Papers

Magnetic resonance elastography (MRE) is a noninvasive imaging modality for quantifying the viscoelastic properties of soft tissues from shear wave propagation. Recovering the complex-valued shear modulus from measured displacement fields leads to a severely ill-posed inverse problem, particularly in the presence of noise and limited boundary excitations. We investigate adjoint-based optimization, convolutional neural network (CNN) reconstruction, and a hybrid framework combining both approaches. The forward model is based on a scalar form of the modified stationary Stokes system with a complex shear modulus. We establish well-posedness of the forward problem, existence of minimizers, and first-order optimality conditions for the adjoint-based formulation, and implement a nonlinear conjugate-gradient method with Armijo line search. While PDE-constrained optimization can accurately refine coefficient reconstructions, its performance depends strongly on initialization. We therefore construct a two-dimensional CNN that maps complex-valued displacement measurements to spatially varying complex shear modulus fields and provides rapid, informative initial reconstructions. The proposed hybrid method uses the CNN reconstruction to initialize the adjoint-based optimization, yielding faster convergence and improved accuracy. The CNN is trained on coefficient fields containing individual perturbations and tested on both individual and previously unseen combined configurations. Numerical experiments demonstrate that the CNN generalizes to these more challenging configurations, while subsequent PDE-constrained optimization further refines the reconstructed coefficient. These results demonstrate the potential of combining data-driven initialization with physics-based optimization for efficient and accurate MRE reconstruction.

Covariance estimation is a key component of many applications in system identification and data-driven control. Although heavy-tailed distributions may lack a covariance matrix to estimate, the shape matrix provides a well-defined, scale-free generalization for the broad family of elliptical distributions. In this setting, practitioners often employ Tyler's M-estimator (TME), which is defined implicitly and is computed using a fixed-point iteration. The spatial sign covariance matrix estimator (SSCM) offers a much simpler alternative: it corresponds to one iteration of TME. Yet, the SSCM has been largely regarded as an inferior estimator due to its statistical inconsistency under fixed-dimensional asymptotics. By contrast, using second-order tools from random matrix theory, we establish that, under standard assumptions, SSCM asymptotically dominates TME in Frobenius risk when dimension and sample size grow proportionally. This advantage arises from implicit regularization, which reduces variance at the cost of a bias that vanishes as the dimension grows. Numerical experiments support the theoretical predictions even at moderate dimensions.

In this paper, we investigate how spatial heterogeneity affects the stability of positive equilibria in one-dimensional predator–prey reaction–diffusion systems with Neumann boundary conditions. Our examples show that spatial heterogeneity can lead to instability even when the positive equilibrium is unique. For the smooth, spatially heterogeneous coefficient families constructed here, this equilibrium loses linear stability as a parameter in the reaction coefficients varies. At the critical parameter value, a simple conjugate pair of eigenvalues crosses the imaginary axis transversely, and a Hopf bifurcation produces a local branch of strictly positive, nonstationary classical time-periodic solutions. The examples include two families with spatially concentrated prey, one with an everywhere positive and the other with an everywhere negative predator intrinsic growth rate. A third family has a spatially concentrated predator population and an everywhere negative predator intrinsic growth rate.

We study the normal-mode stability of a flat interface between compressible Euler flow in the air and incompressible Euler flow in the water, including gravity and surface tension. The base state is quiescent water beneath a hydrostatic, constant-entropy shear flow $U(z)$ with $U(0)=0$, so the atmosphere is neutrally stratified. For a small ratio $\varepsilon$ of the air density at the interface to the water density, and for uniformly subsonic wave-relative flow, we establish a compressible analog of the critical-layer instability of Miles, extending the rigorous incompressible theory of Bühler, Shatah, Walsh, and Zeng. For a real phase speed $c_r$, the singular coefficient at a regular critical level $U(z_c)=c_r$ has a residue proportional to $Q=U''+gU'/a^2=\bar\rho_0(U'/\bar\rho_0)'$, with gravitational acceleration $g$, sound speed $a$, and normalized base-state air density $\bar\rho_0$. Thus the gradient of specific vorticity replaces the curvature $U''$ of the incompressible theory. We prove a limiting absorption principle at regular, non-resonant phase speeds and a residue formula for the imaginary part of the boundary value $Z_{\mathrm a}^+(k,c_r)$ of the air-side pressure response at wavenumber $k>0$. Let $c_k>0$ be the phase speed of the free gravity-capillary wave. If $\mathrm{Im}\,Z_{\mathrm a}^+(k,c_k)>0$, then, for sufficiently small $\varepsilon$, the dispersion relation has exactly one unstable root $c_\varepsilon$ in a neighborhood of $c_k$ that does not depend on $\varepsilon$. A sufficient condition is $Q\le0$ at every critical level and $Q<0$ at the highest one. As the Mach number $M\to0$, $Z_{\mathrm a}^+$ converges to its incompressible counterpart at rate $O(M^2)$. Under the corresponding incompressible sign condition, the unstable root satisfies $|c_\varepsilon(M)-c_\varepsilon(0)|\le C\varepsilon M^2$ with $C$ independent of $\varepsilon$ and $M$.

This work proposes a data-driven framework that synthesizes output-feedback control for linear time-invariant multi-agent systems (MAS). We first develop two algorithms that use offline data samples of state and input trajectories to determine a Nash equilibrium (NE) for MAS. Next, we formulate a semidefinite program that, given the NE solution, computes the static output-feedback controllers for the agents in MAS. Compared to prior work, virtues of our framework include weaker controllability assumptions, the absence of online information and data-driven estimation of states. Next, we use the proposed framework to design control schemes for voltage and frequency regulation for nonlinear multi-nodal electrical networks with grid-forming and grid-following inverters. Thus, compared to prior work that implements their data-driven frameworks in small-scale numerical examples, we validate the performance of our data-driven framework in high-fidelity nonlinear models of MAS. This implementation in realistic MAS provides insights that small-scale numerical examples may fail to capture. Results show successful grid stabilization and provision of regulation services from the inverters in the grid using our proposed framework. It can attain up to 78% less overshoot and 30% faster settling time than traditional strategies such as proportional-integral control.

This paper is concerned with the effect of spatial heterogeneity on the linear stability of unique positive steady states for a nonlocal population equation. The main difficulty is that, because of the nonlocal term, the linearized operator need not be self-adjoint or satisfy the maximum principle, so the usual stability arguments based on the principal eigenvalue do not apply directly. We establish linear stability for sufficiently large diffusion and, under suitable conditions on the maximum set of the local growth ratio, for sufficiently small diffusion. In one space dimension, we also construct unstable positive steady states and obtain Hopf bifurcations for sufficiently small diffusion. Keeping the local coefficients fixed, we obtain either a supercritical or a subcritical Hopf bifurcation by changing the weight in the nonlocal integral. Our results show that uniqueness of a positive steady state does not imply its linear stability.

We study two interacting reinforced occupation processes driven by a common two-state Markov environment whose switching probabilities decrease at the same rate as the stochastic-approximation gain. At this critical scale, the environmental motion persists in the first-order limit, which is a telegraph-driven piecewise deterministic Markov process. Consequently, centering at a deterministic equilibrium does not separate the order-one environmental response from the intrinsic square-root fluctuations. To overcome this obstruction, we introduce an exact discrete random centering that removes the accumulated environmental forcing before rescaling. An exact decomposition into collective and synchronization coordinates then separates the two contraction mechanisms of the system. Under the contraction condition corresponding to the square-root regime, we prove joint functional convergence of the randomly centered coordinates and the first-order background to a stationary hybrid PDMP–diffusion system. Its stationary entrance law is characterized through stochastic convolutions over the remote logarithmic past. Conditionally on the complete environmental path, the two fluctuation modes are independent centered Gaussian processes with a common state-dependent volatility and different contraction rates. We also determine the invariant distribution of the first-order component, derive explicit stationary second moments, and obtain a closed-form correlation formula that recovers interaction information not visible in the synchronized first-order limit.

Problem 73 on Green's list of open problems asks whether a smooth submanifold of codimension two in $\mathbb{R}^n$ must meet some affine two-plane in five points when $n$ is large. In this paper, we answer this question for every $n \geq 6$. Every nonempty relatively open subset of such a submanifold contains the vertices of an affinely regular pentagon, and the bound $n \geq 6$ is sharp for affinely regular pentagons. More generally, for every prime $p \geq 5$, the graph of a continuous map from a ball in $\mathbb{R}^{p-1}$ to $\mathbb{R}^2$ contains an affinely regular $p$-gon. A second argument gives six coplanar points when $n \geq 7$. Both proofs find the points by making the low Fourier modes of the graph map vanish along a circle.

We study derivations and automorphisms of arbitrary non-abelian extensions of $n$-Lie algebras through their full crossed-product data. Their relative center is a canonical coefficient module, and a third cohomology class obstructs completion of data satisfying the mixed Filippov identities. When this class vanishes, a relative second cohomology space classifies the remaining extension data. We prove a non-abelian good criterion, construct the Lie algebra and group of compatible pairs, and obtain derivation and automorphism Wells sequences with explicit lifting equations. The derivation obstruction is a Lie algebra $1$-cocycle; the additive automorphism obstruction is a crossed homomorphism. We distinguish the ordinary center from the relative center, and outer derivations of the kernel from the mixed action required. Sparse crossed products are treated, including the representation class of their second mixed operation and the additional equations governing changes of sparse sections.

We study a three-species competitive Lotka–Volterra system with cyclic dominance, motivated by rock–paper–scissors interactions in ecological communities. The model contains a singularly strong interaction coefficient of order $1/\epsilon$, representing a regime in which one species exerts a much stronger competitive pressure on another. We show that, despite this strong asymmetry, the cyclic dominance mechanism persists. Our main result is a quantitative description of one complete dominance pattern in the singular regime $0<\epsilon\ll 1$. Starting near the state where the third species dominates, we prove that the solution successively passes near the dominance states of the second and first species before returning to a comparable configuration. We obtain logarithmic estimates for the return time and for the population densities at the return section. In particular, although one species becomes extremely small at the beginning and at the end of each pattern, it recovers to an order-one density during the cycle. We then prove that this pattern repeats indefinitely and that the omega-limit set is composed of three heteroclinic orbits on the boundary of the positive orthant, connecting the three single-species equilibria. Thus the long-time behavior is cyclic but aperiodic, with increasingly long excursions near successive single-species states.

We study moving-law random-environment matrix cocycles whose one-step quasiperiodic cocycle law varies stationarily along an ergodic environment orbit. Two fixed-law structures then cease to suffice: a single stationary projective measure does not record transport between successive environment fibers, and pooling the moving supports may create products that are not dynamically admissible along any environment orbit. We replace the first by environment-indexed stationary projective families and prove an environment-wise Furstenberg formula. When the top Lyapunov exponent vanishes, averaged stationarity upgrades to exact equivariance at the support level, \(g_*\eta^\omega=\eta^{\Theta\omega}\), which forces either an environment-wise compact reduction or a finite support-invariant family of nonzero proper measurable subbundles. Environment-wise noncompactness and environment-wise strong irreducibility therefore imply positivity. To address the second obstruction, we compare admissible products along matched environment segments with equal total torus translation. Thus their quotients are genuine returns on a common vector fiber and generate environment-matched monodromy groups. Stable noncompactness and stable strong irreducibility of these groups verify the cocycle-level hypotheses. Applied to stationarily varying random–quasiperiodic Schrödinger cocycles, the framework yields strict positivity of the top Lyapunov exponent at every real energy under the stated assumptions.

We develop a sparse graph method for transporting probability mass toward multimodal target distributions through damped nonlocal Hamiltonian dynamics. The formulation combines logarithmic-mean mobility with symmetric Lévy-type interaction weights, coupling the evolving density to an edge momentum field. A graph constructed from nearest-neighbor connections and sampled long-range edges provides direct mass exchange between spatially separated regions. Once the graph is constructed, the density evolution is deterministic, and each update costs linear in the number of nodes and the long-range sampling budget. Linearization around the target distribution yields a damped oscillator governed by a weighted graph Laplacian. Its spectrum characterizes the interaction between nonlocal connectivity and inertia, with the Lévy exponent alpha tuning the nonlocal connectivity: the spectral gap determines the optimal asymptotic damping, while the largest eigenvalue governs the time-step stability. Experiments on synthetic multimodal distributions demonstrate improved mode balance and more stable mode coverage relative to first-order and MCMC baselines. The resulting framework provides a sparse implementation of nonlocal inertial density transport for sampling problems with low-dimensional spatial structure.

For some integer $\ell\geq 2$, consider $m=2\ell-1$ voters, each of which has a preference ranking over $n$ alternatives. The Smith set is the smallest nonempty set of alternatives each of which defeats every alternative outside the set in a pairwise majority comparison. Under the standard benchmark impartial culture where each voter uniformly and independently chooses a random preference ranking over all alternatives, we study the asymptotic probability that the Smith set has cardinality $s$ for fixed $\ell$ and $n\to \infty$. First, we prove that the probability that the Smith set has constant cardinality $s$ goes to zero at a rate of $\Theta_{\ell,s} (n^{-s(\ell-1)/\ell})$. Next, if $\min\{s,n-s\}\to\infty$, we show that this probability decays superpolynomially. Further, when $s$ and $n-s$ are both $\Theta(n)$, we prove that the probability decays exponentially and determine the exact exponential base. Finally, we prove that the probability that the Smith set contains all alternatives approaches one at a rate of $\Theta_\ell(n^{-(\ell-1)/\ell})$. These theorems resolve the conjectures in [Liu et al, Ann Stat 2026] under impartial culture.

We investigate whether the geometry of a curved conductor–dielectric interface can modify the effective mass and propagation of scalar surface plasmons (SSPs). In the hydrodynamic limit, we describe SSPs as density perturbations in a semiclassical jellium electron gas. Using a thin–layer dimensional reduction procedure, we derive a massive Klein–Gordon equation for SSPs on the curved interface with a geometric potential. We find that under specific conditions, the geometric potential allows SSP propagation with a vanishing effective mass. We then apply the formalism to a well known analogue black hole configuration, determining the corresponding quasinormal mode frequencies. Our results show that interface curvature provides a geometric handle on plasmon propagation and localization, suggesting new possibilities for plasmon transport and analogue gravity experiments with surface plasmons.

We study a divisor-weighted version of Golomb's self-description: the nondecreasing sequence $A$ in which each positive integer $m$ occurs $\sum_{d\mid m}A(d)$ times. This rule has a unique solution, and $$ A(n)\sim Cn^{\varphi-1},\quad C=\left(\frac{\varphi}{\zeta(\varphi)}\right)^{2-\varphi},\quad \varphi=\frac{1+\sqrt{5}}{2}. $$ The golden-ratio exponent of Golomb's sequence survives, while divisor weighting changes the leading constant. We prove the asymptotic without assuming regular variation, by a contraction of upper and lower power envelopes. The argument applies to every nonnegative integer Dirichlet-convolution kernel $w$ with $w(1)=1$ and $\sum_q w(q)q^{-\varphi}<\infty$. For the divisor kernel, a quantitative version gives relative error $O_\gamma((\log n)^{-\gamma})$ for every $0<\gamma<1$. Although the global growth is smooth, the normalized run lengths retain arithmetic fluctuations. A uniform comparison with the divisor sum $\sum_{d\mid m}d^{1-\varphi}$ transfers them to an explicit random Euler product. We prove that its law is singular continuous with full support $[1,\infty)$, determine all real Mellin moments, obtain the sharp maximal order of the fluctuations, and show that sampling sequence positions instead of block labels produces the size-biased law.

Higher-order multivariable Hermite-Kampé de Fériet polynomials provide a unified construction of umbral representations for canonical catastrophe integrals. Expanding the unfolding exponential in these polynomials and integrating against the leading catastrophe kernel gives convergent moment expansions on suitable decay contours. A universal formal umbral operator then expresses the resulting fields through explicit moment ground states. We develop this construction for the fold, cusp and swallowtail, and relate the Euclidean Pearcey field to quartic scalar partition functions. The distinction between formal and analytic umbral frameworks becomes essential when integration is performed before umbral evaluation. For Euclidean Pearcey, the integrated linearized kernel is a rational function whose action is not specified by the original formal rules. The Analytic Umbral Framework supplies a convergent Mellin pairing for this kernel and recovers the independently obtained bivariate Pearcey representation. This establishes, in this example, an exchange of integration and analytic evaluation and identifies a specific extension beyond the formal framework.

A reduced word of the longest permutation $w_0\in S_m$, taken up to commutations, encodes a wiring diagram of $m$ pseudolines, and a braid move $\sigma_i\sigma_{i+1}\sigma_i\leftrightarrow\sigma_{i+1}\sigma_i\sigma_{i+1}$ flips one of its triangles. We prove that deciding whether two wiring diagrams are within a given braid-move distance is NP-complete. Equivalently, deciding flip distance is NP-complete in the higher Bruhat order $B(m,2)$, for triangle flips of marked arrangements of pseudolines, and for rhombic tilings of a $2m$-gon; this answers an open question on flip graphs of pseudoline arrangements (SODA 2024). The reduction is from Vertex Cover: for every graph $G$ we construct two wiring diagrams at braid-move distance $|D|+2\,\mathrm{VC}(G)$, where $\mathrm{VC}(G)$ is the vertex cover number of $G$ and the Hamming distance $|D|$ is the number of triples of wires whose triangles are oriented differently. The construction glues copies of a seven-wire gadget whose braid-move distance exceeds its Hamming distance by $2$; flipping either of two disjoint triangles in both diagrams removes the excess. Adding a line at infinity transfers the construction to rank-$3$ uniform oriented matroids, where two oriented matroids are adjacent in the mutation graph if their chirotopes differ in one basis; so deciding mutation distance is NP-complete in rank $3$. An iterate of Rambau's expansion of signotopes amplifies the excess and gives NP-hardness in every fixed rank $r\ge 4$. By duality we obtain the corresponding results for every fixed corank at least $3$. The same instances show that flip distance in the higher Bruhat orders $B(n,k)$ is NP-complete for every fixed $k\ge 2$.

Lattice constellations can provide significant coding and shaping gains over traditional quadrature amplitude modulations (QAMs), but have not been widely adopted due to the high implementation complexity of modulation, demodulation, and decoding. Voronoi constellations (VCs) enable an efficient modulation map that achieves shaping gain using the Voronoi cell of a lattice. Multilevel coset codes (MLCCs) are a recently proposed coded modulation scheme for VCs that realizes more efficient decoding. However, MLCCs still suffer from a soft demodulation complexity of $O(d2^{sd})$ for a $d$-dimensional lattice with spectral efficiency $s$ bits per real dimension. In this paper, we propose fast quantize-walk demodulation (FQWD), an efficient demodulation scheme for MLCCs that reduces the complexity to $O(sd2^d)$. Our strategy utilizes novel radial projections onto the Voronoi cells of the $D_4$ and $E_8$ lattices. Experimental results demonstrate that FQWD has only negligible loss in block error rate (BLER) relative to maximum likelihood demodulation under the additive white Gaussian noise (AWGN) channel. Using FQWD, we simulate MLCCs on higher-dimensional lattices with $s>2$ bits per dimension under AGWN for the first time, achieving BLER gains over multilevel polar codes on QAMs of up to 0.6 dB for the $D_4$ lattice and 1.2 dB for $E_8$.

We propose a two-level filtered subspace iterative (TLFSI) method for computing all eigenpairs whose eigenvalues in the interval of interest $[a,b]$ after finite element discretization of self-adjoint PDE eigenvalue problems. The method combines a two-level approach with a modern filtered subspace iteration framework. In this paper, we use contour integral filtering as an example. The two-level method constructs a coarse space according to the spectral location of the eigenvalues of interest and obtains from it an approximation to the target invariant subspace. We prove that the distance between the approximate and target subspaces depends only on the separation between the spectral region of interest and the remainder of the spectrum, together with the coarse mesh size \(H\). Numerical experiments on benchmark elasticity problems confirm the theoretical predictions.

Let $G$ be a compact connected semisimple Lie group, let $Q$ be a bi-invariant metric and let $\rho$ be its Cartan three-form. We prove that every left-invariant generalized Ricci flow with $[H_0]=[\rho]$ exists for all positive time and converges smoothly to a Bismut-Ricci-flat pair without rescaling or pullback. The standard pair uniquely maximizes generalized scalar curvature in its torsion class, and every left-invariant Bismut-Ricci-flat pair on a compact simple group is standard, answering questions of Fusi, Lafuente and Stanfield and of Lauret and Will. On the product of two copies of a compact simple group, nonstandard stationary pairs in the Cartan class are known; we construct nonstationary flows with the exact initial form $[H_0]=[\rho]$ and nonstandard limits, disproving the general semisimple case of the convergence conjecture of Garcia-Fernandez and Streets. The proofs use a proper nonincreasing energy and a decomposition of Killing fields that parametrizes the stationary pairs by nilpotent Lie algebra endomorphisms.

Vehicle-to-everything (V2X) sidelink and uplink broadcast must deliver safety-critical and application-layer data reliably under fast time-varying fading, intermittent packet erasures, and exposure to passive eavesdroppers. Fixed-rate physical-layer codes are poorly matched to these conditions because link quality and the number of successfully received symbols vary across vehicles and over time. This paper proposes a unified vehicular communication framework that combines rateless Construction-D' Raptor lattices, nested-lattice physical-layer security (PLS) with direct-sequence spread spectrum (DSSS), and blockchain-assisted trust management for decode-and-forward (DF) cooperative relaying. At the physical layer, Raptor codes provide incremental redundancy over unreliable V2X links; multilevel lattice decoding enables power-efficient joint coding and modulation; nested shaping and spread spectrum limit fine-lattice recovery by eavesdroppers; and a distributed ledger records decoding success at relays to support reputation-aware relay selection. We model the vehicular air interface as a block-Rayleigh fading channel with additive Gaussian noise and random symbol erasures, representative of urban V2X sidelink behavior. The proposed code concatenates Luby-transform (LT) codes with quasi-cyclic low-density parity-check (QC-LDPC) precodes within Construction-D', with girth-4 cycle removal on the Tanner graph. AWGN benchmarks show competitive symbol error rate (SER) and improved frame error rate (FER) versus matched-length LDPC and LDLC lattices. Under block-Rayleigh fading with erasures, for n=995 at average VNR of 1.5 dB and erasure probability 0.15, rateless Raptor reception achieves 0.3% outage with about 2.3 packets on average, outperforming LDLC and LDPC under identical channels, while an eavesdropper with 8 dB combined SNR and spreading loss fails in 62% of sessions.

Let $k$ be a field of characteristic zero and $D_N(a_n)=a_{n+1}a_{n+2}$ the adjacent-quadratic locally nilpotent derivation on $A_N=k[a_0,\dots,a_N]$. We determine the invariant ring at $N=5$, where deleting $a_0$ gives the polynomial base kernel $B^{D_5}=K=k[p,q,J_1,J_2]$, so localization gives the polynomial-line invariant algebra $(A_5^{D_5})_{pq}=K_{pq}[W]$ with $W=F_0/(p^2q^4)$, but this does not by itself determine the global kernel. An invariant $H$ of weight $24$ and $a_0$-degree $2$ lies outside $K[F_0]$, and adjoining it gives the six-generator hypersurface $C_5$; the obstruction is explained by algebraic residue behavior defeating naive denominator descent. An invariant $F_{46}^{int}$ of weight $46$ and $a_0$-degree $4$ lies outside $C_5$ and satisfies $p^2F_{46}^{int}\in C_5$. To pass from this candidate to the full invariant ring, a second slice chart arising from $f_{13}$, together with exact $q$- and $p$-saturation, yields $A_5^{D_5}=C_5[F_{46}^{int}]$, hence finite generation, with a codimension-two complete-intersection presentation and explicit Hilbert series. At $N=6$ the transported $B$-level ring is exact, while the $A_6$ analysis is established through total degree $13$ and the full $A_6$ invariant ring remains open.

We prove that the wedge local model for a ramified unitary group of signature $(r,s)$ ($r>s$) at self-dual level is flat when $p\geq 2s-1$, confirming Pappas's conjecture in this case. The proof first reduces the problem to determining the defining equations of a nilpotent orbit closure of self-adjoint matrices, and then to computing the global sections of the corresponding Kempf resolution. The computation is carried out by reducing to the type $A$ case using the Koszul complex.

We prove a variant of the $L^2$ Fourier restriction inequality for non-degenerate curves in the setting of orthonormal systems. To achieve this, we develop a Hilbert space approach to studying the $L^2$ theory for Fourier restriction for curves. Our arguments also rely on novel $L^p$-improving bounds for associated convolution operators, which take averages over neighbourhoods of cones generated by the curve. The $L^p$-improving bound are established using a variant of the high-low method, in particular using induction-on-scale.

This work proposes a semidefinite programming (SDP) model that synthesizes static output control feedback for networked linear time-invariant multi-agent systems (MAS). We theoretically prove that the output feedback control of MAS can be constructed using the Nash equilibrium previously derived for state-feedback control. The Nash equilibrium considers that agents seek to minimize their individual cost functions subject to the system dynamics of the MAS. The resulting control from the pro- posed SDP model ensures the stability of the closed-loop system and dissipativity when a disturbance is present in the MAS. A set of four case studies is implemented. The first three numerical cases serve to validate and compare cost effectiveness and time-domain performance of the proposed control framework with other state-of-the-art frameworks. The fourth case designs a frequency regulation scheme for a modified IEEE 24-bus power grid. Using the proposed framework, distributed controllers are synthesized for the inverters which are considered agents in the grid. Each inverter uses partial state information of the neighboring inverters. The resulting control scheme is implemented in a high-fidelity non-linear model of 261 states that describes the electromagnetic transient dynamics of the power grid. Hence, compared to prior studies that usually validate their frameworks in small-scale numerical models, this fourth case demonstrates a virtue of our pro- posed SDP: its scalability and practical utility in real-world networked MAS. We implement the proposed framework in an open-access toolbox in Python and MATLAB, enabling practitioners and researchers to readily implement it in their applications.

The off-diagonal Gram map for h vectors in an M-dimensional quadratic space is flat with reduced fibers of rational singularities (FRS) in characteristic zero if and only if M > max_{2 <= r <= h} (2h - r + 1 - 2h/r). Equivalently, this gives the exact FRS convolution threshold of x |-> (x_i x_j){i < j}. For its zero-fiber ideal I{M,h}, we also prove lct(I_{M,h}) = min{ binom(h,2), min_{0 <= k <= h-2} ( binom(k+1,2) + M(h-k)/2 ) }. In the strict range, all truncated coefficient maps have geometrically integral complete-intersection fibers and are faithfully flat over Z[1/2], uniformly over graphs on h vertices. The proof uses a sharp weighted estimate for hollow symmetric matrices over finite local principal ideal rings, paired with isotropic lower bounds. Within the established point-counting approach to rational singularities, this estimate also gives an explicit modulus of continuity for Gram-map densities over every local field of odd residue characteristic.

In this paper, we provide a novel approach to limiting Heesch numbers of a specific family of tiles. In particular, we show how a certain family of three–dimensional prisms has a finite Heesch number. In so doing, we initiate a structural study of ratio $q$ tilings by two prototiles, in which, loosely speaking, one tile appears $q$ times more often than the other. The method works as follows: we take planar intersections with coronas around the prism in two perpendicular directions and compare the frequencies of the resulting prototiles in those patches. If every tiling by these prototiles has limiting ratio at least $q >1$, then sufficiently large coronas of the prism would require incompatible tile counts in the two directions. We also provide some computational results and exhibit $6$ infinite families of notched prisms that have finite Heesch numbers greater or equal to $2$. Finally, we provide some negative computational results about the non–existence of certain prisms with given Heesch numbers.

In this paper, we consider the control of Markov jump linear systems when the active system mode is not exactly known. We rather assume that this true mode is only known to belong to an observed cluster of modes, making both the dynamics and the associated transition probabilities uncertain. We construct a min-max optimal control problem in this setting, with the objective of robust regulation. Using results in regularized least mean-square optimization, we describe how the solution of this problem can be reduced to a Riccati recursion, such that the controller depends only on knowledge of which cluster the mode lies in. We subsequently establish convergence of the recursion and stability of the resulting closed-loop system. A numerical example illustrates the effectiveness of the proposed methodology.

We derive a weighted Kirchhoff diffusion on a smooth Riemannian open book as a scaling limit of regime-switching Brownian motions on boundaryless manifolds. The pages are compact Riemannian manifolds sharing a common boundary manifold and inducing the same metric on the binding, while their second fundamental forms and interior geometries may differ. For each $\varepsilon>0$, we regularize the metric in an $o(\varepsilon)$ collar, pass to smooth doubles of the pages, run Brownian motion on each double, and allow the page label to switch in a collar of width $\varepsilon$ at rates \(q_\varepsilon(r)Q_{ij},\) where $Q$ is an arbitrary irreducible finite-state Markov generator with invariant law $\pi$, while \(q_\eps\) is bounded and supported on an \(\eps\) collar. Denoting \(\Theta_\eps\) the total mass of \(q_\eps\) and \(\Xi_\eps\) its second raw moment, we prove that, if \(\Theta_\varepsilon\to\infty\) and \(\Xi_\varepsilon\to0\), then the glued processes converge, from every deterministic sequence of starting points, to Brownian motion with weigthed Kirchhoff interface conditions. No reversibility of $Q$, pointwise scaling ansatz for $q_\varepsilon$, or first-moment condition is required. Furthermore, we show that the second fundamental form of the pages contributes no additional interface term: curvature remains only through the bulk Laplace–Beltrami operators.

We develop a closure modeling framework for identifying missing components of dynamical systems using Kernel Ridge Regression (KRR). The framework addresses two classes of closure problems: difference-equation closures arising in ODE and PDE settings, and algebraic closures arising from moment closure in kinetic equations. For the first class, we derive an error bound in an ODE setting that quantifies contributions from time integration, approximation of unresolved scales, and interpolation required to couple unresolved-scale effects to the resolved solver. Numerical experiments on the Lorenz-63 system and the Kuramoto-Sivashinsky equation demonstrate accurate long-horizon predictions and substantial improvements over an LSTM-based closure model. For the second class, we consider moment closure for a one-dimensional kinetic equation by modeling discrepancies between kinetic and macroscopic fluxes as a function of the resolved macroscopic variables. We compare global KRR models based on PCA coordinates with spatially local models. While the global model performs well for unimodal initial conditions, its accuracy deteriorates for bimodal initial conditions. Spatially local models with appropriate modeling inputs improve robustness and achieve higher predictive accuracy.