over a global function field k of characteristic p>2 such that Br(X) is finite modulo Br(k), answering a question from our previous paper.

We study controller and control-architecture co-design for output-feedback systems under a hard budget. The architecture activates sensors and actuators and selects among directed communication-service options with specified delivery bounds and costs. Direct optimization over controller transfer matrices and discrete deployments is mixed-integer nonconvex; convex alternatives fix the architecture, use regularization, or impose a quadratically invariant (QI) controller-information pattern. We instead optimize finite impulse response (FIR) output-feedback system-level synthesis (OF-SLS) responses. Binary variables select devices and service options; cumulative binaries record whether the selected service can deliver by each FIR lag. Indicator constraints zero response coefficients that would require unavailable messages. For fixed device and OF-SLS realization-state locations, this yields an exact mixed-integer convex program (MICP) over finite service menus and deployment constraints. Every feasible response admits the standard OF-SLS implementation using only the selected devices and services. In a three-follower platoon, 2736 of 139,968 stabilizable and detectable deployments are QI-compatible. All three actuators are necessary, whereas intermediate-budget optima retain strict subsets of seven sensor packages. At a common budget, the best QI design has 3.86 times the performance loss of the co-design optimum relative to the dense deployment.

We study a planar ODE model for the benthic competition between coral, macroalgae, and algal turf on a reef, extending the classical model of Mumby, Hastings, and Edwards by a nonlinear, density-dependent coral mortality that accounts for crowding. The strength of crowding is set by an exponent $\delta>0$ that reshapes the coral nullcline and enriches the bifurcation structure of the system. We establish positive invariance of the biologically relevant region and the absence of periodic orbits, classify the three boundary equilibria together with their local stability, and reduce the coexistence problem to a single scalar equation whose shape, in particular its concavity, controls the number and local stability of the interior equilibria. The grazing intensity $g$ organizes the dynamics through two thresholds $g_0<g_1$ determining the stability of the coral- and macroalgae-dominated states, and a further threshold $g^\star$ at which two interior equilibria collide. We prove that the system undergoes transcritical bifurcations at the boundary equilibria and a saddle-node bifurcation of interior equilibria, and we discuss the implications for coral reef resilience and hysteresis. We complement these results with numerical simulations that illustrate the bifurcation sequence across the grazing regimes.

We develop a general framework for analyzing representation costs induced by parameter-space regularizers in data-fitting methods. For an arbitrary parametric method, we define its representation cost and native function space, prove existence, and identify conditions under which parameter-space and function-space problems have equal infimal values and minimizers transfer between them. This framework yields representer theorems and recovers classical formulations—including kernel methods and RKHSs, wavelets and Besov spaces, and shallow neural networks and variation spaces—as special cases. Our main new results concern depth-$L$ feedforward ReLU networks with weight-decay regularization. For these networks, we prove that the representation cost is a power of a quasi-seminorm and that, under suitable hypotheses, the native space is a quasi-Banach space with nonconvex unit ball when $L > 2$. These results identify a novel depth-dependent quasi-Banach geometry induced by weight decay.

We study the coherent cohomology of partial flag supervarieties and calculate the sheaf cohomology of the structure sheaves of four new infinite families of flag supervarieties. We introduce compositional varieties, a generalization of determinantal varieties obtained by imposing rank conditions on compositions of a pair of maps. We study geometric properties of compositional varieties and, in cases of interest, compute their $\mathrm{Tor}$-groups. For each of the four families of flag supervarieties, we show that the graded sheaf cohomology is isomorphic to the tensor product of the singular cohomology ring of an ordinary partial flag variety and the graded $\mathrm{Tor}$-groups of a compositional variety.

We derive the dynamic Doppler effects of noninertial observer motion on Dirac spinor fields in 4D spacetime, characterizing how proper acceleration, proper jolt (also known as jerk), observer path curvature, torsion, and hyper-torsion modify the inertial quantum description of free particles. We delineate the impacts of nonlinear observer motion on the amplitude and phase of the Dirac spinor wave field and isolate the spin-induced signatures that have no counterpart in the scalar Klein-Gordon field. For worldlines with constant proper jolt, the wave-field amplitude scales with quadratic rapidity. The complex Frenet-Serret curvature invariants generate both an exponential amplitude factor and a spin-induced phase factor. Spinor structure therefore produces an observable dynamic Doppler signature distinct from any scalar prediction. We analyze the effects of proper jolt and nonlinear rapidity on quantized Dirac fields in the context of acceleration-induced $\beta^+$-decay and derive the instantaneous decay rate, an equivalent of decay rate for non-stationary motions. This work provides new insight for noninertial-proton $\beta^+$-decay and a general framework for testable variable-rapidity Unruh predictions.

Iterative finite element solvers for time-harmonic wave problems are notoriously slow to converge, owing to fundamental properties of these problems. We present a variant of the hybridizable discontinuous Galerkin (HDG) method that is better suited to fast iterative solution. Unlike the standard hybridization strategy, which eliminates physical unknowns by introducing an auxiliary numerical flux on element faces, our approach instead introduces a transmission variable on those faces. For Helmholtz problems, this reformulation, known as CHDG, has been shown to significantly accelerate the convergence of iterative schemes relative to standard HDG. The present work extends CHDG to a general framework covering wave propagation problems with constant coefficients, capable of handling diverse wave types in a unified manner. We prove that the resulting hybridized system is well-posed and amenable to fixed-point iteration. As a practical application, we apply the method to the time-harmonic linearized Euler equations with a uniform subsonic mean flow. The method is illustrated through two-dimensional numerical benchmarks involving both sound and vorticity waves, with a systematic comparison of the convergence behaviour of several iterative schemes across a range of configurations.

We consider system identification for discovering parameterized operators in governing partial differential equations (PDEs) from noisy spatiotemporal data. Building on variational system identification (VSI), which identifies PDEs through Galerkin weak-form residuals, we develop a Bayesian VSI (B-VSI) framework for operator selection, parameter estimation, and uncertainty quantification. The central idea is to define the likelihood directly in weak-form residual space by propagating observation uncertainty through the weak-form residual map. The resulting likelihood captures heteroscedastic and correlated residual errors while avoiding repeated forward PDE solves during inference. For efficient computation, we use lagged-covariance updates that yield generalized least-squares estimates and conjugate posterior approximations when applicable, together with gradient-based and particle-based methods for more general priors and posterior structures. Model-form uncertainty is handled through sequential operator elimination guided by a residual-space Bayesian information criterion. We demonstrate the framework on state-linear and nonlinear PDEs, including the Fokker–Planck equation and a two-field Cahn–Hilliard equation. The results show that B-VSI accurately recovers active operators and coefficients from noisy data, improves robustness relative to classical VSI, and provides posterior uncertainty estimates for coefficients and derived physical quantities.

In this expository paper, we provide an introduction to elliptic functions and Eisenstein series from a classical analytic perspective while highlighting their broader mathematical significance. The exposition assumes familiarity with basic complex analysis, particularly meromorphic functions, contour integration, and power series expansions.

This paper is the final part in a series devoted to the development of numerically stable and efficient algorithms, together with multi-precision Fortran implementations, for computing Bessel functions with real orders and complex arguments. Parts~I and~II presented stable and efficient algorithms for the modified Bessel functions I_nu(z) and K_nu(z); here we treat Bessel functions J_nu(z) and Y_nu(z). The proposed algorithms support complex arguments for both positive and negative real orders and are implemented in native double and quadruple precision. Quadruple precision substantially increases dynamic range and accuracy (by roughly an order of magnitude in reliably computable |nu| and |z|), thereby extending applicability to problems requiring 20-30 digits. Comprehensive accuracy and performance comparisons are carried out against both the widely used Algorithm~644, restricted to double-precision arithmetic, and the more recent Algorithm~912, which supports both double- and quad-precision arithmetic as well as complex orders. In double precision, the present implementation consistently outperforms Algorithm~644, achieving execution times of approximately 35-67% for J_nu(z) and 44-72% for Y_nu(z), while also producing reliable results where Algorithm~644 fails. In comparison with Algorithm~912, the present algorithm achieves comparable accuracy in double-precision computations and significantly higher accuracy in quad-precision calculations. At the same time, it requires only a small fraction of the computational cost (from a few thousandths to a few hundredths) of the time taken by Algorithm~912, depending on the precision and parameter regime. Furthermore, unlike Algorithm~912, whose applicability is restricted to a limited region of the (Re(nu),z) plane, the present algorithm remains stable and accurate over the full tested domain for which reliable reference values are available.

We study scattering on quantum graphs that consist of a channel with periodically attached resonators under scale-invariant vertex couplings. For this model, we derive explicit formulas for the transmission probability and analyse how it depends on the geometric and coupling parameters. Contrary to standard one-dimensional scattering, where the potential barrier becomes transparent at high energy, here the transmission probability does not approach unity; instead, it is periodic or quasi-periodic, with infinitely many energies of complete reflection and of perfect transmission persisting at arbitrarily high energy. We further show that the model exhibits strong transmission suppression near the anti-resonant frequencies, resulting in pronounced spectral gaps. The width and structure of these gaps depend on the number of resonators and the coupling parameters. As a result, such quantum graphs can be used to engineer transport properties and to tune spectral filtering.

We prove that rational homology ribbon cobordism defines a partial order on the set of homeomorphism classes of closed, connected, oriented 3-manifolds. More precisely, if $W_0 \leq W_1$ and $W_1 \leq W_0$, then $W_0$ and $W_1$ are homeomorphic by an orientation preserving homeomorphism.

Boolean models are widely used to characterize the dynamics of gene regulatory networks. However, their coarse state discretization limits their ability to capture complex continuous dynamics and continuous parameter dependencies. In this paper, we present a rigorous mathematical framework that embeds monotone Boolean models into a broader class of multilevel combinatorial models, which in turn embed into the Dynamic Signatures Generated by Regulatory Networks (DSGRN) methodology. We define the DSGRN parameter graph, which encodes the notion of parameter adjacency and is used to map Boolean functions to specific nodes within the DSGRN parameter space. We prove that these multilevel discrete update functions act as a multilevel refinement of monotone Boolean models. We demonstrate that purely Boolean models systematically underestimate network dynamics by missing crucial intermediate behaviors such as higher-order multistability and stable periodic orbits. We show that the DSGRN framework efficiently captures a strictly richer set of dynamics consistent with ordinary differential equations (ODEs), providing a mathematically rigorous and computationally viable bridge between discrete and continuous network modeling.

We analyze diffusion processes ${(\mu_t)}_{t\geq 0}$ on the $2$-Wasserstein space $\mathscr P_2$ over $\mathbb R^d$, which can be classified as Ornstein–Uhlenbeck processes by means of their Fukushima decomposition. Characteristically for this class, $ | \mu_t|_2^2-|\mu_0|_2^2-2ct+2\int_0 ^t|\mu_s|_2^2\,d s,\, t\geq 0,$ is a martingale, where $c\in(0,\infty)$ is a constant and $|\mu_t|_2:=(\int_{\mathbb R^d}x^{\text{T}} x\mu_t(d x ))^{1/2}$ denotes the second moment. Making a particular choice from this class, we obtain a process ${(\mu_t)}_{t\geq 0}$ such that all moments exist and ${\mu_t(\theta)}-{\mu_0(\theta)}-\int_0 ^t\mu_s\big(\Delta \theta-\langle \text{id}_{\mathbb R^d},\nabla \theta\rangle\big)d s$, $t\geq 0$, is a martingale for each multivariate polynomial $\theta:\mathbb R^d\to\mathbb R$.

The cone projection $f_R(z)=z/(1+|z|/R)$ is a radial homeomorphism from $\mathbb{C}$ onto the open disk $D_R$ of radius $R$, obtained by an elementary cone-and-perpendicular construction (independent of the cone's height) and governed by the reciprocal lens identity $1/|f_R(z)|=1/|z|+1/R$. Its main Euclidean feature is the Self-Directrix Theorem: every line $\ell$ not through the origin maps to the focus-side arc of the conic with focus $O$, directrix $\ell$ itself, eccentricity $R/d$, and semi-latus rectum $R$, so the single distance $d=\operatorname{dist}(O,\ell)$ fixes the ellipse/parabola/hyperbola trichotomy. The Confocal–Codirectrix Theorem extends this from lines to every focal polar locus of a fixed focus–directrix pencil, keeping the focus and directrix while lowering the eccentricity by $1/e\mapsto1/e+\delta/R$; the image of a circle, by contrast, is generally a circular quartic rather than a conic. The same lens identity organizes the remaining structure: a curvature-additive composition law and its flow, a raywise cross-ratio structure, and higher-dimensional, metric, and axiomatic results.

In the semigroup $M_2(\mathbb{N}_0)^\bullet$, two-by-two matrices with non-negative integer entries and non-zero determinant, we study the factorization of matrices into atoms, or irreducible matrices. In 2022, Baeth et al. listed some fundamental classes of atoms in $M_2(\mathbb{N}_0)^\bullet$; however, the factorability of most matrices in $M_2(\mathbb{N}_0)^\bullet$ remains unknown. We identify two additional classes of atoms: a class of atoms with determinant $p$, $2p$, or $4p$, for $p$ prime, and a class of atoms in which the main diagonal is much "larger" than the off-diagonal (or vice versa). Finally, we show that bisymmetric matrices with relatively prime entries are a divisor-closed subset of $M_2(\mathbb{N}_0)^\bullet$ and use a factor search algorithm to classify bisymmetric atoms of $M_2(\mathbb{N}_0)^\bullet$ with minimum entry up to 4000.

Hamiltonian partial differential equations (PDEs) often exhibit long-time dynamics governed by conserved quantities such as mass, momentum, and Hamiltonian energy. Standard Fourier neural operators (FNOs) provide efficient data-driven approximations of solution operators, but may not preserve these invariants during autoregressive rollout, and can develop drift in conserved quantities, phase error, and loss of qualitative accuracy. We propose an energy-projection Fourier neural operator (EP-FNO), a structure-informed operator learning architecture that combines a residual FNO time-stepping update with an invariant projection for long-time prediction of parametric Hamiltonian PDEs. We also provide a theoretical analysis showing that EP-FNO can approximate operators associated with PDEs efficiently, we also suggest a stability estimate. We evaluate the approach on the Zakharov–Kuznetsov, Kadomtsev–Petviashvili, and sine–Gordon equations. Numerical experiments show that the projected model improves long-time stability, and gives more accurate propagation of soliton and coherent wave structures compared with a standard FNO baseline. Our results demonstrate that invariant projection improves the reliability of learned surrogates for long-time Hamiltonian PDE simulation.

Based on the framework proposed in Part I, the Part II of this two-part paper investigates two-subarray designs for moving target SAR imaging using planar antenna arrays and the multidimensional Chinese remainder theorem (MD-CRT). In this Part II, we focus on the performance analysis and the detailed two planar subarray designs. In particular, we study a common-scaling two-subarray design, under which the two subarrays share the same scaling factor in the MD-CRT formulation. Under this design, ambiguity resolution can be performed on a common integer frequency vector. As a result, the same unambiguous range as in the general two-subarray framework in Part~I is preserved, while the sufficient conditions for robust recovery become weaker and the corresponding reconstruction error bounds become tighter. Within this common-scaling design, we compare the proposed planar array framework with a conventional separated scheme, in which the motion-induced cross-range shift is recovered by a one-dimensional CRT-based method and the target height is estimated by cross-track interferometric processing. Under the same platform size and minimum antenna spacing constraints, the proposed planar array framework can realize the common-scaling design, whereas the corresponding one-dimensional non-uniform linear array scheme does not admit such a design. With this design, the planar array framework leads to a weaker sufficient condition for robust recovery and thus performs better in moving target imaging. We also compare several planar array designs under fixed platform size and minimum antenna spacing. The analysis shows that recovery performance depends not only on the number of antennas but also on the array geometry. In particular, non-separable planar array geometries can provide better robustness than separable ones when their antenna numbers are comparable.

In this two-part paper, we investigate synthetic aperture radar (SAR) moving target imaging using planar antenna arrays. For a target moving over a three-dimensional terrain, its accurate localization requires the joint estimation of the motion-induced cross-range shift and the target height. In Part I of this two-part paper, starting from the planar array imaging geometry and the corresponding signal model, we show that these two quantities can be unified into a two-dimensional parameter vector and represented, after two-dimensional discrete Fourier transform (2D-DFT) processing across the planar array, through a natural vector remainder formulation. We first develop a general 2D-DFT matrix modulus framework and show that, in the two-dimensional setting, the associated 2D-DFT matrix modulus affects the propagation of vector remainder errors. Under a fixed array geometry and antenna number constraint, we derive an optimal construction of this matrix modulus and adopt it in the subsequent analysis. Under this construction, a single planar array provides only a folded estimate when the true parameter vector lies outside its unambiguous range. To resolve this ambiguity, we develop a multi-subarray framework in which multiple planar subarrays generate multiple vector remainders with different matrix moduli, and the desired parameter vector is recovered through the multidimensional Chinese remainder theorem (MD-CRT). To account for practical errors introduced by 2D-DFT quantization and additive noise, we further introduce an approximate 2D-DFT peak model for non-integer frequency vectors, incorporate robust MD-CRT, and establish sufficient conditions together with explicit reconstruction error bounds for both noiseless and noisy settings. Numerical results verify that the proposed multi-subarray framework enlarges the unambiguous range compared with a single planar array.

In this paper we use the Pirogov–Sinai theory to analyze a class of particle models of Statistical Mechanics on the unit triangular lattice $\mathbb{A}_2$. The models are specified by two parameters: the particle activity $u\in(0,1)$ and a positive Löschian number $t$ interpreted as a maximal squared clearing radius. Admissible configurations are those in which every empty lattice site has an occupied site within squared distance at most $t$, thereby forbidding empty disks of radius exceeding or equal $t$. The Hamiltonian favors configurations with as few occupied sites as possible, while the admissibility condition enforces a positive density of particles. We prove that the periodic ground states of the model, i.e., the configurations achieving an optimal balance of two tendencies, are the ones whose occupied sites form triangular sublattices of explicitly determined squared side-length $d_*(t)$. Furthermore, for sufficiently small values of the activity parameter $u$, some (but not necessarily all) periodic ground states generate extreme Gibbs (DLR) measures. We describe the resulting phase structure and characterize the pure phases associated with stable ground states.

Air traffic growth, advanced air mobility, and increasingly autonomous operations are driving the need for scalable and adaptive airspace design methodologies. Central to this challenge is quantifying how traffic flow structure and demand, governed in part by airspace geometry, influence conflict generation and operational complexity. This paper presents an analytical framework for computing conflict rate and conflict probability in structured airspace using stochastic flow models. Traffic streams are modeled as renewal processes with prescribed inter-arrival time distributions, while interactions between flows are captured through geometry-dependent minimum spacing constraints at merges and crossings. Within this formulation, closed-form upper bounds on the expected conflict rate and conflict probability per aircraft are derived as functions of flow configuration and demand. These metrics are interpreted as complementary measures of airspace complexity, reflecting controller workload and per-aircraft operational risk. The methodology is applied to representative hexagonal cell geometries with varying routing structures and flow distributions. Results reveal non-monotonic tradeoffs between routing flexibility, capacity, and conflict generation, with intermediate flow configurations outperforming both highly constrained and highly distributed cases. The proposed framework provides a tractable tool for evaluating airspace design alternatives and complexity-informed traffic management strategies.

The commute time between two nodes in a network - the expected number of steps for a random walk to travel from one node to the other and then return - is a metric of broad importance arising in community detection, network routing, dimensionality reduction, and diffusion modeling. For random geometric graphs (RGGs), in which nodes are placed at random in a spatial domain and connected pairwise wherever their Euclidean distance is below a threshold radius, the relationship between commute times and the embedding geometry remains poorly understood outside very dense settings (where the role of the geometry disappears and commute times degenerate to a sum of inverse degrees). We develop and numerically validate a model for approximating commute times in sparse RGGs on a torus by combining theoretically motivated geometric contributions with an inverse degree sum. The geometric terms include a universal logarithmic contribution from the Laplacian, a quadratic correction encoding the compact topology of the torus, and a quartic angular term reflecting the square anisotropy of the domain. We fit this model to samples of node pairs across a range of graph sizes and mean degrees, demonstrating good predictive performance and that the geometric terms contribute significantly to model fit. We then study the continuous perturbation of the model from a regular square lattice to a fully random geometric graph, further validating the functional model form through this transition and showing how commute times in sparse RGGs retain meaningful geometric information about the embedding space.

CVXPY is a Python-embedded domain-specific language for convex optimization that lets users express problems in mathematical notation while the system verifies convexity and reduces valid programs to solver-ready form. This paper reports on the major advances from versions 1.1 through 1.9. These include a unified conic quadratic program (CQP) standard form for canonicalization; a stacked-slices backend that accelerates parameterized programs; first-class support for N-dimensional expressions; explicit sparsity for variables; support for multiple variable attributes; cones/atoms relevant to quantum information theory; and the introduction of disciplined nonlinear programming (DNLP). We outline the design, algorithms, and modeling consequences of these features.

We describe two connections between the theory of quasimorphisms and pseudo-Anosov flows without perfect fits on closed hyperbolic 3-manifolds. First we show that for every such flow $X$, there are quasimorphisms whose coarse restriction to each flowline of $\tilde{X}$ (the lifted flow in the universal cover) are uniform quasi-isometries to $\mathbb{R}$ – such quasimorphisms are said to be *adapted* to $X$; and that the space of quasimorphisms $Q_X$ adapted to $X$ is an open convex cone in the space of all quasimorphisms on $\pi_1(M)$. Second, we obtain upper bounds on the exponential growth rate of closed orbits in such flows, both in the hyperbolic metric and in a word metric; quasimorphisms play a key role in obtaining the estimates in the second case.

In this paper, we develop an intrinsic consensus-based optimization framework on Riemannian manifolds with bounded sectional curvature. In contrast to extrinsic approaches based on an ambient Euclidean embedding, our model is formulated directly in terms of the Riemannian structure, using logarithmic and exponential maps induced by the intrinsic geodesic distance. We prove the global well-posedness of the proposed particle system and its associated McKean–Vlasov dynamics. We also establish the global convergence of the mean-field equation toward a global minimizer of the objective function under suitable conditions. Numerical experiments on the sphere, hyperbolic space, and the special orthogonal group demonstrate the effectiveness of the intrinsic CBO dynamics for nonconvex optimization problems on manifolds.