Papers

We outline a theory of type 2 recursion for Infinite Time Turing Machines à la Kleene. We establish a connection between classical descriptive set theory and ittm theory, by calculating the complexity of its halting problem as exactly that of a complete $\Game \Sigma^0_3$ (or $G_{\delta\sigma}$) set. This mirrors exactly what Kleene, Moschovakis et al. achieved for Kleene's type 2 recursion and $\Sigma^0_1$ (or Open) Determinacy.} We ascertain the least ordinal which is not generalised recursive in this sense, and its characterisation {\via}a concept of infinite nestings in Gödel's constructible hierarchy. The results do not require large cardinal axioms, and are all provable within analysis.

We study the ring of Puiseux polynomials with integer coefficients. We prove notably that the order given by the leading coefficient is definable without parameters in the language of rings. This answers a question of R. Robinson.

Linear Quadratic (LQ) control problems are at the heart of linear control theory and Model Predictive Control (MPC). While performant, standard approaches to solving such problems are inherently serial, limiting real-time scalability despite the parallel computing power available on modern multi-core CPUs. Contributing to addressing this challenge and motivated by “divide and conquer” strategies, we present a parallel-in-time approach that solves computationally demanding conic optimal control problems through the use of the alternating direction method of multipliers (ADMM). In particular, we formulate the inner primal update of ADMM as an LQ problem and split the reformulated problem along the time horizon. This enables us to derive a variant of the Riccati recursion using dynamic programming to solve each subproblem in parallel. Numerical benchmarks on two real-world applications demonstrate as much as a 5x speedup compared to existing related approaches on multi-core CPU hardware.

We give a short, self-contained linear-algebra proof of a bound of Hegedüs [Australasian Journal of Combinatorics, 2026; arXiv:2409.07877]: if all pairwise Hamming distances in a family of subsets of $\{1,\ldots,n\}$ equal a fixed value $\lambda\ne(n+1)/2$, then the family has at most $n$ members. Our proof uses the same Gram matrix as in Hegedüs's argument, but reads its eigenvalues in place of its determinant, and keys off of a single fact about vectors of equal norm and equal pairwise inner product. That fact applies verbatim over an alphabet of size $q$, where it yields the bound $n(q-1)$ for $\lambda\ne\bigl((q-1)n+1\bigr)/q$ – the corrected form of a conjecture of Hegedüs, recently established by Hu, Huang, and Yu [arXiv:2504.07036].

Interventions can be varied continuously in many causal models. Differentiating a specified smooth intervention protocol produces vector fields on a statistical model, and their Lie brackets describe the noncommutativity of the corresponding local perturbations. We formulate this differential geometry of interventions on smooth statistical models and call the resulting framework infinitesimal causality (IC). Given a constant-rank distribution spanned by visible intervention fields, we define the normal Lie-bracket residual and show that its vanishing is exactly the involutivity condition in the classical Frobenius theorem. We establish the coordinate invariance of the zero-residual property and characterize its dependence on the intervention protocol, visible span, and metric. Fully observed and latent-variable examples delineate the additional structural assumptions needed to interpret bracket residuals causally. We also distinguish tangent vectors on a statistical parameter manifold from derivatives of stochastic kernels. In the finite-state linearization of a Markov category, normalization and copy compatibility yield well-typed first-order defects. Normalization is automatic for differentiable paths of stochastic kernels, whereas copy compatibility characterizes a more restrictive deterministic or comonoid-preserving perturbation. Together, the geometric and kernel-level constructions make IC a precise foundation for Lie-bracket-based causal diagnostics and identify the assumptions required to pass from local intervention geometry to causal conclusions.

In this work we study two classical problems of (numerical) linear algebra: (i) solving linear systems and (ii) computing eigenvectors, within a constructive framework. Numerical accuracy and indeterminacy are naturally incorporated through Bishop-style constructive mathematics. Our contributions include new results on Gauss-Jordan elimination and on approximating the rank of a matrix. Additionally, we introduce a novel method for constructing approximate eigenvectors, based on a previously unexplored characterization of singular matrices.

In this paper, we derive improved estimates for a class of fully nonlinear parabolic equations with continuous drift and admissible source terms of the form $$ \partial_{t}u - F(D^2u,x,t) + \langle B(x,t), Du\rangle = f(x, t, u^{+}, u^{-}) \quad \text{in}\quad Q_1. $$ Our analysis reveals two distinct regimes. In the first, $f=f(x,t)$ exhibits $\theta$-Hölder decay ($\theta\in(0,1)$), yielding improved gradient regularity at vanishing points via perturbative methods and geometric iteration, as well as nondegeneracy with explicit growth rates under a suitable structural condition. In the second, $f(\cdot,u^+,u^-)=(u^+)^\gamma-(u^-)^\gamma$ with $\gamma\in(0,1)$ (corresponding to an evolutionary semilinear two-phase model), we obtain enhanced regularity at branching points by combining a robust blow-up analysis with local derivative estimates for linear equations. Our results remain relevant even in linear settings with merely continuous data, linking to classical free boundary problems arising in mathematical physics and related areas.

In finite dimensions, every doubly stochastic matrix has the $\ell^p$-operator norm equal to $1$ for all $1 \le p \le \infty$. However, in the infinite-dimensional setting, this property may fail since the norm can be strictly smaller than $1$ when $1<p<\infty$. In this paper, a complete characterization of infinite doubly stochastic matrices for which the norm remains equal to $1$ is obtained. More precisely, for $1<p<\infty$, it is shown that $$ \|D\|_{\ell^p(I)\to\ell^p(I)}=1 \quad\iff\quad \Theta(D^*D)=1, $$ where $\Theta$ measures the maximal average mass of a finite square submatrix. Thus, the norm is equal to $1$ precisely when the matrix contains arbitrarily large finite regions in which it behaves almost like a finite doubly stochastic matrix. The proof uses a Cheeger-type argument, highlighting a natural connection with ideas from spectral graph theory.

In this paper, we study irreducible noncommutative quadrics $S/(f)$ via noncommutative graded matrix factorizations. We show that the line modules over $S/(f)$ are described by the rulings arising from indecomposable noncommutative linear matrix factorizations of $f$ of rank $2$. We study when Zhang twists of a standard smooth irreducible noncommutative quadric are standard. Finally, by identifying all singular central Sklyanin quadrics, we prove that every smooth central Sklyanin quadric is standard.

For a homogeneous one-dimensional random field, positive semidefiniteness of finite Toeplitz correlation matrices imposes non-trivial constraints on admissible correlation coefficients. The widths of the corresponding admissible intervals are closely related to determinants of principal Toeplitz submatrices. Using the classical Desnanot–Jacobi determinant identity, I derive a simple determinantal representation for the widths of admissible correlation intervals. As an immediate consequence, I recover the product expressions for admissible interval widths previously stated by Schneider & Hartlap (2009). The argument places these relations into the general framework of classical Toeplitz determinant theory.

We consider a class of composite optimization problems involving a smooth function and a proper, lower semicontinuous regularizer, which may be nonconvex and nonsmooth. We propose a novel alternating minimization framework that integrates proximal-gradient steps with cubic-regularized Newton updates restricted to a dynamically identified low-dimensional subspace. Under the Kurdyka–Łojasiewicz (KL) property, we establish global convergence of the proposed method to a stationary point. Moreover, by incorporating an adaptive thresholding strategy guided by the KL exponent, we prove a finite identification property without imposing any nondegeneracy assumptions. We further develop a local convergence analysis and show that the proposed method attains a worst-case iteration complexity of $\mathcal{O}(\varepsilon^{-3/2})$ for achieving approximate second-order stationarity. Numerical experiments on both synthetic and real datasets demonstrate the efficiency and effectiveness of the proposed framework.

A general tree-valued dynamics is considered in continuous time: new vertices are added, and the percolation happens on the links, and the connected components can be frozen. The model is an infinite-type branching process. The main result establishes the Perron–Frobenius type theorem on this model, which extends the previous work [Ann. Appl. Probab. 33 (6B) 5233 - 5278]. The proof does not rely on any property of the uniform random recursive tree.

In this paper, we study the precise inversion of adjunction (PIA) conjecture and the lower semi-continuity (LSC) conjecture for hyperquotient singularities. Previously known results for these conjectures in this setting required the singularity to be klt, and without this assumption, a counterexample to the PIA conjecture is known to exist. To resolve this obstacle, we introduce a localized notion of virtually free actions and characterize it via the arc spaces of quotient varieties. Utilizing this characterization, we establish a necessary and sufficient condition for the PIA conjecture to hold for arbitrary hyperquotient singularities, thereby clarifying the mechanism of the counterexample. Furthermore, as an application of this insight, we unconditionally establish the LSC conjecture for arbitrary hyperquotient singularities.

We characterize the asymptotic behavior of time-homogeneous doubly stochastic Markov chains. Our investigation revolves around understanding the dynamics of products of doubly stochastic matrices, which in turn allows us to fully characterize three distinct behaviors: cyclicity, convergence towards a special equilibrium matrix, and divergence. Notably, we introduce a novel and comprehensive sufficient condition for the convergence of an infinite product of doubly stochastic matrices.

Adapting the construction disproving the sum-product conjecture over $\mathbb R$ present in Bloom, Sawin, Schildkraut and Zhelezov, we show the existence of a constant $c>0$ and arbitrarily large finite sets $A \subseteq \mathbb R$ such that $$|AA+A+A| \ll |A|^{2-c}.$$ As a corollary, all of the sets $A+A$, $AA$, $(A+1)(A+1)$, $A(A+1)$ and $AA+A$ are of size $O(|A|^{2-c})$ for this construction.

The binary Type I self-dual [26,13,6] code is classical. We give what appears to be the first direct non-computational proof of its uniqueness. We first determine the weight enumerators of the code and its shadow. Degree-one harmonic MacWilliams identities supply the required 1-designs. Elementary intersection counts then show that the two minimal half-shadows each contain 13 words; these words label the 26 coordinates as 13 points and 13 lines, and the two shadow classes become the point-stars and line-stars. From this structure we give two uniqueness proofs: one reconstructs the projective plane of order 3 and the plane code, including the full automorphism group PGL(3,3):2; the other deletes an intrinsic flag, obtains the odd Golay code together with a deep hole coset, and reconstructs the length 26 code and the size of the automorphism group from this coset datum. Thus the natural length-24 object behind the code is the odd Golay code together with its unique orbit of deep hole cosets.

We construct the Hermitian vector and canonical components of the momentum operator in 3D Euclidean space spanned by general curvilinear coordinates (GCC's) using a simple, natural and unified approach based on identifying the momentum operator in any coordinate system as mass times the velocity operator. When this latter is calculated by applying the Heisenberg equation of motion, it returns ($-i\hbar$ times) the gradient operator plus an additional zero-valued sum, which when distributed among the components of the gradient, it makes each the Hermitian vector component of the momentum operator in GCC's. The canonical components follow immediately upon symmetrizing each of these vector components in the corresponding base vector. For accessability by wider audiences, we first develop the formalism for the simple polar coordinates and then we develop the case for GCC's.

Nilpotent Lie groups with left-invariant metrics provide nontrivial examples of Ricci solitons. Some typical examples are given by the class of two-step nilpotent Lie algebras obtained from simple directed graphs and the class of nilpotent Lie algebras obtained from finite acyclic quivers. In this paper, we generalize the construction of nilpotent Lie algebras that are algebraic Ricci solitons obtained from finite acyclic quivers. We use some special ordered sets to construct nilpotent Lie algebras, which can also be obtained from some special quivers with relations. A transitively and antisymmetrically ordered set (or TAOS, for short) is a set together with a binary relation that is transitive and antisymmetric. Utilizing the concept of incidence algebras of TAOSs, we construct nilpotent Lie algebras. We modify the method introduced by Mizoguchi and Tamaru \cite{MR4941781} and use it to show that the nilpotent Lie algebras with arbitrarily high degrees of nilpotency obtained from some special finite transitively and antisymmetrically ordered sets, called array TAOSs, are algebraic Ricci solitons. We also give some generalizations of this result, which yield more nilpotent Lie algebras that are algebraic Ricci solitons. Moreover, the corresponding simply-connected nilpotent Lie groups admit left-invariant Ricci solitons.

We study the inhomogeneous form of the Jordan–von Neumann quadratic functional equation, in which the right-hand side is a prescribed function $g$ of two real variables. We prove that the existence of a $C^{2}$ solution is equivalent to $g$ being itself of class $C^{2}$ and satisfying a single three-variable cocycle identity, and we exhibit the solution as a closed-form integral expression involving the second partial derivative of $g $ along the first coordinate axis. The construction preserves regularity along the standard scale of $C^{k}$, smooth, and polynomial classes.

We introduce a new family of balance indices for phylogenetic networks: the $H_\alpha$ indices, where $\alpha$ is a positive real number. This family includes the $B_2$ index as a special case ($\alpha = 1$) and provides a natural extension of the Sackin index to phylogenetic networks. We show that the $H_\alpha$ indices share many structural properties with the $B_2$ index, most notably a "grafting property" that makes it possible to express the $H_\alpha$ index of a network in terms of the $H_\alpha$ indices of its biconnected components. These properties allow us to identify networks that minimize / maximize $H_\alpha$ for various classes of phylogenetic networks, and to study its distribution for several models of random trees and networks (in particular, Galton-Watson trees and binary Markov branching trees, with a focus on the Yule and PDA models). Finally, we show how local limits can be used to analyze the asymptotic behavior of $H_\alpha$ for large trees and networks, and we obtain general results for the moments of $H_\alpha$ for a broad class of random phylogenetic networks known as blowups of Galton-Watson trees.

The linear ordering problem (LOP) is a classical NP-hard combinatorial optimization problem. In this paper, we study constructive heuristics for the LOP. We propose a unified recursive framework in which the index set is heuristically partitioned into smaller subsets, sufficiently small subproblems are solved exactly, and the resulting permutations are concatenated. Within this framework, we investigate three different recursive partitioning principles: a rule based on strongly connected components (the level graph method), a cut-based rule (the minimum cut method), and a score-based rule (the recursive Borda method). We first show that the recursive Borda method satisfies the Condorcet criterion, whereas the Borda rule does not. The level graph method and the minimum cut method satisfy a stronger Condorcet-type property, which the recursive Borda method does not satisfy. Computational experiments on xLOLIB instances show that the recursive Borda method provides the best average solution quality among the tested constructive heuristics. They also show that, after applying insertion-based local search, the differences in final solution quality among these heuristics become much smaller.

We consider $D$-equivariant solutions to the harmonic map heat flow from $B^2$ to $S^2$ under general time-dependent smooth boundary data and prove that there is no finite-time blow-up when $D \geq 2$.

Becker and Bergweiler showed that if $f$ is a non-exceptional polynomial, then the Böttcher coordinate $\Psi_f \colon \mathbb D_R \to B_\infty(f)$ associated to $f$ is a transcendental function. In this paper, we study $f$-bialgebraic sets: algebraic subsets of $\mathbb D_R^n$ whose image under the coordinate-wise action of $\Psi_f$ is contained in an algebraic set of the same dimension. We give a complete dynamical classification of bialgebraic sets under the additional assumption that the Julia set of $f$ is either disconnected, or connected and admits a nondegenerate locally connected model. Inspired by the Ax–Lindemann–Weierstrass theorem and the Ax–Schanuel conjecture, we formulate analogs with $\Psi_f$ in place of the exponential function and prove them in the case where the Julia set $J_f$ is disconnected.

We introduce the concept of spectrally $n$-centered operators and study it within the framework of unbounded weighted composition operators in $L^2$-spaces. Based on this, we establish a characterisation of spectrally centered unbounded weighted composition operators, generalising Giselsson's criterion to the unbounded regime. Finally, we analyse various associated conditions, highlighting the delicate measure-theoretic anomalies.

Introduced by R.T. Rockafellar in 2019, variational convexity is a generalized notion of convexity under which stationary points of nonconvex optimization problems can still be guaranteed to exhibit local optimality. In this paper, we provide characterizations of variationally convex functions through their proximal hulls and epigraphs, and investigate operations that preserve variational convexity, including nonlinear and linear composition, summation, and proximal averaging. We further apply these results to identify variational convexity in nonlinear programming problems with possibly nonsmooth objectives, continuously differentiable inequality constraints, and affine equality constraints. This leads to new conditions ensuring local minimizers, rather than merely stationary points, for such problems, extending beyond current state-of-the-art results that typically require twice continuously differentiable objectives and constraints.

Traditional importance sampling (IS) is designed to estimate rare-event probabilities by minimizing estimator variance. However, many applications prioritize rapid discovery: the generation of a trajectory within a rare set $A_n$. This requires a shift from ensemble-based estimation to a design principle focused on the hitting time $\tau_{A_n} := \inf\{t \ge 1 : Y_t^n \in A_n\}$. We formalize a Quality of Discovery problem as the problem of minimizing the description length (surprisal) of the discovered trajectory under the nominal model $p$. We prove that minimizing this description length is equivalent to minimizing the nominal rank exponent $J_{\mathrm{rank}}(q_n) := \lim_{n\to\infty} \frac{1}{n} \log G_n(Y^n)$, where $G_n(x^n)$ is the guesswork of sequence $x^n$. For i.i.d.\ models and type-defined rare sets $\Gamma$, we show that while classical IS targets the mass-dominating type $Q_{\mathrm{IS}}^* \in \arg\min_{Q \in \Gamma} D(Q\|p)$, discovery optimality is achieved by $Q_{\mathrm{GW}}^* \in \arg\min_{Q \in \Gamma} [H(Q) + D(Q\|p)]$. This framework identifies a fundamental rule: minimizing the guesswork exponent ensures the discovered sequence is the "least surprising" representative of the set relative to the nominal model's search order. We further demonstrate that under budgetary constraints, this exponent serves as a lexicographic tie-breaker when the hitting-time minimizer is not unique. This establishes $H(Q) + D(Q\|p)$ as a natural objective for discovery-based importance sampling, providing a formal bridge between randomized sampling and systematic search.

Euler regularized the divergent product of all natural numbers and found beautiful formulas for regularized sums of integer powers of natural numbers. These derivations essentially relied on what is now called the zeta-regularization technique, although analytical continuation had not yet been invented. This classic method is however not applicable to the product of all primes, as the prime zeta function has a natural boundary along the imaginary axis. Muñoz García and Pérez-Marco overcame this obstacle and evaluated the product of all primes to $4\pi^2$ by finding an appropriately regularized value of the derivative of the prime zeta function at the origin, lying on the natural boundary. We extend their approach in two novel directions. First, we show how to make sense of the sum of all primes. This regularization requires going a finite distance beyond the natural boundary. Second, we determine the regularized products of integers and primes in the nine imaginary quadratic fields where integers have a unique factorization into primes, and establish a general power-law relationship between products of integers and primes. Two well-known examples are Gauss and Eisenstein integers. The interest in this approach goes beyond number theory. In a variety of physical situations, the zeta-regularization technique is indeed not applicable because the relevant zeta function has a natural boundary.

We construct explicit elements in the group $K_4^{(3)}$ of the Fermat curves $x^N+y^N=1$ for all $N\geq 3$. The construction, which is uniform in $N$, uses polylogarithmic complexes and a map of de Jeu to $K$-theory. We prove that the elements are non-trivial by showing that their images under Beilinson's regulator map are non-zero. Notably, we obtain explicit formulas for their regulator integrals involving special values of Zagier's trilogarithm function. As a corollary, we show that these regulator integrals are asymptotic to $\frac{3}{2}\zeta(3)N^2$ as $N\to +\infty$. Moreover, we derive formulas for the regulators of our elements in terms of hypergeometric functions, generalizing results of Otsubo for $K_2$ groups of Fermat curves. Finally, we numerically verify some cases of Beilinson's conjectures on special values of $L$-functions at $s=3$ for $N\in \{ 3,4,6 \}$.

We study a vector-space analogue of the Erdős Matching Conjecture. Let $m_q(n,k,s)$ denote the maximum cardinality of a family of $k$-dimensional subspaces of an $n$-dimensional vector space over $\mathbb F_q$ with no $s+1$ members whose sum is direct. Two natural constructions provide lower bounds. The first consists of all $k$-subspaces contained in a fixed $((s+1)k-1)$-dimensional subspace; the second consists of all $k$-subspaces that intersect a fixed $s$-dimensional subspace nontrivially. These constructions motivate the following vector-space analogue of the Erdős Matching Conjecture: for all $n\ge (s+1)k$, $$m_q(n,k,s)=\max\left\{\genfrac{[}{]}{0pt}{}{(s+1)k-1}{k}_q,~\genfrac{[}{]}{0pt}{}{n}{k}_q-q^{ks}\genfrac{[}{]}{0pt}{}{n-s}{k}_q\right\}.$$ We prove this conjecture when $k=2$, when $n=(s+1)k$, and when $n$ is sufficiently large. In particular, the case $k=2$ may be viewed as a vector-space analogue of the Erdős–Gallai theorem. In the large-$n$ range, we also prove a Hilton–Milner-type stability theorem, determining the largest nontrivial families with this property. Finally, we connect this problem with $t$-cover-free families in vector spaces and determine their extremal number up to a lower-order term, extending a recent result of Shan and Zhou for the special case $t=2$. The proofs combine Lovász's minimax theorem for matroid matchings, a high-dimensional Hoffman bound for uniform hypergraphs, and packing-design arguments in vector spaces.

We provide a complete quasisymmetric classification of the Julia sets of postcritically finite McMullen maps $f_\lambda(z)=z^n+\lambda/z^n$ with $\lambda\in\mathbb{C}^*$ and $n\geq 2$, and prove that the quasisymmetry group of each such Julia set is exactly the finite dihedral group generated by the natural symmetries of the map. These results establish quasisymmetric rigidity for all topological classes in this family, including Sierpiński-like carpets, necklaces, and clusters, and provide the first known examples of rigid Julia sets in each of the three classes.