Papers

We study optimal scheduling for remote state estimation over a network with two heterogeneous communication channels: a fast but unreliable channel and a slow but reliable channel. To capture temporal correlations in packet losses, we model the unreliable channel as a Gilbert-Elliott (GE) channel. The remote estimation setup consists of a source, a sensor, and a remote estimator. The source evolves as a discrete-time autoregressive (AR) process, and the sensor decides at each time whether to use the fast unreliable channel or the slow reliable channel. We formulate the scheduling problem faced by the sensor as a Markov decision process (MDP) with a continuous state-space and consider minimizing the infinite horizon average cost criterion, where the cost consists of the squared estimation error and the transmission energy consumed. We establish the existence of an optimal stationary policy. We then characterize the structure of an optimal policy, and show that it has a threshold structure with respect to the estimation error. An optimal policy chooses from amongst the two channels based on whether the error exceeds certain thresholds, where the threshold value depends upon the GE channel state. When the system parameters are unknown, we propose an actor-critic (AC) learning algorithm that exploits the threshold structure of an optimal policy. Numerical results demonstrate that the proposed AC algorithm learns the policy structure effectively and achieves performance close to that of the optimal policy computed using the relative value iteration (RVI).

In this paper, we construct countable uniformly locally finite metric spaces $X$ and $Y$ such that $C_u^*(X)$ is isomorphic to a hereditary $C^*$-subalgebra of $C_u^*(Y)$, but neither space coarsely embeds into the other. This gives a negative answer to the embedding rigidity problem for uniformly locally finite coarse spaces. On the positive side, we prove that a hereditary embedding $C_u^*(X)\to C_u^*(Y)$ implies an injective coarse embedding $X\to Y$ whenever $C_u^*(Y)$ contains no block-rank-one ghost projection. In particular, this strengthens the coarse embedding theorem of Braga, Farah, and Vignati by obtaining injectivity under the same hypothesis that every sparse subspace of $Y$ yields only compact ghost projections.

There is significant interest in the study of calculus on graphs, especially regarding the use of gradient-based methods for applications in data driven problems such as classification, clustering and regularisation for inverse problems. Geometric graphs, whose vertices are take from from a Euclidean domain and whose edge structure is determined by the distance between the nodes in the domain, have been central in theoretical studies. Typical approaches for analysis, such as studying consistency and the existence of continuum limits, rely on $\Gamma$-convergence. This technique has some limitations, as it requires the typical length scale which determines the connectivity structure of the graph to be much larger than the scales frequently used for applications. Moreover, it may fail to provide quantitative results. This paper provides necessary and sufficient conditions on the asymptotic behaviour of this length scale for the existence of a uniform collection of Sobolev inequalities on a sequence of geometric graphs. Furthermore, these inequalities hold when the length scales are much smaller than what is typically assumed for $\Gamma$-convergence results and within the range of what is used for data-driven problems. The Sobolev inequalities provide a quantitative estimate on the $L^q$-regularisation effect of discrete gradients.

We study algebraic geometry codes on hyperelliptic curves of genus $g \geq 2$ equipped with a rational Weierstrass point, with a focus on complementarity properties. Our first contribution is a characterization of non-special divisors of degree $g$ and $g-1$ via the polynomial degrees of their reduced Mumford representation, reducing a classical geometric problem to a degree test on univariate polynomials. Using this, we construct Linear Complementary Pairs (LCP) of codes via polynomial arithmetic on the Jacobian and provide a criterion in terms of Mumford degrees for the resulting codes to be Maximum Distance Separable (MDS). Under a $2$-torsion condition in the Jacobian, equivalently a divisibility condition on the Mumford polynomials, we obtain explicit multipliers that turn these pairs into Linear Complementary Dual (LCD) codes. We give an MDS LCP with parameters $[8,4,5]_{16}$ in characteristic two, an MDS LCD code with parameters $[10,5,6]_{25}$ on the maximal hyperelliptic curve $y^2=x^5+x$, and an LCD code with parameters $[28,14]_{49}$ on $y^2=x^7+x$. Motivated by the explicit $q=5$ computation, we conjecture for odd prime powers $q\ge5$ the existence of MDS LCD codes with parameters $[2q,q,q+1]_{q^2}$ arising from this construction.

We show that large rosy fields are bounded, and substantially simplify the proofs that large stable fields are separably closed and that large simple fields are bounded. Our proofs go through in a general topological setting. We obtain instability, non-simplicity, and non-rosiness explicitly by building the appropriate trees of definable sets. We also show that orders on large rosy fields have several properties of orders on pseudo real closed fields.

By modifying the generating function of the Chebyshev polynomials of the second kind, we obtain a sequence of reciprocal (or palindromic) polynomials, as well as a related companion sequence. Among numerous other properties, we obtain discriminant and resultant identities for these polynomial sequences and prove partial irreducibility results. Throughout, we point to parallels and connections with the Chebyshev polynomials of both kinds.

We consider a novel robust utility maximisation problem under bounded cumulative transition rate uncertainty within the class of non-Markovian marked point processes on a finite state-space. Utility is maximised over the class of admissible controls, while Nature chooses a worst-case biometric scenario from the class of admissible, path-dependent cumulative transition rates restricted by path-dependent upper and lower bounds. We prove a martingale optimality principle and a novel existence and uniqueness result for a non-standard worst-case backwards stochastic differential equation, which allows us to establish existence and uniqueness of worst-case and best-case prospective reserves of life and health insurance contracts with reserve-dependent payments. Finally, we find an explicit solution of a novel robust consumption-insurance problem with power utility preferences.

We prove an analogue of Grünbaum's inequality on the sphere. Let $n \geq 3$ and let $K$ be a convex body on $\mathbb S^{n-1}\subset \mathbb R^n$ with centroid at $\theta\in \mathbb S^{n-1}$. Then for any $u\in \mathbb S^{n-1}$ that is orthogonal to $\theta$ we have $$\sigma(K\cap u^+) \ge \left(1-\frac{1}{n}\right)^{n-1} \sigma(K),$$ where $\sigma$ denotes the spherical measure. The constant in this inequality is optimal.

We develop new adaptive numerical schemes for one- and two-dimensional hyperbolic systems of conservation laws. The methodology relies on the use of a smoothness indicator to automatically partition the computational domain into smooth and nonsmooth (“rough“) regions. We then follow the scheme adaption strategy recently introduced in [S. Chu, P. Feng, V. A. Kolotilov, A. Kurganov, and V. V. Ostapenko, Commun. Comput. Phys., accepted], but instead of the quasi-linear (QL) fifth-order finite-difference scheme used there, we employ the new QL seventh- and ninth-order schemes in the smooth regions. A series of numerical experiments for the Euler equations of gas dynamics demonstrates that the new adaptive schemes contain a smaller amount of numerical dissipation and achieve higher resolution compared with their counterpart that uses the QL fifth-order scheme in the smooth areas.

It is well-known that for a Gabor frame in L2, the canonical dual window is the dual window with the minimal L2-norm. In this paper, we address the problem of finding dual windows for a Gabor frame that minimizes the L1-norm. Since it has been shown that minimizing the L1-norm often coincides with minimizing the L0-norm, we also investigate the support of the dual Gabor window that achieves the minimal L1-norm.

Covariance steering provides an efficient framework for designing linear stochastic feedback policies, but its extension to nonlinear systems relies on a Gaussian surrogate obtained through local linearization. Because this surrogate may differ substantially from the true nonlinear state distribution, risk-sensitive quantities such as collision probability and mean-squared error may be inaccurately estimated. This work develops a distributionally robust covariance-steering framework based on the relative entropy, also known as the Kullback-Leibler divergence (KLD), to account for ambiguity in the propagated probability density function. Using a variational representation of exponential integrals, we derive computable upper bounds on risk-sensitive quantities over a KLD ambiguity set. We then formulate an upper bound on the time rate of change of the KLD between the true nonlinear distribution and a Gaussian reference surrogate. Under some assumptions, this bound is controlled by decision variables within a covariance-steering formulation. The resulting constraints are incorporated into a sequential convex programming algorithm to design stochastic guidance policies that keep the true distribution close to its Gaussian surrogate while enforcing bounds on risk-sensitive performance measures. The proposed approach is demonstrated on a challenging nonlinear spacecraft transfer between two near-rectilinear halo orbits.

We study elliptic superlevel geometry and return dynamics for the class $\mathcal{O}_{C}$. Let $C \subset \mathbb{R}^{N}$ be a compact strictly convex core, and $\Omega \in \mathcal{O}_{C}$ a bounded domain whose boundary is a global normal graph over $\partial C$ with inward normals returning to $\operatorname{int} C$. For $-\Delta u = f$ in $\Omega$ ($u=0$ on $\partial\Omega$), with $f \geq 0$, $f \not\equiv 0$, $\operatorname{supp} f \Subset \operatorname{int} C$, we analyze the superlevel sets $\Omega_{t} = \{u > t\}$ for $t \downarrow 0$. We prove that for small $t > 0$, $\partial\Omega_{t}$ is a global normal graph given by $\partial\Omega_{t} = \{c + d_{t}(c)\nu(c) : c \in \partial C\}$, satisfying $\|d_{t} - d_{0}\|_{C^{1}} = O(t)$ and $d_{t} = d_{0} - t b_{0} + o_{C^{1}}(t)$, where $b_{0}$ depends on the Hopf derivative and transversality. Using normal-graph formulas, we show $\Omega_{t} \in \mathcal{O}_{C}$. The return maps $F_{t} : \partial C \to \partial C$ are well-defined, satisfying $\|F_{t} - F_{0}\|_{L^{\infty}} = O(t)$ and $F_{t} = F_{0} + t \mathcal{A}_{0} + o_{L^{\infty}}(t)$. For any sequence $t_{n} \downarrow 0$, the non-autonomous system $x_{n+1} = F_{t_{n}}(x_{n})$ is asymptotically autonomous with background map $F_{0}$. These results bridge elliptic boundary-layer deformation and return-map dynamics relative to a convex core.

We consider multipartite random graphs with given degree sequences, within and across different partitions. Under general assumptions, we prove the local limit of this graph is a multi-type branching process, establish that a giant component exists only when the local limit survives, and deduce that the typical distance is of logarithmic order in probability in the supercritical regime. Our analysis removes two major assumptions from Gamarnik and Misra (2015), where the giant component problem for this model was first considered. In particular, we do not assume irreducibility of the local limit, and provide a general framework to extract giant components even when the limiting branching process is reducible, which we hope to be useful in other contexts. We also provide a new simpler survival criterion of multi-type branching processes, which we hope to be useful when direct calculation of the spectral radius of the offspring matrix may prove to be difficult.

Let $\mathcal C$ be a symmetric tensor category over an algebraically closed field $\mathbf k$ of characteristic $\ne 2$. We study Clifford and Weyl algebras of objects of $\mathcal C$ with a (skew-)symmetric bilinear form. When the form is non-degenerate, we establish simplicity and the Azumaya property for such algebras under suitable assumptions. We also compute Clifford and Weyl algebras in the Verlinde category ${\rm Ver}_p$ and use them to prove that if $\mathcal C$ is Frobenius exact then the Weyl algebra of a symplectic object of $\mathcal C$ with finite symmetric algebra is Azumaya. Using this, we introduce the symplectic Witt group $\mathcal S\mathcal W(\mathcal C)$, the subgroup of the Brauer group ${\rm Br}(\mathcal C)$ consisting of Morita classes of such Azumaya algebras, and when $\mathcal C={\rm Rep}(G)\boxtimes{\rm sVec}$ for a finite group $G$ of order coprime to ${\rm char}(\mathbf k)$, express $\mathcal S\mathcal W(\mathcal C)$ in terms of second Stiefel-Whitney classes of orthogonal representations of $G$.

We prove a version of the Birkhoff-Kellogg theorem in product spaces, under affine transformations. This is a fairly natural framework that arises when dealing with parameter-dependent systems of functional-differential equations. We provide criteria on the existence of parameter-dependent solutions that have all the components nontrivial. The theoretical results are illustrated in the case of systems of second order functional-differential equations, where the functional part can cover the interesting cases of time and state-dependent deviated arguments. We provide a concrete example, for which we furnish numerically approximated solutions, coherent with our theoretical framework, and in which we compute or estimate all the constants that are required by our abstract results.

This work develops a neural operator preconditioned subspace method for sequences of linear systems arising from the discretization of the two-dimensional Helmholtz equation in transcranial ultrasound applications. The problem involves strongly heterogeneous, patient-dependent velocity fields that induce severe wave distortion and pose significant challenges for standard iterative solvers. Building on neural network preconditioning framework of Giraud et al. (HAL RR-9593, 2025) and the idealized skull dataset used for the learned optimizer of Stanziola et al. (JCP 441, 2021), neural operator preconditioners are trained on six mixed velocity-source datasets combining randomized source configurations and idealized skull-based velocity fields with random noise. The proposed mixed-dataset strategy aims to improve both computational efficiency and generalization across varying configurations. The neural operator is trained on a coarse grid using a physics-informed loss based on the relative residual of the discrete Helmholtz equation and is incorporated as a nonlinear preconditioner within flexible GMRES (FGMRES). Numerical experiments demonstrate that the resulting hybrid method efficiently solves practical transcranial ultrasound problems on grids 64 times larger than those used during training, whereas both classical GMRES and the learned optimizer fail to converge within comparable computational budgets. Moreover, the proposed method achieves arbitrary solution accuracies and exhibits strong out-of-distribution generalization across diverse source and velocity configurations. This work highlights the importance of dataset design in scientific machine learning and provides a practical framework for integrating matrix-free neural operator preconditioning with Krylov subspace methods for solving practical large-scale Helmholtz problems.

Inside the Grazer Burg, two late-Gothic stone flights rise about distinct spindles, overlap, share several treads, and separate again. Their plan is governed not by a coaxial double helix but, to first approximation, by two intersecting circles. This elementary geometry yields a model of recurrent meeting and makes explicit the compatibility conditions that meeting requires. It also leads to a second object that geometers call a double spiral staircase - the helicoid - and to a useful distinction between resemblance and identity. The staircase becomes a meditation on how paths, models, and disciplines can meet without becoming the same.

We establish twisted Siegel-Weil formulas for non-Galois quartic CM fields, identifying the twisted theta integral against a quadratic character with the Doi-Naganuma lift of Hecke's integral. This implies the base change of Jacquet-Langlands correspondence for certain Hecke characters from $\mathbb{Q}$ to real quadratic fields via Doi-Naganuma lift is an isomorphism. As an application, we prove that the twisted CM values of Borcherds forms are algebraic multiple of logarithm of units, explicitly described by the Fourier coefficients of twisted theta integrals.

A truckload carrier must accept or reject each load tender within seconds. The decision depends on fleet state, hours-of-service (HOS) clocks, and appointment windows. We model this as a weakly coupled dynamic program in which the resources relocate and carry clocks: serving a request moves the truck to a new market and depletes its clocks, and whether a truck can serve a request depends on its state. Occupancy-based reusable-resource models do not cover this setting. We build a real-time dual-price policy from the same Lagrangian relaxation that gives the problem's upper bound. Policy and bound come from one object, so every run reports a certified optimality gap. We prove three things. First, the certificate is valid for any duals, any discretization, and any surrogate quality. Second, the policy's same-time spatial-gradient rule is exactly fluid complementary slackness, and the policy is asymptotically optimal in the subcritical fluid regime; the fitted prices are also portable across sample paths, by linear-programming basis stability. Third, certificates have limits: per-resource Lagrangian slack can stay bounded away from zero at every fleet size. We exhibit a three-truck kernel with an exact rational certificate and a replication lemma. On a public closed-loop benchmark with thirty paired seeds, the policy – which needs no rollout labels, only one offline dual solve – beats a rollout-trained surrogate on two of three scenarios (tight: +2.0 pp, 95% CI [+0.5, +3.6], Wilcoxon p = 0.023; mild: +3.5 pp, CI [+2.4, +4.5]) and ties the third. It decides in 0.04-0.09 ms, three orders of magnitude faster than the Monte Carlo rollout teacher. Its certificates are stable across ten bounded instances per scenario, at 57-64% of optimal, within 3-6 points of what the 1000x-slower teacher certifies.

Spread codes are a well-known family of constant-dimension subspace-metric codes. For constant dimension $k$ and ambient space dimension $n$ being a multiple of $k$, these codes have minimum distance $2k$ and a rich geometric structure. In this paper, we study the decoding capabilities of the Nearest Neighbor Decoder for Desarguesian spread codes, establishing that unique decoding is still achievable beyond half the minimum distance. Motivated by this, we develop a new decoding algorithm to uniquely decode Desarguesian spread codes in the presence of both insertions and deletions, which increase and decrease, respectively, the dimension of the transmitted codeword. Even when the sum of the dimensions of insertions and deletions exceeds half the minimum distance, provided that deletions are of dimension at most $k-2$, the algorithm succeeds with a small decoding failure. We also propose two refinements to this algorithm that, empirically, can handle nearly as many insertions as the Nearest Neighbor Decoder.

The neighbourhood complexity $nc(G,k)$ of a graph $G$ is a quantity measuring, for a graph $G$ and an integer $k$, the maximum possible number (over all vertex subsets $S$ of size $k$) $|\{N[v]\cap S, v\in V(G)\}|$ of $S$-neighbourhoods in $G$. This notion is important in structural graph theory and algorithm design (especially in parameterized complexity, in particular model checking and kernelization). While generally $nc(G,k)\leq 2^k$ and this bound can be achieved, it is known that sparse graphs and structured dense graphs have linear neighbourhood complexity, that is, $nc(G,k)\in O(k)$ for any such graph $G$. However, for many graph classes, the best possible constants are not known. We focus on graphs of bounded treewidth and pathwidth, showing that (when $k\geq w+1$) (i) if $G$ has treewidth $w\geq 2$, then $nc(G,k)\leq (k-w+1)2^{w}+w$, and (ii) if $G$ has pathwidth $w\geq 2$, then $nc(G,k)\leq (k-w+2)2^{w-1}+2k-w-2$. Moreover, we provide constructions that reach these bounds, whenever $w\geq 2$ and $k\geq 2w+1$ ($k\geq 2w-1$ for pathwidth). Interestingly, in contrast, we also have the tight bound $nc(G,k)\leq \frac{7}{3}k$, for graphs $G$ with pathwidth 1 or treewidth 1.

We introduce and implement a novel scheme, called PolyChopper, for splitting a convex polyhedron into a finite set of disjoint convex sub-polyhedra. The algorithm cuts the polyhedron with a plane, which can either be prescribed as an external constraint or automatically determined from the inertia tensor. The resulting intersection vertices are then adjusted to control the complexity and quality of the generated elements, introducing additional “notch wedges”, tetrahedra and pyramids, according to a user-defined quality parameter. Experimental results demonstrate that the proposed approach is robust and consistently produces sub-polyhedra that are both simpler and of higher quality than the original polyhedron. Furthermore, recursively applying the algorithm generates a hierarchy of polyhedral subdivisions with progressively smaller elements, all guaranteed to remain convex and predominantly tetrahedral.

We construct a compact metric space with an atomless probability measure and a bounded Lipschitz cost for which the entropic optimal-transport minimisers have no zero-temperature weak limit. More precisely, $P_\varepsilon$ does not converge as $\varepsilon\downarrow0$. In the example, every unregularised minimiser is singular with respect to $\mu\otimes\mu$, so that the entropy on the optimal face is identically $+\infty$. We describe the cluster set by \[ \operatorname{Clust}(P_\varepsilon)=\{P_w:w\in\mathcal W\}, \] where $P_w$ is the mixture of the two zero-cost graph couplings with weight $w$, and where $\mathcal W\subset[0,1]$ is a non-degenerate compact interval. We then compute two explicit points $w^-<w^+$ in this interval. This shows that compactness, atomlessness, and Lipschitz regularity of the cost do not imply zero-temperature convergence. We also present a compactness theorem for the general problem. If $C\in L^1(\mu\otimes\nu)$ is continuous and bounded from below on Polish spaces, then the zero-temperature cluster set is a nonempty weakly compact connected subset of the optimal face. In the proof, we apply the cluster-point theorem of Bernton, Ghosal, and Nutz and the continuity of $\varepsilon\mapsto\pi_\varepsilon$. Finally, we give local and exterior first-order criteria for full convergence and cluster membership. We show that nonconvergence is possible, but only through a connected continuum of optimal plans.

We study variants of graph colouring with parity constraints. More specifically, we consider $q$-colourings $c\colon V(G)\rightarrow \{1,\dots,q\}$ of a graph $G$ where, for every vertex $v\in V(G)$, the number of neighbours $w$ of $v$ with $c(w)=c(v)$ is restricted to be odd, even, positive, zero or a combination thereof. For every colour $i\neq c(v)$ the number of neighbours $w$ of $v$ with $c(w)=i$ is restricted by a constraint of similar type. Many known colouring problems such as proper colouring, defective colouring, exact defective colouring, odd colouring, and strong odd colouring can be described within this framework of constraining graph colourings, and therefore considering variants constitutes a natural generalisation of known colouring problems. We provide a comprehensive study of the computational complexity of different combinations of constraints involving parity.

In this paper, we study the local unitary equivalence of quantum states on $\mathbb{C}^n\otimes\mathbb{C}^n$, which is an important notion in quantum information theory. For the case $n=2$, the local unitary orbits of two-qubit states are completely determined by their local unitary Bargmann invariants. We show that the local unitary Bargmann invariants do not form a complete set of invariants of local unitary orbits for $n\geqslant 3$, which negatively answers a problem proposed by L. Zhang and B. Xie.

We introduce the vehicle routing problem with stochastic demands and outsourcing options (VRP-SDO), in which a logistics service provider partitions customer requests into customers outsourced to a common carrier and customers committed to its fixed fleet. The latter induces a vehicle routing problem with stochastic demands (VRP-SD), solved dynamically. Demands are revealed upon visit; residual demand may be served by other vehicles or after restocking at the depot. Work beyond the regular shift incurs overtime costs, and the unit outsourcing cost decreases with the expected outsourced demand. The objective is to minimize expected travel, overtime, and outsourcing costs. We propose an iterative two-level methodology whose first level partitions customers into committed and outsourced subsets, while the second level estimates the expected VRP-SD routing cost. To avoid solving this problem from scratch at every iteration, we learn an offline routing policy that estimates costs almost instantly for any committed subset. An iterated local search establishes the first-level partitions. We formulate the second level as a Markov decision process and solve it with a deep Q-network whose state is represented by a graph attention network aggregating customer and vehicle information by relevance to the acting vehicle. Trained offline on instances with variable customer cardinality and locations, the policy applies to any daily customer realization; online fine-tuning improves the cost approximation. Experiments show that our policy reduces routing costs by 19.6% relative to a state-of-the-art method and by at least 29.6% over classical heuristics. Our overall algorithm saves 13.7% on average over the version without the attention-based representation and generates high-quality decisions within minutes, whereas benchmarks without an offline-trained estimator require over an hour.

Let $\Omega\subset\mathbb R^n$ be a bounded $C^2$ open set and let $P_t$ be the killed Dirichlet heat semigroup. We prove the sharp fixed-time power-scale modulus of $P_t$ for the Figalli–Gigli boundary-reservoir transport distances $W_{b,p}$. For every $t>0$, $P_t$ is globally Lipschitz with respect to $W_{b,1}$. For every $p>1$, and on every total-mass sublevel $\{\mu:\mu(\Omega)\le m\}$, it is $1/p$-Hölder: \[ W_{b,p}(P_t\mu,P_t\nu)^p \le C_{t,p,m,\Omega} W_{b,p}(\mu,\nu). \] For $p>1$, we show that the exponent $1/p$ is optimal in the scale of power moduli. On the full finite-measure space, $P_t$ is discontinuous at the zero measure. To establish the lower bound, we rely on the amplification of the boundary layer. More precisely, a unit mass initially placed at distance $\varepsilon$ from $\partial\Omega$ has input $W_{b,p}$-distance $O(\varepsilon)$ from zero, whereas after any fixed positive time, its $p$-th boundary moment is bounded below by $c\varepsilon$. As a result, in the quadratic case and in the original finite-measure $W_{b,2}$ metric, there does not exist a standard finite-$\lambda$ $\mathrm{EVI}_\lambda$ semigroup on a $W_{b,2}$-metric domain which would contain the affine constant-boundary data class and could restrict to the affine constant-boundary Dirichlet heat flow. Finally, we also describe the corresponding lower-bound obstruction for smooth uniformly elliptic perturbations in divergence form.

We introduce Treasure Search Optimization (TSO), an interacting particle method for global optimization. Most swarm methods balance exploration and exploitation within a single population, and typically switch between the two by degenerating the noise, annealing a temperature, or tuning a parameter. TSO instead splits these tasks across two kinds of agents. A swarm of explorers stays in exploration mode and a single treasure hunter performs exploitation. The hunter drifts toward an objective-weighted average of the explorers and may teleport to it when the move lowers the objective. The swarm then re-centers around the hunter, creating a feedback loop between search and capture. We model the dynamics as coupled jump-diffusion stochastic differential equations (SDEs). The hunter's jumps are shared by all explorers and act as a common noise. The mean-field limit is therefore a conditional McKean-Vlasov jump-diffusion SDE, whose well-posedness we prove. We also characterize the steady state and prove, via Laplace approximation techniques, that the hunter settles near the global minimum with error of order $1/\alpha$, where $\alpha$ is the weight parameter. Linking the consensus drift to a smoothed free energy, we explain why the swarm ignores spurious local traps and demonstrate how to quantify uncertainty in inverse problems using post-processing Kalman steps after TSO iterations. Numerical experiments on ODE-constrained problems and a low dimensional Bayesian inverse problem demonstrate the effectiveness of the TSO method.

For a graph $H$ denote by $\operatorname{emb}\left(H, m\right)$ the maximal number of labeled embeddings of $H$ in a graph of size $m$. Erdős posed the question of finding $\operatorname{emb}\left(H, m\right)$ for different values of $H$ and $m$. Following related asymptotic and stability results, we determine the value of $\operatorname{emb}\left(H, {\binom{n}{2}}\right)$ for every graph $H$ with fractional independence number $v_H/2$ and all sufficiently large $n$. We also show that for those $H$ (except for matchings) and $n$ the only graph with a maximal number of embeddings is $K_n$. This fully characterizes the graphs for which $K_n$ achieves the maximal number of embeddings.

A hypergraph is linear if every pair of vertices is contained in at most one hyperedge. We study linear Turán numbers of acyclic $r$-uniform hypergraphs, extending earlier work of Gyárfás, Ruszinkó, and Sárközy from uniformity $3$ to general $r$, with emphasis on hypertrees with four hyperedges. For the linear path $P_4^r$, we prove $\operatorname{ex}^{\mathrm{lin}}_r(n,P_4^r)\le (r+1)n/r$, with equality precisely for disjoint unions of Steiner systems $S(2,r,r^2)$, whenever such systems exist. We also give counterexamples to a structural claim used in a previously proposed proof for $r=4$. The same bound and equality characterization hold for the broom $B_4^r$. For the crown $E_4^r$, writing $B=\{v:d(v)\le r\}$ and $C=\{v:r+1\le d(v)\le2r-1\}$, we prove $|E(H)|\le r|B|/(r-1)+(2r-1)|C|/r$, and hence $\operatorname{ex}^{\mathrm{lin}}_r(n,E_4^r)\le(2r-1)n/r$. We also give a lower-bound construction and formulate a sharper conjecture. Finally, for the linear star $S_k^r$, we prove $\operatorname{ex}^{\mathrm{lin}}_r(n,S_k^r)\le n(k-1)/r$, with equality precisely for $(k-1)$-regular linear $r$-uniform hypergraphs, whenever such hypergraphs exist. Under suitable divisibility and existence assumptions, we also construct $T_k^r$-free examples with $n(k-1)/r$ edges for every linear $r$-uniform hypertree $T_k^r$ with $k$ hyperedges.