ReLU neural networks trained as surrogate models can be embedded exactly in mixed-integer linear programs (MILPs), enabling global optimization over the learned function. The tractability of the resulting MILP depends on structural properties of the network, i.e., the number of binary variables in associated formulations and the tightness of the continuous LP relaxation. These properties are determined during training, yet standard training objectives (prediction loss with classical weight regularization) offer no mechanism to directly control them. This work studies training regularizers that directly target downstream MILP tractability. Specifically, we propose simple bound-based regularizers that penalize the big-M constants of MILP formulations and/or the number of unstable neurons. Moreover, we introduce an LP relaxation gap regularizer that explicitly penalizes the per-sample gap of the continuous relaxation at training points. We derive its associated gradient and provide an implementation from LP dual variables without custom automatic differentiation tools. We show that combining the above regularizers can approximate the full total derivative of the LP gap with respect to the network parameters, capturing both direct and indirect sensitivities. Experiments on non-convex benchmark functions and a two-stage stochastic programming problem with quantile neural network surrogates demonstrate that the proposed regularizers can reduce MILP solve times by up to four orders of magnitude relative to an unregularized baseline, while maintaining competitive surrogate model accuracy.

Psychiatric disorders have been traditionally conceptualized as latent conditions producing observable symptoms, but recent studies suggest that psychopathology may emerge from symptoms interactions. Psychometric networking model these relations focusing on pairwise associations but overlooks higher-order dependencies arising among groups of variables. These dependencies may reflect synergistic mechanisms, where joint symptom configurations convey more information than pairwise relations, or redundancy, where information overlaps. We introduce an information-theoretic multiplex hypergraph framework to identify and compare higher-order interactions in eating disorders data, across diagnostic groups (e.g., anorexia nervosa). Higher-order structures are quantified using $Ω$-information, a measure that captures the balance between redundancy and synergy. To address the combinatorial growth of candidate subsets, multiple testing and estimation instability, we propose a structured pipeline comprising: (i) targeted candidate selection based on dyadic network topology and theory-driven subscale information; (ii) a three-stage inferential procedure combining null-model testing with bootstrap robustness assessment; and (iii) the construction and analysis of diagnosis-layered, synergistic and redundant multiplex hypergraphs. Results highlight how synergy captures the emergent, higher-order organization of diagnoses, revealing both a stable transdiagnostic core and diagnosis-specific ways in which these domains combine. By contrast, redundancy is confined to eating and body-image related content, marking reinforcement rather than broader symptom integration.

We investigate the problem of jointly testing a pair of composite hypotheses and, depending on the test result, estimating a random parameter under distributional uncertainties. Specifically, it is assumed that the distribution of the data given the parameter of interest, is subject to uncertainty. Both, a Bayesian formulation and a Neyman-Pearson-like formulation, are considered. It is shown that the optimal policy induces an $f$-similarity that must be maximized to identify the least favorable distributions. Besides the general results, the implementation is investigated using a band-type uncertainty model. For designing the minimax procedures, existing algorithms are modified to increase convergence speed while maintaining numerical stability. The proposed theory is supplemented by numerical results for both formulations.

In this paper, we present the Electric Mobility Dial-a-Ride Problem (EM-DARP), which extends the Electric Vehicle Dial-a-Ride Problem (EV-DARP) to better accommodate human-focused mobility services. The problem involves utilizing a fleet of heterogeneous Electric Vehicles (EVs) to fulfill a set of customer requests with DARP and mobility-related specifications, while incorporating visits to charging stations amid requests. The problem is formulated as a Mixed-Integer Linear Program (MILP) and subsequently solved for a number of curated evaluation scenarios to demonstrate its practical applicability.

Since their rediscovery in the 1990s, multiple zeta values have become ubiquitous in many areas of mathematics and physics. Their standard integral and sum representations can usually be traced back to a single source, namely the iterated integrals on the Riemann sphere with three punctures. We refer to such representations as the \emph{linear} geometry of multiple zeta values, since the denominators of the corresponding integrands factor completely into linear terms. However, there also exist equally important and entirely distinct integral representations for multiple zeta values arising in mathematics and physics, in which matrix determinants appear in the denominator of the integrand. We call this the \emph{non-linear} geometry of multiple zeta values. These lectures trace the origins of this non-linear geometry and provide an introductory journey through a range of topics including tropical geometry, the moduli spaces of tropical curves, Feynman integrals in quantum field theory, the general linear group of integer matrices, and the reduction theory of quadratic forms. In doing so, we propose a geometric framework for multiple zeta values based on such non-linear, determinantal representations and set out a number of open questions for future research.

Several theoretical and astrophysical problems - including gravitational-wave modeling for extreme mass-ratio inspirals - require accurate time-domain solutions of the spin-weight $s=-2$ Teukolsky equation in Boyer-Lindquist coordinates. Because such simulations are performed on finite computational domains, they typically introduce an artificial outer boundary where nontrivial boundary conditions must be imposed. If these conditions are inaccurate, then spurious reflections and slowly-growing unphysical modes may corrupt long-time evolutions. We develop and implement exact radiation outer boundary conditions for the Bardeen-Press equation (a harmonic moment of the $a=0$ Teukolsky equation), making the artificial boundary transparent at any finite radius. We also construct near-to-far field teleportation kernels that map field data recorded at finite radius $r_1$ to the data reaching $r_2 > r_1$. The possible choice $r_2 = \infty$ corresponds to asymptotic waveform evaluation, that is propagation of the data to future null infinity. We show that both boundary and teleportation kernels are well approximated by exponential sums, with associated error bounds. Implemented in a time-domain solver, our kernel-based boundary conditions eliminate unphysical late-time growth and give the correct late-time decay rates, affording efficient long-duration simulations for waveform modeling and related blackhole perturbation calculations.

The set of matrix tuples with invariant subspaces whose dimensions sum up to the dimension of the space, but which do not span the whole space form an algebraic hypersurface. We found the equation of this hypersurface. This generalizes previous joint work.

Component Mode Synthesis methods, such as the Craig-Bampton (CB) approach, are widely used in structural dynamics due to their modularity and compatibility with substructuring workflows. While highly effective for linear systems, extending these methods to geometrically nonlinear structures remains a significant challenge. In this work, we propose a nonlinear extension of the CB method tailored to such contexts. The approach is based on the construction of a quadratic reduction manifold, derived via perturbation analysis, in which high-frequency fixed-interface modes are statically condensed onto a reduced set of low-frequency modes and interface coordinates. This formulation enables the representation of geometric nonlinear effects without increasing the number of reduced degrees of freedom.The resulting Nonlinear Craig-Bampton (NL-CB) reduced-order model is obtained through Galerkin projection onto the tangent space of the manifold and admits a polynomial structure that is efficient for time integration. The formulation preserves the Lagrangian structure of the underlying finite element model, ensuring consistent energetic behavior and numerical stability.The proposed method is demonstrated on representative nonlinear structural systems of increasing complexity. The results show that the NL-CB model captures the essential nonlinear dynamic response while retaining the modularity and computational efficiency of classical substructuring approaches.

This paper presents a method for computing local mean, variance, standard deviation, and effective sample count on incomplete gridded data using boundary-aware spectral operators. The framework combines normalized convolution with explicit boundary-condition modeling: reflective Discrete Cosine Transform (DCT) for non-periodic Cartesian axes and periodic Real Fast Fourier Transform (RFFT) for circular azimuth processing in polar geometry. Stability safeguards (denominator floor, prefill fallback, and variance clamp) are specified for under-supported regions. We evaluate the framework across three targeted scenarios: a Cartesian boundary-condition check demonstrating the mitigation of wrap-around artifacts, a synthetic 3D outlier-identification test, and a real-radar polar application. Results establish bounded, support-aware interpretation of local statistics while preserving a concise reproducibility path through the open-source 'dct\_toolkit' implementation.

We analyse approximation algorithms (greedy heuristics) for the classical domination number and two multiple domination numbers in simple graphs. First, we present a short self-contained proof of the known result that the minimum domination problem in any graph $G$ with maximum degree $Δ$ can be solved within the approximation ratio of ${\ln(Δ+1)+1}$. The proof is based on an analysis of a simple greedy heuristic. Then, by analysing more advanced greedy heuristic techniques and using ideas from our self-contained proof for the classical domination number, we fix a gap in the existing proof of a similar result for the $k$-tuple domination number. That is, we prove that the minimum $k$-tuple domination problem indeed can be approximated within the ratio of $\ln(Δ+1)+1$. The proof of this result is self-contained, direct, and much shorter than the existing proof, which contains the gap. Finally, we show that the known approximation ratio of $\ln(2Δ)+1$ for the minimum $k$-domination problem can be improved to a better ratio.

These lecture notes introduce the statistical analysis of continuous-time generative models built from Markov dynamics. We begin with the stochastic-calculus foundations of score-based diffusion models, including time reversal, score matching, and sampling from learned scores. We then present the broader framework of generator matching, which describes flows, diffusions, jump processes, and discrete generative models through their infinitesimal generators. We then focus on finite-sample guarantees. We explain how errors in the learned drift or generator propagate to the final generated distribution, why stability and regularity properties are essential, and how time-adaptive neural network classes can achieve optimal Wasserstein rates for smooth target distributions. Overall, the notes aim to connect modern generative modeling algorithms with the probabilistic, analytic, and statistical tools needed to understand their worst-case performance.

We extend the refined asymptotics of analytic torsion associated to congruence subgroups of $\operatorname{SL}(n)$ in previous work, to congruence subgroups in a large family of reductive groups. This is applied to give new asymptotics and bounds on the growth of torsion in the cohomology of congruence subgroups of $\operatorname{SL}(2,\mathcal{O}_F)$ for $F$ a number field, and of congruence subgroups in $\operatorname{SO}(n,1)$ with $n$ odd.

We prove a sufficient condition for the existence of a $T$-periodic solution for the planar system $\dot z=F(t,z)$, characterized by the growth to infinity of the rotations made in one period by solutions starting at increasingly large initial values. Our result applies in particular to superquadratic Hamiltonian systems satisfying the Ambrosetti--Rabinowitz condition. The key novelty of the paper is that we do not require any growth condition on the flow to ensure global existence of solutions, allowing finite-time blow-up. Our method is based on a fixed-point theorem which exploits the rotational properties of the dynamics. To conclude, we discuss a family of examples of Hamiltonian systems showing finite-time blow-up.

We conclude an investigation of Abrishami, Esperet, Giocanti, Hamman, Knappe and Möller studying the existence of periodic colourings of locally finite graphs. A colouring of a graph $Γ$ is periodic if the resulting coloured graph has a finite number of orbits under its colour-preserving automorphisms, as such it is natural to consider those quasi-transitive graphs with finite quotient. In the case that the graph is planar and has 1-end we prove that it always permits a periodic proper vertex colouring. This is shown by constructing isometry respecting embedded maps into the Euclidean and hyperbolic planes and leveraging known properties of Euclidean and hyperbolic isometry groups. Moreover, in the case that a graph is Euclidean we show that this can always be done in 5 colours.

We generalize results concerning Gel'fand integration of functions taking values in the space of operators on Hilbert spaces to certain Banach spaces. Building on ideas from \cite{M24} we provide sufficient conditions for the Gel'fand integral to be $(q,p)$-summing and we use the developed techniques to answer a question posed in the mentioned article. Applications to positive operator-valued functions between certain function spaces are also given.

We introduce a class of zero mean curvature surfaces with singularities in the isotropic 3-space, called ZMC-faces. As a main result, we establish three Osserman-type inequalities for a ZMC-face under certain assumptions on both completeness and finiteness of the total curvature. The equality conditions of these inequalities are related to the asymptotic behaviors of the ends. Moreover, we present several examples of ZMC-faces attaining equalities in these inequalities.

In this work, we establish continuity properties of strongly singular integral operators for extreme values of $p$. Particularly, weighted $L^\infty$-$BMO$ boundedness is obtained, generalizing Miyachi's result to the context of Muckenhoupt weights. As an application, we get an alternative proof of Chanillo's weighted $L^p$ estimates via extrapolation techniques.

Khintchine's theorem on the measure dichotomy for the set of $ψ$-approximable numbers has been generalized to inhomogeneous and higher-dimensional settings. Allen and Ramírez conjectured that the monotonicity condition can be removed in the inhomogeneous $nm=2$ cases. In this paper, we resolve the $(n,m)=(1,2)$ case for $ψ$ satisfying a polynomial decay condition $ψ(q)=O(q^{-δ})$ for some $δ>0.$

We study the inverse optimal transport problem of recovering the ground cost from an optimal transport plan. In discrete settings, this problem reduces to inverse linear programming and is intrinsically ill-posed, exhibiting non-identifiability and flat directions. We show that in the continuous setting, the regularity of the marginals fundamentally alters the structure of the inverse problem. Assuming smooth positive densities for the source and target measures, we characterize the second variation of the optimal transport functional with respect to the ground cost in Hölder spaces. In particular, we show that it is non-degenerate modulo the natural transport invariances, yielding a strict curvature property that is absent in discrete transport. As a consequence, we obtain local identifiability and stability results for inverse optimal transport. For the structured family of bilinear costs (i.e. Mahalanobis parametrizations), the ground cost can be uniquely recovered--up to the intrinsic invariances--from a single optimal coupling under a natural spanning condition. We further show that this identifiability property is generic under arbitrarily small perturbations of the marginals, while settings where the optimal transport map is affine (for instance Gaussian or elliptical marginals) remain degenerate. Finally, we establish precise bounds on the bias and statistical variance of inverse optimal transport under entropic regularization. These results reveal a structural parallel between forward and inverse optimal transport: regularity of the marginals ensures smooth optimal maps in the forward problem, while non-degeneracy of the induced transport plan yields curvature and local invertibility in the inverse problem.

In many problems in optimal control, one seeks to minimise an objective function subject to constraints on the velocity of the system. Imposing these constraints directly -- the ``hard-constrained'' approach -- is often analytically and computationally challenging. A natural alternative is to penalise violations of the constraints, solving a sequence of ``soft-constrained'' problems indexed by a penalty parameter $q$, and hoping that solutions converge to solutions of the hard-constrained problem as $q \to \infty$. We show that this approach is justified when applied to a broad class of geometric control problems on a Riemannian manifold $(M,g)$. We first consider the case where there are no autonomous dynamics, and so the control problem reduces to the problem of finding a curve of minimal length or energy between two points, subject to a nonholonomic velocity constraint/penalty determined by the choice of a bracket-generating subbundle $D$ of $TM$. We show that any sequence of solutions to the soft-constrained problem has an accumulation point which is a solution to the hard-constrained problem. Subsequently, we show how to transform a broad class of optimal control problems to the problem of finding a geodesic, by trivialising the inherent dynamics of the system using a change of coordinates inspired by the interaction picture transformation in quantum mechanics.

We study the sharp bounds of $\mathbb{E}[X_1\cdots X_d]$ when the univariate marginal distributions are known, but the dependence structure between them is unspecified. Maximizing products over non-negative variables is straightforward via the comonotonic coupling, but the problem is more subtle when the marginals can take both positive and negative values. Specifically, two negative realizations can be matched to yield a positive product, whereas a single negative realization necessarily yields a negative product. We propose a decomposition of the problem into a magnitude part and a sign part, and show that universal upper and lower bounds for the product expectation follow from the comonotonic coupling of the absolute values and properly chosen sign vectors. Under a mild regularity assumption, we give necessary and sufficient conditions for these universal bounds to be attainable. For the upper bound, the marginal sign-bias vector must belong to the even-parity polytope, while for the lower, the corresponding condition involves the odd-parity polytope. We construct the extremal couplings via measurable selections on the parity polytope whenever these conditions hold. We study the case of identical marginals in more detail and provide examples of non-symmetric extremal coupling that achieve the universal bounds. We explicitly construct the extremal copulas in three dimensions, and use a recursive parity decomposition to obtain higher-dimensional extremal copulas from the trivariate ones.

We prove quenched and annealed statistical stability, linear response, and differentiability of asymptotic moments for parametric families of partially hyperbolic skew products, with random hyperbolic maps on the fibers. The main novelty is that the base maps also depend on the parameter, which leads to different formulas in the linear response and the derivative of the asymptotic moments with respect to the parameter. Our annealed results apply to partially hyperbolic maps that are not covered in \cite{BashCastro26,Dol,DS}.

Recovering latent structure from count data has received considerable attention in network inference, particularly when one seeks both cross-group interactions and within-group similarity patterns in bipartite networks, which is widely used in ecology research. Such networks are often sparse and inherently imperfect in their detection. Existing models mainly focus on interaction recovery, while the induced similarity graphs are much less studied. Moreover, sparsity is often not controlled, and scale is unbalanced, leading to oversparse or poorly rescaled estimates with degrading structural recovery. To address these issues, we propose a framework for structured sparse nonnegative low-rank factorization with detection probability estimation. We impose nonconvex $\ell_{1/2}$ regularization on the latent similarity and connectivity structures to promote sparsity within-group similarity and cross-group connectivity with better relative scale. The resulting optimization problem is nonconvex and nonsmooth. To solve it, we develop an ADMM-based algorithm with adaptive penalization and scale-aware initialization and establish its asymptotic feasibility and KKT stationarity of cluster points under mild regularity conditions. Experiments on synthetic and real-world ecological datasets demonstrate improved recovery of latent factors and similarity/connectivity structure relative to existing baselines.

We introduce a framework for pulling back Cartier modules and their associated invariants along regular $F$-finite morphisms. To achieve this, we construct a relative Cartier isomorphism and operator for an arbitrary regular $F$-finite map of locally noetherian schemes. As an application, we obtain new results on the constancy regions of mixed test ideals, based on the work of Felipe Pérez.

We give an independent eta-product derivation of the level-8 Apery limit lim B_n^{(8)}/s_n = (7/32) zeta(3), where s_n = sum_{k=0}^n C(n,k)^2 C(2k,n)^2 and B_n^{(8)} is the rational companion sequence satisfying the same cubic recurrence with initial values B_0^{(8)}=0, B_1^{(8)}=1. This value was identified numerically by Almkvist-van Straten-Zudilin and was proved by Golyshev via Beukers's Atkin-Lehner modular method; it was later recomputed by Golyshev-Kerr-Sasaki in the motivic/normal-function framework. The continued fraction PCF((2n+1)(3n^2+3n+1),-n^6) = 8/(7 zeta(3)) already appears in Batut-Olivier and was later rediscovered by the Ramanujan Machine as conjecture Z1. The contribution of the present paper is an explicit rederivation, in the eta-product normalization, of the already-known level-8 Apery limit. We spell out the eta-product verification of the Wronskian identity, the normalization of the Eichler integral, the residue computation of the Fricke period polynomial, and the elementary continuant conversion.

Given a cooriented branched surface $\mathcal B$ fully carrying a foliation $\mathcal F$, we use the dual graph of $\mathcal B$ to define a simplicial 1-cycle $Γ_m(\mathcal B)$ representing the Poincaré dual of the Euler class of $\mathcal F$ relative to the boundary. As an example, we complete the classification of which homology classes in the Whitehead link exterior are realisable as relative Euler classes of taut foliations. We also show how our formula generalises previous results of Lackenby and Dunfield. Finally, we observe that cooriented branched surfaces whose complement is a union of balls satisfy a Combinatorial Transverse Surface Theorem, in the sense of Landry--Minsky--Taylor.

We prove tight lower bounds for online multicalibration, establishing an information-theoretic separation from marginal calibration. In the general setting where group functions can depend on both context and the learner's predictions, we prove an $Ω(T^{2/3})$ lower bound on expected multicalibration error using just three disjoint binary groups. This matches the upper bounds of Noarov et al. (2025) up to logarithmic factors and exceeds the $O(T^{2/3-\varepsilon})$ upper bound for marginal calibration (Dagan et al., 2025), thereby separating the two problems. We then turn to lower bounds for the more difficult case of group functions that may depend on context but not on the learner's predictions. In this case, we establish an $\widetildeΩ(T^{2/3})$ lower bound for online multicalibration via an $O(\log^3 T)$-sized group family constructed from an orthonormal basis, again matching upper bounds up to logarithmic factors.

In this work, we investigate the bi-Lipschitz invariance of two fundamental local invariants in singularity theory: the Łojasiewicz exponent and the local Euler obstruction. We draw inspiration from Bivià-Ausina and Fukui, whose framework we extend to ideals in rings of analytic functions defined on affine toric varieties. We establish conditions under which these invariants remain unchanged under bi-Lipschitz equivalence. We also provide an answer, to a particular case, to the open question of whether the local Euler obstruction is a bi-Lipschitz invariant. For hypersurfaces with isolated singularities, we show that the Euler obstruction is preserved under non-degeneracy conditions. These results contribute to the understanding of metric invariants in complex analytic geometry.

Hidden Markov Models (HMMs) are fundamental for modeling sequential data, yet learning their parameters from observations remains challenging. Classical methods like the Baum-Welch algorithm are computationally intensive and prone to local optima, while modern spectral algorithms offer provable guarantees but may produce probability outputs outside valid ranges. This work introduces Belief Net, a differentiable filtering framework that learns HMM parameters by formulating the forward filter as a structured neural network and optimizing it with stochastic gradient descent. This architecture recursively updates the belief state, which represents the posterior probability distribution over hidden states based on the observation history. Unlike black-box transformer models, Belief Net's learnable weights are explicitly the logits of the initial distribution, transition matrix, and emission matrix, ensuring full interpretability. The model processes observation sequences using a decoder-only (causal) architecture and is trained end-to-end with standard autoregressive next-observation prediction loss. On synthetic HMM data, Belief Net achieves faster convergence than Baum-Welch while successfully recovering parameters in both undercomplete and overcomplete settings, whereas spectral methods prove ineffective in the latter. Comparisons with transformer-based models are also presented on real-world language data.

We study the space of $A_\infty$-natural transformations between braiding functors acting on the Fukaya category associated to the Coulomb branch $\mathcal{M}(\bullet,1)$ of the $\mathfrak{sl}_2$ quiver gauge theory. We compute all cohomologically distinct $A_\infty$-natural transformations $\mathrm{Nat}(\mathrm{id}, \mathrm{id})$ and $\mathrm{Nat}(\mathrm{id}, β_i^-)$, where $β_i^-$ denotes the negative braiding functor. Our computation is carried out in a diagrammatic framework compatible with the established embedding of the KLRW category into this Fukaya category. We then compute the Hochschild cohomology of the Fukaya category using an explicit projective resolution of the diagonal bimodule obtained via the Chouhy-Solotar reduction system, and use this to classify all cohomologically distinct natural transformations. These results determine the higher $A_\infty$-data encoded in the braiding functors and their natural transformations, and provide the first step toward a categorical formulation of braid cobordism actions on Fukaya categories.