Papers

CBCT (Cone Beam Computed Tomography) scans provide detailed three-dimensional images, widely used in dentistry for diagnostic and treatment planning tasks. While invaluable, analyzing and documenting these scans is labor-intensive, prompting efforts to automate key steps like the classification and segmentation of anatomical structures to identify tooth types and associated pathologies. In this article, we propose an approach to automation that leverages persistent homology, a framework from topological data analysis that studies the shape of data by identifying features like connected components, holes, and voids across multiple scales. Persistent homology, together with a support vector machine, allows us to classify teeth in a CBCT scan and to perform diagnostics. Our method advances the state of the art, reaching average accuracy scores of 97.67% for tooth-labeling and 96.77% for diagnostic tasks, outperforming a CNN trained on the same data with accuracy of 70.27% and 86.67%, respectively.

We introduce a stochastic process for modeling the evolution in time of compositional measurements (i.e., a vector of non-negative values that add up to a total of 1). This model is a diffusion, as it is defined as the solution for a stochastic differential equation in the Ito sense, and it has a Dirichlet distribution as its steady distribution. We have named this process Dirichlet Diffusion (DD). As the process is confined to a manifold and the coefficients of the equation are not globally Lipschitz, the usual theorems do not apply directly and establishing the existence and properties of solutions requires a somewhat delicate analysis. We establish the existence of strong solutions under the assumption that all the parameters of the Dirichlet distribution are greater than 2; for the general case we were only able to establish the existence of weak solutions, but also that these solutions remain confined to the closed simplex without need for reflecting boundaries. A useful feature of DD that it inherits from the Dirichlet distribution is the property of aggregation: If components are combined to create a coarser composition, the resulting process is also a DD. This makes it useful, for example, for jointly modeling the evolution of a microbiome grouping the microbe species at different taxonomic levels.

A coefficient duality first encountered for formal Bernoulli series is shown to be equivalent to a general Möbius covariance law for formal power series. We obtain a structural characterization, an eigenspace interpretation, and a weighted form of this duality. The Catalan convolution and Chebyshev identities from the motivating Bernoulli setting extend to arbitrary Möbius-covariant families and yield a general Ramanujan-type summation formula encompassing consecutive half-integer powers. The framework also recovers classical Bernoulli and Euler recurrences and produces recurrence families for colored matchings and generalized central trinomial coefficients, with further realizations from reflection-symmetric Appell sequences and Gorenstein Hilbert series.

Neurofinance shows that financial valuation depends on evolving neural states, while temporal experience is itself state dependent. Yet intertemporal models typically treat time as exogenous and ask how delay affects valuation. This paper examines the converse question: can valuation-related neural dynamics generate subjective financial time? We develop a continuous-time model on a joint financial–neuro-evaluative state space in which subjective financial time is accumulated valuation along dynamically admissible histories. We characterise realised temporal-rate dispersion and show that, under matched financial dynamics, distinct neuro-evaluative trajectories can generate different subjective financial times. Financial equivalence therefore need not imply temporal equivalence. Valuation curvature further determines local path dependence. A proof-of-concept behavioural–fMRI analysis uses \(1{,}183\) observations, \(1{,}126\) valuation transitions, and \(798\) financially matched pairs. Using a medial-prefrontal valuation-state coordinate based on vmPFC and dmPFC responses, the strongest participant exhibits a positive association between neural-state separation and subsequent valuation divergence (\(\rho=0.253,\ p_{\mathrm{perm}}=0.001\)), robust to residual financial distance (\(\rho=0.251\)) and tighter matching (\(\rho=0.207\)). Participant effects are heterogeneous, although combined evidence gives \(p=0.0216\). The data support the neural state-separation premise rather than directly identifying subjective financial time. The framework establishes a theoretical and empirically grounded basis for endogenous subjective time in neurofinance. \keywords{Neurofinance $\cdot$ Subjective financial time $\cdot$ Neural valuation $\cdot$ Relative subjective value $\cdot$ Intertemporal choice $\cdot$ Temporal dispersion $\cdot$ Path dependence

Let $X$ be a complex Banach space with an unconditional Schauder basis. We prove that the monomials associated with this basis, endowed in each homogeneous degree with the square order and globally with any compatible ordering, form a Schauder basis for the space $(\calH(X),\tau_0)$ of entire holomorphic functions on $X$ with the compact-open topology. The proof relies on two main ideas. First, coordinate-tail conditions yield a fundamental system of compact solid subsets of $X$. Second, Fourier projections in the coordinates give the uniform estimate $c_{K,n}\leq n+1$ for the basis constant of the degree-$n$ monomials with respect to the supremum seminorm on each such compact set $K$. We derive applications to Banach sequence lattices with dense $c_{00}$, including classical, Lorentz, Orlicz-heart and Schreier-type sequence spaces, and to general vector-valued $E$-sums. In particular, the result covers mixed $c_0$- and $\ell_r$-sums of finite-dimensional $\ell_p$ spaces and the Lorentz predual $d_*(w,1)$.

We develop a general framework for constructing pseudometrics on objects of Abelian categories using valuations, which assign non-negative costs to morphisms. Specializing to finitely presented vector space representations of the poset $[0,\infty)^\ell$, a natural setting for multiparameter persistence modules, we show that a large class of these distances is governed by contours, right-continuous lax actions of $[0,\infty)$ on the parameter poset. Properties of contours translate into geometric properties of the induced distances, including conditions under which the resulting pseudometrics are metrics and allowing us to identify a rich family of compact subsets of finitely presented representations.

Linear and nonlinear diffusion equations arise in a range of phenomena of mathematical interest, including slow and fast diffusion, sandpile dynamics, height-constrained transport, the two-dimensional Navier-Stokes equation, and dynamics for sampling probability measures. In recent years, blob methods have attracted significant interest as an approach for numerically simulating these types of PDEs. To address the $O(N^2)$ computational bottleneck of blob methods, we consider stochastic discretizations of space and time. We compare the random batch method with a new approach, which we call the random multirate method. Both of these methods build on classical stochastic methods in the optimization literature, with many similarities to stochastic gradient descent and random coordinate descent. We find that, for linear and nonlinear diffusion equations, in the tradeoff between computational complexity and accuracy, the random multirate method has the best performance. On one hand, we prove that the random multirate method converges to the underlying ODE system at a rate of $O(k \Delta t)$, matching forward Euler, and show by example that this rate is theoretically sharp. On the other hand, we observe even better rates of convergence for the random multirate method when applied to ODEs arising from blob methods for diffusive PDEs. Finally, due to its ability to simulate a wide range of nonlinear diffusion equations, including height-constrained transport and sandpile dynamics, our method succeeds in capturing key features of PDEs for which few numerical approaches exist.

A convex 3D-polytope is said to have Rupert's property if it can pass through a copy of itself. The Nieuwland number of a convex polytope $P$ is the largest $\nu \in \mathbb{R}^+$ such that $\nu P$ can pass through $P$. We reduce showing $P$ passes through $Q$ to a feasibility problem over a quadratic constraint set. Using this, we prove that the Nieuwland number of the octahedron is $\frac{3\sqrt2}{4}$ and that the computation of the Nieuwland number of a convex polytope can be reduced to polynomially many semialgebraic optimization problems in a fixed number of variables.

Let $\mathfrak{X}_k\subseteq{\sf Mat}_{m, n}$ denote the variety of $m\times n$ complex matrices with rank at most $k$. The power series and rational expressions for the Hilbert series of $\mathfrak{X}_k$ are known by geometric arguments, and equating these expressions yields a family of formulas generalizing the classical Cauchy and dual Cauchy identities. We pose the problem of giving a direct combinatorial proof of these formulas for $0 < k < \min\{m, n\}$. When $k=1$ or $k=\min\{m, n\}-1$, we give such a proof via an explicit sign-reversing involution on certain sets of Littlewood–Richardson tableaux.

In convex geometry, a number of inequalities for the volume of coordinate projections of convex bodies share a common structural property. We formalize them as inequalities of log-submodularity type, providing a definition that stems from the notion of a submodular function. The Loomis-Whitney inequality and some of its generalizations are the most prominent examples. Some inequalities of log-submodularity type are valid for zonoids but not for general convex bodies. They have a natural meaning in statistics and information theory. We propose a new inequality of log-submodularity type for zonoids. In dimension 3, our inequality is equivalent to the log-submodularity conjecture for zonoids formulated by Fradelizi, Madiman, Meyer, and Zvavitch, which is true in dimension $3$ and false in higher dimensions. In contrast, our inequality is true in every dimension. We reduce our inequality to copositivity of a special polynomial, which we call the hypercube polynomial. For 4-dimensional zonoids, we verify the copositivity of the respective hypercube polynomial algebraically by providing a sum-of-squares certificate. In every dimension, a proof of copositivity of the hypercube polynomial that uses probability theory and well-known matrix inequalities has been produced by LLMs and brought to our attention. We present a detailed and easy-to-read version of this proof.

Pathological neuronal synchrony provides one practical motivation for studying sparse desynchronisation control in networked oscillatory systems, particularly in applications where actuation and communication are resource constrained, as in deep brain stimulation. Motivated by this challenge, we study how to stabilise desynchronisation in coupled oscillatory dynamical systems without continuously updated control, even when the oscillators' phase is unavailable. We develop a general event-triggered control framework for stabilising the desynchronised state of coupled limit-cycle oscillatory dynamical systems. Our analysis establishes a unified theoretical result showing that desynchronisation can be achieved by various feedback controllers that act sparsely and depend only on an order parameter observable. The proposed controllers admit a gradient-descent interpretation and stabilise the desynchronisation state of general phase-reduced oscillator networks. We further prove the controlled systems under event-triggered mechanisms possess a strictly positive lower bound of the inter-event dwell-time, excluding Zeno behaviour. To address the practical unavailability of exact phase reductions, we introduce a pseudo-phase construction that yields a computable order parameter from state measurements alone. Numerical studies on representative oscillator networks demonstrate the effectiveness and robustness of the proposed framework.

These notes are devoted to the kinetic Fokker–Planck equation on the tangent bundle of a compact Riemannian manifold, with motivations coming from relativistic kinetic theory of gases (à la Debbasch) and geometric analysis (à la Bismut). They are developed in particular from collaboration with Fabrice Debbasch and Yann Ollivier. The presentation is mostly self-contained, intended for readers without much prior familiarity with geometric analysis. Once the kinetic equations are reviewed, I expand the basic toolbox from functional and spectral analysis. Then the main focus is on global hypoelliptic $C^\infty$ regularization and hypocoercive exponential equilibration, both of which are addressed both in $L^2$ and $L^1$ settings. Various types of interpolation inequalities, spectral gap and entropic inequalities are established and used. Several existing results are simplified and unified, along with new ones. Some directions of research are mentioned.

Since the 1980s, the Bogomolov multiplier of a finite group has been known to obstruct rationality in complex algebraic geometry, and more recently it is understood to be responsible for any torsion in the oriented and stable unitary 2-dimensional $G$-equivariant bordism groups $\Omega_2^{SO,G}$ and $\Omega_2^{U,G}$. In this note, as a small step toward building a bridge between these two far-flung roles, we discuss the vanishing of Bogomolov multipliers of two specific families of finite groups. First, we revisit Kunyavskiĭ's result that the Bogomolov multipliers of all finite simple groups vanish, taking inspiration from the low-dimensional topological interpretation of the Ore conjecture. Second, in lieu of arguments in complex birational geometry (such as the hard direction of the Chevalley-Shephard-Todd theorem), we combine cut-and-paste combinatorial-topological techniques with elementary calculations of Ihara-Yokonuma to show that all finite Coxeter groups have vanishing Bogomolov multiplier.

We compute the minimal exponents of semi-quasihomogeneous and Khovanskii non-degenerate complete intersections.

We obtain estimates for the roots of the Bernstein-Sato polynomial in terms of a log resolution by an orbifold.

We study rigidity phenomena for maps on vector and normed lattices arising from isometric, order-theoretic, and modulus-preserving conditions. We first investigate the role of strict convexity in Baker's nonsurjective version of the Mazur-Ulam theorem. In the process, we develop a theory of midpoint injective functions and investigate their connections with convexity and monotonicity. Using this framework, we prove a generalization of Baker's theorem. As our first main application, we show that every modulus isometry, i.e., a map $T:X\to Y$, where $X$ is a sublattice of a vector lattice $Y$, satisfying $$|T(x)-T(y)|=|x-y|, \qquad x,y\in X,$$ is affine and has a disjointness-preserving linear part. When $X$ is an ideal, the linear part is moreover a bijection of $X$ onto itself and an involution. In the normed lattice setting, we also show that every positive norm isometry into a strictly convex normed lattice is not only affine, but has a linear part that is a lattice homomorphism. Finally, we prove a nonlinear order-isometric rigidity theorem. After shifting to the origin, every order-preserving norm isometry between normed lattices whose codomain has a strictly monotone norm, preserves suprema and infima and is both disjointness preserving and disjointly additive. These conclusions do not force linearity, as shown by an explicit family of nonlinear isometries on $L_1[0,1]$.

This study presents a hybrid computational framework for simulating non-reacting inviscid high-speed flows of nitrogen gas (N$_2$) around a circular cylinder. Owing to the strongly convection-dominated nature of the compressible Euler equations, the compressible-flow streamline-upwind/Petrov–Galerkin (SUPG) formulation is combined with the YZ$\beta$ shock-capturing technique to stabilize the finite element discretization in the presence of strong discontinuities. Building upon the stabilized solution, a physics-informed neural network (PINN) is employed as a post-processing correction stage (\underline{P}INN-\underline{A}ugmented \underline{S}UPG with \underline{S}hock-\underline{C}apturing—PASSC). The network is anchored to the finite element solution through a shock-weighted data-consistency loss, while the governing equations are enforced in a conservative space–time control-volume form supplemented by macroscopic conservation windows, an entropy-admissibility penalty, and the boundary conditions of the underlying problem. Two-dimensional simulations are performed for free-stream Mach numbers ranging from $2.0$ to $12.0$, and the results are assessed against analytical normal-shock and stagnation relations, the semi-empirical Billig correlation, and reference solutions from the literature. The correction is designed to improve the numerical representation of shocks by reducing localized discretization-induced oscillations and mesh-scale serrations while preserving the agreement of the stabilized solution with the analytical and semi-empirical reference quantities.

Let $V$ be an $F_2$-vector space with a given nondegenerate symplectic form and a circular basis. We define a partition of $V$ into subsets each of which has cardinal equal to a product of Catalan numbers. This can be viewed as a partition of the set of unipotent representations of a finite symplectic group corresponding to a certain two-sided cell.

Let $(V_i,\omega_i)$ be real symplectic vector spaces and let $\Omega_i\in\mathcal U^+(V_i)$ in the sense of Haiden. We prove that $p_1^*\Omega_1\wedge p_2^*\Omega_2$ belongs to $\mathcal U^+(V_1\oplus V_2)$ whenever one factor has real dimension at most six. There are forms $\Omega\in\mathcal{U}^+(\mathbb{R}^6)\setminus\mathcal{U}_{\mathrm{ag}}(\mathbb{R}^6)$ for which $p_1^*\Omega\wedge\cdots\wedge p_k^*\Omega$ belongs to $\mathcal{U}^+$ for every $k\geq 1$; hence a conjecture of Kontsevich does not hold in complex dimension three. We also give a criterion for a product to belong to $\mathcal{U}$ and show that, for every $N\geq 26$, there are two forms in $\mathcal{U}^+(\mathbb{C}^N)$ whose exterior product does not belong to $\mathcal{U}$. The main content of this paper is generated by ChatGPT 5.6 and verified by the author.

We consider quantum group generalizations of the action of monomial matrices on complex Hadamard matrices. This gives rise to various notions of quantum symmetries and quantum equivalences of Hadamard matrices. We show that if one acts by a certain largest monomial quantum group, then all Hadamard matrices of a given size become quantum equivalent. Taking a more restrictive quantization leads to a notion of $s$-quantum equivalence. We exhibit examples of Butson matrices of the same size and order that are not $s$-quantum equivalent for any choice of $s$. We also show that $s$-quantum equivalence is operationally modeled by a synchronous non-local “Hadamard equivalence” game. Our methods are largely graphical calculus based, using categories generated by complementary spiders. We use these same tools to study quantum affine equivalence of quantum groups, and prove that all finite quantum groups of the same size are quantum affinely equivalent. We also provide a short graphical proof of a result of Kasprzak–Sołtan–Woronowicz asserting that quantum symmetries of finite quantum groups must be classical.

We prove a reciprocity formula for a twisted second moment of the Riemann zeta function over a short interval of length $T^{\delta}$, where $\delta \in (\frac{1}{2},1)$, centered at height $T$ on the critical line. This result extends Khan's reciprocity formula to the short interval setting and makes explicit the relation between the twisting parameters and the interval length.

The Zaporozhets–Tarasov conjecture is considered, which states that the mean distance between two random points inside a convex body does not exceed the mean distance between two random points on its boundary. For centrally symmetric bodies the question is settled completely: the conjecture is proved for planar bodies and disproved in every dimension at least three, for all moments of the distance; an explicit family of counterexamples is constructed together with a dimension-lifting construction. It is also shown that for sufficiently high moments an analogous inequality holds for any planar convex body. For circumscribed bodies exact relations between the mean distances are obtained — analogues of Kingman's formula derived via the projective distributions introduced here; moreover, the conjecture is proved for triangles in a stronger per-projection form.

We propose a novel regularity-informed filtering framework for data assimilation in the context of hyperbolic conservation laws and other time-dependent partial differential equations. We focus on systems whose states exhibit steep gradients and jump discontinuities. While filtering is widely used to improve numerical simulations by incorporating observational data, traditional filtering methods lack awareness of the spatial regularity of states produced in these systems. As a result, data assimilation often produces unphysical state estimates, introducing spurious oscillations in smooth regions and smearing sharp features. To address this limitation, we introduce a filtering framework that incorporates edge-preserving regularization into the filter's analysis step; this framework balances simulation forecasts, observational data, and structural prior knowledge. We formalize this approach using the ensemble Kalman filter (EnKF) and a class of hierarchical generalized sparse Bayesian learning (GSBL) priors, which adaptively infer spatially varying hyperparameters to promote non-oscillatory behavior in smooth regions while preserving discontinuities. We demonstrate the effectiveness of the resulting GSBL-EnKF method on challenging benchmark problems governed by hyperbolic conservation laws. Our results show that enforcing regularity in the analysis step yields sharper, less oscillatory state estimates and lower errors of the ensemble mean. This sometimes comes at the cost of ensemble spread, which we quantify and discuss.

We give an effective algorithm for determining the transportation distance between two given probability density functions defined on the vertices of a graph $G=(V,E)$ by analyzing an associated polytope. The vertices of the polytope correspond to feasible flows on spanning trees in $G$, and the $1$-skeleton of the polytope is a projection of the spanning tree state graph associated with the Glauber dynamics on $G$. The optimal value of this transportation problem, known as the $1$-Wasserstein distance, can be computed by tracing the transportation cost along the vertices of this polytope. We show that a local minimum of the transportation cost is also a global minimum, and this leads to a steepest descent algorithm for solving the transportation problem. If the probability density functions take discrete values in $\delta \mathbb{Z}$ for some $\delta>0$, then the optimal transport cost can be reached in at most $\frac{|V|-1}{\delta}$ steps. As an application, we give an efficient algorithm for computing the Ollivier–Ricci curvature of a graph.

Let \(0\leqslant k<n\), and let \(\mathcal N(k,n)\) denote the supremum of the cardinalities of finite families of affine \(k\)-planes in \(\R^n\) whose pairwise Euclidean separation distances are all equal to one. For \(k=1\), the finiteness of \(\mathcal N(1,n)\) was recently established by Solymosi and Zahl. We extend this result to affine subspaces of arbitrary dimension and prove the fully explicit bound \[ \mathcal N(k,n) \leqslant 2^{\,2^{\,10(k+1)(n-k)}}. \] The main difficulty in higher dimensions is that the rational distance formula degenerates along several possible rank strata of \(\dim(U+V)\), rather than only along the parallel locus. For each fixed rank \(r\), we replace the choice of a nonvanishing Gram minor by a global sum-of-squares expression involving all minors. This produces a polynomial relation \(H_r(x,y)=0\) which encodes both the unit-distance equation and an upper-rank condition, while also vanishing on the diagonal. We propagate this relation by means of the restricted Zariski closure and decompose the resulting algebraic set into an explicitly bounded number of connected Nash submanifolds. An analytic diagonal approximation argument shows that, within each Nash piece, the maximal possible value of \(\dim(U_x+U_y)\) must decrease. Iterating this rank descent over all possible ranks and combining the finitely many standard affine Grassmann charts yields the stated double-exponential upper bound.

Recent major breakthroughs in $q,t$-combinatorics include the introduction of the Dyck path algebra $\mathbb{A}_{q,t}$ by Carlsson and Mellit and of the Catalanimals by Blasiak et al., both of which led, among other things, to independent proofs of different extensions of the rational shuffle conjecture of Bergeron et al. The first main contribution of this paper is a simple, explicit formula inside the algebra $\mathbb{A}_{q,t}$ for the Negut operators, yielding a direct, elementary connection between the original operators of the rational shuffle conjecture and the corresponding Catalanimals. Our formula bypasses the elliptic Hall algebra, turning these operators into transparent, workable tools whose action we can compute exactly and efficiently on any symmetric function, not just constants. Our second main contribution consists of a series of explicit formulas relating the Negut operators to the Theta operators introduced by D'Adderio et al. To prove these formulas, we provide an extension of the aforementioned Theta operators to the entire algebra $\mathbb{A}_{q,t}$, allowing us to obtain a series of new combinatorial results. The algebraic computations underlying this extension have been formalized in Lean. To showcase the power of our results, we give a proof, also partially formalized in Lean, of the Theta conjecture of D'Adderio et al., first stated in 2019.

Let $\sigma$ be a Poisson-finite measure on the real line. If its logarithmic integral diverges, then the functions from the classical Hardy space $H^2$ with compactly supported Fourier transform are dense in $L^2(\mathbb{R}, \sigma)$. We give a quantitative version of this result, adapting the ideas from the 2020 paper by Borichev, Kononova and Sodin, where a similar question was studied in the setting of polynomial approximation.

We prove that the mean distance between two independent uniformly distributed points in an arbitrary planar convex body is strictly smaller than the mean distance between two independent uniformly distributed points on its boundary. Equality is impossible, although the difference between these two mean distances tends to zero along a sequence of thin rectangles. The proof reduces the problem to a one-dimensional comparison of two Gini mean differences. The main new ingredient is a moment lemma for a random pair $(\varepsilon,R)\in\{-1,1\}\times[0,1]$. The only finite algebraic part of the proof is given by exact rational certificates in the Bernstein basis; the verification script and all 6492 coefficients are available in the supplementary materials.

Existence of canonical metric on a Sasakian variety was a long standing conjecture and the major part of this conjecture is about varieties which do not have definite basic first Chern class . Most of the Sasakian varieties do not have definite basic first Chern class. We give a program by using Fujino's Minimal Model program to find canonical metric on the canonical model of Sasakian varieties.

We give a complete proof of the 2011 Tu–Deng conjecture. We begin from its original modular pair-count formulation, prove an equivalent cyclic Hamming weight-drop formulation, and establish the exact transfer identity that connects this count with a two-variable matrix polynomial. The proof then reduces the conjecture to normalized inequalities for the coefficients of that polynomial. A 2011 conjecture by the author which came to be called the Cusick Conjecture (it is a consequence of the Tu–Deng Conjecture) was proved by K. Cheng in 2026. The proof in the present paper extends the cyclic deletion ideas of Cheng. The new ideas might be applicable to other problems.