Papers

We study the structure of the centralizer of a non-elliptic holomorphic self-map of the unit disc and its relation with continuous dynamical structures. We prove that if the identity is not isolated in the centralizer of a non-elliptic self-map $\varphi \in \mathrm{Hol}(\mathbb D)$, then $\varphi$ is univalent and its centralizer contains a non-trivial continuous one-parameter semigroup of holomorphic self-maps. This result provides a general criterion for the existence of continuous dynamics arising from discrete iteration. Our approach is based on the theory of holomorphic models and on simultaneous linearization techniques for commuting self-maps. In particular, we exploit recent developments on simultaneous linearization for parabolic self-maps with positive hyperbolic step. As a consequence, we obtain new sufficient conditions for the embeddability of holomorphic self-maps into continuous semigroups.

We study helical shear of a thick-walled tube made of an incompressible, isotropic hyper-anelastic solid carrying a radially symmetric distribution of axial screw dislocations. Through the Bilby–Kröner–Lee decomposition, the dislocations are encoded in a Weitzenböck material manifold, whose Riemannian metric alone enters the elastic response. Equilibrium reduces at each radius to an algebraic system for the azimuthal and axial shear strains, whose cross terms are proportional to the dislocation strength and to the dependence of the strain energy on the second invariant; the two shear modes therefore couple only in the presence of dislocations. Exact solutions are obtained for neo-Hookean, Fung–Demiray, Gent, Knowles, and Mooney–Rivlin materials, and numerical ones for Gent–Gent materials. The universal relations of Horgan and Saccomandi are extended to dislocated generalized neo-Hookean tubes. Dislocations stiffen the tube when the strain energy stiffens with the first invariant, couple the two shear modes chirally when it depends on the second, and may restrict admissibility: they cap the dislocation strength in Gent and Gent–Gent materials, and contract the admissible traction region in Knowles materials with power-law index below n = 1/2. A weakly nonlinear analysis to third order in the applied tractions and dislocation strength shows that, at first order (linear elasticity), the dislocations leave the shear response unaffected while carrying a residual stress. The chiral coupling enters at second order, through the dependence of the strain energy on the second invariant, so that every dislocated material couples the modes as the Mooney–Rivlin solid with the same second- and third-order moduli. At third order, no new coupling arises; the modes interact through stiffness corrections involving the fourth-order modulus, which the dislocations further correct mode by mode.

We propose that mathematics is done largely as a means of both coping with and celebrating our physicality in the natural world. Inspired by David Abram's "Becoming Animal", we suggest that – in opposition to the popular image of an otherworldy Platonic realm– the spiritual component of doing mathematics is grounded on and in the Earth. We explore this as a means of questioning the widespread adoption of AI in the workflows of research mathematics.

We study orthogonal $(4,3)$-equipartitions of finite measures absolutely continuous with respect to Lebesgue measure: every three of four mutually orthogonal affine hyperplanes must divide each measure into eight equal parts. For every number $m$ of measures, we determine the exact dimension threshold of Musin's polynomial criterion and express it in terms of the binary expansion of $m$. The threshold coincides with the geometric lower bound precisely when $m=2^j-1$. Consequently, $\Delta^*(2^j-1,4,3)=7\cdot2^{j-1}-2$ for every $j\geq1$.

We consider nonparametric regression with random design and non-Gaussian noise, where the regression function has the multi-index form $F(x)=f(\Theta x)$ with an unknown link $f:\RR^m\to\RR$ in an anisotropic Hölder class and an unknown matrix $\Theta$ with $m\le d$ orthonormal rows. For estimation of $F$ at a fixed point we prove that the minimax risk is bounded from below by a multiple of $L^{1/(2\gamma+1)}(n^{-1}\ln n)^{\gamma/(2\gamma+1)}$, where $\gamma$ is the harmonic mean of the smoothness indices of the link divided by $m$. For $m<d$ this extends a Gaussian white noise result to regression. The main new case is $m=d$, where no dimension reduction takes place: the logarithmic factor still appears as soon as the link is anisotropic and the orientation of its axes is unknown. In contrast, when the link is isotropic with smoothness index at most one, the class is rotation invariant up to a constant and the classical rate without logarithm is attained. The logarithmic loss is therefore caused by the unknown orientation of the directions in which the function is least smooth

Selecting a budget-constrained set of documentary fragments, for instance the knowledge base of a retrieval-augmented generation system, raises a statistical question: how much expected query coverage is lost relative to the best feasible selection when the query distribution is unknown? We construct finite-sample, distribution-free upper confidence bounds for this coverage regret that remain valid after data-dependent selection. A pilot stage may select the document set, screen competitors, build comparison networks and learn an annotation policy. Then an independent audit acquires relevance labels with probabilities that are constant within observable strata. The main result is a simultaneous bound for all retained coverage contrasts. The bound is made observable by empirical-Bernstein variance calibration or by a one-stage construction without calibration, and it accounts for estimated stratum proportions. For maximum-type coverage we prove variance contractions for swaps of redundant fragments, give a condition under which multiscale radii improve on a direct union bound, and show that such redundancy cannot be certified from samples alone. A simulation study confirms validity and quantifies the gains from variance calibration and certified redundancy.

Let $f,g\in\mathbb{Z}[t]$ be products of cyclotomic polynomials such that either $g/\gcd(f,g)$ or $f/\gcd(f,g)$ (or both) is squarefree. We give explicit formulae for the invariant factors of $f(C_g)$, where $C_g$ is the companion matrix of $g$, expressing them in terms of the indices and multiplicities of the cyclotomic factors of $f$ and $g$. A more compact formula holds for the circulant case $g(t)=t^n-1$, assuming that every cyclotomic factor shared by $f$ and $g$ has multiplicity $1$ in $f$. We discuss applications to several topics, including abelianizations of cyclically presented groups, homology groups of cyclic branched covers of knots, adjacency matrices and shifted graph Laplacians, $K$-theory, and Smith forms of matrix powers. For example, for two families of hyperbolic knots we give the first integral homology in every covering degree; or, when $g$ is squarefree, we determine the least eventual period of the Smith forms of powers of $f(C_g)$.

We compute the Euler characteristic of Bestvina-Brady kernels $\operatorname{BB}(\Gamma)$ for all Artin groups of spherical type. Our method involves discrete Morse theory to describe the cellular structure of a classifying space for the Bestvina–Brady kernel, through the action of $\operatorname{BB}(\Gamma)$ on non-integer level sets of the universal cover of the Salvetti complex. We extend the method of computation of $\chi(\operatorname{BB}(\Gamma))$ to Artin groups that satisfy the $K(\pi,1)$-conjecture and whose nerves are rationally acyclic, and we provide an example by realising the projective plane as the nerve of a Coxeter group.

This note grew out of two questions arising in different settings. The first concerns recognizing holomorphic functions among continuous functions on an annulus. The second concerns what self-reproducing subsets can tell us about semigroups of functions arising in analysis. Our analytic starting point is a variation on the classical relation between holomorphy and invariance of contour integrals. It identifies the holomorphic functions as the largest class containing the constant function 1 for which a contour convolution is independent of the chosen curve for every pair of functions. On this class, convolution amounts to multiplying Laurent coefficients. We then study families $B$ satisfying $B\diamond B=B$. We show that expressing a function as a convolution product of $m$ holomorphic functions corresponds precisely to its extension to a wider annulus determined by $m$. This identifies the largest self-reproducing family as the functions extending holomorphically to the punctured plane. We also prove that uniform boundedness on a fixed subannulus surrounding the unit circle forces all members of such a family to have Laurent expansions supported on one common finite set. A general semigroup theorem describes the minimal self-reproducing families.

For a Chern nontrivial Bloch bundle over the three torus, we construct an orthonormal Wannier basis with the optimal pointwise decay $\mathcal{O}(|x|^{-5/2})$. All but one of the generators decay exponentially. This resolves the question left open in arXiv:2505.01999.

Optimal Morse vector fields on closed surfaces, i.e.\ fields with the minimal possible number of critical points (a single source and a single sink), are classified up to topological trajectory equivalence by one-cycle signed chord diagrams in Kybalko, Prishlyak and Shchurko (2018). We recall this construction and the associated duality, which corresponds to reversing the direction of the flow, in a unified encoding in which both endpoints of a chord carry the same label, and we make it algorithmic. We give an explicit traversal procedure that detects one-cycle diagrams, a canonical form of an equivalence class under cyclic shifts, reversal and relabeling of chords, and a construction of the dual diagram. On this basis we describe an algorithm that enumerates all one-cycle signed chord diagrams with a given number of chords, both for orientable and for non-orientable surfaces, groups them into equivalence classes, pairs each class with its dual, and identifies the self-dual ones. The algorithm is implemented in Python and archived in Zenodo; its output is validated against the Harer–Zagier numbers in the orientable case and against orbit-size counts in both cases. We also note that, when a one-cycle diagram is read as a cellular embedding of a one-vertex graph, its dual diagram describes the embedding of the dual graph.

We construct an infinite family of solutions to the Mondrian art problem with absolute defect bounded above by $280$, of side lengths $n_0 = 315$, $n_1 = 106785$, and $n_{j+2} = 340n_{j+1} - n_j$, where $n_j$ is congruent to $315 \pmod{630}$. This absolute bound shows that the minimum defect does not necessarily increase with $n$, providing a counterexample to a conjecture which has been open since the problem's original proposal. Our design is derived from a well-known seven-rectangle dissection over the real quadratic field $\mathbb{Q}(\sqrt{19})$, with non-integer side lengths; we tailor the side lengths to be integers via an infinite Pell family. Using the same method, we devise a second family with defect $560$ that contains infinitely many side lengths which are divisible by any given positive integer, implying that for any $k$ there are infinitely many side lengths $n \equiv 0 \pmod{k}$ with defect bounded above by $560$. This constant bound offers a dramatic improvement on the previous quadratic bounds.

We analyze the KdV–ghost system arising from the BRST formulation of BF-type topological field theory, its symmetries, Lax pair, spectral problem, zero-curvature formulation and Darboux formalism. The system is invariant under shifts of the ghost field, whose generator does not commute with the BRST symmetry of the KdV–ghost system. Their generators allow the introduction of a systematic procedure for generating new solutions from a given one, and the corresponding expressions in the Grassmann algebra are obtained explicitly. We introduce a Grassmann-valued Lax pair and prove an isospectral theorem relating the spectral problem of the spatial Lax operator for the KdV equation to that of its KdV–ghost extension. We also construct a triangular zero-curvature representation and show that the Grassmann-valued spatial Lax operator is related to the KdV spatial Lax operator by a nilpotent conjugation. The natural variable for a local Darboux formulation is the differentiated ghost $v=C_x$, for which the ghost equation coincides with the Fréchet linearization of the KdV flow. We then construct a nilpotent lift of the scalar KdV Bäcklund–Darboux transformation. Using Riccati variables and off-shell factorization identities for the KdV residue and its Fréchet derivative, we prove that the transformation closes on the local KdV–ghost system. Successive Darboux transformations yield a recursive ghost extension of the bosonic Darboux chain, with a new odd homogeneous contribution introduced at each step. For the solitonic examples, the resulting local ghost fields are naturally identified with tangent directions associated with the soliton phase parameters. The construction provides a Darboux-theoretic interpretation of the ghost sector while remaining distinct from the standard supersymmetric KdV hierarchies.

Dantzig-Wolfe reformulation applied to two-stage stochastic mixed-integer programs (SMIP) has been shown empirically to yield a formulation with strong bounds that can be solved via column generation. However, the time to solve the restricted master problem grows significantly with the number of scenarios due to the increasing number of nonanticipativity constraints (NACs). Moreover, the associated high-dimensional dual space can cause the dual multipliers to oscillate between iterations, further slowing convergence. We propose an aggregated column generation framework for solving a Dantzig-Wolfe reformulation. Applied to SMIP, this method partitions the NACs into groups and replaces each group with a weighted surrogate constraint, leading to a reduced size and providing implicit stabilization by restricting the dual space. While aggregation leads to a relaxation, we develop two strategies for recovering the standard Dantzig-Wolfe reformulation bound. The first iteratively refines the partition by splitting groups that contain violated constraints. The second keeps the partition fixed and adaptively updates the aggregation weights. We prove convergence of the two approaches and study their computational performance on two benchmark problem classes.

The secondary zeta function is defined by $$ Z(s)=\sum_{\gamma>0}\frac{1}{\gamma^{s}}, $$ where $\tfrac12+i\gamma$ runs over the non-trivial zeros of the Riemann zeta function with positive ordinate, and the Riemann hypothesis is assumed throughout. In this article, we derive convergent integral representations of $Z(s)$ at positive even and odd integer argument, expressed through a differential operator acting on the logarithm of the Riemann $\xi$-function. Define $$ L(x)=\log\left(\frac{\xi(x+\tfrac12)}{\xi(\tfrac12)}\right), \qquad \mathcal{E}=x\frac{\mathrm{d}}{\mathrm{d}x}, \qquad \mathcal{D}_k=\mathcal{E}\prod_{j=1}^{k}(\mathcal{E}-2j). $$ Then, for $k\geq1$, $$ Z(2k)=\frac{(-1)^k}{2^k(k-1)!}\int_{0}^{\infty}\frac{\mathcal{D}_k L(x)}{x^{2k+1}}\,\mathrm{d}x $$ and $$ Z(2k+1)=\frac{(-1)^k}{\pi\,(2k-1)!!}\int_{0}^{\infty}\frac{\mathcal{D}_k L(x)}{x^{2k+2}}\,\mathrm{d}x. $$ In the last part, we write these formulas out explicitly for $Z(k)$, $2\leq k\leq 7$, both in terms of $L(x)$ and in terms of $\xi(x+\tfrac12)$ for reference. We also derive a limit formula for $\gamma_1$, the ordinate of the first nontrivial zero, and demonstrate its convergence numerically.

Motivated by recent advances in the synthesis of effective one-dimensional mechanical metamaterials, we develop a direct theory of strain-gradient special Cosserat rods based on the principle of virtual work. The internal virtual work is formulated intrinsically in terms of variations of the classical Cosserat strains and their spatial derivatives, without requiring a three-dimensional parent material or a stored energy. Objectivity and insensitivity to rigid virtual displacements characterize the admissible internal-work density, while localization yields the dynamic field equations, higher-order contact interactions, natural boundary conditions, and balances of linear and angular momentum. For inextensible and unshearable rods, the constrained principle determines a unique multiplier functional, although its representation by reactive force and double-force fields possesses a gauge freedom absent from classical rod theory. We also formulate objective elastic, hyperelastic, and viscoelastic constitutive laws, extend transverse hemitropy, transverse isotropy, and flip symmetry to strain-gradient rods, and derive invariant representations of symmetric stored energies. As an application, we study a planar inextensible cantilever under a compressive terminal force. We prove that the eigenvalues of the linearized problem are simple and coincide precisely with the loads at which bifurcation from the straight state occurs. In addition, each bifurcation point generates a connected global branch of nonlinear equilibria that is locally analytic near its origin.

We develop a homological model for hamiltonian reduction of Courant algebroids, parallel to the BFV formulation of hamiltonian reduction in symplectic geometry. To a hamiltonian action on a Courant algebroid, we associate a differential graded Courant–Dorfman algebra, obtained from the BFV algebra of the corresponding graded symplectic manifold; under usual regularity assumptions, its cohomology recovers the Courant–Dorfman algebra of the reduced Courant algebroid.

The automorphism groups of smooth cubic fourfolds over the complex numbers, together with actions on defining equations, are classified in previous works. In this paper, we complete the calculation of group actions on the primitive integral cohomology lattice for all of the $156$ families in the classification. We distinguish similar families, determine the remaining action-maximal cases, and use restriction to complete the list. We also determine the generic algebraic lattices for each family. The calculation depends mainly on OSCAR.

We study the spaces of bounded multilinear operators $\mathcal L^i(X,Y)$ endowed with $L^p_\mu$-norms, where $X,Y$ are normed spaces and $\mu$ is a measure with $p$-moments. These are norms without taking a quotient, under suitable support conditions. The topology induced coincides with the strong operator and compact-open topologies on operator-norm-bounded subsets. Furthermore, we identify the completion of these spaces with spaces of Hilbert–Schmidt operators on the reproducing kernel Hilbert space (RKHS) for $p=2$; with iterated $\gamma$-radonifying multilinear operators on the RKHS in the Gaussian case, for general $p$.

For $1\le p < \infty$, $\varphi_p$ is the function of minimal $L^p$ norm among all functions $f$ of exponential type at most $\pi$ for which $f(0)=1$. We start from the ansatz that the positive zeros $\tau_n$ of $\varphi_p$ are close to \(n+1/p-1/2\) and use a weighted convex correction to first place them in an \(\ell^2\)-neighborhood of this sequence. A bootstrap argument then improves this to a {Poincaré asymptotic} expansion of $\tau_n-(n+\frac1p-\frac12)$ in terms of odd powers of $(n+\frac1p-\frac12)^{-1}$. We also get precise information about the Fourier transform $\widehat{\varphi}_p$, a result that in particular implies that for all integers $n\ge 2$, $\widehat{\varphi}_{2n}$ is an integrable function which, when convolved with itself $2n-2$ times, is constant on the interval $[-\pi,\pi]$. In particular, we identify the extremizer in an autoconvolution problem posed by B. Green in 2001.

The Chan–Robbins–Yuen polytope $\mathrm{CRY}_{m+1}$ has normalized volume $\prod_{j=1}^{m-1}C_j$, a product of Catalan numbers. Zeilberger proved the formula using the Morris constant-term identity; finding a combinatorial proof remained an open problem. We give such a proof by counting nonnegative integer flows. We add an extra edge from each vertex to a common sink and count flows according to how many of these edges carry positive flow. A double count proves a special case of a recurrence of Morales and Shi and gives these counts in terms of Narayana numbers. Deleting the first vertex then yields the Catalan product. We also give an explicit bijection for the recurrence and prove the corresponding reduction formula posed by Postnikov.

We carefully construct the profinite iterated monodromy group of a rational map as a quotient of the étale fundamental group of the punctured projective line. In doing so, we prove a number of folklore results regarding profinite IMGs; in particular, their topological generation by inertia concentrated in the post-critical orbit, and their relationship with the discrete iterated monodromy group over $\mathbb{C}$.

We prove the uniform convergence of Plateau borders to Plateau singularities in the vanishing-volume limit of a soap film model based on capillarity theory. The results are formulated in the context a new type of local area minimizer that takes into account how a soap film locally separates its surrounding space.

Let $\pi$ be a genuine irreducible cuspidal automorphic representation of the $n$-fold cover $SL_2^{(n)}(\mathbb{A})$ of $SL_2(\mathbb{A})$, $n \geq 3$ odd, and let $\tau$ be the genuine representation of $GL_1^{(n)}(\mathbb{A})$ obtained from a unitary Hecke character of $\mathbb{A}^\times$. We prove that, for a certain choice of data, a new way integral for $SL_2^{(n)}(\mathbb{A})$ represents the partial $L$-function $L^S(ns+\tfrac12,\pi\times\tau)$. The integral is derived from the generalized doubling integral of Cai, Friedberg, Ginzburg and Kaplan, following the idea of Ginzburg and Soudry.

A $u\hbox{-}v$ separator in $G$ is a set $S\subseteq V(G)\setminus\{u,v\}$ such that there is no $u\hbox{-}v$ path in $G\setminus S$. A minimal separator in $G$ is a set $S$ such that for some $u,v\in V(G)\setminus S$, $S$ is a $u\hbox{-}v$ separator in $G$ but no proper subset of $S$ is. A set $S$ is bisimplicial if it is the union of at most two cliques. Recognizing graphs in which every minimal separator is the union of at most $k$ cliques is polynomial-time solvable for $k=1$ and NP-hard for every fixed $k\geq 3$. We settle the remaining case $k=2$ by giving a polynomial-time algorithm to decide whether every minimal separator in a graph is bisimplicial. A set $S$ is triad-free if its independence number $\alpha(S)$ is at most two. Given a graph $G$ and two vertices $u,v$, the problem of finding a $u\hbox{-}v$ separator $S$ with $\alpha(S)\leq k$ had the same complexity gap: it was known to be polynomial-time solvable for $k=1$ and NP-hard for every fixed $k\geq 3$. We resolve the remaining case $k=2$ by giving a polynomial-time algorithm that finds a triad-free $u\hbox{-}v$ separator or determines that none exists.

We propose a method for solving partial differential equations on complex domains by learning smooth, invertible mappings from a fixed reference domain using Neural Spline Flows (NSFs). Monotone rational-quadratic spline coupling layers ensure a strictly positive Jacobian determinant by construction, eliminating the need for Jacobian-sign penalties. A diffeomorphic pullback transfers the weak form of the PDE to the reference domain, enabling finite element analysis on reusable structured meshes. To generalize across shapes, a shared NSF is conditioned on low-dimensional latent codes learned through an auto-decoder framework; a subsequent encoder predicts these codes for unseen geometries in a single forward pass. For a given learned map, the pullback preserves the mathematical structure of the PDE and is exactly equivalent to solving on the domain represented by that map. Because the learned domain generally approximates the target geometry, geometric mapping error remains distinct from finite element discretization and quadrature errors. Numerical benchmarks show that sufficiently accurate learned mappings yield reliable finite element solutions. By avoiding repeated generation of unstructured meshes, the method addresses a bottleneck in simulation workflows. The fixed reference mesh also creates opportunities for efficient assembly, memory-aligned data access, and precomputed quadrature. The resulting framework assigns geometric complexity to the neural network while retaining conventional finite element discretization and convergence properties on the learned domain, with overall accuracy also governed by geometric fidelity.

Assuming $\mathrm{AD}^+ + V = L(P(\mathbb{R}))$, we show that the Ultrapower Axiom holds in $\mathrm{HOD}_s$ for every $\omega$-sequence of ordinals $s$. We use this result to analyze the structure of Nairian models, showing that they contain many measurable cardinals. As an application, we establish the consistency of the existence of infinitely many consecutive measurable cardinals with ZF, starting with a Woodin limit of Woodin cardinals. This solves a question of Henle, Mathias, and Woodin. From the same large cardinal hypothesis, adapting a forcing technique of Adkisson and Ben-Neria, we produce a model of ZF in which every uncountable cardinal is singular. Gitik's original construction requires a proper class of strongly compact cardinals.

In this paper we prove the sharp rate of convergence for stationary states of the porous medium equation with an external drift in the p-Wasserstein metric under very mild assumptions on the potential and for any $1\leq p\leq +\infty$.

We study functional Blaschke–Santaló inequalities in which the inner product is replaced by a cost. An inequality on each orthant, combined with Steiner symmetrization, gives a sufficient condition for a cost inequality to hold for even convex functions. We apply this principle to a multiplicative cost defined on all orthants. Its second variation gives a weighted improvement of the Brascamp–Lieb inequality in the directions for which differentiation is justified. We also prove an $L^p$ version of the Fradelizi–Meyer theorem from a weighted volume-product inequality and derive the corresponding radial cost and Brascamp–Lieb inequalities. Explicit examples include anisotropic power and Cauchy densities, with dilation equality cases and a sharp obstruction below the coordinate exponent two. For the spherical logarithmic cost we compute an exact entropy–logdet deficit decomposition and the second-order deficit transverse to ellipsoids. We also explain the transport-entropy formulation and distinguish the restricted spectral question from the scalar $(B)$ property.

We investigate a discrete analogue of the Fisher information. By using a discrete Laplacian operator, we can construct a discrete analogue of the score function and hence of the information. This provides a natural framework for establishing several information-theoretic inequalities in the discrete setting.