Papers

Given a set $R$ of positive integers, an $R$-graph $H = (V, E)$ is a hypergraph where the cardinality of each hyperedge belongs to $R$. If $R = \{r\}$, we sometimes refer to the hypergraph as an $r$-graph rather than an $R$-graph. For a set $S \subseteq V$, let $d_H(S)$ denote the number of hyperedges of $H$ containing $S$. Given a nonnegative integer $s$, the minimum $s$-degree $\delta_s(H)$ is the minimum of $d_H(S)$ over all $s$-vertex subsets $S$ of $V$. Let $r$ and $t$ be positive integers with $r < t$. We denote by $C_t^r$ the $t$-vertex $r$-uniform tight cycle, which is an $r$-graph with at least three hyperedges whose vertices admit a cyclic ordering such that every $r$ consecutive vertices form a hyperedge. In particular, $C_t^2$ is the classical cycle $C_t$ in $2$-graphs. For hypergraphs $F$ and $H$, we say that $H$ is a Berge-$F$ if there exist an injection $f \colon V(F) \to V(H)$ and a bijection $g \colon E(F) \to E(H)$ such that $\{f(v): v \in e\} \subseteq g(e)$ for all $e \in E(F)$. Lu and Wang [Discrete Math. 344 (2021), 112462] proved that every $[3]$-graph $H$ on $n \geq 6$ vertices with $\delta_2(H) \geq 1$ contains a Berge-$C_t$ for all $3 \leq t \leq n$. In this paper, we prove that for any positive integer $r$ and any set $R \subseteq [k]$ with $k \geq 2$, there exists an integer $n_0 = n_0(k,r)$ such that every $R$-graph $H$ on $n \geq n_0$ vertices with $\delta_r(H) \geq 1$ contains a Berge-$C_t^r$ for all $r+1 \leq t \leq n$. In particular, when $k = 4$ and $r = 3$, we show that every $[4]$-graph $H$ on $n \geq 9$ vertices with $\delta_3(H) \geq 1$ contains a Berge-$C_t^3$ for all $4 \leq t \leq n$. We also characterize all the counterexamples when $4 \leq n \leq 8$.

Let $R$ be a complete noetherian local ring of prime characteristic $p$, and let $R^\infty$ denote its perfect closure. We prove that a finitely generated \(R\)-module $N$ has finite injective dimension if and only if $\operatorname{Ext}_R^i(R^\infty, N) = 0$ for all $i > 0$. As a Gorenstein counterpart, we show that finiteness of the Gorenstein injective (or projective) dimension of $R^\infty$ forces $R$ to be Gorenstein, and we relate this to a non noetherian Cohen factorization of $R \to R^\infty$. Applications include preservation of Gorensteinness under weakly etale extensions, structural results on F-coherent and weakly F-nilpotent rings, along with the ascent of Frobenius closure of a parameter ideal to its powers. Assuming $\operatorname{Ext}^{i}_{R}(\frac{R}{\mathfrak m},R^{\infty})=0$ for some $i>\dim(R)$, we show $R$ is regular. This has some applications. We study the rationality problem of Hilbert–Kunz multiplicity by linking it to $R^\infty$. Recall that the global dimension can be viewed as a uniform bound on the injective dimension of modules. Finally, we determine $\operatorname{gldim}(R^\infty)$ by presenting a new bound.

Let $\epsilon_{ij}, i,j\geq 1$ be independent Rademacher variables. We prove \begin{equation*} \mathbb{E} \max_{1\leq j\leq p}\left|\frac{1}{n}\sum_{i=1}^n\epsilon_{ij}\right| \geq \min\left\{\frac{255}{256},\frac{1}{\sqrt{2\log 2}}\sqrt{\frac{\log(2p)}{n}}\right\}. \end{equation*} The equality is attained, for instance, by $(n,p)=(2,1)$ and $(n,p)=(2,8).$ We also discuss the optimality of the numerical constants.

Cohen, Manin, and Zagier recovered the Rankin-Cohen bracket for modular forms from an action of the modular group on pseudodifferential operators whose coefficients are holomorphic functions on the Poincaré upper half plane. We investigate pseudodifferential operators on the Jacobi upper half space with respect to the elliptic variable instead of the modular variable typically considered. We introduce a family of actions of the Jacobi group and show that a space of invariant pseudodifferential operators is isomorphic to the space of Jacobi forms by producing an equivariant map. Our construction arises from the explicit action of a Casimir operator for the complexified Lie algebra of the real Jacobi Lie group. As an application, we identify new families of Rankin-Cohen brackets with geometric origin indexed by a complex parameter. In particular, we isolate a subvariety of lines of Rankin-Cohen brackets in each degree of expected dimension $1$ reflecting the geometry of the Jacobi upper half space.

In this paper, the usage of preprocessing to achieve physical-layer security in a wireless THz-MIMO scenario is investigated. The goal is a reliable and secure communication. Optimization of the preprocessing is done either based on the error performance or the transmission rate. For both criteria, we present a variant that is based only on the legitimate receiver or also includes the eavesdropper. For each variant, linear and lattice-reduction-aided approaches are considered. Numerical simulations are used to assess the resulting secrecy rates and error ratios. A comparison between all variants is compiled and the possible trade-offs are discussed.

We present the entropy-degree theorem for Lipschitz maps between Alexandrov spaces with curvature bounded below, extending the classical Besson–Courtois–Gallot entropy-rigidity results to this singular setting. The proof requires a new degree theorem for Alexandrov spaces, developed using the Ambrosio–Kirchheim theory of integral currents, showing the equivalence between analytical and topological degrees. Applications include geometric obstructions for negatively curved Einstein metrics on 4-orbifolds, volume bounds for cone-manifolds, quantitative inequalities for hyperbolic convex cores, and lower bounds on the asymptotic translation lengths of end-periodic surface homeomorphisms. We show that entropy-volume minimization under uniform lower curvature bounds obstructs to the formation of metric singularities in Gromov–Hausdorff limits, prove an Alexandrov boundary rigidity theorem, and establish volume minima for cone manifolds and cone orbifolds.

The marginal degree of sums in dimension \(n\) is the smallest integer \(k\) such that the joint distributions of all subcollections of at most \(k\) coordinates of a real-valued random vector \(\left(X_1,\ldots,X_n\right)\) determine the value of \(\E\left(X_1+\cdots+X_n\right)\), whenever this expectation is defined. For every \(n\ge2\), we prove that this marginal degree is \(\left\lceil n/2\right\rceil\). The upper bound follows from a theorem of Simons (1977). The lower bound is proved by constructing, for every \(1\le k<\left\lceil n/2\right\rceil\), two joint laws whose marginals of dimension at most \(k\) agree, but for which the corresponding expectations of \(X_1+\cdots+X_n\) are defined and unequal.

This paper is concerned with the time-harmonic wave scattering problems in three dimensional poroelastic media. By introducing an intermediate variable $p$, the original $\mathbf{u}-\mathbf{w}$ system is equivalently transformed into a $\mathbf{u}-p$ system with fewer degrees of freedom, which facilitates the derivation of the fundamental solution, Green's identity and positivity of the complex wave numbers. A perfectly matched layer (PML) method is then introduced in the spherical coordinates to truncate the unbounded scattering problem. Under certain assumptions on the poroelastic and PML parameters, we prove the existence and uniqueness of solutions to the PML problems both in the truncated domain and layer. Moreover, the exponential convergence of the PML method is established in terms of the thickness and parameters of the PML layer. The proof is based on the PML extension and the exponential decay properties of the stretched fundamental solution. As far as we know, this is the first convergence result of the PML method for poroelastic scattering problems.

Farkas, Pandharipande, and Sammartano constructed non-rational irreducible components of Hilbert schemes of points in affine space $\mathbb{A}^n$ for all $n \geq 12$. Their construction starts from Hilbert schemes of curves in $\mathbb{P}^3$, adjoins two auxiliary variables in order to apply Jelisiejew's TNT frame construction, and then doubles the number of variables. We give a one-variable variant of the construction. The new input is a local-cohomology replacement for the depth-three step in Jelisiejew's negative tangent computation. It uses the vanishing of the low-degree Hartshorne–Rao module for the complete $g^3_9$ curve source. As a consequence, over a field of characteristic zero, $\operatorname{Hilb}(\mathbb{A}^n)$ has non-rational irreducible components for all $n \geq 10$.

Recently a series of papers introduced and investigated the maximal covering location problem with customer preference ordering, a variant of the classical maximal covering location problem (MCLP). In these papers, mixed-integer bilevel optimization models and single-level reformulations were presented for this problem, as well as various heuristics such as a GRASP, a Tabu search and a variable neighborhood search. In this short note we show that instances of this new problem can actually be easily transformed into instances of the classical MCLP and this transformation even reduces the size of the instance. Thus, existing algorithms for the classical MCLP can be used to solve it. We provide a short computational study to show that this transformation leads to speed-ups of at least a magnitude when considering exact algorithms.

We study multiple testing under continuous global–local shrinkage priors, with a focus on the horseshoe prior in high-dimensional sparse settings. While such priors provide adaptive shrinkage and computational scalability, they do not induce exact zeros and hence do not directly yield posterior inclusion probabilities, making principled false discovery control nontrivial. We propose posterior–based decision rules for signal detection that are applicable across a broad class of continuous shrinkage priors and are calibrated to control the false discovery rate (FDR) while retaining high power. In the sparse normal means model, we show that the proposed procedures attain the optimal detection boundary and achieve frequentist asymptotic control of both FDR and false negative rate (FNR). The method is readily implementable via standard posterior sampling, and empirical studies indicate that the realised FDR and FNR closely track their theoretical targets. Applications to high-dimensional regression and Gaussian graphical models further illustrate the scope and practical effectiveness of the approach.

We determine “cumulant-type” upper bounds of the non-commutative Wasserstein distance for certain classes of distributions $\mu$ and $\nu$, which are infinite divisible with respect to the Boolean, classical and free convolutions. The main contribution of the manuscript is an estimation of the non-commutative Wasserstein distance between $\mu$ and $\nu$, expressed in terms of the difference between cumulants of order less than $2m+4$.

This paper studies how a risk holder should combine self-protection and self-insurance strategies when market insurance is absent. Self-protection reduces loss frequency, while self-insurance reduces loss severity. The risk holder incurs a joint risk-reduction cost that allows technological interaction between the two strategies and evaluates residual risk using either Value-at-Risk or Tail Value-at-Risk. In a Bernoulli model, we show that Value-at-Risk leads to a threshold-driven solution in which the optimal strategy is either no risk reduction, pure self-protection, or pure self-insurance, thereby exhibiting a substitution-type structure between the two risk-reduction strategies. By contrast, although Tail Value-at-Risk also admits a left-region/right-region decomposition, its left-region problem creates a direct residual frequency-severity interaction, making the local problem non-convex even in the Bernoulli setting. We solve this problem using an isoquant geometry method based on the marginal-balance curves for self-protection and self-insurance. The analysis identifies boundary, extreme constrained, touching, and crossing candidates, and shows how the confidence level and the cost technology determine whether self-protection and self-insurance behave as substitutes or complements. Illustrative examples compare the Value-at-Risk and Tail Value-at-Risk strategies, show how the confidence level changes the relevant isoquant geometry, and demonstrate that multiple crossings may generate non-unique optimal joint risk-reduction strategies.

We study a quasi-exactly solvable singular sextic oscillator and its algebraic spectrum. For a distinguished range of parameters, we prove that the discriminant of the characteristic polynomial of the matrix determining the algebraic spectrum admits a natural factorization into three factors. One of these factors is the square of a generalized Hermite polynomial $H_{m,n}$, whose zeros are poles of a rational solution of the fourth Painlevé equation. Hence, the spectral degeneracies (level crossing points) corresponding to a component of the discriminant locus are in exact correspondence with the zeros of generalized Hermite polynomials, providing an exact Painlevé IV analogue of the Shapiro–Tater asymptotic correspondence originally conjectured for the quartic oscillator and Painlevé II. We also characterize the values of the parameters for which the sextic oscillator admits simultaneously two quasi-polynomial eigenfunctions with opposite exponential behaviour at infinity, and show that this phenomenon is also governed by generalized Hermite polynomials. Our result also yields a new determinantal representation of $H_{m,n}$ as the resultant of the characteristic polynomials of two complementary blocks of the matrix determining the algebraic spectrum.

In this paper we prove a generalisation of the concentration-of-measure phenomenon in the discrete cube. In this setting, the concentration-of-measure phenomenon states that for every subset $\mathcal{A}$ of the discrete cube, its sum with a Hamming ball of suitably large radius $r$ – or equivalently, its $r$-expansion – results in a substantial increase in measure. We define a notion of `$(\gamma,C)$-well-spread' for subsets of the discrete cube $\{0,1\}^n$ for which the following holds: for all $\epsilon$, there exist constants $\gamma$ and $C$ such that for every $\mathcal{A}$ with $|\mathcal{A}| \geq \epsilon2^n$ and every $(\gamma,C)$-well-spread $S$, $|\mathcal{A} + S|$ is at least $(1-\epsilon)2^n$. We use this result to prove new non-trivial upper bounds to two intersection problems: how many subsets (or subgraphs) can one take from $[n]$ or $[\binom{n}{2}]$ such that every pair's intersection contains some given substructure? We prove non-trivial upper bounds for the $C_4$-intersection problem and the $4$-AP-intersection problem. We also give upper bounds that tend to $0$ for the $H$-intersection problem and $k$-AP-intersection problem as the number of edges and $k$ tend to infinity. Previously, non-trivial upper bounds were only known for non-bipartite $H$ and nothing was known for the $k$-AP-intersection problem.

The vector calculus in non-integer dimensional space (NIDS), including the NIDS version of the standard vector differential operators (gradient, divergence, and curl) is well-known. A deformation of the quaternionic Moisil-Teodorescu operator, written in terms of NIDS derivatives is the main purpose of this article. Along similar lines, we consider the NIDS reformulation of the quaternionic Bitsadze operator and the Lamé-Navier operator of the three-dimensional elasticity theory. Also, a quaternionic reformulation of a NIDS time-harmonic Maxwell system is introduced, whose solutions are directly related with those of the perturbed NIDS Moisil-Teodorescu operator. Finally, a generalized approach to the study is addressed.

We study the point process formed by the normalized logarithms of the distinct prime factors of a harmonic random sample. We prove a quantitative convergence result, in a Wasserstein-type metric over decreasing sequences, toward the atom sequence of a Dickman Poisson cloud conditioned to have total mass at most one, equivalently a uniformly attenuated Poisson-Dirichlet law. The proof is based on the conditioned geometric representation of harmonic samples, a Poisson approximation chain for the associated point processes, monotone couplings of Poisson point processes, and Kolmogorov estimates for the Dickman approximation of weighted geometric sums.

In this technical note, we study a class of infinite-horizon linear-quadratic difference games with coupled affine inequality constraints involving both state and control variables. We derive necessary conditions for the existence of open-loop generalized Nash equilibria and establish their sufficiency under additional assumptions. Sufficient conditions are characterized in terms of the existence of square-summable solutions to associated infinite-horizon linear complementarity systems. We further reformulate these conditions and show that computing open-loop generalized Nash equilibria reduces to solving a large-scale linear complementarity problem together with verifying additional conditions. Finally, we illustrate our results using a vehicle platooning example with constraints.

We formulate a Bayesian framework for reconstructing doping profiles in pn-junction semiconductor devices from boundary flux measurements. The unknown doping field is modeled as a piecewise-constant function characterized by an unknown interface and two plateau concentrations, leading to a nonlinear ill-posed inverse problem governed by a Poisson-Boltzmann-type equation. To represent this structure while enabling efficient gradient-based inference, we introduce a pushforward prior constructed by mapping a latent Gaussian field with Matérn-type covariance through a sigmoid transformation. The latent field is parameterized by a truncated Karhunen-Loève expansion, while the two piecewise-constant levels are represented by scalar plateau parameters. The prior yields differentiable approximations of piecewise-constant fields with controllable interface sharpness. We establish well-posedness of the Bayesian formulation by proving Lipschitz continuity of the forward map and Hellinger stability of the posterior. We then sample the posterior using the No-U-Turn Sampler (NUTS) with gradients computed by the adjoint method. Numerical experiments show that the combination of the proposed prior and NUTS provides more efficient posterior exploration than the dimension-robust preconditioned Crank-Nicolson (pCN) sampler, yielding one to two orders of magnitude larger effective sample sizes. In the known-plateau setting, the method reconstructs both planar and curved interfaces and provides spatially resolved uncertainty quantification (UQ). When the interface geometry and plateau concentrations are inferred jointly, posterior correlations reveal structural non-identifiability. These results demonstrate the effectiveness of combining pushforward priors with adjoint-gradient-based sampling for reliable UQ in nonlinear partial differential equation-constrained inverse problems with sharp interfaces.

External evaluations are becoming increasingly central to the governance of AI systems. In practice, however, independent auditors often have limited access to deployed models and must rely on query-based interactions. Most existing fairness evaluation methods assume static datasets and fixed-sample statistical tests, making them poorly suited to real-world auditing scenarios in which evidence must be collected sequentially under query constraints. In this work, we formulate fairness auditing as a tolerance-aware sequential hypothesis-testing problem under limited model output access. We develop a sequential generalized likelihood-ratio framework that allows auditors to accumulate evidence from a finite audit pool and stop once sufficient support for compliance or violation has been obtained. The framework is instantiated for decision-based Statistical Parity and Equal Opportunity audits, and extended to score- and logit-based proxy audits when richer observables are available. Our results show that both the fairness metric and the level of model access significantly affect audit efficiency, and that the benefits of richer output information are not uniform across auditing settings. In particular, richer outputs can substantially reduce the number of queries required for some fairness metrics and operating regimes, while offering limited gains in near-threshold cases. This work provides a practical statistical framework for sequential fairness auditing under realistic deployment constraints.

The generalized Ising problem captures a broad spectrum of hard combinatorial problems, including MAX-CUT, Number Partitioning (NPP), and Maximum Independent Set. In this work, we consider the notion of one-flip local minima for this problem. We construct a polynomial relaxation and prove the landscape equivalence theorem: there exists a one-to-one correspondence between the local minima of the relaxation and the one-flip minima of the original Ising problem. This guarantee reduces the Ising problem to finding the local minima of a smooth function, allowing us to leverage gradient-based optimizers such as ADAM. We demonstrate that our method is scalable and it achieves strong performance across challenging benchmarks, including spin-glass models, MAX-CUT, and NPP.

We consider the periodic behavior of value functions $b\mapsto\min\{f(x)\ \vert\ Ax=b,\,x\in\mathbb Z_{\ge0}^n\}$ of integer programs. We show that there exists a positive integer $M$ depending only on the constraint matrix $A\in\mathbb Z^{m\times n}$ so that the value function is convex extensible on any subdomain of the form $r+M\mathbb Z^m$ for any $r\in\mathbb Z^m$ and any separable convex objective function $f$. With this, we extend the known periodic convexity of such functions for linear objective functions $f$, as established by Eisenbrand and Rothvoss (SODA 26), to the broader class of separable convex objective functions. We derive our main periodic convexity result by first showing that periodic convexity along lines is equivalent to the integer decomposition property of dilated polyhedra. Subsequently, we use Graver basis techniques to extend the $1$-dimensional periodic convexity to domains of arbitrary fixed dimension. We apply this periodic convexity to show that value function reformulations of block-structured integer programs become periodically convex, which yields fixed-parameter tractable (FPT) algorithms. More specifically, we optimize two-stage stochastic integer programs and $n$-fold integer programs in FPT time when parameterized by the block dimensions and coefficient size of the local blocks of the constraint matrix, allowing the coefficients of the global blocks to be large. In the setting of this parameterization, which was recently introduced by Cslovjecsek, Koutecký, Lassota, Pilipczuk, and Polak (TheoretiCS 2025), our algorithms exponentially improve on the running times of the previous optimization algorithms and extend the class of objective functions that can be optimized from linear to separable convex.

Rapid prototyping of algorithms is a critical step in modern machine learning. Most algorithms exploit linear algebra, creating a need for lightweight numerical routines which – while potentially sub-optimal for the task at hand – can be rapidly implemented. For the numerical solution of ill-conditioned linear systems of equations, the standard solution for prototyping is Tikhonov-regularised inversion using a nugget. However, selection of the size of nugget is often difficult, and the use of data-adaptive procedures precludes automatic differentiation, introducing instabilities into end-to-end training. Further, while data-adaptive procedures perform multiple linear solves to select the size of nugget, only the result of one such solve is returned, which we argue is wasteful. This paper aims to circumvent the above difficulties, presenting autonugget; a Python package for automatic and stable numerical solution of linear systems suitable for rapid prototyping, and fully compatible with automatic differentiation using JAX. autonugget combines multiple linear solves using Richardson extrapolation to determine the solution of the ill-conditioned system, improving in accuracy over approximations based on a single nugget.

In this paper we investigate the stability properties of a fundamental mechanism of biological cells called phosphorylation. The system is a chemical reaction network (CRN) for which a chemical species, the substrate, can be sequentially transformed into two phosphorylated forms, by the activity of two types of enzymes, one type for phosphorylation, the other for dephosphorylation. We investigate a stochastic representation of this model, under the mass action kinetics. The total mass of the substrate is fixed at $N$, while the total mass of enzymes scales proportionally to $N$. The asymptotic behavior, when $N$ is large, of the concentrations of all chemical species is studied. We investigate the possible stable subsets of chemical species for the kinetics of the law of mass action. A stable subset is such that, with a convenient initial state, the number of copies of the species of this subset remains $O(1)$ on any finite time interval as $N$ gets large. The role of the twelve reaction rate constants, the catalytic constants of the CRN, is investigated from this point of view. An averaging principle of the corresponding Markov process is established for several regimes of the CRN. It is shown in particular that there exists a regime with three equilibrium points, with two of them stable. The proofs of the results rely on stochastic calculus with Poisson processes, convenient couplings of subsets of coordinates of the Markov process, technical results on $M/M/\infty$ queues, and a stability analysis of a dynamical system in $\mathbb{R}_+^4$.

We study continuous-time information design for emergency evacuation, where an Emergency Management Agency (the Stackelberg leader) steers strategic evacuation zones via two levers: public advisory precision (information design) and a tiered release schedule. The latent storm is a jump-diffusion process with publicly observed rapid-intensification epochs tracked by an exact finite-dimensional belief filter. Zones play a capacity-constrained congestion game on shared corridors with belief-weighted hazard exposure. The running cost couples beliefs to a convex congestion externality, making disclosure double-edged: sharper information reduces false-alarm departures but synchronizes genuine ones, and convex congestion penalizes that synchronization. We prove that: (i) the followers' game admits a potential reduction to a convex control problem; (ii) the leader's distributionally robust relative-entropy problem is characterized by an Isaacs equation whose value is the unique viscosity solution, with verification valid for non-smooth bang-bang feedback; and (iii) without transfers, the leader's first-order condition retains an equilibrium-response term, positioning optimal information design as a second-best congestion toll. Structurally, we show that a staggered evacuation order dominates simultaneous advisories; phased evacuation emerges endogenously as optimal information design. Furthermore, public-signal precision is sign-ambiguous due to an informational Braess effect, where vague advisories are optimal unless complemented by a staggered order. Calibrated to Hurricane Rita using NHC archives, TxDOT capacities, and HRRC surveys, the model reproduces the observed gridlock along the Interstate 45 (I-45) evacuation corridor in Texas. The optimal policy removes essentially all in-transit congestion exposure, reducing social cost by 89%, while staggered disclosure alone yields a 70% reduction.

Anti-Gaussian formulas represent an efficient tool for a dynamical estimation of the error of the underlying Gaussian rule. When applied to the Jacobi weight function it is known that such formulas are not always internal. In this work we show how to overcome this problem by using the so called modified anti-Gaussian rule with suitable parameter {\theta} = {\theta}(n), that depends on the number n of quadrature points of the Gaussian formula. Next we study theoretically the asymptotic rate of convergence of the corresponding modified averaged Gaussian formulas. We conclude by showing the benefits of this approach via numerical experiments. All the Matlab codes used in this work are available as open-source software.

We study the evolution of plane closed curves with fixed area moving by the negative $L^2$-gradient of their elastic energy. For smooth initial data, we establish local and global existence of the flow. By imposing a simplicity assumption and an initial energy bound, we show that the length of the evolving curve remains uniformly bounded. This yields subconvergence to a critical point, which is then improved to full convergence by utilizing a Łojasiewicz–Simon inequality. Conversely, an analysis of the energy profile curve, which maps a given length to the minimal energy among all curves with that length and fixed area, reveals that the length diverges to infinity for initial data satisfying specific length and energy criteria. We visualize our findings through numerical simulations.

In this manuscript, we provide an independent equational basis for the variety of reflexive Nelson algebras, a generalization of the variety of SNA-algebras. The proof of this result relies on a substantial number of technical arguments and computational examples. The result complements a broader study of reflexive Nelson algebras by showing that the variety admits an independent equational axiomatization.

In this paper we will present neighborhood semantics for non-normal modal extensions of $\clon$, which is a sublogic of FDE. Our framework is built upon earlier work on FDE-based non-normal modal logics and employs two different neighborhood functions for each modal operator. Despite being a logic with a very weak negation operator, we will show that with the right definition of the rejection sets of the modal operators, we can validate non-trivial axioms that contain the weak negation operator. The philosophical aim of our approach is to construct the basis for deontic logics that are able to accommodate both the usual deontic principles and moral dilemmas, without resulting in trivialization of the system.

Differential equation models are widely used to describe, interpret, and predict dynamical phenomena across science and engineering. In practice, however, the governing dynamics are rarely fully known and must be inferred from observational data. Traditionally, inverse problems in differential equation modelling have focused on estimating unknown parameter values. In this setting, structural identifiability determines whether parameter values can, in principle, be uniquely recovered from ideal observations and is, therefore, a prerequisite for meaningful inference. More recently, the integration of machine learning with mechanistic modelling has enabled the discovery of unknown equations, functions, and constitutive relationships, substantially expanding the space of admissible models. This raises a fundamental question: under what conditions can unknown functional components be uniquely recovered from data? In this paper, we generalise the classical notion of structural parameter identifiability to functional identifiability. We first identify broad classes of models for which unique functional recovery is impossible. We then show how functional identifiability can be assessed for differential equation models using differential algebra-based techniques which are well-established as a means of assessing structural identifiability for ordinary differential equation-based models. Our framework reveals new phenomena that arise in the transition from parametric to functional inference and have no analogue in the classical setting. Finally, we characterise functional identifiability in several common model classes. Taken together, our results demonstrate that functional identifiability provides a theoretical foundation for modern inverse problems in differential equation modelling, particularly those that use machine learning representations of unknown system components.