Papers

In this paper, we study the dimension of the correlated equilibrium polytope of finite games. Under the oriented-matroid notion of genericity, we prove that if a generic game is not full-dimensional, then there exists a subgame whose correlated equilibrium polytope is affinely isomorphic to that of the original game. This settles and generalizes an earlier conjecture of Brandenburg, Hollering, and Portakal (2024). Moreover, we show that the existence of a correlated equilibrium whose slices are all non-zero implies that the correlated equilibrium polytope is either full-dimensional or a singleton.

We study the prime geodesic theorem for congruence subgroups of indefinite quaternion orders. We prove that the geodesic analogue of the Chebotarev density theorem for these groups holds with exponent $25/36 + \varepsilon$. In particular, we deduce that the prime geodesic theorem holds with exponent $25/36 + \varepsilon$ for any congruence subgroup of any indefinite quaternion order over $\mathbb{Q}$. The idea of the proof is to reduce the problem to the split case, which has been handled previously by the author.

Extracting total correlations from a quantum system usually requires reconstructing its state, whereas many experiments access only a few measurement settings. A possible shortcut is to add the mutual informations obtained from complementary measurements; in dimensions above two, however, this procedure can count the same classical correlation twice. We establish that qubits are protected from such overcounting. For every two-qubit state, the correlations observed in two complementary local bases are bounded by the premeasurement quantum mutual information. The proof traces this protection to binary-entropy curvature on the Bloch ball and combines a qubit information-exclusion tradeoff with data processing under local dephasing. Consequently, two correlation tables give a tomography-free lower bound on total correlation. A score above one bit also certifies a quantitative one-way entanglement-distillation rate; when applied to the Choi state of a qubit channel, the same data lower bound its quantum capacity. The theorem therefore identifies both an operational use of complementarity and the trusted two-dimensional setting in which its correlation accounting is valid.

We provide a new approach to the classification, up to equivalence, of abelian complex structures on nilpotent Lie algebras. As an application, we recover the already known classifications in dimensions 4 and 6 and provide a classification in dimension 8 for 1-abelian complex structures. Some results about the general case are given.

Network experiments are used throughout the social and medical sciences to investigate causal effects under the presence of interference. While a large body of work has developed improved statistical procedures, the fundamental limits of statistical estimation in these settings is less well understood. In this paper, we develop and investigate a design-based theory of minimax risk for network experiments under an arbitrary neighborhood interference model. Our notion of minimax risk describes the optimal precision among all statistical procedures for investigating a particular causal effect on the observed interference network. We show that the minimax risk is a function of the corresponding conflict graph, which captures inherent unobservability of estimand-relevant potential outcomes given the observed interference network. Our main contribution is a series of upper and lower bounds on the minimax rate in terms of local and global connectivity properties of the conflict graph. To illustrate their utility, we apply these general results to obtain minimax analyses for two commonly studied effects: the direct treatment effect and global average treatment effect.

We study local minimizers of a sextic-potential Landau–de Gennes energy for nematic liquid crystals in the small-elastic-constant limit. Under the uniform energy and $L^\infty$ bounds, these minimizers converge to a locally energy-minimizing harmonic map $\mathbf{Q}_0$ into a biaxial vacuum manifold. The main result of this paper is that such a limiting map $\mathbf{Q}_0$ has no interior point singularities. The proof relies on a geometric identification of the lifted Frobenius metric on the universal cover $\mathbb{S}^3$ with a rescaled Berger metric. For a hypothetical tangent cone at a point singularity, its link is a nonconstant harmonic two-sphere into the Berger sphere. We construct a smooth variation field adapted to the Hopf direction and show that it induces a quantitative instability estimate, in contradiction with the shifted stability inequality inherited from local minimality. This replaces the usual round-sphere test fields by a target-specific construction and rules out interior point defects in the biaxial setting. The result is sharp in view of the well-known existence of point defects in the uniaxial theory.

Learned spectral models fail at symmetry forced degenerate sectors for two distinct reasons. Where symmetry forces levels to coincide exactly, the per level observable one would normally fit is not well defined, since every unit vector of that shared space is an eigenvector; and near a symmetry protected crossing, the eigenvector observable gradient carries a factor 1/(lambda_i - lambda_j) that is genuinely singular as the gap closes. The usual response is to regularize the divergence or threshold the gap, and both carry a real cost: a fixed gap cannot both protect a forced multiplet and keep two genuinely distinct levels apart. We show a different fix, on synthetic operator families with a known symmetry answer key. A gate reads the symmetry structure directly from the observed operators, as the linear commutant of the family, one singular value decomposition nullspace, following the simultaneous block diagonalization of Maehara and Murota. Block identity is read from the centre of that commutant rather than from eigenvalue clustering, which makes the forced versus accidental distinction structural rather than metric, and the objective switches between a projector trace through forced blocks and a per level target elsewhere. Under operator estimation noise the gate classifies correctly to epsilon of about 0.3, where energy clustering already fails by 0.02. Gated fitting reaches the truth at machine precision in both the symmetric and the symmetry breaking regime, in the latter converging to a truth that is a singularity of the ungated objective, and observable bias sits at the noise floor. Validation off the regular representation, where multiplicity and dimension separate, eliminated two defective estimators that all regular representation tests had passed. The demonstrated object is a gated estimator; a full parametric matrix model, with the matrices learned, is the next experiment.

We prove a Hoeffding-type moment generating function bound for weighted sums of bounded exchangeable random variables centered by their finite-population average. The bound reduces the excess inflation above one in a recent weighted exchangeable Hoeffding inequality from order $(\log N)/N$ to the rate-optimal order $1/N$, with an explicit constant. The proof reduces the problem to Hamming slices, identifies two-level extremizers for the relevant symmetric variational problem, and applies a hypergeometric martingale bound. We also give a lower bound showing that an excess inflation of order $1/N$ is unavoidable.

We develop a comprehensive theory of dualizable additive categories. We provide several equivalent characterizations, notably identifying them as separated Grothendieck prestable categories satisfying the $\mathrm{AB4}^*$ and $\mathrm{AB6}$ axioms. We establish a connection to almost mathematics by demonstrating that they arise precisely as the categories of connective almost modules over connective $\mathbb{E}_1$-rings. Furthermore, we prove that dualizable additive categories are generated by flat objects, and that the passage to flat objects yields an equivalence between dualizable additive categories and compactly assembled additive categories. As a primary application within analytic geometry, we characterize the category $\mathrm{Nuc}(R)_{\geq 0}$ of connective nuclear modules (in the sense of Clausen–Scholze) over an adic $\mathbb{E}_\infty$-ring $R$ via a universal property, identifying it as the additive rigidification of the category of connective complete $R$-modules. Finally, we construct the universal finitary stable localizing invariant for dualizable additive categories, the presentable stable category $\mathcal{M}\mathrm{ot}_{\mathrm{pst}}$ of prestable motives, and demonstrate that its unit corepresents nonconnective algebraic $K$-theory. We prove that the motives of small additive categories and those of dualizable additive categories generate the same presentable stable subcategory.

Let $K$ be a complete nonarchimedean valued field with $v(K^\times)=\mathbf R$, and let $V=\mathcal O_K$. We prove that $K$ is spherically complete if and only if $V[[T]]$ is coherent, and that this is also equivalent to $V[[T]]$ being a GCD domain. If $K$ is perfect of characteristic $p$, the same characterization holds for the Witt vector ring $W(V)$. Thus, this settles the previously unresolved full-real-value-group case in the coherence problems for both formal power series and Witt vector rings. In particular, this result also gives affirmative answers to Questions~9 and~10 of Anderson–Kang–Park. The proof combines a coherence criterion for complete rings with a spherically complete valuation quotient and a uniform construction of non-finitely generated intersections of two principal ideals from an empty ball chain.

Minimal linear codes have significant applications in secret sharing schemes, secure multi-party computation, and cryptography. In this paper, we propose a generic construction of a new family of minimal binary linear codes with dimension n+4 from a special class of Boolean functions. By leveraging the geometric properties of partial spreads in finite fields, we determine the explicit weight distribution and weight enumerator of the constructed codes. Furthermore, we derive a necessary and sufficient condition for these codes to be minimal, and establish that the proposed family yields minimal codes that structurally violate the well-known Ashikhmin-Barg condition, making them highly desirable for advanced communication systems.

We develop accelerated algorithms for convex $(H_0,H_1)$-smooth optimization, where $\|\nabla^2 f(x)\|\le H_0+H_1(f(x)-f^*)$. This class generalizes standard smoothness and contains the $(L_0,L_1)$-smooth class. Combining a Nesterov-type accelerated gradient scheme with small-dimensional relaxation and phase restarts, we obtain a full-gradient method with iteration complexity $\widetilde O(\sqrt{H_0\widetilde R^2/\varepsilon}+\sqrt{H_1\widetilde R^2}\log(F_0/\varepsilon))$. We extend the same approach to randomized coordinate optimization, obtaining a coordinate method with uniform sampling whose iteration complexity carries the standard factor $d$, and a coordinate method with non-uniform sampling whose iteration complexity is governed by $S_{1/2}^{(j)}=\sum_i\sqrt{H_{j,i}}$. These results provide, to our knowledge, the first accelerated full-gradient and coordinate guarantees for this convex class. We also provide practical implementation recommendations. Experiments confirm the predicted acceleration, gains from non-uniform sampling, and the viability of inexact relaxation.

We introduce a variational approach to a finite-temperature continuous-spin perceptron trained on a Gaussian mixture. The model allows for a broad class of concave utilities and log-concave separable prior measures on the spins. By combining the interpolation method with log-concavity and concentration estimates, we derive lower and upper minimax variational bounds for the limiting quenched pressure. Remarkably, the two bounds differ only in the order of optimization of two variational parameters, while all remaining extrema are controlled by the concave–convex structure of the variational potential. Whenever the two optimizations commute, the two bounds match and identify the solution of the model. The same potential yields the fixed-point equations as stationarity conditions and provides a unified route to the computation of the ground-state energy, training loss, and generalization error.

We characterize the analytic self-maps of the unit disk inducing isometric composition operators on \(\BMOA\) with respect to the Möbius-invariant \(H^p\) norm for \(p\ge1\) and the corresponding quasi-norm for \(0<p<1\). In particular, this gives a complete answer to Laitila's question on the characterization of isometric composition operators for the Möbius-invariant \(H^p\) norms, while also covering the quasi-Banach range. As a consequence, the isometric property of \(C_\varphi\) is independent of the exponent \(p\in(0,\infty)\).

We prove a quantitative version of the convergence case of Khintchine's celebrated theorem in metric Diophantine approximation, but where the approximated points are restricted to lying on the parabola. A novel feature of our result is that unlike other results in literature, the approximating function is no longer required to be monotonic. This requires us to obtain explicit constants in classical number theoretic results, most notably in Burgess' bound for character sums in short intervals.

We study the effective, connective, and very effective filtrations in the $\mathbb{C}$-motivic, $2$-complete, cellular, stable homotopy category. We do so by using the filtered spectrum model for this category due to Gheorghe-Isaksen-Krause-Ricka, and in particular the motivic analogue functor $\Gamma_\star$. Then we can express the covers making up the respective filtrations of a nice motivic analogue $\Gamma_\star(X)$ via filtered spectra, and use these to compute the slices. Applying this in the case of $X$ being the sphere spectrum, $\text{MU}$, $\text{ku}$, or an Eilenberg-MacLane spectrum recovers a number of conjectures due to Voevodsky. Applying it to $\text{ko}$ recovers a computation of Ananyevskiy-Röndigs-Østvær. We can also apply it to $\text{tmf}$ and compute the effective slices of the motivic modular forms spectrum $\text{mmf}$. We also study the effective slice spectral sequence for $\Gamma_\star(X)$, which turns out to contain the same information as the classical Adams-Novikov spectral sequence for $X$.

This paper provides a deeper analysis of Rump's correspondence between near-rings and skew braces. We show that, under a topological assumption on the structure of a near-ring, Rump's correspondence is invertible and it is always possible to construct an associated skew brace. Furthermore, we provide several examples where this topological assumption is fulfilled in well-known algebraic structures such as the Nottingham group, groups of triangular functions, iterated wreath products of arbitrary groups, and groups of IA-automorphisms of free nilpotent groups.

Let $D$ be a negative discriminant, and let $K=\mathbb{Q}(\sqrt{D})$. Let $\mathcal{Q}(D)$ denote the set of primitive positive definite binary quadratic forms over $\mathbb{Z}$ of discriminant $D$. We introduce the set of adelic framed forms \begin{equation*} \widehat{\mathcal{Q}}(D)= \left\{(Q,\,\gamma)\in \mathcal{Q}(D)\times\mathrm{SL}_2(\widehat{\mathbb{Z}})~|~ Q\left(\gamma\begin{bmatrix}1\\0\end{bmatrix}\right)\in \widehat{\mathbb{Z}}^\times\right\} \end{equation*} and its orbit space $\widehat{C}(D)$ under the natural action of $\mathrm{SL}_2(\mathbb{Z})$. We define an explicit adelic analogue of the Gauss-Dirichlet composition law on $\widehat{C}(D)$ and endow $\widehat{C}(D)$ with the quotient topology induced by the subspace topology on $\widehat{\mathcal{Q}}(D)$ inherited from the product topology on $\mathcal{Q}(D)\times\mathrm{SL}_2(\widehat{\mathbb{Z}})$, where $\mathcal{Q}(D)$ is discrete and $\mathrm{SL}_2(\widehat{\mathbb{Z}})$ has its profinite topology. We then prove that there is an isomorphism of topological groups \begin{equation*} \widehat{C}(D)\simeq\mathrm{Gal}\left(K^\mathrm{ab}(\mathfrak{t}^{1/\infty})/K(\mathfrak{t})\right), \end{equation*} where the Galois group is endowed with the Krull topology, $\mathfrak{t}$ is a positive transcendental real number, and $\mathfrak{t}^{1/\infty}=\{\sqrt[N]{\mathfrak{t}}~|~N\geq1\}$. Moreover, we identify an explicitly defined subgroup of $\widehat{C}(D)$ with $\mathrm{Gal}(K^\mathrm{ab}/K)$ and describe the corresponding Galois action on special values of modular functions. In this way, classical Gauss composition, finite-level form class groups, and Shimura reciprocity are brought together within a single adelic framework. Finally, we show that the abstract group structure of $\widehat{C}(D)$ uniquely determines the imaginary quadratic field $K$.

The classical normally hyperbolic invariant manifold theorem asserts that a \(C^1\) normally hyperbolic invariant manifold persists under sufficiently small \(C^1\) perturbations. For a family of standard-like dissipative twist maps with Jacobian determinant \(\lambda\in(0,1)\), we prove that \(C^1\) perturbations of Lipschitz seminorm at most \((1-\sqrt{\lambda})^2\) preserve a unique \(C^1\) invariant graph, including at equality. Under the strict bound, this graph is normally hyperbolic. Conversely, for every prescribed \(a^*>(1-\sqrt{\lambda})^2\) and every rational frequency, we construct a \(C^\infty\) perturbation of \(C^1\)-norm exactly \(a^*\) whose unique invariant graph is Lipschitz but not differentiable everywhere. We also give an analytic perturbation at the critical norm whose invariant graph is \(C^1\) but not normally hyperbolic. Thus the closed bound for \(C^1\) regularity and the strict bound for normal hyperbolicity are both sharp.

This paper develops procedures for nonparametric goodness-of-fit testing under covariate shift, where labelled data are drawn from a source population but goodness-of-fit is evaluated for a target population. The distribution mismatch is quantified by either a bounded moment condition or a sub-exponential tail condition on the target-to-source density ratio. Our method combines truncated importance-weighting kernel ridge regression with a multiplier bootstrap to construct confidence sets for the regression function. The truncation stabilizes the importance- weighting kernel ridge regression as well as the bootstrap calibration, making our approach applicable even when the density ratio has heavy tails. We prove nonasymptotic validity and sharpness of the resulting confidence sets under suitable operator compatibility conditions, and establish explicit error rates for coverage probability under specific conditions on the target- to-source density ratio and on the spectral decay of the kernel integral operator. Numerical experiments corroborate our theoretical findings.

Extreme flows realize the largest possible growth, either instantaneously or in finite time, of certain quantities of interest which is achieved by a suitable choice of the initial condition or the applied forcing. The quantities of interest usually measure some small-scale properties and therefore provide information about the regularity of the flow. Extreme behavior is at the heart of several open problems in fluid mechanics including the dissipation anomaly in turbulence and formation of singularities in various models of fluid flow. In this essay we describe a framework making it possible to study such extreme behavior systematically by combining mathematical analysis, scientific computation and physics. As a first step, one aims to deduce rigorous upper bounds on the growth of the quantities of interest in the solutions of a given model. These inequalities express fundamental limitations on the most extreme behavior possible among all admissible solutions. However, given how they are obtained, these bounds may be conservative and overestimate the growth actually realizable in the system. In order to probe this possibility, as the next step, we set up variational optimization problems where the growth of the quantity of interest is maximized under suitable constraints. Solution of such problems is enabled by modern methods of numerical optimization. When properties of the thus obtained maximizers match the bounds, the bounds are declared sharp and therefore cannot be fundamentally improved. Finally, properties of the solutions saturating the bounds reveal insights about the physical mechanisms realizing the extreme behavior. We survey problems where this research program has produced sharp bounds together with extreme flows saturating these bounds. A collection of open problems is then presented and we close the essay with a discussion of possible methodological improvements.

Grayson and Gage proved that an immortal curve shortening flow of simple closed curves on a closed surface converges subsequentially to a closed geodesic, and they asked whether this limiting geodesic is unique. We answer this question negatively by constructing a smooth Riemannian metric on $\mathbb{S}^2$ and an immortal simple closed curve shortening flow that converges along different sequences of times to every geodesic in a one-parameter family of distinct simple closed geodesics.

We study restrictions of stationary kernels on locally compact abelian groups $G$ to closed subgroups $H$. For a nonnegative spectral density $\hat k$, we derive an explicit fibrewise Fourier representation of the minimal-norm extension operator from the reproducing kernel Hilbert space of the restricted kernel on $H$ to the original space on $G$. We characterize when the canonical Fourier formula extends boundedly from $L^2(H)$ to $L^2(G)$, identify its exact operator norm and lower norm, and obtain bounds on the associated interpolation spaces. When $G$ is compact, the extension is a contraction and, for stationary Gaussian random variables admitting a measurable continuous version, maps the observed restriction to the conditional expectation. We also give a counterexample to a previously asserted supremum-norm contraction and illustrate the theory through cardinal interpolation and conditioning on one-dimensional subgroups of the torus.

We prove a number of Jacobson/solvable/nilpotent-radical (co)invariance results for finite-dimensional associative or Lie (co)module-algebras over Hopf algebras $H$, provided the characteristic, if positive, is large relative to the dimension of the algebra and the (co)actions satisfy trace-preservation conditions automatic when $H$ is involutory. This generalizes a number of such radical-invariance criteria in the literature, due to A. Gordienko, V. Linchenko, Pagon-Repovš-Zaicev and others, providing a uniform enriched-categorical framework for those results.

Although Arnoldi reduction of a generally non-Hermitian Hamiltonian yields an upper Hessenberg matrix rather than the tridiagonal form of Hermitian Lanczos theory, we show that a closed Toda sector survives in its diagonal and subdiagonal coefficients. For a fixed finite-dimensional Hamiltonian and a cyclic state vector deformed holomorphically, the Krylov Gram determinants are $\tau$ functions of the finite two-dimensional Toda lattice, whose Flaschka variables coincide exactly with these Arnoldi coefficients. The Toda dynamics therefore closes on this sector without determining the remaining upper Hessenberg entries. The subdiagonal part of the same sector also has a direct geometric meaning: the squared subdiagonal coefficients determine both the Fubini–Study metric and the Berry curvature of holomorphic Krylov subspaces, whereas the geometric quantities associated with subspaces lost at Arnoldi breakdown cease to be defined. Along a smooth real path in the cyclic region, the Arnoldi-frame connection further provides a Hermitian tridiagonal generator of exact isospectral transport. When added to the Arnoldi matrix, this generator cancels transitions between instantaneous eigenspaces and realizes counterdiabatic driving whenever the matrix is diagonalizable with a nondegenerate spectrum.

We present a new proof of the theorem of Dickson that gives generators for the groups $SL_{2}(q)$ where $q$ is an odd prime power.

In optimization, one often encounters problems of the form $\min_x F(x)+E(x)+G(Kx)$. In this work, we combine primal-dual algorithms with multigrid techniques for their solution. To link the the fine-grid and coarse-grid problems problems, we introduce a nonsmooth primal-dual coherence condition, and an efficient partially linearized line search procedure. Our work is motivated by total variation regularized inverse imaging problems, on which we demonstrate the efficacy of the method, being able to solve problems not previously possible with forward-backward multigrid methods.

The main objective of this work is to investigate the traveling wave solution and dynamic characteristics of the (2 + 1)-dimensional seventh-order Caudrey-Dodd-Gibbon-KP (sCDG-KP) equation. Applying the ($\frac{G^{\prime}}{G^{\prime}+G+A}$) method, we examine the exact solution of the (2 + 1)-dimensional seventh-order Caudrey-Dodd-Gibbon-KP (sCDG-KP) equation by altering it into a reduced ODE via a suitable wave transformation. Graphical representations, such as 2D, 3D, and a heat map of the ascertained solution, are present to facilitate comprehension of the empirical relevance of the obtained solutions. As a result, we acquired a bright and anti-kink soliton solution. Next, we alter the ODE into a 2D system of equations to analyze the dynamical behavior of the reduced system via bifurcation analysis, phase portrait, and attractor analysis. During this process, we portray the graphical visualization of the bifurcation phase portrait, 2D phase portrait, 3D phase portrait, time series, chaotic attractor, sensitive analysis, fractal dimension, recurrence plot, and power spectrum of the dynamical system.

Let $H$ be a fixed graph with $e(H)\ge3$ that contains two adjacent edges, and let $\ell\ge e(H)$ be fixed. We establish an entropy-transference principle for rainbow-$H$-free edge-colourings of the binomial random graph at the natural scale $p=n^{-1/m_2(H)}$. Writing $R_{H,\ell}(G)$ for the number of such colourings and $\lambda(H,\ell)$ for the rainbow entropy–Turán density on complete graphs, we show that, with high probability, the per-edge logarithmic counting rate can be made arbitrarily close to $\log\ell$ below a sufficiently small constant multiple of this scale, and arbitrarily close to $\lambda(H,\ell)$ above a sufficiently large constant multiple. Thus the dense-side counting rate on a sparse random host is governed exactly by a deterministic entropy–Turán parameter on complete graphs. We further investigate this parameter, obtaining partial exact evaluations, corresponding counting-stability results, and its first-order asymptotic behaviour as the number of colours tends to infinity. This extends the random Gallai-colouring transition from triangles to every fixed non-matching graph containing at least three edges, and provides a general mechanism for transferring complete-graph template entropy to sparse random hosts.

We generate infinite binary exchangeable sequences by sequential comparison of data points against a latent benchmark. Assuming a prior distribution of the benchmark rank \(R_0\) within an unobserved group of players, we set up the Bayesian machinery which determines the posterior distribution of the running rank \(R_n\) in purely combinatorial terms, by counting insertion slots in a uniformly random ranking. This yields an explicit predictive probability of winning against the benchmark, which is a rational function of the counts. We identify some cases where the predictive probability has particularly simple form, notably a tournament related to the Topp–Leone distribution. The running benchmark rank is a Pólya urn with random initial composition, and its normalised value converges to a latent strength variable with polynomial density. In a projective time scale, the fluctuations of the normalised rank about this limit are Brownian, with the Wright–Fisher coefficient frozen at the limit, while the associated recurrent up-down Markov chain on a high level $n$ of the Pascal graph is approximated by a Wright–Fisher diffusion with a nonlinear drift expressed through the prior density.