Papers

This paper investigates a stochastic linear-quadratic (SLQ) control problem for a regime-switching jump-diffusion system. Unlike traditional regime-switching diffusion systems that couple a diffusion process with a Markov chain, we incorporate the jumps of the Markov chain into the state equation. This modeling methodology effectively captures potential gains or losses of the system during the regime transitions. It should be noted that the introduction of Markov chain jumps into the state equation leads to increased complexity in the corresponding coupled differential Riccati equations (CDREs), thereby rendering the solvability of the control problem more challenging. Under the assumption that the cost functional is uniformly convex, we establish the unique solvability of the corresponding CDREs. Building upon this foundation, we derive a closed-loop representation for the unique open-loop optimal control. Finally, we apply our theoretical results to the mean-variance portfolio selection problem in a regime-switching financial market with Markov chain jumps and obtain its efficient frontier.

A coherent radar observes sea clutter through a non-injective multiplicative combination of latent texture and coherence variables. We study a class of partially observed slow–fast diffusions for which the projected quadratic covariance has the form \(Q=S+qyy^\top\), using Field's sea-clutter diffusion as a solvable benchmark. For fixed-shape fast covariance, we derive an exact generalized spectral-gap criterion for recovery of the latent slow coordinate and a stability bound. When the fast covariance changes shape, we obtain an invariant scale–shape decomposition and exact conditions for reinforcement or cancellation of the latent quadratic signature. We then quantify the loss of resolvability caused by finite sampling and receiver noise. Controlled CIR–OU simulations verify the analytic identities and exhibit regimes in which structural observability does not yield stable reconstruction.

We study generalized Volterra companion operators on Bergman spaces over smoothly bounded convex domains of finite type. A derivative Carleson embedding gives boundedness and compactness criteria between reflexive Bergman spaces. When the target exponent is smaller than the source exponent, boundedness already implies compactness. In the other range, we also obtain an essential-norm formula. Local masses on McNeal polydiscs describe these criteria. Their norm estimates require an extra weight because radial derivatives vanish at the origin. On the Hilbert Bergman space, we characterize Schatten-class membership at and above the Hilbert–Schmidt threshold and prove a sufficient condition below it. We also give a Hilbert–Schmidt kernel test. For bounded symbols and self-maps with relatively compact image, singular values decay exponentially in a power of their index. An ellipsoid example gives a sharp power law governed by boundary type and dimension. The results extend to positive radial shifts.

A de Bruijn sequence packs all $n$-bit binary words into a cycle of length $2^n$. A de Bruijn torus is the two-dimensional analogue in which each word appears exactly once in a rectangular window. Here we consider the natural analogue for permutations using their shorthand representation (i.e., each permutation's final redundant value is omitted from the window). We show that these tori exist when $n = 2m + 1$ is odd and the torus and windows have two rows (i.e., the torus is a "tworus"). These twori can be constructed with a high degree of symmetry. More specifically, there are twori that can be partitioned into $2^{m-1}$ matching blocks where each block contains the same sequence of unordered columns. Furthermore, given one such block we can generate each successive column of a tworus in amortized $\mathcal{O}(1)$-time. We also prove non-existence results for certain sizes of tori and provide algorithms for constructing multiversal cycles (perfect necklaces) of unlabeled binary words.

Let $\Ga$ be a connected prime-valent $X$-arc-transitive Cayley graph of a finite characteristically simple group $G\cong T^k$, where $k\geqslant2$. We obtain a precise structural characterization of such graphs and their arc-transitive automorphism groups. In the cubic case, every connected symmetric Cayley graph of $T^k$, where $T$ is a finite nonabelian simple group, is normal.

Distributed intelligence concerns systems in which semi-autonomous components with local dynamics and partial observations coordinate through information exchange to maintain a shared function. This paper proposes an operational way to study such systems: measure information at the interface where a message changes a receiving action, then connect that measure to function by intervention and disturbance evaluation. We instantiate this proposal in DI-Walker, a two-dimensional four-limb embodied plant controlled by frozen Cross-Entropy-Method policies. We compare a controller using each limb's own realized-force sensor with one using the realized-force sensors of peer limbs. Under limb loss, limb slip, and weak central-control dropout, Peer-Sensor has lower late tracking error in several conditions. A corrected finite-history action-predictive estimator shows a substantially larger peer-message gain under compound failure. A future scalar functional-prediction estimator does not show the same stable advantage. We interpret this discrepancy as a methodological result: information useful for an intermediate control action can be hidden by later plant dynamics, redundancy, and context. The paper relates this result to Predictive Information, Transfer Entropy, Directed Information, information-to-go/IT-PAC ideas, empowerment, and the robust control data-rate perspective, while explicitly distinguishing operational predictive gains from exact Directed Information, channel capacity, and a formal data-rate theorem.

In this paper, we introduce fractional Cesàro–Morrey function spaces over $p$-adic fields and investigate their fundamental properties. We first study the behavior of dilation operators on these spaces and establish a Minkowski-type integral inequality. We then prove the boundedness of $p$-adic Hardy–Hilbert–type integral operators on fractional Cesàro–Morrey function spaces. As an applications, we derive the $p$-adic Hardy inequality, the Hilbert inequality, and the Hardy–Littlewood–Pólya inequality. Finally, we establish the boundedness of the Erdélyi–Kober fractional integral operator and the Hadamard fractional integral operator on these spaces.

To each modular knot, we attach a weight $2$ modular integral for $\mathrm{SL}_2(\mathbb{Z})$ whose homogenized cycle integrals recover its linking numbers with other modular knots in $S^3$. This construction answers Choie-Zagier's question of explicitly constructing modular integrals with prescribed rational period functions in weight $2$. Our formula applies to individual modular knots without symmetrization, thereby realizing an approach discussed but not pursued by Duke-Imamoḡlu-Tóth. This gives an answer to Ghys' question on pairwise linking numbers in terms of modular integrals, as sought by Simon.

Consider the space of $C^\infty$-functions on the two-dimensional sphere $S^2$ and its decomposition $\oplus\mathcal H_n$ into a direct sum of minimal rotation-invariant spaces. We consider elements of $\oplus\mathcal H_n$ as functions of two variables, a nonnegative integer variable $n$ and a complex variable $u$ (a restriction of such function to the set $n=k$ is a polynomial in $u$ of degree $\le 2k$). For operators of multiplication by $x_1$, $x_2$, $x_3$ in $C^\infty(S^2)$ we obtain the corresponding operators in $\oplus\mathcal H_n$, they are differential-difference operators in the variables $u$, $n$ (including second derivatives in $u$ and shifts $n\mapsto n\pm1$). We obtain the similar correspondence for operators of differentiation along conformal vector fields on $S^2$.

In this paper, we investigate the growth, distortion, pre-Schwarzian and Schwarzian norms of functions in the exponentially convex class \(\mathcal C_{e^\lambda}\), \(0<\lambda\le\pi/2\), defined by \(1+zf''(z)/f'(z)\prec e^{\lambda z}\). By representing the associated Schwarz function explicitly, we derive parameter-dependent estimates for \(f\), \(f'\), and the pre-Schwarzian derivative. We further obtain Schwarzian norm estimates under both the general normalization and the additional condition \(f''(0)=0\). The corresponding extremal problems are analyzed through suitable Schwarz functions, and the dependence of the resulting bounds on the exponential parameter is made explicit.

In this paper, we prove that every rationally connected projective manifold admits a smooth uniformly RC-positive Hermitian metric on its tangent bundle, answering Yang's question and giving a characterization of rational connectedness by uniform RC-positivity. We find an example to show that RC-positivity alone does not characterize rational connectedness. We also prove that uniform RC-positivity of the tangent bundle is preserved under blow-ups along connected smooth centers on compact complex manifolds. In the non-Kähler setting, we construct such metrics on all Hopf and Kato surfaces and obtain classification results for compact complex surfaces.

We investigate the semi-discrete Active Flux numerical method on Cartesian grids applied to two-dimensional linear and nonlinear shallow water equations with Coriolis source terms. These hyperbolic systems admit non-trivial stationary solutions governed by a geostrophic equilibrium. In this setting, we analyze the stationarity-preserving properties of our semi-discrete Active Flux formulation. To do so for the linear system, we apply a discrete spatial Fourier transform to the semi-discrete method and analyze the resulting evolution matrix. We show that its discrete kernel is a non-trivial discretization of the geostrophic equilibrium, which implies that the method preserves a discrete geostrophic equilibrium state. This is an inherent feature of Active Flux and is achieved without any additional modifications. Numerical experiments confirm the method's ability to maintain geostrophic stationary states for the linear system. Additionally, we numerically investigate the performance of the Active Flux method for the nonlinear shallow water system.

Magnetic navigation has emerged as a promising alternative for navigation in Global Positioning System (GPS)-denied environments, leveraging geomagnetic field maps in conjunction with onboard magnetometer measurements. However, its performance is highly sensitive to trajectory-dependent observability, which limits its practical effectiveness under conventional flight paths. This paper proposes an optimal trajectory design framework for magnetic navigation that maximizes information content along the flight path. The trajectory generation problem is formulated as an optimal control problem that minimizes the posterior Cramér–Rao lower bound on the position estimation error, subject to a penalty on path length. The resulting trajectories are non-intuitive and significantly enhance the observability of the navigation system. Simulation results demonstrate that the proposed optimal trajectories yield substantial reductions in estimation error compared to conventional straight-line trajectories, highlighting the critical role of trajectory design in enabling high-accuracy magnetic navigation. These findings suggest that trajectory optimization can substantially improve the viability of magnetic navigation as a robust alternative for aerospace applications in GPS-denied environments.

We give a unified treatment of definite Albert algebras over $\mathbb{Z}$, with exceptional Siegel–Weil identities as the main arithmetic tool. Starting from the standard octonion order alone, its first scalar theta coefficient saturates the genus average and forces every remaining class to be rootless. A cubic-ring coefficient of the $G_2$ theta lift then gives the mass of the remaining scale-$2$ framed orders, also recovered from shell geometry. A uniform Niemeier–Golay stabilizer bound saturates this mass, proving that there is exactly one further class and determining both automorphism orders, frame transitivity, and the $2457$ frames on $819$ rank-one elements. The associated generalized hexagon identifies the nonstandard group as ${}^3D_4(2):3$ only after its order has been proved. An independent weighted-theta argument gives the trace-zero root-system alternative $\varnothing$ or $A_2E_8^3$. The construction relates positive masses of framed objects to exceptional Eisenstein coefficients.

This article is the second of three articles whose goal is to generalize the Cartier transform of Ogus and Vologodsky to the logarithmic setting. We generalize a topos-theoretic version of this transform, due to Oyama. Let $k$ be a perfect field of positive characteristic $p$ and equip $S=\operatorname{Spec}k$ with the trivial log structure. For a log smooth morphism of logarithmic schemes $X \rightarrow S,$ we construct crystalline-like ringed topoi $\mathcal{E}'$ and $\underline{\mathcal{E}}$ and subcategories of crystals of quasi-coherent modules $\mathcal{C}'$ and $\underline{\mathcal{C}},$ equivalent respectively, under some lifting assumption, to modules with Higgs fields and integrable connections, both satisfying certain nilpotence conditions, and a morphism of topoi $\underline{\mathcal{E}} \rightarrow \mathcal{E}'.$ We then prove that the pullback functor of this morphism of topoi preserves quasi-coherent crystals and hence induces a functor $\mathcal{C}' \rightarrow \underline{\mathcal{C}},$ generalizing the Cartier transform. We finally use a log flat descent theorem for morphisms, that we proved in the first article, to prove that this functor is fully faithful.

We propose a time-stepping minimum-residual (MinRes) framework for transient coercive variational problems, applicable to finite-element and neural-network trial approximations. For the Backward Euler scheme, we measure the residual at each time level in the dual norm induced by the steady test-space norm, scaled by the time step. This choice yields stability estimates controlling a discrete parabolic energy quantity, up to the time-discretization defect. We develop conforming and broken-test formulations and specialize the analysis to diffusion-advection-reaction problems. For finite-dimensional test spaces, we derive fully discrete reliability estimates by supplementing the computable discrete residual with a complementary contribution that accounts for residual components the test space does not resolve. We then extend the framework to neural-network trial classes and obtain computable residual decompositions for conforming and broken polynomial test spaces. Numerical experiments confirm the expected finite-element convergence rates, examine the effect of test-space refinement for a global space-time neural approximation, and demonstrate residual-driven spatial refinement for a transient problem with a moving localized feature.

Score-based diffusion models are increasingly considered in settings where the underlying data distribution may differ from the training distribution, yet existing theoretical guarantees largely focus on the no-shift setting. In this work, we study robust score estimation under Wasserstein perturbations of a reference distribution. For the Ornstein–Uhlenbeck diffusion, we show that robust estimation decomposes into two fundamental components: the statistical cost of learning the reference distribution and the intrinsic cost of distribution shift. The latter scales quadratically with the Wasserstein radius, and this dependence is minimax optimal. We construct an explicit finite-sample estimator achieving the resulting robust minimax rate without knowing the shift radius. When the reference distribution lies on an unknown low-dimensional subspace, the statistical term adapts to the intrinsic dimension while the shift cost remains unchanged. Finally, we show that the same decomposition governs positive-time reverse sampling and obtain matching minimax guarantees in KL divergence. Together, these results characterize how finite data, intrinsic dimension, and distribution shift affect the robustness of score-based diffusion models.

In data-driven contextual stochastic optimization, existing approaches are predominantly frequentist: they commit to a single predictive model and treat it as ground truth, yielding prescriptions that are fragile to model uncertainty and sampling variability in small- and moderate-sample regimes. We propose Bayesian contextual optimization (BCO), a framework that maintains a Gibbs posterior over the parameter space. This decision-focused posterior weights candidate models by their empirical decision quality rather than statistical fit, thereby avoiding commitment to a potentially misspecified likelihood while encoding the full distribution of plausible models consistent with the data. A Bayesian contextual policy is then derived by minimizing the expected decision cost under the posterior predictive distribution, hedging prescriptions against model uncertainty by aggregating over the posterior. We establish three theoretical guarantees: (i) the Gibbs posterior concentrates exponentially fast around the frequentist best-in-class parameter set; (ii) BCO can strictly improve over the frequentist best-in-class policy when model uncertainty is non-negligible; and (iii) BCO attains an $O(n^{-1/2})$ excess risk rate against the oracle over all probability measures up to a misspecification term and an oracle aggregate gap. Computationally, we tailor a variational inference scheme that has the same per-iteration cost as frequentist alternatives and a gradient-free Metropolis-Hastings algorithm that handles nondifferentiable problems. Numerical experiments on two-stage shipment planning, contextual newsvendor, and return-constrained portfolio problems confirm that BCO consistently reduces out-of-sample cost and variance relative to kernel estimators and decision-focused baselines, with the most pronounced gains in small- and moderate-sample regimes under substantial model uncertainty.

The Tutte polynomial is a bivariate polynomial that has been extensively studied in graph and matroid theory. Gordon was the first to study the Tutte polynomial $T(P;x,y)$ of the greedoid induced by a poset $P$, including the special case of series-parallel posets. In particular, Gordon and McMahon conjectured that, for any two series-parallel posets $P$ and $Q$, the equality $T(P;x,y)=T(Q;x,y)$ holds if and only if $P\cong Q$. In studying this conjecture, Gordon introduced a subclass $\mathcal{P}$ of series-parallel posets and proved that this equivalence holds for all $P,Q\in\mathcal{P}$. In this paper, we introduce a new subclass $\mathrm H$ of series-parallel posets and prove that $\mathcal P\subseteq\mathrm H$. Moreover, we show that, for every $P\in\mathrm H$ and every $Q\in\mathrm{SP}$, the equality $T(P;x,y)=T(Q;x,y)$ holds if and only if $P\cong Q$, thereby extending Gordon's result.

We analyze two fixed-point iterations for computing the principal square root of an M-matrix $A$. Although these iterations, with customary initialization, converge sublinearly when $A$ is a singular M-matrix, we show that, under suitable mild conditions on the initial approximation, the convergence is linear. Moreover, we provide component-wise accurate versions of these iterations, which allow us to approximate the principal square root with a component-wise relative error uniformly bounded by a small multiple of the machine precision. Numerical experiments demonstrating the effectiveness of the proposed algorithms for certain classes of problems are presented.

Block error rate is a standard metric in finite blocklength (FBL) communication, yet it conflates two qualitatively different failure modes: block confusions, where the decoder selects a wrong codeword, and block erasures, where it declares a loss. Higher-layer protocols treat physical-layer failures as erasures, but this cross-layer assumption lacks FBL justification. We derive a rigorous upper bound and a companion lower-side estimate on the block confusion probability (BLCP) and block erasure probability (BLEP) for bounded-distance decoders over additive white Gaussian noise (AWGN) channels at finite blocklength, recasting the coding problem as a geometric sphere packing one. We analyze the sensitivity of these bounds to blocklength and signalto-noise ratio, characterize the envelope of the upper bound, and derive closed-form Chernoff approximations. Extending the model to idle transmission blocks, we bound the false alarm probability (FAP) and show that a block z-channel abstraction emerges from the bounded-decoding geometry. Numerical results confirm that confusion and false alarm probabilities lie far below the error rate constraint, providing quantitative physicallayer support for the block erasure channel and block z-channel abstractions assumed in protocol design.

We characterise C*-irreducible regular C*-inclusions using a number of different conditions considered by different authors. In particular, we show that all C*-irreducible regular inclusions $A\subseteq B$ are modelled by outer Fell bundles $(B_{g})_{g\in G}$ over discrete groups with a simple unit fibre $A=B_1$. In this case, we prove a bijection between intermediate C*-algebras $A\subseteq C \subseteq B$ and subgroups $H$ of $G$. This extends the Galois correspondence for reduced crossed products by discrete group actions established by Cameron-Smith. We relate it to the Galois correspondences of Izumi and Mukohara for fixed-point algebras of actions of compact abelian groups, and the mixed inclusion of a fixed-point algebra in a reduced crossed product considered by Echterhoff-Rørdam. In addition, using a recent result of Geffen-Ursu, we show that the inclusion of a fixed-point subalgebra $A\subseteq B$ of an action of $\mathbb{T}$ or $\mathbb{Z}/p$ for a square-free number $p>0$ is aperiodic if and only if $A$ detects ideals in $B$. We apply this to give examples of C*-irreducible inclusions coming from Cuntz-Pimsner algebras, including crossed products by endomorphisms or transfer operators. In particular, we characterise when a core subalgebra of a graph C*-algebra is C*-irreducible. Lastly, we show that a general regular topologically graded C*-inclusion $A\subseteq B$ is aperiodic and has a unique pseudo-expectation provided $A$ detects ideals in all intermediate C*-algebras of $B$. This partially answers a question by Pitts-Zarikian.

We prove that every minimizing $p$-harmonic map from $B^3$ into $\mathbb S^3$ is locally $C^{1,\alpha}$ for some $\alpha\in(0,1)$, for every $p>p_0$, where $p_0=\frac{7-\sqrt{17}}{2}\approx 1.44$. This closes the gap between the previously established regularity ranges $[2,2.642]\cup[2.961,3]$ and, in particular, yields full interior regularity for all $p\ge2$. The result also extends regularity to the subquadratic range $p_0<p<2$.

We investigate the fast-advection limit ($\varepsilon \to 0$) of the stochastically forced two-dimensional incompressible Navier-Stokes equations on the torus. Numerical simulations are performed at resolution $N = 128$ for four different forcing shells on both the square torus and a thin torus, with the advection parameter $\varepsilon$ varied over approximately one decade. We find that the stationary distributions become increasingly concentrated on the lowest Fourier modes as $\varepsilon \to 0$, with the energy-to-enstrophy ratio $E/\Omega$ and the fraction $R$ of enstrophy in the lowest modes both increasing monotonically. At finite $\varepsilon$ the degree of concentration depends on the forcing shell through the effective spectral value $B_1/B_0$, in agreement with the condensation bound of Sznitman and Widmayer. Linear extrapolation to $\varepsilon = 0$ suggests that the limiting values may be forcing-independent, with $B_1/B_0$ governing only the rate of convergence. The same qualitative behaviour persists on the thin torus.

We revisit the classic 1959 paper of L. D. Landau on possible singularities of the integral associated to a Feynman graph. Focusing on the real setup, we relate it to convex geometry by representing a collection $p$ of incoming momenta as the weighted normals of a convex polytope $Q$ via the Minkowski problem. Then, a regular polyhedral subdivision $\mathcal{P}$ of $Q$ exhibits $p$ as a Landau singularity labelled by the dual graph of $\mathcal{P}$ with masses being the areas of the faces. Positivity of the Landau/Feynman/Schwinger multipliers is interpreted as strict convexity of a PL-function. This gives an interesting class of “polyhedral” Landau singularities. In the planar case going back to the original 1959 paper, Landau graphs can also be identified with plane webs of Gaiotto-Moore-Witten that provide a language dual to that of regular polygonal subdivisions.

Following the work [27], we further study spacetime observability for the wave equation on a one-dimensional interval. The main task is to characterize the observable symmetry condition (OSC) under both Dirichlet and Neumann boundary conditions, which is more complex than the torus setting due to boundary reflection, and to discuss the relation between OSC under different settings. We prove that OSC, together with the geometric control condition (GCC), is necessary and sufficient for observability. We also provide a necessary and sufficient condition for the related unique continuation property.

In this paper, we consider a nonlocal elliptic equation involving a nonnegative parameter $a$ that quantifies the strength of the nonlocal term, posed in a smooth bounded domain with homogeneous Dirichlet boundary conditions and a given nonnegative source term $f$. This class of problems arises in several applications, including the modeling of thermistor behaviour and population dynamics with crowding effects. We focus on the breakdown of the maximum principle as the parameter $a$ increases. We characterize the two types of degeneracy that may occur, possibly simultaneously, at the critical parameter $a^*$: the solution may vanish at an interior point of the domain, possibly forming a dead-core region, or its normal derivative may vanish on the boundary. In particular, we obtain sufficient conditions for the emergence of dead-core solutions, thereby answering an open question raised in the recent literature. Several one- and two-dimensional examples are constructed to illustrate each possible scenario. Additionally, we study an optimization problem motivated by biological considerations, where the goal is to maximize the critical threshold $a^*$ over an admissible class of resource distributions. For sources with prescribed total mass, a uniform upper bound, and an interior-contact condition, we show that the critical threshold is unbounded and that the corresponding total population can be arbitrarily small. We also establish the existence of a maximizer in a class with a prescribed positive lower bound on the total population, a uniform bound on the sources, and the same interior-contact condition.

For positive integers $n,r,t$, let $\delta(n,r,t)$ denote the maximum possible minimum degree of a balanced $r$-partite graph with parts of size $n$ and chromatic number at most $t$. Lo, Treglown and Zhao established a general upper bound for this parameter and used it, together with explicit constructions, to determine the corresponding multipartite clique threshold up to an additive constant in a broad parameter range. I determine the chromatic parameter throughout the range $r=mt-a$, $m\ge2$, $t\ge3$, $2\le a\le \min\{m,t-1\}$. The answer differs from the Lo–Treglown–Zhao upper bound by at most one. I give an explicit arithmetic criterion deciding when this one-unit correction occurs. The proof reduces the problem to an integer matrix extremum. In the boundary case, equality forces the supports of all mixed rows to form a spanning star, after which the only remaining obstruction is a divisibility condition. Combining this formula with the Andrasfai–Erdos–Sos theorem sharpens the known equality range for $f(n,r,t+1)=\delta(n,r,t)$. In particular, for $t=3$ it removes the remaining size restrictions at $r=10$ and $r=13$. Together with the $r=7$ result in arXiv:2609.19177, the classical $r=4$ case, and the known congruence classes, this gives a formula for the multipartite $K_4$-free problem for every admissible $r\ge4$ and every $n\ge1$.

We investigate the optimal damping design and exponential stabilization for a class of higher-dimensional weakly degenerate damped wave equations. In weighted Sobolev spaces, we establish well-posedness and uniform exponential energy decay, and verify the existence of long-time optimal damping potentials. To overcome analytical difficulties arising from system non-self-adjointness, we use finite-time approximation via semigroup operator norms. Through linearization and adjoint-state analysis, we derive first-order necessary optimality conditions and show finite-time optimal dampings possess a bang-bang structure.

Algorithms in multi-agent systems such as federated learning, mobile robotic swarming, and consensus control can be designed and analyzed as distributed stochastic approximation algorithms. Such algorithms involve information exchanges between agents for various computations. The freshness of the information can be quantified using the Age of Information (AoI) metric. Consider robotic teams operating in highly obstructed geographical settings, such as subterranean or dense urban environments. Because of spatial disconnections, AoI has empirically been observed to be heavy-tailed with unbounded moments. However, most analyses assume AoI with bounded moments, creating a gap between theory and practice. To the best of our knowledge, ours is the first analysis under general heavy-tailed AoI with potentially infinite mean. We study the stability (almost sure boundedness of the distributed iterates) and convergence of multi-agent systems that are strictly dissipative in the scaling limit (system at “infinity”). Examples include most gradient-based and consensus algorithms under the Robbins-Monro step-size regime.