Papers

We study the Iwasawa cohomology complex attached to a pseudorigid family of $(\varphi, \Gamma)$-modules and prove coadmissibility of its cohomology groups, perfectness and compatibility with base change. We apply this result to give another proof of perfectness of the Herr complex in pseudorigid family. We also use it to prove perfectness of Selmer complexes and obtain a result on variation of ranks of Selmer groups in analytic family.

We establish the first partial regularity results for minimizers of a class of suitably quasiconvex variational integrals defined on anisotropic Sobolev spaces. These variational integrals are related to higher gradient continuum models for pantographic metamaterials.

Let $F/\mathbf Q_p$ and $E/\mathbf Q_p$ be finite extensions, and let $G=\mathrm{GL}_2(F)$ or $D_F^\times$, where $D_F$ is the quaternion division algebra over $F$. Let $\Pi$ be an admissible Banach $E$-representation of $G$ whose locally $\mathbf Q_p$-analytic vectors $\Pi^{\mathrm{an}}$ have an infinitesimal character. If $p>2$ or $F=\mathbf Q_p$, we prove that $\Pi^{\mathrm{an}}$ and $\Pi$ have finite length in their respective admissible categories.

For a smooth locally reductive stack admitting an adequate moduli space, we show that there are finitely many open substacks preserving the adequate moduli space, and give a complete classification of such opens. For $\mathbb{G}_m$-actions, this becomes an explicit description in terms of the codimension-one Bialynicki-Birula strata. Moreover, for a stable adequate moduli space, we show that the number of preserving opens are bounded by 3 raised to the relative Picard rank. As an application, the number of fixed components for $\mathbb{G}_m$-actions is at least the relative Picard rank.

Motivated by recent work on $\lambda$-fold near-factorizations by Kreher, Li, and Stinson (Designs, Codes and Cryptography, 2026), we introduce and systematically investigate $(\lambda,\mu)$-fold factorizations as combinatorial structures with prescribed representation multiplicities. Such a factorization consists of two subsets of a finite group whose products represent each nonidentity element exactly $\lambda$ times and the identity exactly $\mu$ times. These structures generalize $\lambda$-fold near-factorizations and provide a common framework for several families arising in combinatorial design theory, including strong external difference families with two equal-sized subsets, as well as binary sequence pairs with two-level correlation. Their associated $\{0,1\}$-matrices satisfy the Bridges-Ryser equation $XY=\lambda(J-I)+\mu I$, linking the factorization problem to a classical matrix framework for combinatorial designs. We establish the foundational theory of $(\lambda,\mu)$-fold factorizations, provide two general constructions, and prove several nonexistence results for finite abelian groups.

Sequential statistical procedures are increasingly embedded in scientific systems in which observations arrive incrementally from heterogeneous sources. In such systems the statistical update is only one part of the computation: observations must also be acquired, aligned across inputs, ordered, delivered, persisted, and replayed, and results must be passed to systems that display them or act on them. Sequential methods are nevertheless missing from the general software toolkit. They run inside proprietary experimentation platforms and commercial statistical packages, or exist as offline functions over complete data, and the batch fit interface that dominates machine-learning libraries has no place for a procedure that listens, keeps state, and decides when to stop. We propose an architecture that separates statistical inference from data topology, execution runtime, run lifecycle, persistence, and external action. Its central abstraction is the Sequential Procedure: a state-transition function that maps an inferential state, a valid statistical input, and a statistical random state to a new state and a scientific output. The procedure does not define how observations are acquired, paired, scheduled, stored, or acted upon. We specify the architecture semantically rather than through an object model, derive seven conformance invariants that any implementation can be tested against (among them synchronous/asynchronous, checkpoint, replay, and batch equivalence), and describe a dependency-free reference implementation in which all seven hold bitwise. Lorden's 2-SPRT for the Kiefer-Weiss problem serves as the reference procedure; the same contract accommodates sequential estimators, Bayesian filters, change-point detectors, and stateful predictive models.

In this paper, we discuss equivariant formality for spaces equipped with a primary cyclic group action. We study the behavior of equivariant formality with some topological constructions. The algebraic complexity needs to deal with injectivity. In the end, we discuss equivariant intrinsic formality.

This paper proves the weak type (1,1) boundedness for maximal truncated singular integral operators with rough non-convolution kernels of the form $\Omega(x-y) K(x,y)$, where $\Omega$ is homogeneous of degree zero with $\Omega \in L\log L(\sS^{d-1})$ and $K$ is a standard Calderón-Zygmund kernel. This extends the recent resolution of the classical convolution case by Lai \cite{Lai2025weak} to a wider class of operators.

This article takes steps to show that an equivariant version of the well known phenomena of Koszul Duality can be formulated in a manner not considered prior: via dg (co)algebra valued functors on the orbit category. As such techniques used in this article mix traditional and modern categorical methods in homotopy theory.

Recent work shows that ordinary Reed–Solomon codes over prime fields admit polynomial-size lists up to list-decoding capacity on every evaluation set. We show a contrasting phenomenon over binary extension fields. For every fixed rate $0<R<1$, there exists $\varepsilon_R>0$ such that, for all sufficiently large $h$, the full-field Reed–Solomon code over $\mathbb{F}_{2^h}$ has a received word with $n^{\Omega_R(\log n)}$ codewords within relative distance $1-R-\varepsilon_R$. To our knowledge, these are the first superpolynomial ordinary Reed–Solomon list-size lower bounds at fixed positive rate and a fixed positive agreement gap above the rate. Thus capacity-approaching polynomial list size for Reed–Solomon codes is not uniform over the underlying field, and the obstruction occurs at every fixed rate. As a puncturing consequence, the same holds for primitive Reed–Solomon codes; the obstruction is also robust under controlled dense puncturing. The proof begins with an exact correspondence that turns certain binary Bose–Chaudhuri–Hocquenghem weight-enumerator coefficients into sizes of single Reed–Solomon lists. A classical binary weight family gives the initial quarter-rate construction, while a quadratic-form enumeration supplies its superpolynomial multiplicity. A common-kernel quotient/descent argument and common-root padding then extend the construction to all rates. We also determine the optimized agreement envelope achieved by the construction.

Bugeaud, Haynes and Velani (2011) resolved a metric version of the $p$-adic Littlewood conjecture. We establish a fully inhomogeneous generalisation of their result, as well as a partially inhomogeneous generalisation of a result of Harrap and Haynes (2013) in the more general mixed setting.

We develop an $N$-adic modification of the construction of David-Jeznach-Julia for compact subsets of the real line $\mathbb{R}\subset\mathbb{C}$ for which harmonic measure for the complementary domains has no dimension drop. In particular, we show that for every $s\in(0,1/2)$, there exists an $s$-Ahlfors regular compact set $K\subset\mathbb{R}\subset\mathbb{C}$ such that $\mathrm{dim}\omega^\infty=s$, where $\omega^\infty$ denotes harmonic measure for $\mathbb{C}\setminus K$ with pole at infinity. Combined with a result of the third author, this shows that the threshold $1/2$ is sharp.

Let $X$ be a smooth compact irreducible hyperkähler manifold of dimension six. We prove that the second Betti number satisfies $b_2(X) \le 23$. Furthermore, we show that the third cohomology vanishes whenever $b_2(X) \ge 19$, and that all odd cohomology vanishes when $b_2(X) = 23$. If the second Chern class lies in the subalgebra generated by $H^2(X,\mathbb{Q})$, then $b_2(X) \in \{5,8,23\}$. The proof uses the Looijenga–Lunts–Verbitsky decomposition, curvature inequalities for Segre classes, and the Hodge–Riemann bilinear relations on the projective cotangent bundle.

We investigate the problem of uncertainty quantification in the mean estimation framework. Given an i.i.d. sample from a distribution with unknown mean $\mu$ and variance $\sigma^2$, we want to construct an estimator $\wh\mu_n$ and provide a computable and size-optimal upper bound for the error $|\wh \mu_n - \mu|$. While estimators possessing strong deviation guarantees are well known, no data-dependent non-asymptotic upper bounds for their performance exist in general. We show that this gap can be bridged by introducing a parameter that characterizes the “effective sample size” available for uncertainty quantification. This parameter captures the difficulty of the problem, interpolating between “easy” cases where sub-Gaussian confidence intervals exist and “hard" cases where their construction is impossible. Using this characterization, we design confidence intervals of optimal length that are fully adaptive to the unknown variance. Numerical experiments confirm that our approach maintains nominal coverage even in asymmetric and heavy-tailed regimes where other existing methods fail.

We study the rank-twelve potential modular datum denoted $12^{39,406}_{4,1276.}$ in the low-rank classification tables. Its global dimension is $(1287+351\sqrt{13})/2$, and its fusion ring agrees with that of the known rank-twelve Haagerup modular category. The datum satisfies the modular relations, has nonnegative integral Verlinde coefficients, and passes several familiar genus-one necessary tests. We prove, nevertheless, that it cannot be realized by a unitary modular tensor category. The obstruction already occurs in the representation of the three-strand braid group on $\operatorname{Hom}(C_{[10]},B^{\otimes3})$. Generalized Frobenius–Schur indicators determine the eigenvalues of the two-strand generator on this twelve-dimensional space. The central relation in $B_3$ then reduces the problem to a product of three prescribed unitary conjugacy classes. A sixfold symmetry reduces the $1302$ determinant-compatible cases to $217$ representatives. Rank-one inequalities leave only $17$ of these, and ten explicitly displayed degree-zero rank-two Schubert inequalities exclude the remaining cases. A violated inequality with gap $2/117$ is worked out explicitly. All calculations used in the obstruction are exact. The argument uses unitarity essentially and leaves non-unitary categorification open.

Ordinal optimization selects designs that are good enough rather than optimal when evaluation is expensive and the budget is small. The alignment probability measures the chance that the selected set contains at least a prescribed number of truly good designs. We formulate quantum ordinal optimization as a class of problems that maximize this probability under a fixed oracle budget. We study exact, coherent pairwise comparisons with one selected design as a first instance, with the budget counted in comparison queries. For every classical comparison budget smaller than the number of designs, we prove that a knockout among uniformly sampled designs attains the optimum. The sample is drawn without replacement and contains one more design than the number of comparisons. For a uniformly random order, the selected design's rank has a polynomial probability law. At the same comparison budget, the maximum quantum degree is at most twice the classical maximum, a factor attained whenever there are more than twice as many designs as comparisons, so a quantum algorithm can realize rank laws that no classical algorithm with the same budget can. We determine the exact quantum optimum with one comparison for every number of designs and every good-enough set. For selecting the best of n designs, the optimal success probability with one quantum comparison is approximately 3.30/n for large n, compared with 2/n for one classical comparison. When the best design is sought, one quantum comparison gives a higher alignment probability than two classical comparisons whenever there are more than nine designs.

In this paper, we develop an asymptotically normal estimator for the value of policy improvement (VPI) under adaptively collected data. This value, which is defined as the gap between the value of the best treatment policy in a class and that of a baseline policy, gauges whether or not it is worth deploying a personalized treatment policy. Building an estimator for the VPI is naturally more difficult than estimating the value of a fixed policy. This is because, in addition to contending with the correlated data and decaying propensities associated with adaptive experiments, any estimator must also handle the non-regularity, or lack of pathwise differentiability, of the underlying estimand when the optimal policy is non-unique. Our proposed estimator works by sequentially fitting a weighted empirical welfare maximizing policy on historical data and then evaluating it using inverse propensity weighting on future samples. We neither assume that the optimal policy is unique nor make a margin assumption on the underlying heterogeneous treatment effects. A key difficulty in analyzing our estimator is showing that the regret incurred through policy learning is asymptotically smaller than the standard error of the estimator. We accomplish this by showing faster regret rates for policy learning under decaying propensities, which may be of independent interest. We not only evaluate the coverage of our estimator on a mix of synthetic and semi-synthetic data, but also employ it to estimate the VPI in an adaptive experiment evaluating the efficacy of job assistance programs in Jordan due to Caria et al. Although our estimates are generally more conservative than their hierarchical Bayesian analysis, which assumes that the optimal policy is unique, we still corroborate the benefits of personalization amongst Syrian refugees.

Frank and Loss proposed critical Sobolev inequalities for the Dirac operator and for vector fields. They conjectured optimality of their spinorial family and wrote that it is conceivable that the higher-dimensional Loss-Yau potential is an optimizer. Wang and Zhang recently disproved both optimizer claims using direct comparison and instability arguments. We compute closed-form quotients for two elementary families of explicit competitors with strictly smaller quotients. For every $d\geq2$, the spinorial competitor is a constant spinor multiplied by a scalar Aubin-Talenti optimizer. For every $d\geq3$, the vector competitor is the planar azimuthal field \[ A_d(x)=\frac{d}{(1+\lvert x\rvert^2)^2} (-x_2,x_1,0,\ldots,0). \] It lies in the nonlinear gauge and its quotient is computed in closed form. For every odd $d\geq3$, it gives an explicit proof of nonoptimality of the higher-dimensional Loss-Yau-Dunne-Min potential. In dimension three, this field is a specific member of the competitor family of Wang and Zhang: it is a linear combination of the Loss-Yau potential and its reflected twin. Its axisymmetry gives a negative answer to their question about minimization within the $\mathrm{SO}(2)\times 1$-invariant class.

The periodic tiling conjecture asserts that if a bounded set $\Omega\subset \mathbb R^d$ of finite measure tiles $\mathbb R^d$ by translations, then it admits at least one fully periodic tiling. This conjecture was proved to hold in $\mathbb R$ by Lagarias and Wang, and was recently disproved by Greenfeld and Tao in sufficiently high dimensions. Its discrete analogue in $\mathbb Z^2$ was proved by Bhattacharya, whereas the conjecture in $\mathbb R^2$ remains open. In this paper, we analyze the structure of tilings by axis-parallel polygonal sets: bounded open sets in $\mathbb R^2$ whose boundary is a finite union of line segments parallel to the axes. We prove that every such tiling that is topologically minimal is weakly periodic. Moreover, we show that if an axis-parallel polygonal set $\Omega$ tiles $\mathbb{R}^2$ then it admits a quasi-periodic tiling (i.e., a finite union of translated lattices). The proofs combine geometric decompositions of tilings with rational approximation and discrete tiling theory. As a consequence, we obtain that the translational tiling problem for axis-parallel polygonal sets with real algebraic vertex coordinates is decidable.

Linnik's approximation to Goldbach's conjecture concerns the representation of sufficiently large even integers as the sum of two primes and a bounded number of powers of two. We prove that $6$ powers of $2$ suffice unconditionally and that $5$ suffice under the GRH, improving previous works.

Minimum product-set problems ask how small a product of two subsets of prescribed cardinalities can be in a finite group. Although the problem is completely understood for finite abelian groups, exact results in the nonabelian setting are available only for particular classes. We determine the minimum product-set function $\mu_G$ for every finite metacyclic group, without assuming that the relevant cyclic extension splits. Our proof converts the extremal problem into a finite recursive arithmetic description of the feasible cardinality vectors of subset trios. The recursion combines the subgroup structure of finite metacyclic groups with the structural classification of maximal critical subset trios. The resulting characterization recovers the previously known formulas for abelian and nilpotent groups, dihedral groups, and nonabelian groups of order $pq$. It also yields metacyclic examples in which the minimum product-set size lies strictly between the classical lower and upper arithmetic bounds for finite solvable groups. For faithful split metacyclic groups, we further show that requiring one factor to be symmetric, contain the identity, and generate the whole group can force a strictly larger product set than the unrestricted minimum. The latter constraint admits a vertex-isoperimetric interpretation on connected Cayley graphs.

Horizontally explicit, vertically implicit (HEVI) time integration removes the acoustic time-step restriction imposed by fine vertical grids in nonhydrostatic atmospheric models, but requires fast solvers for the resulting vertical linear systems, especially on GPUs. We present GPU-accelerated nested Schur-complement solvers for these systems for a split-form discontinuous Galerkin spectral element method on Legendre-Gauss-Lobatto (LGL) nodes applied to the compressible Euler equations with gravity. In time, we use a Rosenbrock-W method, a linearly implicit Runge-Kutta scheme that retains its order and stability with approximate Jacobians. This allows a sparse vertical Jacobian that captures vertical acoustic waves and gravity on curvilinear meshes with orography and decouples into independent column systems. The solvers are nested: a global Schur complement (static condensation) exploits the LGL sparsity to condense each column onto a few element-edge unknowns, and a local Schur complement eliminates all variables but the vertical momentum within each element, exploiting that the pressure depends only on the potential-temperature density. We propose and analyze two condensation strategies, which lead to banded and block-tridiagonal reduced systems. The resulting HEVI schemes conserve total mass and potential-temperature density to round-off. The solvers are implemented for CPUs and GPUs in TrixiAtmo.jl, the atmospheric extension of Trixi.jl. We verify high-order convergence with linear inertia-gravity waves in a 2D channel and demonstrate robustness on the sphere with the baroclinic instability, a mountain-induced baroclinic instability, and the DCMIP-2025 gap-flow and vortex-shedding cases. For the baroclinic instability without orography, the scheme achieves 18.8 and 2.9 simulated years per day at 100 and 50 km resolution on one NVIDIA GH200 GPU in double precision.

We consider singular Schrödinger operators with an attractive $\delta'$ interaction of strength $\beta$ supported by a star graph and prove that the ground-state eigenvalue is uniquely maximized by stars of the maximum symmetry. Two different proofs are presented, one for an even number of edges using a comparison to the $\delta$ interaction star, the other general based on Birman-Schwinger technique.

Let $(T_n)_{n\ge1}$ be the Tribonacci sequence, defined by $T_1=T_2=1$, $T_3=2$, and $T_n=T_{n-1}+T_{n-2}+T_{n-3}$ for $n\ge4$. In this paper, we study the Tribonacci numbers whose decimal representation is a palindromic concatenation of two distinct repdigits, that is, numbers of the shape $\overline{d_1\cdots d_1\,d_2\cdots d_2\,d_1\cdots d_1}$ with $1\le d_1\le9$, $0\le d_2\le9$, $d_1\ne d_2$. We prove that no Tribonacci number is a palindromic concatenation of two distinct repdigits. The proof combines lower bounds for linear forms in logarithms, the Baker-Davenport reduction method, and a finite computational verification.

Levenshtein's classical theorem shows that any code that can uniquely decode from \(t\) deletions can also uniquely decode from \(t\) insertions. This symmetry, however, does not generally hold in the list-decoding regime. Several prior results on the capacity and radii of list-decoding from insertions and deletions suggest that insertions are easier to handle than deletions. In an effort to better understand this asymmetry between insertions and deletions under list-decoding, we study an analogous version of McEliece's average list size problem. More precisely, given a binary code \(\mathcal{C}\) of length \(n\) that uniquely corrects \(t'\) insertions (or deletions), we are interested in the average number of codewords returned by a \(t\)-insertion (or deletion) list decoder, where \(t' < t\). Focusing on the case of \(t'=1\) and \(t=2\), we find that under two insertions, this quantity is at most \((n+12)/4\) for any VT code of length \(n\), whereas under two deletions it is at least \(n/2 - 10\) when averaged over all length-\(n\) VT codes. The nearly factor-of-two gap between these upper and lower bounds serves to confirm the relative ease of list-decoding of insertions compared to deletions from an average-case perspective.

Let $\mathcal O$ be the ring of integers of a non-Archimedean local field with uniformizer $\varpi$, and let $\mathcal O_\ell=\mathcal O/\varpi^\ell\mathcal O$ for $\ell\geq1$. In this article, we study representations of $\mathrm{GL}_n(\mathcal O_\ell)$ parabolically induced from characters of the Borel subgroup $\mathrm{B}_n(\mathcal O_\ell)$. For $n=3$, we establish a criterion for the irreducibility of these induced representations in terms of the relative conductors of the inducing characters. Motivated by this result, we formulate a conjecture characterizing the irreducibility of such representations for arbitrary $n$. We further show that when all inducing characters are equal, the resulting induced representation is not multiplicity-free, implying that the corresponding triple $(\mathrm{GL}_n(\mathcal O_\ell),\mathrm{B}_n(\mathcal O_\ell),\chi)$ is not a Gelfand pair. Finally, we study the irreducible decomposition of these representations for $\mathrm{GL}_3(\mathcal O_\ell)$ when two relative conductors attain their maximal values. Using the decomposition theory for $\mathrm{GL}_2(\mathcal O_\ell)$, we describe the irreducible constituents and determine their dimensions.

For every $0<s<1$ we construct a compact $s$-Ahlfors regular set $K\subset\mathbb C$ for which harmonic measure of $\mathbb C\setminus K$ with pole at infinity is comparable to $\mathcal H^s|_K$. In particular, there is no dimension drop for any dimension $s<1$. The construction is $N$-adic and based on configurations close to the $N$-th roots of unity. We adjust the positions of the children by small polynomial deformations so that the logarithmic potential of the natural Bernoulli measure becomes constant in the limit. A first-mode correction gives the range $s<2/3$, while corrections of arbitrary finite order yield the full range $0<s<1$.

We establish two quantitative results concerning additive decompositions of the set $\mathbb{P}$ of primes. First, we prove that, if $A,B\subseteq\mathbb N$ have at least two elements and $A+B$ differs from the primes by only finitely many elements, then $$ \frac{\sqrt{x}}{\log x}\ll A(x),B(x) \ll \sqrt{x}. $$ Second, we show that, if $A,B,C\subseteq\mathbb N$ have at least two elements each, then $$ ((A+B+C)\mathbin{\bigtriangleup}\mathbb P)(x) \gg \frac{\sqrt{x}}{(\log x)^2}. $$ An analogous estimate is obtained for sumsets of $k\ge 3$ sets. The proofs use Montgomery's large sieve, Gallagher's larger sieve, a mean-value argument for a certain multiplicative function, and a submultiplicative sumset inequality.

We prove orbital stability of minimizing solitary waves at sufficiently small positive speeds for an Airy equation with a quadratic interaction filtered by a Bessel potential of order one. The stability hypotheses are verified: constrained minimization gives one negative direction, the small-speed KdV limit and operator-norm perturbation of the profile resolvent give a one-dimensional translation kernel, and a differentiable rescaled branch with uniformly bounded derivative gives a positive slope of the quadratic invariant. A conserved Lyapunov functional then proves orbital stability in the three-halves-derivative energy space and the perturbation also gives uniqueness of centered minimizers at small speed. The dynamical framework is supplied by a direct Bourgain-space bilinear estimate, local well-posedness in the square-integrable class, unconditional uniqueness from one-quarter derivative onward, and global well-posedness in the half-derivative energy space. For every prescribed positive speed, constrained minimization constructs smooth positive profiles that are symmetric, strictly monotone away from their center, and exponentially localized. At general positive speeds the stability criterion remains conditional on nondegeneracy and the slope condition.

We consider symplectic analogues of the fractional Fourier transforms and obtain convenient representations in terms of Bargmann transforms and Wigner distributions. We show that they act as homeomorphisms on Schwartz spaces, the space of square integrable functions, the space of tempered distributions, and on Fourier-invariant Gelfand-Shilov spaces and their corresponding distribution spaces. More generally, we establish their homeomorphism properties on a broad class of spaces defined through (Wigner-)Hermite expansions. In addition, we prove that these transforms preserve norms on Schatten-von Neumann symbol classes associated with the Weyl calculus.