Papers

Indirect signal production is often treated as a higher-order variant of classical Keller-Segel chemotaxis, but its critical structure depends strongly on how the signal cascade is closed. This paper separates two asymptotic regimes of a two-stage signalling mechanism. In the parabolic–elliptic–elliptic limit, the chemoattractant is generated by the self-adjoint fourth-order operator $K_\tau=(I-\tau\Delta)^{-1}(I-\Delta)^{-1}$. We prove its spectral positivity, entropy-dissipation structure, fourth-order principal scaling, and logarithmic kernel singularity in four dimensions. Consequently, the correct critical space is $L^{N/4}$, and $N=4$ is the mass-critical dimension. A concentration calculation identifies the natural threshold candidate $M_* = 64\pi^2\tau/\chi$, while the sharp threshold theorem is formulated as an Adams/logarithmic-HLS open problem. In contrast, the mixed elliptic–parabolic cascade cannot be reduced to a static fourth-order kernel. Its eliminated signal is a Volterra memory operator whose near-diagonal multiplier has the same order as the classical Keller-Segel drift. Thus its critical theory must be based on mixed space–time estimates, not static elliptic scaling. Numerical experiments support our operator-level distinction.

Hydrogen supply chains are expected to play a central role in future low-carbon energy systems by enabling renewable energy integration, long-duration storage, and decarbonization of industrial and transportation sectors. However, their operation is challenged by renewable generation variability, electricity price fluctuations, uncertain hydrogen demand, and engineering constraints associated with electrolyzers, energy storage, and grid interaction. As hydrogen infrastructure expands toward commercial deployment, operational strategies must balance economic performance, reliability, and sustainability under dynamic and uncertain conditions. This paper investigates and compares four control approaches for a renewable-powered hydrogen supply chain: a rule-based controller (RBC), model predictive control (MPC), reinforcement learning without forecasts (RL-NF), and reinforcement learning with forecast-augmented observations (RL-F). All methods are evaluated within a unified, physically realistic framework incorporating electrolyzer minimum-load and ramp-rate constraints, battery and hydrogen storage dynamics, grid import limits, and consistent economic assumptions, enabling a fair comparison under identical operating conditions. Simulation results show that MPC achieves the highest economic performance by exploiting short-term forecasts to coordinate storage, reduce grid dependence, and improve efficiency. RL-NF demonstrates robust and competitive performance without future information, highlighting the capability of learning-based methods to discover effective policies from experience. RL-F does not consistently outperform its no-forecast counterpart, suggesting that forecast uncertainty and increased state complexity can limit forecast-augmented learning. The results provide guidance for selecting operational control strategies in future hydrogen energy systems.

We study the kinetic action that d'Alembert's functional equation induces on positive paths in $\Rplus$, and prove it strongly convex. Calibrated d'Alembert forces the cosh cost $\Jcost(x)=\tfrac12(x+x^{-1})-1$, i.e.\ $\Jlog(\xi)=\cosh\xi-1$ in the log coordinate $\xi=\log x$. Evaluating this log-cost at the log-velocity $\dot\xi$ rather than the log-position – a single postulate (Postulate~\ref{post:step}) – yields $\actionA[\gamma]=\int_a^b(\cosh\dot\xi-1)\,dt$, strongly convex under geometric (log-space) interpolation. This convexity has three consequences, none requiring an Euler–Lagrange equation, a Fréchet derivative, or a second variation. First, a one-sided chord condition characterizes global minimality. Second, the unique fixed-endpoint minimizer is the uniform-log-velocity path. Third, the action gap obeys an exact Bregman / Pythagorean identity $\actionA[\gamma]-\actionA[\gamma_*]=\int D_\Kkin(\dot\xi\,\|\,\dot\xi_*)\,dt$, sharpened by a quantitative Friedrichs–Poincaré bound on $\log(\gamma/\gamma_*)$. It has a dually-flat / Hessian-manifold reading in the additive coordinate $\xi$. \\ This theorem is purely mathematical, and we delimit it. The bridge to Newtonian and rapidity mechanics is conditional, requiring structure beyond Postulate~\ref{post:step}: a kinematic embedding, a mass coupling, a time calibration, and a Hamiltonian-primary Legendre structure. Granted these, the cosh action recovers the Newtonian small-step limit and the rapidity profile $\Kkin_m(\phi)=m(\gamma_L-1)$; yet the cosh-dual Hamiltonian is not the special-relativistic free-particle Hamiltonian (Proposition~\ref{prop:not-SR}), the agreement being one of profile, not an identity of Hamiltonians. Global minimality is a free-sector phenomenon: once a non-affine strictly convex potential is added, joint convexity is lost and the classical stationary-action picture returns.

Ribbons are thin elastic bodies whose thickness $t$ is much smaller than their width $w$, which is in turn much smaller than their length. Starting from a three-dimensional model, we derive a one-dimensional limit theory for ribbons with rough prestrain, by means of $\Gamma$-convergence. Our model shows that for narrow ribbons, the energy minimizer coincides with the reference (effective) second fundamental form along the midline, while for wide ribbons this is generically not the case. This proves the existence of shape-transitions in such ribbons, as observed in many experiments, generalizing recent results by Maor & Mora to rough prestrains, which more accurately model many of the relevant physical systems. Our analysis combines techniques from the study of Euclidean ribbons due to Freddi et al., the work of Schmidt on dimension reduction of prestrained plates, and a new structure theorem on the scaled limiting strain due to Maor & Mora.

The rapid growth of AI workloads and renewable energy resources exacerbates supply-demand imbalance in power systems, making traditional load regulation designed for efficient allocation inadequate and motivating demand response (DR) mechanisms to enable load controllability in smart grids. However, existing DR-oriented approaches either focus on optimizing electricity cost or occupant comfort with limited benefit to system-level balance. Others overlook the diverse and dynamic consumption patterns of heterogeneous energy entities, leading to significant over- or under-regulation. Therefore, we propose DRP-FLR. First, DRP-FLR achieves accurate short-term load forecasting by embedding exogenous knowledge (e.g., entity information, prediction time) into historical load representations. Next, it constructs entity-specific load-pattern profiles by clustering historical load curves, and estimates DR potential by matching forecasted loads with pattern profiles. Finally, DRP-FLR formulates flexible load regulation as a mixed-integer optimization problem and solves it with an MILP solver to jointly optimize DR utilization, participant economic benefit, and renewable accommodation, while enforcing supply-demand balance and economic feasibility. Experiments on a regional grid and a campus microgrid show that DRP-FLR reduces regulation deviation by 36.63%-91.87% and improves participant benefit by 44.66% on average.

We study constitutive conditions of hyperelastic potentials for incompressible material behavior in three dimensions. By means of a counterexample, we show that polyconvexity does not imply true-stress-true-strain monotonicity. Thus, polyconvexity alone is not strong enough to guarantee a physically reasonable response for idealized elasticity.

Metastable reaction rates are commonly inferred from transition-state fluxes, mean first-passage times, or fitted kinetic models. We show that they are directly encoded in the plateau gap of an occupation-time Poisson corrector. For a centered basin-occupation observable, the Poisson corrector develops metastable plateaus in the reactant and product basins, and their separation determines the forward and backward transition rates. This construction requires only the generator, stationary measure, and metastable partition, and therefore does not rely on a predefined transition-state surface. In overdamped and underdamped double-well dynamics, the plateau-gap rate recovers the Kramers, Grote-Hynes, and Pollak-Grabert-Hänggi hierarchy. The same corrector-martingale decomposition yields a reactive-noise density, revealing where stochastic forcing contributes to transitions in configuration or phase space. Thus, reaction rates and their fluctuation sources emerge from a single corrector field.

The purpose of this paper is to provide a constructive method for generating the moments of Laguerre-Hahn linear forms of class zero. The approach relies on a general algebraic system on the moments, equivalent to the Laguerre-Hahn functional equation for forms of arbitrary class, from which the restriction to class zero yields explicit recurrences allowing the entire moment sequence to be built recursively from the first moment. These recurrences are applied to each of the ten canonical class-zero families, and the corresponding first moments are derived explicitly via a {\it Mathematica$^{\circledR}$} implementation.

Imploding solutions to the radially symmetric, isentropic, compressible Euler equations have been well-studied, inspired by the work of Guderley. However, these smooth imploding solutions are shown to be numerically unstable and difficult to compute in practice. On the other hand, the imploding solution of Kidder has a closed form solution and is numerically computable. But, it is unbounded in the far field. We consider Kidder's formulation in one dimension in which the unbounded far field condition is replaced with a constant density cutoff of the initial data. Strikingly, a non-centered rarefaction emerges from the cutoff and suppresses the implosion. We present an exact analytic solution to the problem with the cutoff and support our theoretical predictions with numerical simulations.

We study Prandtl's classical problem on minimising the induced drag for a finite wing with fixed span. The induced drag is given by a singular quadratic functional of the circulation, with admissible functions satisfying the prescribed lift and second-moment conditions. We formulate the problem in the fractional Sobolev space \(H^{1/2}\), which is the natural energy space for the functional, prove existence and uniqueness of minimisers by variational methods, and derive the corresponding Euler–Lagrange equation. % Passing to a periodic formulation on the one-dimensional torus, we identify the drag functional with the quadratic form of the half-Laplacian and solve the resulting singular integral equation explicitly and recover Prandtl's bell-shaped circulation profile.

We formulate a conditional scale-critical defect-cascade reduction for the local regularity problem of the three-dimensional incompressible Navier–Stokes equations. The theorem concerns a potential singular point for which no sufficiently small dyadic scale enters the Caffarelli–Kohn–Nirenberg smallness regime. Under the structural hypotheses of the framework, such a point cannot be explained by an undifferentiated concentration of energy or pressure. It must lead either to non-effective moving-window observability or to an NS-realizable, cleaned, scale-critical defect cascade invisible to the combined active-pressure, flux, energy, and adjoint-trace tests. The reduction is built from dyadic rescaling, coarse graining, active/harmonic pressure splitting, Reynolds covariance positivity, pressure compatibility, and local energy-flux identities. Finite-window observability reduces possible invisible directions to explicitly defined residual kernels, while budget compatibility and sign coherence convert visible pressure–flux activity into depletion. Consequently, within a controlled window class where dyadic defect extraction, observable depletion, and moving-window growth control hold, effective observability together with exclusion of NS-realizable combined-invisible cascades yields a CKN scale and hence local regularity. The final component interprets the remaining obstruction as a critical recurrence problem and proposes diagnostics for vortex stretching, active pressure work, and interscale flux. Spatially harmonic pressure is retained as a physical local pressure component; only purely time-dependent pressure functions are treated as gauge.

We examine upper bounds on Nordhaus-Gaddum type problems for parameters related to graph connectivity. Our main result is that for a graph $G$ on $n$ vertices where both $G$ and its complement $G^c$ are connected, then the sum of the algebraic connectivity of $G$ and the algebraic connectivity of $G^c$ cannot exceed $n-3$ (with finitely many exceptions with a small number of vertices). We obtain similar results for the isoperimetric number of a graph, and explore similar Nordhaus-Gaddum type questions for the Cheeger constant and the second eigenvalue of the normalized Laplacian matrix.

We give a detailed, corrected presentation of some fundamentals of the constructive theory of locally convex spaces that appear without proofs in <cite>BVtech</cite>. This suffices for some important functional analytic theorems that are stated in our final section.

We prove a large-diameter fundamental-gap lower bound for compact horoconvex domains in real hyperbolic space of curvature \(-1\). The geometric part reduces large horoconvex domains to a fixed-width radial-height problem in all dimensions. The analytic part proves the needed radial-height theorem by comparing the low-energy Dirichlet form with a limiting angular operator on the sphere, while the radial complement is separated by a one-dimensional branch gap and endpoint Green estimates. The result gives the polynomial \(D^{-3}\) scale matching the Nguyen–Stancu–Wei large-diameter upper bound.

We study the relationship between almost toric base diagrams, perfect exceptional classes, and optimal ellipsoid embeddings for $H_b=\mathbb{CP}^2_1 \# \overline{\mathbb{CP}\!}\,{}^2_b$ and $P_b=S^2_1\times S^2_b$. Starting from a quadrilateral almost toric base diagram with one Delzant corner, we encode the three non-Delzant corners by a recursive triple. We show that every quadrilateral obtained via a well-defined sequence of mutations from the initial diagrams is encoded by a recursive triple in the same way. Moreover, geometric mutation of these diagrams corresponds to algebraic mutation of the associated triples. These algebraic mutations are the recursive operations used to generate the $(p,q)$-perfect classes for $H$. We apply this dictionary to realize every $(p,q)$-perfect class for $H$ by an explicit sequence of almost toric mutations for suitable values of $b$. We also prove the analogous realization result for triples of quasi-perfect classes for $P$, showing that these classes are in fact $(p,q)$-perfect. Finally, we apply these results to ellipsoid embedding problems, including visible embeddings, visible obstructions, and ATF-visible staircases.

We develop a theory of partial matchings between finite-dimensional subspaces $A,B$ in a field extension $K \subsetneq L$, providing structural, quantitative, and extremal results for their deficiency. Our main results include (1) a characterization of those pairs $(A,B)$ that are partially matchable up to a specified defect, (2) a decomposition theorem for pairs $(A,B)$ having positive deficiency, (3) an existence criterion for pairs having a prescribed dimension and satisfying a deficiency bound, and (4) a formula for the maximum attainable deficiency at each dimension, assuming $1 \notin B$ and the extension has a proper nontrivial intermediate field of finite $K$-dimension. We use these results to recover and extend various parts of this area of matching theory. Our approach blends algebraic techniques with tools from matroidal transversal theory, and utilizes a linearized version of the $e$-transform from additive number theory.

We study Cauchy singular integral operators on planar wedges whose opening angle is a rational multiple of $\pi$. For $\theta=p\pi/q$, the covering $w=\zeta^q$ yields an exact finite-sheeted factorization of the wedge Cauchy transform into $2q$ interval Cauchy transforms with explicit algebraic recombination coefficients. The factorization is formulated on weighted conormal Hölder spaces. We prove that the lifting operator preserves conormal order, lowers the Hölder exponent from $\beta$ to $\beta/q$, and has sharp $\ell^1$ sheet norm $q$. Combining this operator factorization with a Mellin model for interval Cauchy transforms, we derive a mode-by-mode propagation rule for polyhomogeneous endpoint expansions. Nonresonant powers preserve their logarithmic order, while integer exponents raise it by one. The results also give a local singular decomposition for Cauchy operators on piecewise analytic curves with rational corner angles.

We argue that dependent versions of McDiarmid's inequality are a useful but underutilized tool in mathematical statistics, learning theory and theoretical computer science. To make this point, we first highlight that approximate tensorization of entropy (ATE) implies McDiarmid's via the Entropy Method. Second, we derive McDiarmid's inequality for non-isotropic Gaussian random vectors $X \sim \mathcal N(\mu, \Sigma)$ through ATE with a constant of the order of the condition number of $\Sigma$. We both independently obtain this ATE through a simple application of stochastic localization and also discuss how a more general ATE for the Gibbs sampler due to Ascolani et al., 2026 generalizes McDiarmid's-like concentration to strongly log-concave and log-smooth probability measures. We then apply the resulting concentration inequalities to resolve a question on the concentration of $\operatorname{sign}(X)$ posed by Simone Bombari, investigate Erdős-Rényi graphs under dependence and prove a Dvoretzky-Kiefer-Wolfowitz-type inequality for observations from a joint measure fulfilling ATE and continuous marginal CDFs. For the class of strongly log-concave and log-smooth measures, this result improves upon a prior Dvoretzky-Kiefer-Wolfowitz-type inequality for non-i.i.d. observations due to Bobkov and Götze, 2010, by establishing the expected $1/\sqrt{n}$-rate of convergence under weak dependence instead of $n^{-1/3}$.

We present an asymptotic analysis and high-order boundary integral method for the three-dimensional Neumann Green's function in general closed and smooth geometries. The Neumann Green's function is a fundamental quantity which arises in numerous fields of science and engineering. In the application of singular perturbation methods to strongly localized reactions and diffusive transport, the Green's function plays the key role in mediating global dynamics. However, this essential quantity can only be determined in closed-form for a limited set of geometries. The Green's function for the Laplacian is an elliptic problem with a Dirac forcing term. Accurate resolution of the solution requires a careful decomposition into a singular and a regular part. The bulk scenario is where the source is placed off surface and the singularity is given by the free-space function. In the surface case, where the source is placed at a curved point on the boundary, we use asymptotic analysis to determine a three-term singularity structure. With explicit knowledge of these singularities, we develop a high-order boundary integral method for the determination of the remaining regular part. To resolve the singular boundary data, our integral method uses a custom discretization with Duffy patches near the source. We validate our method using several test cases in which closed-form solutions can be developed, including spheres, prolate spheroids and constructed domains. We demonstrate the applicability of our method to address some open problems in narrow capture theory.

Diffusion models provide expressive data-driven priors for Bayesian inverse problems, but many diffusion posterior samplers rely on heuristic guidance approximations that can fail for nonlinear operators and multimodal posteriors. In this work, we develop a stabilized path-space framework for diffusion-based posterior sampling. Starting from a base diffusion process whose terminal marginal represents the prior, we define a likelihood-weighted target measure on trajectories and cast posterior sampling as learning a controlled stochastic process whose path measure matches this target. This formulation connects diffusion posterior sampling to stochastic optimal control while preserving the Bayesian structure needed for uncertainty quantification. We introduce a time reparameterization that makes the path-space control problem well posed by removing the bias induced by the unknown initial value function, without auxiliary training. We then learn the control via a trust-region path-space optimization method with log-variance objectives. The path-space perspective also unifies our learned control approach with existing guidance-based samplers, quantifies the sampling error induced by approximate controls, and yields importance sampling corrections for asymptotically exact posterior expectations. We evaluate the proposed framework on a suite of benchmark inverse problems with analytically characterized or high-quality reference posteriors, enabling principled assessment of sampling accuracy and uncertainty quantification. These experiments provide insight into the behavior of diffusion-based posterior samplers and demonstrate improved accuracy and robustness over leading approaches.

We study a two-node stochastic resource system operating over a finite horizon. Each node experiences uncertain supply and demand and is equipped with finite storage. The objective is to ensure that resource levels remain within prescribed limits with high probability. To this end, we formulate a chance-constrained capacity-design problem in which resources can be exchanged through a capacity-limited transport link. We characterize the minimum storage required at each node, derive the optimal transport policy, and quantify the trade-off between storage and transport capacities. Our results show the existence of a critical transport-capacity threshold that enables full risk pooling between the nodes. Moreover, this threshold decreases with the operating horizon, implying that full-pooling performance can be achieved with progressively smaller transport capacity over longer horizons.

In this note, we exhibit a group which has a right-angled Artin group as a finite index subgroup, but no finite index normal subgroup isomorphic to any right-angled Artin group. This answers a recent question of Vankov.

Quantum random access codes (QRACs) ask how well N classical bits can be encoded into M qubits while allowing any single bit to be recovered. Although the Nayak bound remains the standard general upper bound on the decoding probability, numerical evidence suggests a stronger upper bound in the small-qubit regime. In this work, we formulate the optimal decoding probability in terms of decoding measurements, reformulating QRAC design as a spectral problem for noncommuting measurements. Using this formulation, we give an elementary proof of the Nayak bound by simplifying the Chernoff-bound argument. Moreover, we refine the argument to obtain upper bounds that improve over Nayak's bound in the entire finite-size regime. The equality conditions of our bounds justify defining mutually unbiased projector-valued measurements (MUPVMs), a generalization of mutually unbiased bases. We show that decoding measurement of any two-qubit QRAC attaining the conjectured bound must form MUPVMs. We also show that any MUPVM, assisted by one ancillary qubit, yields a QRAC with optimal N-scaling decoding probability. Finally, we propose a new MUPVM-based construction for the (M+2,M)-QRAC family attaining the conjectured bound.

The Milnor map is the homomorphism from the unitary bordism ring to the unoriented bordism ring, halving the dimension, that maps the unitary bordism classes of the complex Milnor hypersurfaces to the unoriented bordism classes of their real points. In this work, we propose to generalize this construction to the equivariant setup and we show the existence of such a map for the equivariant unitary groups of the circle and the cyclic group of order two. Furthermore, we relate the kernel of these Milnor maps to the magnetic unitary equivariant bordism groups of free conjugations.

We give a simple, unified, and nearly tight bound for sampling arbitrary logconcave distributions from a warm start using the In-and-Out algorithm along with exponential lifting. The main new ingredient in the analysis is an improved bound on the Poincaré constant of a lifted distribution. As a consequence, the resulting convergence rate is nearly tight for both constrained settings (e.g., Gaussian restricted to a convex body) and well-conditioned settings (e.g., strongly logconcave and smooth densities).

Auto-regressive models have emerged as powerful tools for sequential data, from language to video. Understanding how and why these models learn latent representations remains an open theoretical question. In this work, we demonstrate that when trained by empirical risk minimization on data from partially observed linear dynamical systems, two-layer linear auto-regressive models naturally learn to approximate Kalman filtering. In particular, we show that the learned hidden representation coincides, up to a similarity transformation, with the state estimates produced by the optimal (Kalman) filter, even though the model has no explicit knowledge of the underlying dynamics or state. The result follows from three main insights. First, we establish that the Kalman filter is well approximated by an auto-regressive model with bounded truncation error. Second, we show that despite non-convexity, the two-layer optimization landscape is benign, i.e., all stationary points are either strict saddles or global minima. Finally, as our main contributions, we provide finite-sample guarantees on prediction error, parameter estimation error, and latent state recovery. Numerical simulations support the theoretical results and demonstrate that the latent representations of auto-regressive models recover state estimates.

Neural assemblies, transiently coordinated groups of neurons, observed in the hippocampus are thought to underlie the formation of episodic memories. Acetylcholine (ACh), a neuromodulator, that is received by the hippocampus, plays a critical role in memory and learning. A well supported hypothesis suggests that high levels of ACh during active exploration and rapid eye movement (REM) sleep promote memory encoding, while low levels during quiet waking and slow-wave sleep (SWS) support memory consolidation. We study this bidirectional role of ACh in neural assembly formation through its effect on the synchrony among neurons. We consider a network model of pyramidal neurons, each equipped with a slow, voltage-dependent, non-inactivating potassium current (M-current), which is downregulated in the presence of ACh. Neural assemblies are represented as cluster solutions to this system. Using a one-dimensional phase model reduction of a pair of weakly coupled pyramidal neurons under different levels of the M-current, we predict the symmetric cluster solutions that may emerge in larger networks equipped with all-to-all globally homogeneous, symmetric distance-dependent and nearest-neighbours coupling architectures. We find that under low ACh conditions, the network can fully synchronize, whereas high levels can desynchronize the network into multiple stable symmetric cluster solutions representing distinct neural assemblies.

We demonstrate how supersymmetries of Hamiltonians for coupled Bose-Fermi systems can be used to place the Hamiltonians of the Jaynes-Cummings model and Dicke model under the rotating wave approximation in matrix form and provide explicit analytic solutions for their eigenvalues. We then use this supersymmetry to place the Liouvillians of the associated Markovian open systems in matrix form and provide explicit solutions for their eigenvalues. These results are a consequence of the fact that the Hamiltonian of the Jaynes-Cummings model commutes with the linear Casimir invariant of the superalgebra $u(1|1)$ and that the Hamiltonian of the Dicke model commutes both with the linear invariant of $\sum_{i} u_{i}(1|1)$ and with the invariant of an additional $su(2)$ algebra. Our methods apply to various coupled Bose-Fermi systems with $u(1|1)$ and more generally with $u(n|m)$ dynamical superalgebras, and may provide efficient tools for studying more complicated examples.

In this paper we consider the Dirichlet boundary value problem driven by the weighted Laplacian equation and considered on the Sierpiński gasket. Under suitable growth conditions imposed on the nonlinear term, we provide existence and multiplicity results employing the direct method, mountain pass theorem and some related multiplicity result.

We study the communication complexity of convex decentralized optimization over time-varying networks, where $n$ nodes hold private functions and must agree on the global minimizer using only synchronous exchanges with neighbors. The cost is the number of communication rounds to reach accuracy $\varepsilon$ – a measure akin to round complexity in the LOCAL model, but constrained by nodes sharing only oracle responses. We prove a new lower bound of $\Omega\!\left(\chi_{\mathcal G} \sqrt{\kappa_g}\,\log\frac{n}{\chi_{\mathcal G}}\log\frac1\varepsilon\right)$ communication rounds, where $\chi_{\mathcal G}$ is the condition number of the network Laplacians and $\kappa_g$ that of the global objective, showing the round complexity attainable under uniform regularity cannot be matched in the nonuniform regime. The construction rests on spectral graph theory: we embed time-rotating star gadgets into the edges of an expander and patch them to preserve spectral connectivity.