Papers

This paper develops Visibility Theory (VT), a framework in which visibility is taken as the primitive semantic notion and truth is defined in terms of visibility together with a structural condition called step-identity. The theory is formulated in a classical setting and supplied with a revision-theoretic semantics. Visibility theory includes a provability logic KDC, characterized by seriality and confluence and shown to be strongly complete by standard Sahlqvist-theoretic methods. The framework distinguishes between visions, propositions, sets, and truths. Propositions are identified with minimal visions, while sets are characterized by step-identity. Truth is defined as visible sethood. This yields permanence results for truth and a recovery theorem establishing familiar truth principles whenever the relevant proposition satisfies the set condition. The theory blocks semantic paradoxes without abandoning classical logic. Liar and revenge constructions are shown to fail through violations of step-identity rather than through restrictions on inference. The final part of the paper introduces the beginnings of a Visionary Set Theory based on the set-theoretic visions and restricted set quantifiers. The resulting framework provides a unified setting for truth, provability, and set-theoretic reasoning grounded in the primitive notion of visibility.

We derive an exact integral representation of the response to an impulsive boundary excitation in a semi-infinite, homogeneous, linear viscoelastic medium governed by the Zener model. Starting from the Bromwich integral, we analytically invert the relevant Laplace transforms and express the response in terms of modified Bessel functions of the first kind and Macdonald functions of half-integer order. The representation reduces to the known Maxwell solution in the appropriate limit. As an independent computational approach, we characterize the saddle points of the phase function and construct the associated steepest-descent path (SDP). Its branches satisfy a sixth-degree polynomial equation in the imaginary part of the complex variable, which is cubic in its square. Numerical comparisons for several model parameters confirm the agreement of the two methods. The integral representation reveals the structure of the response, whereas the SDP approach requires no closed-form inverse Laplace transform and can therefore be applied to more general viscoelastic models.

We study neural approximation of elliptic PDE solutions from a variational perspective. The central point is the distinction between the geometry of neural parameters and the convergence of the corresponding physical states. Even when the original elliptic energy is coercive and strictly convex in the natural energy space, its restriction to a nonlinear neural ansatz may fail to be coercive in parameter space. This failure is caused by non-closedness of neural approximation manifolds and by condensation of neurons, which may generate limiting profiles outside the fixed ansatz class. Nevertheless, the associated state functions may remain bounded and converge strongly to the exact PDE solution. We prove this mechanism for Gaussian wave-packet approximations of a prototypical elliptic model in the whole space, derive convergence rates, and explain how the same state-level stability principle applies to residual minimization methods of PINN type, and HYCO-type hybrid methods. We also discuss relaxation and Tikhonov regularization.

We give an alternative formulation of Stein's maximal inequality for generalised spherical averages in terms of derivatives of standard spherical means: if \[ k \ge 0, \qquad d \ge 2 k + 3 , \qquad \frac{d}{d - k - 1} < p < \frac{d - 1}{k} , \] and $\sigma$ is the normalised surface measure on the unit sphere $\mathbb S$, then the maximal operator \[f \mapsto \sup_{r > 0} \, \biggl\lvert r^k (\tfrac{d}{dr})^k \int_{\mathbb S} f(\cdot + r y) \sigma(dy) \biggr\rvert\] is bounded on $L^p$, with a constant that is independent of the dimension $d$.

We study an infinite-horizon optimal consumption-investment problem for an investor with Epstein-Zin stochastic differential utility in an incomplete market with stochastic investment opportunities. Risk aversion and intertemporal substitution are separated, and we work in the regime $\theta\in(0,1)$, where there exists a unique generalised utility process for arbitrary non-negative progressively measurable consumption streams. Our main contribution is a variational characterisation of the value function. We show that the value function is the unique minimiser of a functional whose Euler-Lagrange equation coincides with the Hamilton-Jacobi-Bellman equation. Although the functional may be non-convex, the direct method yields existence, and we prove that every minimiser is a strictly positive, bounded classical solution. A verification theorem identifies any minimiser with the value function and gives feedback representations for optimal consumption and investment policies. The proof combines a change of measure to the myopic probability with uniqueness results for Epstein-Zin BSDEs and a perturbation argument for optimality. Examples with stochastic volatility, Gaussian excess returns, and fat-tailed excess returns illustrate the scope of the framework and its implications for intertemporal hedging.

Mixed structures and their weight filtrations appear in various contexts, prominently Hodge theory and the theory of Galois representations. In the setting of André's formalisation of slopes, we propose an abstract framework explaining why mixed structures have shared characteristics.

We study local finiteness for modal $K4$ algebras via the tunability of their dual general frames. In particular, we provide a sufficient condition for modal $K4$ algebras to be locally finite by identifying a structure which must be present in non-locally finite modal $K4$ algebras. We then show that this condition becomes both necessary and sufficient for $\sigma$-complete modal $K4$ algebras. Next, we translate this condition into a pair of order-theoretic conditions on transitive Kripke frames, providing a classification of local finiteness on their dual modal algebras. We further show that the logic of any class of well-founded transitive relations with no infinite antichains has the finite model property, and conclude that the logic of the class of well-quasi orderings has the finite model property.

This paper develops a data-driven realization of the generalized Koopman operator (GeKo) [1], in which states and inputs are lifted independently and the dynamics are expressed as a tensor bilinear system. The first contribution is a time-sequenced multi-step Khatri-Rao kernel regression formulation that exposes the operator to evolved snapshots along trajectories rather than only single one-step pairs, which reduces compounded prediction error. Secondly, we develop a kernel- and input-agnostic structured SVD reduction that compresses the lifted state and input spaces while preserving the Khatri-Rao realization. We instantiate the framework with random Fourier features and describe a complete predictive-control pipeline, including a multi-step roll-out diagnostic that guides the choice of MPC horizon. The framework is validated on the chaotic Lorenz system, where the learned reduced-order GeKo model reduces the long-horizon prediction error by up to a factor of five relative to one-step and lifted linear baselines fitted on the same data, and stabilizes an unstable equilibrium.

Public scientific and metrology releases can leak the hidden settings that produced them. We formalize and quantify this risk as a profiled statistical side-channel audit: a release map exposes finite-band statistics of a power spectral density (PSD), a profiled observer trains labeled template spectra under an explicit budget, and a challenge release is drawn from one of two utility-equivalent recipes separated by a protected coordinate. Averaged PSD bins follow a gamma channel, replaced by a covariance-weighted log-spectrum channel when the bins are correlated; this yields exact Kullback-Leibler divergences, Chernoff exponents, protected-bit advantage bounds, and finite-training, finite-library, finite-compute, and model-mismatch corrections. Our headline result is a finite-band transport-leakage law: after amplitude and blur are eliminated, the protected acid-transport information obeys $I_{\lambda|\alpha,\beta}(K) = (64/1225)\, w \lambda^{6} K^{9} + O(w \lambda^{8} K^{11})$ for $K\lambda \ll 1$, a ninth-order exponent with a closed-form safe band. A step-by-step protocol turns a measured release into these numbers, and a fixed-seed reproducibility package regenerates every table and figure. We instantiate the audit on screened extreme-ultraviolet (EUV) roughness spectra as a model-conditioned case study, with deployment on measured releases the next step.

We present a new method for the efficient imaginary time evolution of quantum many-body wavefunctions represented by matrix product states (MPS). We first show that logarithmic time grids are sufficient to resolve long imaginary time dynamics, yielding an exponential reduction in the number of time steps compared with standard approaches. We then show that A-stable implicit time-stepping methods for ordinary differential equations allow stable propagation for any time step size. The resulting scheme requires only matrix-vector products and linear solves, standard operations in the MPS toolbox. We validate our approach with two examples: a Heisenberg spin chain, which we use to demonstrate a speedup of several orders of magnitude over the standard time-dependent variational principle method with uniform time steps, and a single-site Anderson impurity model with a metallic bath, for which propagation to large imaginary times allows one to observe the exponential dependence of the Kondo temperature on the interaction strength.

The one-shot approach is a powerful simultaneous optimization framework for design tasks governed by computationally expensive steady-state systems. While previous formulations mainly focused on additional equality constraints, this work extends the one-shot framework to optimization problems with inequality constraints using slack variables embedded into a doubly augmented Lagrangian formulation. After elimination of the slack variables, the resulting formulation contains nonsmooth active-set dependent terms while preserving the characteristic coupled one-shot matrix structure. The resulting generalized gradient system admits a representation of the form $ \nabla \DLag = -Ms, $ analogous to the equality-constrained one-shot framework. To analyze the nonsmooth active-set transitions, generalized second-order subdifferentials in the sense of Clarke and Rockafellar–Wets are employed. We derive explicit positivity conditions ensuring positive definiteness of all admissible generalized Hessians and hence strict local optimality. Furthermore, we give an outlook on the construction of preconditioners based on generalized Hessian approximations.

We prove that if $\theta$ is a continuous unital homomorphism of an operator algebra $A$ into $B(\mathcal{H})$, and $\beta$ is in the dual space of $A$, then the completely bounded norm of $\theta$ is less than or equal to the maximum of $1$ and the completely bounded norm of $\theta + \beta I $. As an application, we give another proof of the Okubo–Ando theorem.

Let $L$ be a subspace of a complex vector space $V$ and fix $s \leq \dim{L}$. The (type A) Kalman variety consists of all endomorphisms of $V$ that have an $s$-dimensional invariant subspace in $L$. We introduce a generalization where $V$ and $L$ are symplectic vector spaces. We fix an isotropic subspace $W \subseteq V$ satisfying $W^\perp = W \oplus L$. The isotropic (type C) Kalman variety consists of symplectic morphisms of $V$ that have an invariant coisotropic subspace of a prescribed dimension inside $W^\perp$. We are mainly interested in studying the Lagrangian case. In type C, we prove analogues of results known for type A Kalman varieties; in particular, we determine the defining equations, compute geometric invariants, and analyze their singularities. We conjecture the existence of a long exact sequence relating the structure sheaves. Based on the results in the symplectic case, we describe Kalman variety analogues with respect to endomorphisms of odd orthogonal (type B) and even orthogonal (type D) vector spaces.

The Lipschitz-free space $\mathcal{F}(M)$ is a canonical linearization of a complete metric space $M$ whose topological dual is the space of Lipschitz functions on $M$. We review the properties of $\mathcal{F}(M)$ when the underlying space $M$ is purely 1-unrectifiable, that is, it contains no bi-Lipschitz copy of a subset of $\mathbb{R}$ with positive measure. For compact $M$, this is equivalent to several Banach space properties of $\mathcal{F}(M)$, including the Radon-Nikodým and Schur properties or admitting a predual. We shall see how the study of locally flat Lipschitz functions on $M$ reveals these equivalences, and describe a technique that allows most of them to be transferred to the non-compact setting. This manuscript is an expository text based on results by the author in collaboration with C. Gartland, C. Petitjean and A. Procházka, originally published in a Trans. Amer. Math. Soc. paper, and corresponds to a lecture delivered at the Second Winter School in Geometric Measure Theory at Westlake University, Hangzhou, on February 2026.

In this paper, we study optimal distributional estimates for the multiple Hilbert transform. We obtain pointwise upper and lower distributional estimates of the multiple Hilbert transform in terms of the $d$-fold composition of the Calderón operator with itself. This extends the fundamental results by A. P. Calderón, D. Boyd, and Ch. Fefferman for arbitrary $d\in\mathbb N.$

We give a completely explicit Lerch, Chowla–Selberg formula without absolute values of the Dedekind eta function, and give some applications, in particular to the argument of individual CM values of the eta function. We also give precise conjectures on these CM values and a large table supporting them.

We consider supertori which are quotients of affine superspace by translations by algebraically independent odd parameters. Specifically, we describe the ring structure of its space of global sections by generators and relations and completely determine the coherent cohomology groups of its structure sheaf by reducing it to a problem of Lie algebra cohomology. We also show that Poincaré duality on sheaf cohomology is compatible with that of the group cohomology of the translation group and give explicit tables of examples. Finally, we compute the Picard groups of these supertori.

In this work, we introduce a new macroscopic model for crowd motion inspired by the celebrated Hughes' model \cite{Hughes2002, Hughes2003}, which couples a nonlinear conservation law for the pedestrian density with an Eikonal equation describing the shortest path to the target. Our approach can be viewed both as a modification of Hughes' original formulation and as a refinement of the prediction-correction framework proposed in the recent work \cite{ennaji2023prediction}. The resulting model incorporates anticipatory behavior and dynamic route adjustment, offering a more realistic representation of crowd dynamics in complex environments. We present the mathematical formulation of the model, discuss its well-posedness properties, and illustrate its qualitative behavior through numerical simulations. Ultimately, we show, at least from a numerical perspective, that this variant provides a promising avenue towards establishing the well-posedness of the classical Hughes' model, which has remained a challenging open problem for a long time.

We isolate a simple preservation principle governing when it is absolute, between transitive models of set theory, that a given algebraic or topological-algebraic structure has a standard form $F(X)$ indexed by a set $X$. The principle is: if the index $X$ (or a proxy for it) can be recovered from $F(X)$ by a uniform definable construction, then the class of structures isomorphic to some~$F(X)$ is downward absolute from forcing extensions. Answering a question raised by Noah Schweber, we deduce in particular that no group that fails to be a full symmetric group in the ground model can become one after forcing; the result holds already in ZF. The same mechanism applies to full transformation monoids, powerset Boolean algebras, full relation algebras, full clones, full partition lattices, products $R^X$ of finitely generated centrally indecomposable rings, the commutative $C^*$-algebras $\ell_\infty(X)$ and $c_0(X)$, full endomorphism rings, the operator algebras $\mathcal{B}(H)$ and $\mathcal{K}(H)$, and $\ell_1(X)$ as a real Banach lattice. In the motivating symmetric-group case, the same reconstruction gives more than descent: it yields a uniform $\Pi^1_1$ definition of fullness over transitive ZF-models. We then exhibit clean torsor obstructions, in the standard symmetric-model situation: finite covers $Y \times n$ already separate ZF-failure from ZFC-descent without any completeness caveat, and the finite-support normed space $c_{00}(I)$ provides the analogous Banach example. Bare-Banach-space isomorphism with $\ell_1(\Gamma)$ exhibits a genuine ZFC-descent. We conclude with the corresponding, relative, obstructions to $\Pi^1_1$-definability of standardness over transitive ZF-models.

We introduce disciplined nonlinear programming (DNLP), a syntax for specifying nonlinear programming problems. DNLP is inspired by disciplined convex programming (DCP) and allows smooth functions to be freely mixed with nonsmooth convex and concave functions, with rules governing how the nonsmooth functions can be used. Problems expressed in DNLP form can be automatically canonicalized to a standard nonlinear programming (NLP) form and passed to a suitable NLP solver. As in DCP, the canonicalization relaxes nonsmooth convex and concave functions in a lossless way, allowing them to be handled by NLP solvers that require smooth functions. In addition to extending NLP to include useful nondifferentiable convex and concave functions, transforming the original problem to an equivalent NLP form offers several advantages, including simpler problem initialization. We describe the language and our open-source implementation of DNLP as an extension of CVXPY, a parser for DCP.

Parametric resonance – periodic variation of a system parameter driving exponential growth of oscillations – is among the most fundamental instabilities in physics and engineering. The nondissipative LC circuit with harmonically varying capacitance is one of its simplest realizations: the modulation renders the circuit equation a Hill equation with either bounded or exponentially growing solutions. Identifying the governing equation as a special case of Ince's four-parameter Hill equation yields two main results. First, a sharp structural theorem: instability occurs only at the odd sub-harmonics of the natural frequency, while every even resonance is exactly stable at all modulation amplitudes. This selectivity, invisible to the Mathieu approximation, follows from Krein's collision theory: at odd resonances the colliding Floquet multipliers carry opposite Krein signatures, opening a tongue; at even resonances the signatures agree and the tongue collapses. Second, closed-form formulas for the widths and boundary curves of all surviving tongues, derived by a continued-fraction and Magnus–Winkler method, confirmed against Cambi's 1950 numerics and recovered via the Yakubovich–Starzhinskii series. The continued fraction also gives the Floquet exponent as an exact power series in the modulation amplitude with rational coefficients, and finite-product formulas for all Fourier coefficients of the periodic Floquet factor. The tongue boundaries consist entirely of exceptional points of degeneracy (EPD) of the monodromy matrix, enabling hypersensitive capacitance sensing: at an EPD a small perturbation splits the coincident frequencies by the square root of its size, diverging relative to linear sensing as it shrinks. A closed-form splitting formula is derived, and a work-point strategy shifting slightly into the stable zone keeps the scheme robust while preserving square-root sensitivity.

Despite the prevalence of nonautonomous systems in applications, their statistical properties are much less understood than in the autonomous setting. Building on recent results on response theory for nonautonomous systems, we study the approximation of equivariant families and of their linear response by Ulam-type finite-dimensional reductions. First, we show that coarse-graining procedures associated with the classical Ulam method, and more generally with suitable finite-element projections, such as those often associated with the Koopman/Kolmogorov operator-informed analysis of dynamical systems, provide rigorous approximation of equivariant families for sequential systems with memory loss. Second, for systems whose transfer operators are regularising, we prove that the linear response of the reduced finite-state Markov model converges to the projected linear response of the original system. We illustrate the results with numerical experiments on simple but representative time-dependent diffusive models. These results provide a rigorous foundation for the use of Markov approximations in the study of statistical properties of nonautonomous complex systems which almost invariably relies on finite-scale and finite-precision descriptions of their states and dynamics.

Symmetric nonnegative matrix factorization (Symmetric NMF) approximates a matrix as $WW^T$ with nonnegative rectangular factor $W$. It has broad applications in graph clustering and machine learning. In contrast to the NMF, projected gradient methods for the symmetric problem had been associated with slow convergence. To address this, we introduce SNMPBB, the first adaptation of nonmonotone projected Barzilai-Borwein methods to Symmetric NMF, demonstrating that gradient algorithms are significantly more effective than previously understood. We further extend SNMPBB to graph clustering using the graph Laplacian regularization (Graph-SNMPBB) and to large problems with low-rank approximations (LAI-SNMPBB). For all variants we prove global convergence to first-order stationary points and also that Barzilai-Borwein curvature information is preserved with randomized approximations. On synthetic data, SNMPBB achieves 6 times speedup over the alternative SymANLS for similar residuals, with advantages growing at higher ranks. Across six real-world clustering benchmarks, Graph-SNMPBB matches or exceeds SymANLS accuracy. Lastly, LAI-SNMPBB outperforms state-of-the-art LAI-SymPGNCG on 34 SuiteSparse matrices in both runtime and residual quality.

Deep learning weather models now match numerical weather prediction accuracy while running orders of magnitude faster, but produce deterministic forecasts without uncertainty estimates, a critical gap for high-stakes decisions during extreme weather events. This paper proposes Neural Tangent Kernel-based uncertainty quantification (NTK-UQ) using last-layer empirical features. Theoretical analysis predicts that UQ quality is architecture-dependent through two mechanisms. First, a variance collapse mechanism explains when UQ fails: when the eigenvalue truncation rank approaches the effective rank of the feature space, the GP correction term consumes nearly all prior variance, destroying discrimination between tropical cyclones and routine conditions; architectures with concentrated spectra (spectral operators) require aggressive truncation ($k \leq 10$), while attention-based models tolerate full-rank computation. Second, decomposition performance depends on the non-Gaussian, heavy-tailed structure of extreme weather: Independent Component Analysis exploits higher-order statistics (kurtosis, negentropy) to isolate heavy-tailed extreme-event features, achieving higher discrimination than singular value decomposition, which captures only second-order variance. A data-driven selection rule chooses ICA or SVD from the feature eigenspectrum concentration ratio, correctly prescribing the superior decomposition for all four evaluated architectures. Compared to split conformal prediction (the natural post-hoc baseline), NTK-UQ achieves 31–37% sharper prediction intervals at 90% coverage, and uniquely produces adaptive intervals that scale with extreme event severity, which conformal prediction cannot achieve by construction. The framework requires no retraining; inference-time uncertainty requires only a single matrix-vector product per sample.

We consider $K3$ surfaces collapsing to a three-dimensional affine base. We show that certain affine lines on the base lift to degenerating sequences of special Lagrangian two-spheres and tori in the collapsing $K3$ surface. In particular, we construct special Lagrangian two-spheres connecting pairs of Taub-NUT bubbles. These examples fit into the broader program of reconstructing special submanifolds from graphs and combinatorial data on a collapsed affine limit.

Pál's isominwidth inequality (1921) answered the Kakeya needle problem (1917) for convex sets. It states that among convex bodies of fixed minimum width $w$ in the Euclidean plane, the regular triangle has minimal area. The isominwidth inequality was generalized to the $2$-dimensional sphere by Bezdek and Blekherman and Freyer and Sagmeister (arXiv:2411.11462). Interestingly, in hyperbolic space, no minimizer exists, as shown by Böröczky, Freyer and Sagmeister (arXiv:2502.04427). The stability of the Euclidean Pál inequality with respect to the Hausdorff metric and the symmetric difference metric was proved by Lucardesi and Zucco (arXiv:2405.18294). Fodor, Robock and Sagmeister (arXiv:2602.19300) proved $r$-ball convex analogs of the isominwidth inequality in all three constant curvature planes connecting Pál's theorem with the Blaschke–Lebesgue inequality. In this paper, we prove optimal stability versions of this statement with respect to the Hausdorff distance and the symmetric difference metric in all three constant curvature planes.

We prove a sparse transference theorem for induced Ramsey graphs. The theorem transfers the weighted random-host proof of Aragão, Campos, Dahia, Filipe, and Marciano to the sparse random setting. It follows that, for every fixed graph $H$ with no isolated vertices and at least two edges, and every $\eta>0$, there is $C>0$ such that, whenever $N\ge r^{Cr}$ and $N^{-1/m_2(H)+\eta}\le p\le \frac12$, with high probability every $r$-colouring of the edges of $G(N,p)$ contains a monochromatic induced copy of $H$. Here, $m_2(H)$ denotes the usual maximum 2-density of $H$.

The maximal entries of Dynkin friezes over the positive integers have recently been determined for all finite Dynkin types except $B_n$ and $D_n$. In this note, we explicitly construct large positive integral points on affine cluster varieties of type $B_n$ (resp. $D_n$), giving rise to friezes of types $B_n$ (resp. $D_n$) over the positive integers with largest entries $F_{n+1} F_{n+2} - 1$ (resp. $F_n F_{n+1} - 1$) where $F_k$ is the $k$-th Fibonacci number. We conjecture that these are the maximal possible entries for their respective Dynkin types.

Identifying ideal Perron similarities is a problem of central interest in the longstanding nonnegative inverse eigenvalue problem (NIEP). A normalized ideal Perron similarity is called totally extremal if every entry has modulus one. Recently, Gershnik et al. [J. Algebra 694 (2026), 782–800] proved that the character table of a finite Abelian group is totally extremal and conjectured the converse. In this paper, we settle this conjecture in the affirmative by first showing that the rows of a totally extremal normalized ideal Perron similarity form a group under the Hadamard product. Then, it is shown that the rows of a nonsingular matrix form a group under the Hadamard product if and only if it is the character table of a finite Abelian group, and we further show that this group is isomorphic to the underlying group. These results extend the classical theorem due to Romanovsky and Karpelevič on the unimodular eigenvalues of stochastic matrices in the complex unit disk to the setting of spectratopes in the unit ball of complex Euclidean space.

We prove that stationary random metric measure spaces have 0,1,2 or a Cantor space of ends. This notion includes stationary random graphs, manifolds and discrete subgroups. In the case of surfaces, we classify all possible homeomorphism types, in analogy with the work of Biringer and Raimbault on unimodular Riemannian manifolds. Our approach relies on a general "no geometric core" principle and an analysis of finite versus infinite expected return times.