We classify fillable contact structures on all negative-definite star-shaped plumbings. Along the way, we show that such Seifert fibred spaces admit a unique negative maximal twisting number, and compute it explicitly using the Alexander filtration in lattice cohomology. In particular, we show that the negative-twisting tight structures on these manifolds are induced by the Stein structures on the minimal resolution of the underlying complex surface singularity. As an application, we provide a necessary condition for a negative-definite Seifert fibred space to admit a separating contact-type embedding in a strong symplectic filling of a generalised $L$-space.

We introduce a refined immersed boundary (IB) methodology that is better-than-first-order accurate in practice, while preserving key properties of "continuous-forcing" IB approaches that retain a singular source term in the governing equations. Our method leverages a smoothed indicator (Heaviside) function, following ideas from multiphase flow and immersed layers formulations, to recast the IB solution as a composite of distinct interior and exterior fields. We demonstrate that, when cast through this composite-solution lens, prior continuous-forcing IB methods can be seen as neglecting terms in the governing and constraint equations that restrict the solution to first-order accuracy. We incorporate these terms to systematically improve accuracy without the need for heuristic corrections. In canonical Poisson problems, we empirically demonstrate second-order convergence, and in incompressible Navier-Stokes simulations the method achieves slightly sub-second-order performance. While our present study focuses on these cases, the framework suggests a path towards second-order accuracy or higher, with further extensions. This perspective reframes accuracy limitations typically attributed to IB schemes. Although continuous-forcing IB methods are often reported to be only first-order accurate, we show that neither smoothing nor interface interpolation inherently restricts attainable order. Moreover, we naturally incorporate this higher-order formulation into a projection-based solution process. The resulting algorithm simultaneously mitigates the spurious surface stresses produced by ill-conditioned linear systems and reduces sensitivity to geometric resolution, addressing both conditioning and accuracy concerns within a unified approach.

We study how long the SIRS process persists or how quickly it reaches extinction across various network topologies. Our results provide a three-part characterization of this process: In finite sparse graphs, we prove the existence of a regime where the process survives for an exponentially long time. In heavy-tailed networks with power-law-like exponents, we show that for all range of parameters, the survival time is exponential. Finally, for infinite trees, we find sufficient conditions for strong survival, showing the root is re-infected infinitely often even for light-tailed distributions like the Poisson distribution.

Efficient exploration is a central problem in reinforcement learning and is often formalized as maximizing the entropy of the state-action occupancy measure. While unconstrained maximum-entropy exploration is relatively well understood, real-world exploration is often constrained by safety, resource, or imitation requirements. This constrained setting is particularly challenging because entropy maximization lacks additive structure, rendering Bellman-equation-based methods inapplicable. Moreover, scalable approaches require policy parameterization, inducing non-convexity in both the objective and the constraints. To our knowledge, the only prior model-free policy-gradient approach for this setting under general policy parameterization is due to Ying et al. (2025). Unfortunately, their guarantees are limited to weak regret and ergodic averages, which do not imply that the final output is a single deployable policy that is near-optimal and nearly feasible. In this work we take a different approach to this problem, and propose Policy Gradient Penalty (PGP) method, a single-loop policy-space method that enforces general convex occupancy-measure constraints via quadratic-penalty regularization. PGP constructs pseudo-rewards that yield gradient estimates of the penalized objective, subsequently exploiting the classical Policy Gradient Theorem. We further establish the regularity of the penalized objective, providing the smoothness properties needed to justify the convergence of PGP. Leveraging hidden convexity and strong duality, we then establish global last-iterate convergence guarantees, attaining an $ε$-optimal constrained entropy value with $ε$ bounded constraint violation despite policy-induced non-convexity. We validate PGP through ablations on a grid-world benchmark and further demonstrate scalability on two challenging continuous-control tasks.

Denote by $p(k)$ the limit, as $n \rightarrow \infty$, of the probability that a random permutation on a set of size $n$ has an invariant set of size $k$. We give an asymptotic formula for $p(k)$, showing that it is asymptotically $f(\{\log_2 k\}) k^{-δ} (\log k)^{-3/2}$ where $δ= 1 - \frac{1 + \log \log 2}{\log 2} \approx 0.086$ and $f$ is a smooth, positive, function on $\mathbb{R}/\mathbb{Z}$, which we will describe explicitly. The function $f$ satisfies $\frac{\max f}{\min f} < 1 + 2 \times 10^{-7}$ and we conjecture that it is not constant. Estimating $p(k)$ is a model for the more well-known question which asks for an estimation of $M(n)$, the number of distinct elements in the $n$-by-$n$ multiplication table. By elaborating on the techniques in this paper, we will give an asymptotic for $M(n)$ in forthcoming work.

We review the quantization of scalar field theory on $λ$-Minkowski space using the Batalin--Vilkovisky (BV) formalism. We consider $ϕ^3$-theory in two different quantization schemes: standard and braided. While standard BV quantization is based on an ordinary $L_\infty$-algebra, braided BV quantization is based on a braided $L_\infty$-algebra. We compare the tree-level three-point and four-point correlation functions in the two approaches. For the four-point function, standard quantization leads to two inequivalent classes of diagrams with different noncommutative contributions, whereas braided quantization yields only a single class of diagrams with noncommutativity entering solely through an overall phase factor depending on the external momenta.

In this paper, we propose a scaled gradient modified non-monotone line search method for solving constrained minimization problems, and explore several specific properties of this method, namely, its convergence analysis. We discuss the linear convergence rate of the sequence generated by the proposed algorithm to a solution of the constrained minimization problem where the objective function is strongly quasiconvex. We consider numerical examples of large-scale fractional programming and quadratic programming for the function of pseudo convex and strongly quasiconvex and compare the performance of the proposed algorithm with the existing ones for these examples.

We provide a unified framework for independence and mean independence tests based on the Hilbert-Schmidt independence criterion, extending some previous results in the literature to hold in general topological spaces. We also present a complete theoretical analysis of the test statistic asymptotic behavior when the observed sample corresponds to a partial sample path of some stationary and ergodic stochastic process under near epoch dependence assumptions. In particular, we explore the test statistic consistency and limit distribution under both fixed and local hypothesis. The finite sample performance of the test(s) is illustrated with a succinct simulation study involving functional data.

We construct bounded, commuting projections for the three-dimensional de Rham complex with the additional property that the projections preserve the trace of functions/fields if the latter is a piecewise polynomial in the appropriate trace space. The projections are locally defined and stable in the graph norm. More precisely, the part of the graph norm involving the exterior derivative only involves the oscillation of this derivative in a narrow strip of elements touching the boundary and weighted by the local mesh size. Moreover, the projections are $L^2$-stable locally when acting on functions/fields whose exterior derivative is a piecewise polynomial in the appropriate space. We present two salient applications of the present bounded, commuting, discrete-trace preserving projections: the construction of stable liftings of piecewise polynomial data and an optimality result on the discrete versus continuous extension of piecewise polynomial data.

In 1967 Hajnal and Juh{á}sz showed that the cardinality of a first-countable Hausdorff space with the countable chain condition has cardinality at most $\mathfrak{c}$, the cardinality of the real line. We give an improvement of this celebrated theorem by replacing ``first-countable" with the weaker condition ``each point has a countable centered local $π$-base". Given a point $p$ in a topological space $X$, a \emph{local} $π$-\emph{base} $\scr{B}$ at $p$ acts like a neighborhood base at $p$ except that $p$ may not be in any member of $\scr{B}$. A local $π$-base $\scr{B}$ has the \emph{finite intersection property} if any finite intersection of members of $\scr{B}$ is nonempty. We call this type of local $π$-base \emph{centered}. A centered local $π$-base behaves even more like a neighborhood base in a sense. A space has the \emph{countable chain condition} if every family of pairwise disjoint open sets is countable. We also improve a theorem of Pospi{\v s}il from 1937 using centered local $π$-bases. As is customary, examples are given to demonstrate these improvements are strict. Compact Hausdorff spaces are also explored in this connection, along with variations on the notion of a centered local $π$-base.

This paper investigates the existence and uniqueness of solutions for a nonlinear evolution equation governed by an m-accretive operator A in a Banach space, presenting a perturbation term that does not satisfy the Lipschitz condition.

In this article we show that the semi log canonical abundance for compact Kähler varieties fails in dimension $3$. More specifically we construct a counterexample of a compact Kähler (irreducible) slc threefold $(X, 0)$ such that $K_X$ is nef and $κ(\tilde X, K_{\tilde X}+\tilde D)=0$, where $μ:(\tilde X, \tilde D)\to X$ is the normalization morphism, but $K_X$ is not semiample. On the other hand, we show that if we start with a compact Kähler semi-dlt pair, then the abundance does hold, i.e., if $(X, Δ)$ is a compact Kähler sdlt pair of dimension $3$ such that $K_X+Δ$ is nef, then it is semiample. We also show that if $(X, Δ)$ is a compact Kähler slc pair of dimension $3$, $K_X+Δ$ is nef, and $κ(X'_i, Δ'_i+D'_i)>0$ for all $i$, where $μ:\sqcup(X'_i, Δ'_i+D'_i)\to (X,Δ)$ is the normalization, then $K_X+Δ$ is semiample.

This paper studies a key research question: how to achieve perfect privacy in over-the-air computation (AirComp)? The problem is particularly intriguing due to a dilemma. Real-field operations can ensure invertibility but generally introduce statistical dependence, resulting in inevitable privacy leakage. In contrast, modulo operations can decorrelate the output from the original message, but suffer from the ill-posed invertibility when applied over non-prime groups (e.g., the real field). This raises a subtle yet fundamental question: Does perfect privacy intrinsically conflict with AirComp? We show that the answer is no. By carefully leveraging the interplay between real-field and modulo operations, perfect privacy and accurate computation can, in fact, be achieved simultaneously, enabling perfectly private aggregation.

Building on the genus-3 reduction $C_A : w^2 = λ^8 + A λ^4 + 1$ established in our companion paper (arXiv:2604.09328), we give an unconditional proof of the perfect-cuboid conjecture ("Conjecture B") on $1{,}072$ explicit master-tuple fibers, excluding all rational $(a,b)$-specialisations on each such fiber. Our three main contributions are: (i) a structural classification theorem showing that every primitive Euler-brick arises from the standard $(a,b,m,n)$-parametrisation up to scaling; (ii) a torsion-intersection argument applied to the elliptic quotients $E_A'$ and $E_A''$: whenever the rank-zero hypothesis and the appropriate torsion condition hold for one of them, $|H_{m,n}(\mathbb{Q})| = 8$ is forced, with the eight points all corresponding to degenerate bricks; (iii) two complementary techniques to verify the rank-zero hypothesis algorithmically -- PARI's ellrank (2-descent) and, where this is ambiguous, Sage's exact rational evaluation of $L(E,1)/Ω_E$ via modular symbols, which combined with the modularity theorem, Kolyvagin's theorem, and Edixhoven's bound on the Manin constant for semistable curves yields an unconditional rank-zero certificate -- together with an explicit lift count refining the naive torsion-intersection bound when the torsion is larger than the leading case. We exhibit $1{,}072$ such fibers with $\max(m,n) \le 100$ on which Conjecture B is thereby established unconditionally.

We investigate the dynamics of dissipative systems with stochastic forcing and focus in particular on mean-square stability. First we show, under a natural condition on the drift and diffusion, that the stochastic system is mean-square dissipative. Next we examine the linearised system and state conditions ensuring that perturbations of a linear system with affine noise are bounded. We then relate the mean-square dynamics of the nonlinear and linearised systems. The approach gives a straightforward deterministic method to examine the effects of stochastic forcing on the stability of equilibria of deterministic systems and to obtain bifurcation diagrams that can be included into standard numerical continuation packages. The technique is illustrated numerically on some standard and non-standard examples.

The impossibility of eliminating hallucination, understood here as incorrect definite answers, in sufficiently expressive yes-or-no formal domains is an immediate consequence of classical undecidability theorems. This note does not revisit that forced-answer obstruction as its main claim. Instead, it attempts to formally describe the corresponding limitation for abstaining systems. Abstention can trivially avoid hallucination if the system is allowed to abstain on every input; the substantive question is how large the domain of guaranteed correct non-abstaining answers can be. We formulate this question using separation in the arithmetical hierarchy. Given disjoint sets $A$ and $B$, any system that answers Yes on all queries indexed by $A$ and No on all queries indexed by $B$ induces a separator of $A$ from $B$. By combining this observation with the classical existence theorem of $Δ_{n}^{0}$-inseparable pairs of $Σ_{n}^{0}$-sets, we yield a computability-theoretic trade-off between avoiding hallucination by abstention and maintaining a large domain of guaranteed coverage.

We study Turán-type extremal problems for distance graphs, motivated by work of Csikvári, Bollobás, Tyomkyn, and Uzzell. We determine the maximum number of vertex pairs at distance three in an $n$-vertex graph with no triangle formed by these pairs, resolving the first case of a conjecture of Tyomkyn and Uzzell. We also determine the maximum number of vertex pairs at distance two in an $n$-vertex graph with no triangle formed by these pairs and give a complete characterization of the extremal graphs, settling another problem of Tyomkyn and Uzzell.

We address the problem of numerically approximating the velocity and pressure governed by the Stokes system when the boundary conditions are only partially known and thus do not uniquely determine the velocity-pressure couple. We propose an algorithm that takes advantage of available linear measurements of the velocity and pressure to construct a numerical approximation. This approximation is guaranteed to be near-optimal in the sense that it approximates the velocity-pressure couple that minimizes, in the energy norm, the distance to all other solutions satisfying the measurements and the Stokes system.

Exponential families encompass the distributions central to modern machine learning -- softmax, Gaussians, and Boltzmann distributions -- and underlie the theory of variational inference, entropy-regularized reinforcement learning, and RLHF. We isolate a simple identity for exponential families that expresses the KL difference $\mathrm{KL}(q \| p_{λ_2}) - \mathrm{KL}(q \| p_{λ_1})$ in terms of the log-partition function $A(λ)$ and the moment $μ_q$. Remarkably, this identity together with the single fact that $\mathrm{KL} \geq 0$ (with equality iff $p = q$) suffices, by direct substitution and rearrangement, to derive a cluster of results that are classically obtained by separate, heavier arguments: a generalized three-point identity for arbitrary reference distributions, Pythagorean theorems for I-projections and reverse I-projections, convexity of the log-partition function, identification of its Legendre dual in KL terms, the Gibbs variational principle, and the explicit optimizer in KL-regularized reward maximization, including the exponential tilting formula underlying entropy-regularized control and RLHF. Beyond these purely algebraic consequences, standard analytic arguments recover the gradient formula for the log-partition function, the Bregman representation of within-family KL divergence, and the surjectivity of the moment map. The note is self-contained.

For a graph \(G\), let $avm(G)$ denote the average size of its maximal matchings. This parameter was introduced by Engbers and Erey in the study of extremal problems for maximal matchings, and they asked for extensions from trees and unicyclic graphs to \(k\)-cyclic graphs. In this paper, we solve the first non-unicyclic case by determining the minimum value of $avm(G)$ over all connected bicyclic graphs with \(n\) vertices and \(n+1\) edges. We prove that, for every connected bicyclic graph \(G\) of order \(n\ge 5\), \[ \operatorname{avm}(G)\ge \frac{4n-11}{2n-5}. \] Moreover, equality holds uniquely for the graph obtained from two triangles sharing a common edge by attaching all remaining \(n-4\) pendant edges to one of the two vertices of degree \(3\). The key point is to translate the minimization of \(\operatorname{avm}(G)\) into structural restrictions on small maximal matchings, which are then analyzed through the three possible bicyclic core types.

Fairness in machine learning remains challenging due to its ethical complexity, the absence of a universal definition, and the need for context-specific bias metrics. Existing methods still struggle with intersectionality, multiclass settings, and limited flexibility and generality. To address these gaps, we introduce MIFair, a unified framework for bias assessment and mitigation based on mutual information. MIFair provides a flexible metric template and an in-processing mitigation method inspired by the Prejudice Remover, defining group fairness as statistical independence between prediction-derived variables and sensitive attributes. We further strengthen its information-theoretic foundation by establishing equivalences with widely used fairness notions such as independence and separation. MIFair naturally supports intersectionality, complex subgroup structures, and multiclass classification and employs regularization-based training to reduce bias according to the selected metric. Its key advantage is its versatility: it consolidates diverse fairness requirements into a single coherent framework, enabling consistent benchmarking and simplifying practical use. Experiments on real-world tabular and image datasets show that MIFair effectively reduces bias, including previously unaddressed multi-attribute scenarios, while maintaining strong predictive performance across the evaluated settings.

In this paper, we prove that if the Frobenius traces agree at all but finitely many places, then two $l$-adic Galois representations, associated to rank-$2$ non-CM Drinfeld modules of generic characteristic, are isomorphic. As a generalization, we show that under a mild dimension hypothesis, the "Frobenius trace equality at all but finitely many places forces isomorphism" between two Galois representations over a local field of positive characteristic only holds under an absolute irreducibility assumption. Moreover, we formulate and prove a function field analogue of strong multiplicity one property for semisimple Galois representations over a local field of positive characteristic.

We investigate the lifetime of dynamical regimes under the impact of noise motivated by low-dimensional models of the atmosphere. One may expect that the inclusion of noise tends to make the system leave prescribed regions of the state space faster. However, for relevant systems with complexities ranging from phenomenological toy models to reduced models of atmospheric dynamics, this intuition has proven misleading. As long as the noise is sufficiently small, the noisy system stays in regimes of interest on average longer than its deterministic counterpart, an effect we call ``stochastic inertia''. This phenomenon has been observed through extensive numerical simulations for different noise levels. We propose a numerical technique for testing the occurrence of stochastic inertia, constructing, for any fixed noise level, a Markov chain on the set of points given by a sufficiently long trajectory of the system without noise. The method is shown to correctly predict the presence of stochastic inertia in simple systems, and its utility is demonstrated on a paradigm model of atmospheric dynamics.

We show that every minimally generically globally rigid graph in $\mathbb R^d$ which contains a subgraph isomorphic to $K_{d+2}$ is itself isomorphic to $K_{d+2}$, confirming a conjecture by Garamv{ö}lgyi, Jackson, and Jord{á}n. The proof is entirely generated by ChatGPT 5.5.

Let $M$ be a smooth manifold without boundary and let $\mathcal{T}$ be a countable collection of manifolds with corners, each equipped with a smooth map to $M$. We show that the singular simplicial set $\mathrm{Sing}(M)$ of $M$ deformation retracts onto the simplicial subset $\mathrm{Sing}^{\mathcal{T}}\!(M)$ of smooth singular simplices that are transverse to every element of $\mathcal{T}$.

Viscoelastic rate-type fluid models constitute a fundamental framework for the mathematical description of complex materials exhibiting coupled elastic and viscous effects, with a wide range of applications in engineering, biomaterials, and medicine. In realistic regimes, thermal effects are essential and lead to strongly coupled systems in which heat conduction and temperature-dependent constitutive laws play a decisive role. In this paper, we develop a thermodynamically consistent model for heat-conducting viscoelastic rate-type fluids. We establish the existence of a global weak solution in the full three-dimensional setting. In contrast to the existing literature, no smallness, regularity, or structural restrictions on the initial data are imposed beyond natural energy and entropy bounds, and no additional regularising mechanisms such as artificial stress diffusion are required. The analysis is based on a weak-strong framework combining energy and entropy balances with compactness tools, allowing us to treat the full nonlinear coupling between the fluid velocity, the temperature, and the elastic stress.

We study the Liouville equation $\triangle u+e^{2u} =0$ in a Riemannian surface $(M, g)$ with nonnegative $Ricci$ curvature. Under some asymptotic lower bound assumptions, we classify all the solutions to this equation, meanwhile we obtain the rigidity results for the ambient manifold. Note that our assumptions are optimal in some sense and differ from the classical assumption of finite total curvature.

We derive a sharp upper bound on the residuals produced by the flexible GMRES (FGMRES) method. The bound shows that FGMRES exhibits two phases of convergence depending on the residual tolerance of the inner preconditioner. For small tolerances, the convergence of FGMRES is practically geometric with a constant rate throughout, while for looser tolerances the two-phase behavior becomes more pronounced. We also show that the derived bound cannot be improved and construct an example for which it becomes an equality.

Clustering is a fundamental collective phenomenon in agent-based models (ABMs) of opinion dynamics. To study clustering in systems with co-evolving social and opinion variables, we derive stochastic partial differential equation (SPDE) models that describe the evolution of clusters on a reduced state space. We consider two settings: one in which opinions do not affect social interactions, and another one in which a feedback mechanism couples the two. Our approach extends reduced PDE modelling to a stochastic framework, which is essential for capturing long-term cluster behaviour. Numerical experiments demonstrate that the proposed reduced SPDEs substantially decrease computational cost compared to full-state SPDE models, such as the Dean-Kawasaki equation, while still accurately reproducing the clustering behaviour of the underlying ABM. As a result, these reduced models provide an efficient tool for studying systems with large populations, including those arising in the analysis of real-world data: in particular, we provide an application related to the large-scale General Social Survey (GSS), which comprises opinion and social data of the US population since 1972.

Many stochastic systems are built by wiring typed components together, but the wiring is often neither purely sequential nor type-homogeneous. This paper develops categorical semantics for such systems using ordered polycategories whose morphisms are Markov kernels. The basic operation is kernel slotwise composition, which connects one output slot of a many-output kernel to one input slot of another and marginalizes the internal wire. We prove its structural laws by assigning trace semantics to finite acyclic diagrams. We then introduce colored Markov polycategories, where objects and kernels carry colors and typed connections are realized by coherent interface kernels. This gives a colored kernel slotwise composition and trace semantics for typed stochastic diagrams. To describe systems whose structure changes, we co-index colored Markov polycategories and parameter spaces over an indexing category. Finally, for finite acyclic parameterized diagrams, we prove a diagrammatic differentiation result. The derivative of an expected scalar objective is obtained from local reverse-mode contributions at the parameterized vertices, with stochastic and deterministic kernels handled through admissible local gradient operators. The construction gives a typed, compositional language for finite acyclic stochastic systems and their parameter sensitivities.