Papers

We disprove the expectation stated by Xin, Yu, and Ronney that the physical positive part strain $G$-equation should possess an effective burning velocity in cellular flows. For the standard cellular flow in dimension two $V_A(x_1,x_2)=A(-\sin x_1\cos x_2,\cos x_1\sin x_2)$, if $0<d<20/399$ and $\sqrt{1+4d^2}<Ad\le1+d/10$, then for every unit planar slope the periodic correction develops oscillations at least linearly in time. The solution remains bounded below on an explicit horizontal channel through $(\pi,0)$, while at $(\pi/2,0)$ it decreases at rate at least $CA/\log A$, with $C>0$ universal. The same conclusions hold for arbitrary continuous periodic perturbations of planar initial data. Under the physical scaling $V_A(x/\varepsilon)$ and $d_\varepsilon=\varepsilon d$, an order one value gap persists between points at distance $O(\varepsilon)$ at every positive macroscopic time, so the rescaled solutions have no locally uniformly convergent subsequence. The proof uses the Hamiltonian sandwich $H_{\mathrm{unc}}\le H_+\le\widehat H$. The upper comparator $\widehat H$ is a rectangular support function, equivalently an upper expectation over a state-dependent credal set, whose reversed control dynamics possess an invariant comparison channel. We also prove that for any $C^2$ incompressible periodic flow, every $\varepsilon$-outward barrier certificate has covering radius at most $2d\varepsilon$ for all sufficiently small $\varepsilon$. We further discuss implications for statistics and machine learning: rectangular, time-consistent local uncertainty need not imply forgetting of the initial state in the long run, so additional global stability or ergodicity conditions are needed in robust sequential decision making. Two Lean 4 appendices record conditional formalizations of a sufficient $p=e_1$ subregime and of the logical assembly of the rigidity theorem for barrier certificates.

We consider for an arbitrary fixed $\rho$ and for each positive integer $n$ a multilayer feedforward artificial neural network with $\rho$ layers, $n$ neurons in the first layer (the input layer) and only one neuron, the output neuron, in the last layer. Very roughly formulated, the main result is that if the distribution of weights of connections from a layer to the next are, for all large $n$, approximated well by a fixed continuous (but otherwise arbitrary) curve which does not depend on $n$, and if the values of the $n$ input neurons are independently and identically distributed with a continuous probability density function, then there is a number $\psi$ such that for all $\varepsilon > 0$ the probability that the value of the output neuron is in $[\psi - \varepsilon, \psi + \varepsilon]$ tends to 1 as $n$ tends to infinity.

We propose a family of control variate estimators for variance reduction in design-based survey sampling and causal inference, with and without interference. In these settings, inverse probability weighting (IPW) estimators are widely used, but may have large variance when sampling, treatment, or exposure probabilities are small. Building on the observation that several common estimators, including the Hajek estimator, the normalized estimator, the augmented inverse probability weighting (AIPW) estimator, and the targeted maximum likelihood estimator (TMLE), all correct the Horvitz-Thompson estimator by canceling part of its randomness, we provide a unified interpretation of these estimators as special cases of a general control variate estimator. We then construct optimal control variates that can reduce the finite sample variance compared to these common estimators. We parameterize the proposed control variates by their bases and characterize the optimal bases through a stochastic optimization formulation. In survey sampling and causal inference without interference, the optimal bases are characterized by leading eigenvectors of matrices that depend on both the design-based sampling structure and the model-based outcome uncertainty. In causal inference under network interference, the optimal bases solve a nonconvex quadratic optimization problem; we provide a $\frac{1}{2}$-approximate solution and an alternating local search heuristic. We apply the control variate estimators to the Swiss Environmental Panel survey data and the Chinese social network data, and conduct extensive simulations to show that the proposed control variate estimators can achieve substantial variance reduction.

We propose a categorification of the Lusztig integral form of the quantum Cartan subalgebra for $sl(2)$ via a colimit of categories of equivariant coherent sheaves on finite-dimensional projective spaces. The nonequivariant version of our construction yields a nonsemisimple abelian categorification of the ring of integer-valued polynomials.

This paper presents an iterative model predictive control algorithm that stabilizes constrained nonlinear systems without evaluating a single plant derivative. By factoring the exact nonlinear dynamics into a pseudo-linear form using state- and control-dependent coefficients (SCDCs), we replace the standard nonconvex optimization with a sequence of constrained linear-quadratic programs. Refreezing the coefficient matrices along the previously predicted trajectory drives the iteration. Near the origin, we prove this sequence contracts to a unique fixed point. We explicitly bound the number of iterations required to reach any stopping tolerance, and we quantify the distance from the fixed point to a true Karush-Kuhn-Tucker point, showing this optimality gap vanishes quadratically as the state approaches the origin. Inflating the discrete algebraic Riccati equation generates terminal ingredients that guarantee recursive feasibility and asymptotic stability, even when the solver terminates early. We adapt the terminal penalty online, proving it remains uniformly bounded, and we secure output feedback through the block-observable canonical form, which extracts the exact system state directly from past inputs and outputs. Retaining the block-banded structure of the subproblem forces the computational cost to scale linearly with the horizon length $\ell$. This $O(\ell)$ complexity matches the iterative linear quadratic regulator (iLQR) but sharply undercuts the $O(\ell^3)$ scaling of dense sequential quadratic programming (SQP). Numerical studies on a saturated quadrotor, a nonholonomic integrator, and a nonminimum-phase plant illustrate the theoretical bounds and map how the algorithm compares with iLQR, SQP, and linear-parameter-varying MPC.

Stochastic geometric mechanics (SGM) is known for its potential utility in quantifying uncertainty in global climate modelling of the Earth's ocean and atmosphere while also preserving the fundamental advective transport properties of ideal fluid flow. This paper is a pedagogical review of the recent developments of the mathematical framework of stochastic geometric mechanics obtained from Lie group-invariant stochastic variational principles in the context of model building for upper ocean dynamics, The paper is divided into the following five parts. Part I discusses the origins of geometric mechanics applications in deterministic fluid dynamics. Part II focuses on the example of the deterministic 3D Euler Boussinesq (EB) equations. Part III adds stochastic transport to the 3D Euler Boussinesq (EB) and derives its SALT equations. (SALT is the abbreviation of Stochastic Advection by Lie Transport.) Part IV focuses on Lagrangian Averaged Stochastic Lie Transport, abbreviated as LA-SALT. LA-SALT treats atmospheric `climate' as the ensemble expectation, while the atmospheric `weather' is treated as a field of pathwise fluctuations, as discussed in Ed Lorenz's famous 1995 lecture. Part V applies SALT and LA-SALT to create stochastic Ocean–Atmosphere Models, abbreviated as SOAM.. The SOAM approach brings us back to Hasselmann's 1976 paradigm, which decomposes a general climate model into its deterministic and stochastic parts.

Decorrelating a multivariate system need not destroy source-variable identity. We introduce Constrained Oblique Rotation with Anchored Loadings (CORAL), which minimizes residual cross-correlation while guaranteeing a declared minimum correlation between each transformed variable and its designated source. For a p-variable correlation matrix $R$, we show that every exact decorrelator can be written as $R^{-1/2}Q$ for some orthogonal matrix $Q$, and define $\rho_\star(R)$ as the maximum common source fidelity compatible with exact decorrelation. Constructive lower bounds and rigorous analytical upper bounds tightly bracket $\rho_\star$ at [0.972,0.976], [0.959,0.962], and [0.949,0.950] in simulations with p={6,18,50}, respectively, compared with PCA's largest achievable minimum correlation between distinct principal components and matched source variables of 0.358, 0.329, and 0.172. Corresponding intervals are [0.751,0.758] for World Development Indicators and [0.826,0.835] for wine chemistry data sets. Thus, loss of source-variable identity is not inherent to exact decorrelation but depends on the decorrelator selected. CORAL uses constrained Riemannian optimization and extends to exact support restrictions.

An Euler magic matrix is an integer matrix M with MM^t = gamma I for some gamma != 0, whose squared entries sum to gamma along both main diagonals; it is proper if its squared entries are pairwise distinct. Euler gave a proper example of order 4; Müller settled orders 3 (none exists) and 8; and Kominers settled order 5. We give an order-6 construction, a case not addressed by Müller or Kominers, exhibiting a proper Euler magic matrix of order 6 with gamma = 18500 together with a second, independent one with gamma = 43290. The proof is the explicit matrix and a finite exact verification. We also record an elementary counting bound gamma >= 2485 for proper order-6 examples.

We study fractional-Sobolev Tikhonov regularization for linear inverse problems on a bounded Lipschitz domain. The regularization penalty is generated by the restricted Dirichlet fractional Laplacian, and the associated variational problem is shown to admit a unique minimizer that depends Lipschitz continuously on the data. Identifying the positive self-adjoint operator $$A_s=I+(-\Delta)^s,\, D(A_s^{1/2})=H_0^s(\Omega),$$ we transform the problem isometrically into a classical Hilbert-space Tikhonov problem with observation operator $B=KA_s^{-1/2}$. This yields explicit mean-square error bounds and an order-optimal a priori and a posteriori parameter rules under Hölder-type source conditions. The framework is illustrated by partial observations and by the backward fractional heat equation. In the latter case, $$ B^*B=A_s^{-1}e^{-2tA_s}, $$ which permits a mode-wise description of the source condition, the singular-value decay, and the effective reconstruction bandwidth. We also study the local limit $s\to1^-$: after Bourgain–Brezis–Mironescu normalization, the fractional functionals $\Gamma$-converge in $L^2(\Omega)$ to the classical $H_0^1$-Tikhonov functional, and the corresponding minimizers converge strongly in $L^2(\Omega)$. Numerical experiments for the backward fractional heat problem illustrate the reconstruction procedure and the influence of the penalty order, and confirm the predicted mean-square convergence rate to within a few percent via Monte Carlo simulation, with Morozov's discrepancy principle attaining the same order-optimal rate a posteriori.

High-throughput single-cell and spatial transcriptomic technologies provide high-resolution snapshots of heterogeneous cellular states, but their destructive nature prevents repeated measurements of the same cells over time. Consequently, temporal and spatial dynamics must be inferred from independently sampled, unaligned cell populations, making it challenging to reconstruct developmental trajectories. Optimal transport (OT) offers a geometric framework for aligning cell populations and inferring developmental trajectories, but many existing approaches focus on modeling the evolution of distributions of cells in gene expression space rather than the relational structure encoded by gene expression networks. To address this limitation, we introduce a geometric framework for analyzing the spatiotemporal evolution of gene expression networks through embeddings in Gromov–Wasserstein (GW) space. By representing each developmental stage as a graph combining gene expression and spatial proximity, our approach enables comparisons of network structure across time, continuous interpolation between developmental stages via GW geodesics, and quantification of network-level changes using Ollivier-Ricci curvature. We evaluate our framework on a spatiotemporal transcriptomic Drosophila dataset and show that GW geodesic interpolations reproduce main trends in curvature dynamics observed in empirical gene expression networks. Agreement with higher-order Co-Optimal Transport (COOT) distances, which jointly represent spatial and temporal information, further validates the framework and suggests that hypernetwork representations successfully record salient biological changes across time. In general, our approach provides a unified geometric approach to study dynamically evolving biological networks.

Scheduling in multiuser multiple input multiple output (MU-MIMO) systems is essential for efficient resource allocation and overall performance enhancement. In this work, a multiuser scheduling problem is formulated to maximize the product of user equipments' (UEs) aggregate satisfactions, which maintains user fairness. Solving such a combinatorial problem using exhaustive search (EX), which requires evaluating all possible multiuser groups within a massive number of resource blocks (RBs), is prohibitive. Instead, we propose an efficient users' satisfaction based scheduling approach (US-SA). In our US-SA, a low dimension sub-grouping matrix is constructed {at each frame}, which is used to schedule the best multiuser group in each time slot; satisfied users are eliminated from the scheduling process. Our US-SA performs close to the optimal EX method in terms of satisfaction, transmitted data amount, spectral efficiency, latency, and fairness with lower computational cost. Moreover, our experiments demonstrate that the proposed scheme outperforms competing techniques.

Kühn, Sauermann, Steiner and Wigderson recently disproved the Odd Hadwiger Conjecture, even for graphs with independence number 2. For this class of graphs the conjecture is known to be equivalent to the following: every $n$-vertex graph $G$ with independence number 2 contains $K_{\lceil \frac n2 \rceil}$ as an odd minor. While this does not hold, we prove that every graph $G$ with independence number 2 contains $K_{\lceil \frac n2 \rceil}$ as an odd minor or as a totally odd immersion.

Receptor occupancy models quantify inhibition of cancer signalling, often treating target occupancy as a proxy for downstream activity. Although this simplification yields low-dimensional models, it cannot represent delayed pathway shutdown, transient resistance, or non-monotone viability responses generated by intracellular networks. We formulate a fast–slow framework that distinguishes drug–target occupancy from downstream signalling activity. The fast variable $X$ represents occupancy, the activity variable $A$ represents pro-survival signalling, and the slow variable $B$ represents adaptive feedback including phosphatase induction, stress adaptation, or signalling rewiring. Rapid occupancy relaxation permits a quasi-steady reduction, while weak feedback guarantees global convergence to a unique equilibrium. Stronger feedback may bring the frozen activity subsystem near a saddle-node fold, producing a ghost regime in which trajectories linger near the remnant of a vanished high-activity equilibrium. We determine when slow adaptive passage preserves the inverse-square-root ghost law or, under transverse crossing, produces the dynamic delay scale $O(\varepsilon^{-1/3})$. Coupling activity to viability translates these delays into shoulders in dose–response curves, while adaptive lag may generate early-time overshoot without ad hoc forcing. A proof-of-concept fit to time-resolved viability data reproduces patterns across measured concentrations and observation times and reveals practical identifiability limitations. The framework also predicts exposure-time-dependent shifts in apparent potency, including $IC_{50}(t)$, because viability integrates signalling activity over time rather than receptor occupancy alone.

Nicomachus' identity $\sum_{k=1}^{n} k^3 = \left(\sum_{k=1}^{n} k\right)^2$ is familiar, but the structure behind it is less often emphasized. We begin with Nicomachus' original pattern, in which successive cubes are expressed as successive blocks of consecutive odd integers, and show how triangular numbers naturally enter the resulting sum-of-cubes identity. We then reverse the usual triangular-number argument and start from a general difference of squares representing $n^p$ as a sum of $n$ consecutive odd integers. This construction is possible for every $p\geq 2$, but only for $p=3$ do the two squares correspond to consecutive triangular numbers, which explains why Nicomachus' odd-number blocks fit together without gaps or repetitions. Finally, using the leading terms of the Faulhaber polynomials, we show that, apart from the trivial case, Nicomachus' identity is the unique relation of the form $S_s(N)=S_1(N)^r$ for positive integer exponents.

The space-time Galerkin discretization of retarded layer potentials leads to a linear system where the coefficients are expressed in terms of integrals over the ansatz and test elements. They require carefully designed quadrature schemes because the kernel is singular in the origin and is discontinuous across the hyperbolicity cone. This paper introduces a new integration approach that leads to a scheme that converges exponentially with the number of quadrature points. The key here is to consider the integration domain as a convex polytope and to devise a decomposition into the convex hulls of simpler polytopes which are parameterized such that the singularity and discontinuity occurs in a single variable. The method is implemented for piecewise constant elements and tested on a scattering problem with known analytic solution.

For $1\leq k<n$, let $P(n,k)$ be the directed overlap graph whose vertices are the $k$-permutations of $[n]$ and whose arcs are the $(k+1)$-permutations. Isaak proved that $P(n,n-2)$ has no directed Hamiltonian cycle for $n\geq4$ and asked whether it nevertheless has a directed Hamiltonian path. We answer this question affirmatively by showing that $P(n,n-2)$ has a Hamiltonian path.

Integrated sensing and communication (ISAC) is emerging as an important direction in the Third Generation Partnership Project (3GPP) evolution toward 6G because it allows cellular networks to provide environmental awareness in addition to connectivity. This paper surveys the current 3GPP trajectory from Release~19 feasibility studies to Release~20 radio, protocol, and architecture studies, while distinguishing established requirements, ongoing study assumptions, and possible forward directions. The survey covers service requirements, sensing topologies, channel model evolution beyond 3GPP Technical Report (TR)~38.901, Radio Access Network Working Group~1 (RAN1) physical layer design, Radio Access Network Working Groups~2 and~3 (RAN2 and RAN3) system implications, and the role of sensing-assisted communication. It also synthesizes the main unresolved issues in waveform and reference signal design, multi-node coordination, sensing data reporting, service exposure, privacy, and implementation constraints. By connecting service-level motivations to physical layer, protocol, and architecture implications, the paper provides a standards-centric reading of how 3GPP may evolve toward practical 6G ISAC support.

Mizzi conjectured that every nontrivially unstable graph contains cycles C_k and C_{2k} for some odd k. We give a connected, nonbipartite, vertex-determining counterexample on ten vertices. Its instability is certified by an explicit nontrivial two-fold automorphism, and its complete set of simple-cycle lengths is {5,5,6}.

Intelligent control of wireless propagation environments is crucial for future network capacity and reliability. Unlike circuit-controlled reconfigurable intelligent surfaces (RIS), mechanically actuated specular reflecting surfaces (SRS) offer a simpler and potentially more cost-effective alternative. In this paper, based on the tractable ray-based cascaded channel model with power-projection correction, we investigate the fundamental operational behaviors of an ideal SRS in free space. Specifically, in the angle-aligned near field, edge reflections cause non-constructive combining, resulting in a damped oscillatory convergence of the gain to an aperture-independent constant. We further obtain the far-field behavior, unbounded-aperture asymptotics, an optimal aperture size and reflection angle, and a gain-based near/far-field boundary. For misalignment, we provide accurate approximations for small and large apertures via center-point and stationary-point analyses. We also define the SRS beam pattern, derive analytical 3-dB beamwidths, and quantify the effective region where a main lobe exists. Finally, we derive a closed-form achievable-rate for an SRS-aided communication system. Numerical results validate the proposed expressions, reveal distinct near-/far-field behaviors of specular reflection, and show that SRS can outperform RIS in the far field due to continuous aperture and angular-resolution control and stronger power projection.

Let $G$ be a simple graph with cover ideal $J(G)$ in a polynomial ring $S$ with $|V(G)|$ variables. We prove that the local $v$-number of the symbolic powers $J(G)^{(t)}$ is linear for all $t \ge 1$ when $G$ is bipartite, and quasi-linear with period two for all $t \ge 2$ when $G$ is non-bipartite. Furthermore, we provide explicit formulas for these invariants in terms of the combinatorial data of $G$.

Dimension reduction removes variables, but a reliable reduction should not erase their dynamical influence. We study this principle for linear block evolution systems on resolved and unresolved Hilbert spaces. Exact elimination of the unresolved component yields a Volterra equation with a memory kernel and an effective forcing term carrying hidden initial data and forcing; in the Laplace domain the same operation is a dynamic Schur complement. We introduce finite-horizon trajectory-stable reduction and derive perturbation estimates for operator-valued kernels, then lift them to perturbations of the hidden propagator, couplings, hidden initial state, and hidden forcing. For passive skew-adjoint couplings, the reduced memory operator has positive type and an exact storage identity, and norm-convergent passive realisations preserve both trajectories and the storage–dissipation functional. Compatible finite spectral truncations provide structure-preserving internal-variable reductions. Two elementary examples separate positive type from instantaneous $L^2$-coercivity and show that small hidden amplitude need not imply small long-time influence. The framework isolates the controlled approximation of unresolved influence, rather than elimination alone, as the central reduction problem.

We investigate long-time learning in ergodic, potential, monotone mean-field games (MFGs) via a self-fictitious-play (SFP) dynamics coupling an optimally controlled diffusion with a slowly evolving belief. At each time, the state follows the optimal feedback associated with the current belief, while the belief is updated using the player's own empirical occupation measure rather than the population distribution. For ergodic monotone potential MFGs on the torus, we prove that the SFP dynamics is contractive and admits a unique invariant law. Moreover, we show that this invariant law is quantitatively close to the MFG Nash equilibrium, with an error of order equal to the square root of the belief-update rate. The proof combines uniform-in-time regularity estimates for the ergodic Hamilton-Jacobi-Bellman equation with an energy argument based on the Lasry-Lions divergence. The linear-quadratic example shows that this rate is sharp, and the numerical experiments illustrate the predicted scaling.

For every integer $k\ge2$, Chen and Raspaud conjectured that each graph $G$ with odd girth $\og(G)\ge2k+1$ and maximum average degree $\mad(G)<2+1/k$ has a $(2k+1:k)$-coloring. In this paper, we prove the conjecture.

In 1970, Rankin and Swinnerton-Dyer showed that the non-elliptic zeros of Eisenstein series $E_k$ in the fundamental domain all lie on the lower arc $\{ e^{i\theta}: \frac{\pi}{2} < \theta < \frac{2\pi}{3}\}$. Very recently, Sugibayashi showed that the same property also holds for the Serre derivative $\vartheta_k(E_k)$ of Eisenstein series. In this paper, we first give very precise estimates for where exactly these zeros are located on the lower arc. These location estimates then allow us to prove four main results. First, we show that the zeros of $\vartheta_\ell(E_\ell)$ Stieltjes interlace with the zeros of $\vartheta_k(E_k)$ on the lower arc for all $\ell > k$. Second, we classify precisely when the zeros of $\vartheta_\ell(E_\ell)$ (standard) interlace with the zeros of $\vartheta_k(E_k)$ on the lower arc. Third, we show that the zeros of $\vartheta_k(E_k)$ always (standard) interlace with the zeros of $E_{k+2}$ on the lower arc. Fourth, as an application of the third main result, we show that the zeros of the cuspidal projection of $\vartheta_k(E_k)$ all lie on the lower arc, extending a result of Xue and Zhu.

In this article, we introduce a novel approach to solving the ergodic stochastic optimal control problem whose dynamics is driven by a controlled stochastic differential equation. The coefficients of this stochastic differential equation are non-autonomous but periodic in time. We first prove that the infinite horizon average stochastic optimal control problem is ergodic, i.e., the value function equals a constant $\rho$ which is independent of the initial conditions. Based on the result of ergodicity, we construct an auxiliary function $w(t,x)$ that is well-defined and periodic in time. We can prove that the pair $(w,\rho)$ satisfies the dynamic programming principle and is a viscosity solution of the associated Hamilton-Jacobi-Bellman equation. Finally, we apply our results to the study of ergodic backward stochastic differential equations. Our method is even new in the homogeneous case.

The discrete cylinder phase space offers a natural setting for twisted photons, Josephson junctions, and certain bosonic codes. In that context, we build a number-of-zeros hierarchy for the stellar distribution of states on the discrete cylinder phase space, tied to usefulness for quantum computation. The orbital angular momentum (OAM) and phase eigenstate, having no zeros, sits at the bottom of the hierarchy and share mathematical properties with the coherent state on the plane phase space, both sharing zero rank and a positive Wigner distribution. This reveals a structural peculiarity of the cylinder: its coherent state—the wrapped coherent state—is instead the most quantum, carrying infinitely many zeros, so "coherent state" need not signal classicality across different phase-space geometries. We identify the operations that preserve or change the zero count, giving explicit access to every rank of the hierarchy. We then ask whether this hierarchy also governs localization. Using the $L^r$ norm, the inverse participation ratio, and the Wehrl entropy as complementary measures of spread in the Husimi density, we find that the two hierarchies need not track one another: the wrapped coherent state can be more localized than the OAM eigenstate even though it carries infinitely many zeros.

Bondarko and Luzgarev constructed the Chow weight structure on the homotopy category of $KGL$-modules; we show that the heart of this weight structure is equivalent to the category of $K_0$-motives introduced by Gillet and Soulé, which is the $K$-theoretic counterpart of the category of relative Chow motives defined by Corti and Hanamura. The proof is achieved via a detailed study on the Borel-Moore theory in $KGL$-modules, as well as explicit formulas for products in algebraic $G$-theory in terms of $\operatorname{Tor}$-operations.

We show that for a finite group $G$, the $G$-category $\underline{\mathrm{Sp}}_{G,\mathbb{Q}}$ of rational genuine $G$-spectra is parametrized semiadditive, in the sense of Cnossen–Lenz–Linskens, for morphisms of genuine $G$-spaces whose fixed point maps have $\pi$-finite fibers. In particular, this means that the canonical norm map between a colimit and a limit of rational genuine $G$-spectra indexed by such $G$-spaces is an equivalence. This extends the Wirthmüller isomorphism from $G$-orbits to a broader class of $G$-spaces and combines the theory of ambidexterity of Hopkins and Lurie with parametrized category theory.

We study a variant of Cops and Robbers in which the robber attempts to visit as many vertices of the graph as possible without being captured, while the cop aims to keep the robber confined to a small set of vertices. The damage number of a graph $G$, introduced by Cox and Sanaei in 2019, is the maximum number of vertices the robber can visit in a game of Cops and Robbers on $G$. In this paper, we determine damage numbers for several classes of graphs, including hypercubes, Hamming graphs, Johnson graphs, incidence graphs of projective planes, and Erdős-Rényi random graph $G(n,p)$, for $p \gg \log^{2/5}(n) / n^{1/5}$. We also show that the problem of determining the damage number of a graph is PSPACE-complete.

In the late 1980s, Baulieu and Grossman demonstrated that the supersymmetric formulation of Langevin stochastic differential equations (SDEs), proposed earlier by Parisi and Sourlas, belongs to the family of Witten-type topological field theories (TFTs). From a certain angle, this finding may appear puzzling: TFTs have no local degrees of freedom, whereas SDEs do exhibit local fluctuations. In this paper, we address this apparent contradiction in the context of the supersymmetric theory of stochastic dynamics (STS), a generalization of the Parisi-Sourlas-Baulieu-Grossman approach to SDEs of arbitrary form. We further discuss how, within STS, dynamical chaos can be identified with the spontaneous breakdown of the corresponding topological supersymmetry, while 1/f noise can be understood as a consequence of the Goldstone theorem. The butterfly effect, in turn, calls for an effective field-theoretic description that may itself possess a hidden topological structure, as we speculate on the basis of the Ginzburg-Landau approach and insights from AdS/CFT duality.