Papers

This paper presents a novel design approach for movable antenna (MA)-enabled wireless sensing systems by jointly minimizing the Cramér-Rao bound (CRB) and the maximum sidelobe level (MSL) of the ambiguity function via antenna position optimization. In particular, the mean squared error (MSE) of angle-of-arrival (AoA) estimation is decomposed into a local estimation error within the mainlobe of the ambiguity function (i.e., CRB) and an additional ambiguity error caused by its sidelobes. Since the MSE is dominated by the CRB in the high-signal-to-noise ratio (SNR) regime but by the sidelobes of the ambiguity function in the low-SNR regime, our analysis reveals a fundamental trade-off between CRB minimization and MSL minimization in the moderate-SNR regime. Specifically, minimizing the CRB prefers a narrower mainlobe, where antennas are concentrated near the two edges of the one-dimensional (1-D) movement region; whereas minimizing the MSL favors a wider mainlobe, where antennas are distributed more densely near the center of the movement region. Inspired by this and to ensure robust sensing performance across different SNR regimes, we formulate an optimization problem to minimize the CRB subject to a prescribed MSL constraint via antenna position optimization. An efficient successive convex approximation (SCA) algorithm is developed to optimize the antenna position vector (APV), and a 1-D linear search method is proposed to determine the optimal MSL threshold that minimizes the actual MSE for any given SNR. Numerical results demonstrate that the proposed scheme effectively balances the trade-off between MSL and CRB minimization, thus achieving a significantly lower AoA estimation MSE across the entire SNR range compared to conventional uniform and non-uniform fixed-position antenna (FPA) arrays.

A control system admits a positive gain of entrainment (GOE) if entrainment to a periodic input yields a larger output, on average, than the output generated by the corresponding constant input with the same mean value. We analyze GOE in continuous-time stable linear control systems with a static nonlinear output map. Although linear systems with linear outputs have zero GOE, we show that a nonlinear output may generate a nontrivial GOE through the mismatch between the average output along the entrained periodic orbit and the output evaluated at the corresponding averaged equilibrium. We derive a second-order characterization of GOE for smooth output maps revealing that the leading-order contribution is determined by the curvature of the output map. We then show that if the output is convex (concave) on the controllable subspace, then GOE is nonnegative (nonpositive) for every periodic input. Furthermore, GOE admits a natural geometric interpretation as the average Bregman divergence between the entrained periodic orbit and the equilibrium associated with the averaged input. For the special case of quadratic output functions, we derive explicit frequency-domain formulas for GOE. These yield necessary and sufficient conditions guaranteeing the sign of GOE, characterize the contribution of individual input harmonics, and lead to an optimal periodic excitation that maximizes GOE under an energy constraint. The theoretical results are illustrated using an electrical RLC circuit and a compartmental pharmacodynamic model with a nonlinear drug-effect map.

Let $N\in\mathbb{N}\cap[2,\infty),$ $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^N$, and $X(\Omega)$ be a ball Banach function space on $\Omega.$ In this article, under some mild assumptions, we establish a compactness theorem for Sobolev spaces associated with $X(\Omega)$. Different from the fractional Sobolev space, our proof is based on an elaborate decomposition of bounded Lipschitz domains and its corresponding weighted fractional Poincaré inequality on each piece. As applications, we obtain the fractional Poincaré inequality in $X(\Omega)$ that, for any $s\in (s_0,1)$ and $f\in X(\Omega)$, \begin{align*} \|f-f_\Omega\|_{X(\Omega)} \lesssim(1-s)^{\frac 1q} \left\|\left[\int_\Omega \frac{|f(\cdot)-f(y)|^q}{|\cdot-y|^{N+sq}}\,dy \right]^{\frac{1}{q}} \right\|_{X(\Omega)}, \end{align*} where $s_0$ is a given positive constant and the implicit positive constant is independent of $s$ and $f$. Using this, we further establish the well-posedness of a weighted Triebel–Lizorkin type nonlocal variational problem. These results are of wide generality and, even when they are applied to Morrey spaces, weighted Lebesgue spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces, and Orlicz-slice spaces, the obtained results are also new.

We introduce generalized Milnor manifolds (GMM), extending the classical Milnor manifolds over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$. We compute their integral cohomology algebras in the complex and quaternionic cases and their mod-$2$ cohomology algebras in the real case. We compare GMM with partial flag manifolds, investigate when they are homotopy equivalent, and obtain a necessary and sufficient condition in the complex and quaternionic cases. We further prove that complex GMM are Kähler manifolds. As an application, we determine their higher topological complexities, obtaining exact values in the complex and quaternionic cases and bounds in the real case. Along the way, for a fibre bundle satisfying the Leray–Hirsch hypothesis, we establish a lower bound for the zero-divisor cup-length of the total space in terms of base and fibre.

Deep learning models, particularly Transformers and Neural Operators, exhibit a well-documented "spectral bias," effectively acting as low-pass filters that smooth out high-frequency information. While benign in fluid dynamics, this bias is catastrophic for Chaotic Dynamical Systems, where the underlying strange attractor is characterized by fractal geometry and infinite spectral density. We introduce the Fractal Neural Operator (FNO), a novel architecture that utilizes a non-resonant prime number basis to approximate continuous dynamical systems. Unlike geometric encodings ($2^k$), which suffer from spectral gaps and resonance, our Harmonic Weierstrass Encoder injects infinite spectral resolution into the latent space. We demonstrate that FNO extends the valid prediction horizon of the Lorenz-63 system to 347 Lyapunov times, exceeding state-of-the-art Reservoir Computing baselines by a factor of 2.3x. These results suggest that "chaos" is not inherently unpredictable to neural networks, but rather requires non-differentiable, fractal embedding manifolds.

Menger's theorem on $A$–$B$-paths and Gallai's theorem on $A$-paths are among the most useful results in structural graph theory. Many variants and extensions are known. We add to this line of research and prove results that relate the maximal number of vertex-disjoint paths between vertex sets and a wall to the minimum number of vertices meeting all these paths. We also include types of paths that start and end in the wall.

In this paper, we develop a fundamental analytical framework for integrated sensing and communications (ISAC) enabled by the Enormous Fluid Antenna System (E-FAS), which transforms a collection of coordinated intelligent surfaces into a gigantic reconfigurable electromagnetic aperture, with particular emphasis on the limits of angular sensing.We begin by developing a bidirectional sensing channel model that explicitly captures the complete sensing process, including surface-wave (SW) routing, distributed reradiation, target scattering, and echo propagation. Based on this channel model, we formulate a parametric observation model for target sensing and derive the associated Fisher information matrix (FIM) and Cramer-Rao bound (CRB) for angular estimation. The analysis demonstrates that E-FAS gives rise to a fundamentally different sensing regime compared with conventional array-based and reconfigurable-surface-aided ISAC architectures. Our analysis uncovers that maximizing coherent routing gain does not necessarily maximize sensing performance, exposing a fundamental trade-off between SW routing gain and sensing diversity in programmable propagation environments. Numerical results validate the developed framework and demonstrate that E-FAS-enabled ISAC systems can achieve substantial angular sensing gains over conventional architectures under the same transmit-power budget. The results further underscore the importance of jointly optimizing propagation routing and sensing functionality, positioning E-FAS as a new paradigm for ISAC.

This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix.

We extend the classical theory of affine processes to a path-dependent setting by introducing path-dependent coefficients and provide analytic formulas for their Fourier–Laplace transform in terms of generalized Riccati-type equations. In the proposed framework, we define path-dependent affine processes through their exponential-affine Fourier–Laplace transform on the path space and establish a characterization theorem. Conversely, for path-dependent stochastic differential equations with affine path-dependent coefficients, we also provide explicit exponential-affine representations of the Fourier–Laplace functional in terms of those Riccati equations. Moreover, we derive a condition ensuring non-negativity of the path-dependent diffusion coefficient, guaranteeing well-posedness of the model. Finally, we apply these results to a path-dependent volatility model and a path-dependent extension of the Heston model, including a delayed Heston model as a special case.

We give a concrete construction of a natural extension of $(-\beta)$-transformation when $\beta$ is greater than the golden mean. Our construction relies on its Markov diagram and the eigenvectors of the associated countable Markov shifts. Its positive recurrence can be shown by path counting using a special property of the diagram. Our down-to-earth construction elucidates the result of Bruin-Kalle \cite{BruinKalle} by examples.

Minimax risk and regret are expectation-based criteria and do not capture rare but consequential failures. To address this concern, we develop a $\delta$-explicit minimax-quantile theory for interactive statistical decision making (ISDM). We first provide structural relations between minimax quantiles, lower minimax quantiles, and minimax risk. This includes a quantile-to-expectation conversion and an equivalence between strict and lower minimax quantiles outside a countable set of confidence levels. We then derive two converse tools for ISDM: a high-probability interactive Fano's method and a high-probability interactive Le Cam's method. Then, we show that mutual-information (MI) privacy can be handled in the same framework by restricting the admissible decision class. For coordinatewise Gaussian privatization, we derive a two-point template that isolates the privacy-induced variance inflation. We instantiate this template for Gaussian mean estimation, and use the same two-point strategy directly for two-armed Gaussian bandits. We then derive a minimax quantile lower bound for the $K$-armed Gaussian bandit problem, showing that the interactive Fano method captures the exploration cost over multiple possible best arms. The resulting lower bounds are explicit in the confidence level $\delta$ and in the privacy budget for the private problems. They yield $\log(1/\delta)/n$ scaling for squared-error Gaussian mean estimation, $\sqrt{T\log(1/\delta)}$ scaling for two-armed bounded-mean Gaussian bandits, and $\sqrt{KT\log(1/\delta)}$-type scaling for the $K$-armed bandits, with privacy appearing through a Gaussian variance-inflation factor for the private problems.

The evolution of dispersal traits is a fundamental topic in evolutionary ecology, where natural selection may drive the trait distribution toward concentration in the rare-mutation regime. This singular behavior poses a serious numerical difficulty, since direct discretizations of the population density require very fine trait grids to identify the fittest trait and to resolve the concentrated profile accurately. In this paper, we develop a WKB-based numerical framework for a dispersal evolution model. By separating the exponentially concentrated trait dependence from a smoother amplitude variable and combining this WKB representation with dual trait-grid implementation and other specially designed techniques, the method recovers the selected trait and the associated concentration structure accurately and efficiently on fixed trait grids. We establish a semi-discrete stationary fixed-grid asymptotic-preserving structure for the rare-mutation limit of the steady-state problem. Numerical experiments compare the proposed method with direct density discretizations and confirm its advantage in the small-mutation regime.

We revisit Fenn's table theorem from a differential-topological point of view. We prove a zero-existence theorem on a cylinder, which gives a short proof of the horizontal square-table theorem under Fenn's boundary conditions, and We establish a theorem under more general boundary conditions. We also discuss square tables on saddle surfaces and conjecture that every sufficiently small square table can be placed horizontally on a saddle surface. We further conjecture that any prescribed rectangular table can be placed horizontally on a Fenn graph.

In this paper, using the example of a two-dimensional nonlinear sigma model with the Heisenberg group, we compare two variants of chiral field decomposition into a background part and a fluctuation. It is shown that only one of these methods is consistent with the construction of the generating functional by introducing a background field. Furthermore, we perform a one-loop renormalization of the quantum action, calculate power-law singularities in the two-loop approximation, and consider transition to an extended classical action. Finally, we study the consistency of the cutoff with special functional relations within the framework of the background field method.

In this article, we introduce a power-scale variation (PSV$_{a,p}$) with two tunable parameters: the sparsity-inducing exponent $p\in(0,1]$ and the scaling factor $a\in(0,\infty)$. By minimizing the PSV$_{a,p}$, we establish stable reconstructions in both the gradient and the image domains under the restricted isometry property (RIP) framework. Furthermore, we design an iteratively re-weighted least squares algorithm IRLSPSV to solve the unconstrained PSV$_{a,p}$ minimization. Numerical experiments demonstrate its superior performance and broad applicability. The main novelties are: (i) the PSV$_{a,p}$ minimization enjoys great flexibility and wide applicability due to its two tunable parameters $a$ and $p$, (ii) as $a\to\infty$, the PSV$_{a,p}$ minimization reduces to the $p$-th power total variation (TV$_p$) minimization and, even in this limiting case, the established RIP condition for image reconstruction is also new, (iii) the derived RIP upper bound $\overline{\delta}$ is proved to be asymptotically optimal in $a$ for gradient recovery, (iv) sensitivity analysis confirms the distinct roles of $a$ and $p$, thereby motivating a practical parameter tuning scheme for the proposed model.

This work presents a method for constructing nonlinear reduced-order models from input-output time-domain data. The proposed approach, termed Mixed Interpolatory Inference with Variable Projection (MIIvp), exploits the fact that the considered class of nonlinear state-space models is linear in the output equation parameters. By applying the Variable Projection (VarPro) algorithm, the optimization is restricted to the state equation parameters alone, while the output equation parameters are recovered via linear least squares. As a consequence, the output dimension does not enter the nonlinear optimization parameter vector, making the method well suited for systems with very high-dimensional outputs, a setting where many other approaches become computationally prohibitive. Under mild assumptions, it is shown that MIIvp can recover the true model parameters up to similarity. The method is first validated on a synthetic bilinear system, where it achieves machine-precision accuracy and recovers the true eigenvalues. MIIvp is then compared with existing methods on two experimental benchmarks from the nonlinear system identification literature. These numerical experiments showcase both the validity and the limitations of the proposed approach. Finally, directions for improvements and future work are outlined.

We review the construction of a pseudo-braided category structure on the $C_1$-cofinite module category of a vertex operator algebra using conformal blocks and analytic continuation along paths in configuration spaces. In the rational $C_2$-cofinite case, the pseudo-braided category is represented by tensor products and becomes a balanced braided tensor category. We then compute all four-point conformal blocks of the Virasoro vertex operator algebra of central charge $1/2$ in terms of hypergeometric functions. We explain how analytic continuation of these blocks determines the braiding and associator, and identify the resulting module category with the Tambara–Yamagami category over $\mathbb{Z}_2$ as a balanced braided tensor category.

Stochastic programming has become a popular tool for supporting decision-making under uncertainty in the long-term planning of energy systems. Existing scenario reduction methods, however, are naive about the long-term temporal nature of scenarios, which limits their efficiency in reducing model size. In this paper, we overcome this inefficiency by proposing a novel time-variant scenario reduction framework that explicitly allows for varying scenario aggregations over time. As a result, scenario probabilities become time-variant, enabling not only the accurate capture of scenario realizations but also their probabilities at the time steps that drive investment decisions. This substantially increases flexibility compared to traditional time-invariant methods, which we demonstrate on a two-stage stochastic generation expansion planning problem with uncertain renewable power production.

Non-normality can induce large transient growth in linearly stable systems. Determining whether this growth triggers a transition in the underlying nonlinear system, however, requires understanding the interaction between non-normality and nonlinearity. Here, we develop a weakly nonlinear theory for linearly-stable, non-normal systems subject to harmonic forcing, enabling a systematic analysis of this interaction. Following Ducimetière et al. (J. Fluid Mech., vol. 947, 2022, A43), we define a formal small parameter $\varepsilon$ as the reciprocal of the system's maximum linear amplification. However, we ensure asymptotic consistency by providing a framework that naturally adapts to the underlying structure of the system. The approach is applied to a harmonically forced channel flow and to a two-dimensional model mimicking the structure of the Orr-Sommerfeld-Squire equations. Unlike classical weakly nonlinear analysis near bifurcation points, the resulting amplitude equations are non-universal. In fact, a single linear mode amplified by the non-normality can nonlinearly excite a multi-modal and multi-frequency response at leading-order, which is system- or even regime-specific. Nevertheless, the method yields asymptotically consistent amplitude equations that capture this complexity provided a limit in which $\varepsilon\rightarrow0$ can be identified. As the forcing amplitude increases, the reduced equations capture stable nonlinear states emerging from the laminar flow, their subsequent bifurcations, and their eventual collision with the boundary of their basin of attraction. Thus, the amplitude equations can capture subcritical transitions driven by forcing and varied initial conditions and enable the identification of critical parameters beyond which no stable weakly nonlinear state exists.

Drawing on the observed regularity of typical trips, this study proposes a method for identifying commuting patterns without using the individual identifiers of public bike-share users. The approach combines an analysis of trip duration distributions, mixture models and logistic regression based on the posterior probabilities of belonging to the ___fast' and ___typical' journey categories.

Inductive proofs can be represented by proof schemata, a formalism that represents infinite sequences of proofs by recursive definitions. Since proof schemata avoid the explicit application of induction rules, they admit novel applications, one of which is the realization of Herbrand's theorem in the presence of induction. In this paper, we develop a new type of proof schema based on point transition systems. For skolemized proof schemata without quantified cuts, so-called Herbrand systems, that is, schemata of Herbrand instances of quantified formulas, can be computed. Herbrand systems also allow the representation of schemata of Herbrand sequents, thereby realizing Herbrand's theorem for proof schemata. We compare proof schemata with cyclic proofs and define a transformation from a large class of cyclic proofs to proof schemata. Finally, we show that proof schemata based on point transition systems are capable of proving the 2-Hydra statement, a well-known example that is provable by the cyclic proof system CLKID\omega but not in LKID.

In this article, we define and study the statistical physics of planar Carrollian systems. While it has been shown recently that, for general dimensions, the Carroll limit of Poincaré statistical physics typically does not converge, we show that thanks to the central extensions of the Carroll algebra in the plane, and by considering systems with angular momentum, there exists a well defined notion of planar Carrollian statistical physics. Using Souriau's geometric thermodynamics, we compute the partition function for particles on a uniformly rotating disc, and show that rotation is inevitable for thermal equilibrium of planar Carroll systems, with one of the central charges determining the direction of rotation. We derive thermodynamic quantities in particular entropy, which scales logarithmically with the disc area, and pressure, which follows the two-dimensional ideal gas law. Though all results are obtained from symmetry considerations, we also derive the corresponding effective Hamiltonian.

We establish a two-mode stability theory for Monge solutions of the multi-marginal optimal transport problem with barycentric quadratic cost. The associated tuple of maps splits into an external barycentric mode and internal relative modes. A quadratic lower bound for the Kantorovich defect controls the internal modes and yields a square-root estimate without invoking any two-marginal map-stability theorem. The external mode is the optimal transport map from the fixed source to the Wasserstein barycenter. Combining the resulting two-mode estimate with Mérigot's sharp theorem gives a $\frac{1}{4}$-Hölder estimate for general perturbations, while barycenter-preserving perturbations satisfy a $\frac{1}{2}$-Hölder estimate. We prove that both exponents and the dependence on the weights are optimal. We also examine the scope of such decomposition beyond the barycentric cost: collective-coordinate perturbations and uniformly concave costs of the sum retain the two-mode estimate, whereas the analyses of graph interactions, hedonic costs, and translation-invariant costs identify the two possible obstructions – loss of relative coercivity and lack of stability for the remaining external modes.

Functional data are often modeled through one likelihood-linked curve, while the scientific target is a larger state containing rates, accumulated quantities, boundary values, or nonlinear functionals of several linked levels. These targets require more than smoothing the observed curve: derivative uncertainty, cross-level covariance, and integration constants must be handled jointly. We introduce anchored Gaussian process differential ensembles, embedding an anchor \(f_0\) in a joint Gaussian state with its mean-square derivatives and repeated integrals. Integral levels add explicit Gaussian integration constants. This separates the anchor-induced covariance from finite-dimensional boundary uncertainty and clarifies why anchor-only observations do not identify independent integration constants. For stationary one-dimensional kernels, we compute the ensemble with a transformed Hilbert space Gaussian process approximation that applies derivative and integral operators to Laplacian–Dirichlet basis functions while retaining the integration-constant covariance exactly. We establish operator-level approximation bounds and conditional finite-grid posterior convergence. We introduce TARTARE, a target-aware calibration procedure for finite-rank differential ensemble approximations, to address derivative under-resolution by anchor-calibrated bases. In second-order simulations, derivative-aware calibration improves derivative posterior recovery relative to anchor-only calibration while preserving anchor and integral summaries. A motorcycle crash analysis illustrates coherent posterior inference on a coupled kinematic state and short-horizon turning-point functionals.

We study a deterministic filtering problem formulated directly on the Wasserstein space of probability measures with finite second moment. Motivated by the Mortensen minimum-energy observer, we consider the reconstruction of an evolving probability density from partial observations by minimizing an action functional combining a kinetic transport cost and a time-dependent observation mismatch. The resulting value function is defined on the infinite-dimensional manifold $(P_2(R^d), W_2)$ and satisfies a Hamilton-Jacobi-Bellman equation involving the Wasserstein gradient. Under suitable regularity and growth assumptions on the observation functional, we establish dynamic programming principles, continuity of the value function, existence of minimizing trajectories, and viscosity solution properties of the associated Hamilton-Jacobi equation. We provide two complementary notions of viscosity solutions: a geometric formulation based on subdifferentials in Wasserstein space, and a Hilbertian formulation inspired by Lions' lifting approach. This allows us to prove a comparison principle and uniqueness of solutions. Extensions to transport equations with drift are also discussed. Finally, we introduce a semi-Lagrangian scheme in order to approximate the value function, and show $\Gamma$-convergence of the scheme.

Can the topology of an unknown reflecting domain be recovered from one reflected Brownian trajectory, and how long must it be observed? We formulate this as a minimax homological inference problem based on the reflected Wiener sausage. The optimal observation time is, up to constants, the sum of three terms: a spectral-access or burn-in time set by the Neumann spectral gap; an inverse-capacity detection time for small topological features, showing that Brownian capacity rather than volume controls discovery; and a logarithmic factor in the number of features to be resolved. Matching minimax lower bounds prove that all three contributions are necessary. The proof combines a finite-target spectral hitting estimate, expressed through killed eigenvalues, with a uniform small-hole eigenvalue theorem comparing the principal killed eigenvalue of an epsilon-scale feature with its Brownian capacity, uniformly over interior and boundary targets. The resulting framework distinguishes full reconstruction, obtained from an intrinsic epsilon-net and yielding two-sided homology isomorphisms, from faster feature detection by witness regions, which gives the corresponding surjective homological recovery.

Over a field $\mathbb K$ of characteristic zero, we determine the $St_n$-elements and $n$-centers, defined by standard polynomials, of principal subalgebras of path algebras. Our description expresses each $n$-center as the direct sum of a path ideal and explicit subspaces associated with isolated circles. We prove that these $n$-centers are subalgebras and stabilize for $n\geq2|Q_0|$. We also obtain explicit descriptions of the center and the $3$-center, and determine the least degree of a standard polynomial identity in the acyclic case. Further applications concern algebras of block formal languages and rational Hilbert series of the $n$-centers.

We settle the case $n=4$, $k=2$ of the Bojanov–Naidenov problem for algebraic polynomials. Let $P$ be a real polynomial of degree at most four with $\left\lVert{P}\right\rVert_{C[-1,1]}\leq 1$, and let $T_4(x)=8x^4-8x^2+1$. We prove that, for every $t\geq0$, \[ \int_{-1}^{1} \bigl(|P''(x)|-t\bigr)_+\,dx \leq \int_{-1}^{1} \bigl(|T_4''(x)|-t\bigr)_+\,dx . \] This tail estimate implies \[ \int_{-1}^{1}\varphi(|P''(x)|)\,dx \leq \int_{-1}^{1}\varphi(|T_4''(x)|)\,dx \] for every nondecreasing convex function $\varphi:[0,\infty)\to\mathbb{R}$. If $\varphi$ is strictly increasing and convex, equality can occur only for $P=\pm T_4$. The proof is elementary and finite. We interpolate at the five extremal points of $T_4$; convexity then reduces the problem to the $32$ sign choices at these nodes. At each vertex the second derivative is a quadratic polynomial, so the remaining work is an explicit comparison of level sets.

Sidorenko's conjecture asserts that every bipartite graph has at least the expected homomorphism density in every graph of a given edge density. Motivated by Cayley-type formulations of Sidorenko-type inequalities, we study a two-sided correlation construction on finite groups. Let $\Gamma$ be a finite group and let $f:\Gamma\to\mathbb{R}$ be a real-valued function. We define a directed kernel on $\Gamma$ by $$\mathcal C_f(x,y)=|\Gamma|^{-1}\sum_{a_1,a_2\in\Gamma:\, xa_1=a_2y} f(a_1)f(a_2)=\mathbb{E}_{z\in\Gamma} f(x^{-1}z)f(zy^{-1}).$$ When $f=\mathbf{1}_A$, this is the normalized size of the intersection $xA\cap Ay$. We prove that, for every finite directed graph $F$, $$t(F,\mathcal C_f)\geq t(\overrightarrow{K_2},\mathcal C_f)^{e(F)}=(\mathbb{E}_{g\in\Gamma}f(g))^{2e(F)}.$$ Equivalently, if $W_f^\times(x,y)=f(xy)$ is the directed product Cayley kernel on $\Gamma$, then the directed $1$-subdivision of every finite directed graph satisfies the same homomorphism-density lower bound in $W_f^\times$.

In spatially extended systems, tipping does not necessarily lead to a uniform, abrupt transition typical of low-dimensional conceptual climate models. Instead, spatially structured forms of tipping can emerge, for example, through front propagation and pinning, coexistence between alternative states, and pattern formation. Tipping can therefore remain partial or spatially heterogeneous rather than affecting the entire system. Existing work on spatial tipping in conceptual models remains mostly descriptive, and a general framework to influence tipping dynamics through spatial interventions is still lacking. Here, we introduce a constrained optimisation framework that systematically identifies spatial interventions designed to maximise resilience or promote recovery of a desirable state, subject to dynamical and resource constraints. The framework is general and applicable to a broad class of spatially extended systems. We illustrate it using the one-dimensional Allen-Cahn equation with spatial heterogeneity, a minimal bistable model in which the key spatial tipping processes in gradient systems can be analysed explicitly. In this setting, optimisation acts by shifting local tipping thresholds and controlling front dynamics. Our results show that small-scale local, targeted modifications can determine large-scale system outcomes: optimal interventions can prevent tipping, induce recovery of the desirable state, or confine collapse to limited regions of the domain.