Papers

Hybrid beamforming (HBF) reduces the hardware cost of massive MIMO base stations, but its channel acquisition requires recovering the full spatial-domain channel from highly compressed beam-domain observations. This paper proposes a learned score-based turbo message passing (L-STMP) framework for uplink HBF channel recovery under extreme spatial compression. By deeply unfolding the turbo message-passing loop and introducing trainable calibration parameters and learnable score priors in each layer, L-STMP liberates STMP from the idealized compression-matrix requirements of right-rotational invariance, and reduces inference complexity to only several unfolded layers. In addition, to preserve dominant channel components that random discrete Fourier transform (DFT) selection may miss, we introduce precoding matrix indicator (PMI)-assisted beam selection as a low-overhead directional prior. Simulation results demonstrate that L-STMP consistently outperforms existing baselines and obtains substantial reconstruction gains under PMI-assisted compression in channel reconstruction.

For a $d$-dimensional Noetherian local ring $(R,\mathfrak{m})$, with $d>0$, let $c_{\infty}(R)$ be the limit, as $N\to\infty$, of the suprema of $e_R(I)/(d!\ell_R(R/I))$ over ideals $I$ with $\sqrt{I}=\mathfrak{m}$ and $\ell_R(R/I)>N$. We prove that if $R$ is complete, reduced, equidimensional, and Cohen–Macaulay on the punctured spectrum, with $d\geq2$ and infinite residue field, then every nonzerodivisor $x\in\mathfrak{m}$ satisfies \[ c_{\infty}(R)\leq \max\left\{c_{\infty}(R/xR), \sup_{\mathfrak{p}\ne\mathfrak{m}}e(R_{\mathfrak{p}})\right\}. \] As an application, we prove the asymptotic Lech conjecture of Huneke, Ma, Quy, and Smirnov in arbitrary characteristic: $c_{\infty}(R)=1$ whenever $\widehat{R}$ has isolated singularities. This also settles the conjecture of Ma and Smirnov that the Lech–Mumford constant $c_{\mathrm{LM}}(R)$ is attained when $\widehat{R}$ has isolated singularities and $c_{\mathrm{LM}}(R)>1$. For reduced equidimensional Cohen–Macaulay standard graded algebras of dimension $\geq2$ over infinite perfect fields, we prove attainment and rationality of $c_{\mathrm{LM}}$ for the completed vertex local rings of sufficiently high Veronese subrings. In characteristic zero, a refinement of our method produces nonisolated klt affine cones in every dimension $d\geq3$, with $d-2$-dimensional singular loci, whose completed vertex local rings have $c_{\infty}=1$ and arbitrarily large $c_{\mathrm{LM}}$.

We develop an algebraic theory of local intertwining periods for symmetric pairs $(G,H,\theta)$ over non-archimedean local fields $F$ of characteristic different from $2$, and finite-length representations of Levi subgroups of $G$ with coefficients in an algebraically closed field $R$ of characteristic different from the residual characteristic of $F$. For open orbits this algebraic version is essentially contained in the work of Blanc and Delorme. The algebraic theory relies on a crucial new uniqueness result, extending a result of Blanc and Delorme for open periods, which asserts that an admissible intertwining period is the only $H$-equivariant linear form extending the corresponding integral on the space of well-supported sections. Moreover the algebraic construction does not require the unimodularity hypothesis imposed in the work of Matringe–Offen–Yang. When $R=\mathbb{C}$, we also remove this assumption in the classical non-archimedean analytic setting, and prove convergence of intertwining periods for all symmetric spaces. At the same time we extend the context of the whole theory in the following manner: we allow $F$ to be of positive odd characteristic, we allow twisting characters in the periods, and we allow $H$ to lie between $(\mathbb{G}^\theta)^\circ(F)$ and $G^\theta$. Finally, in the algebraic setting, as in the work of Blanc and Delorme on open periods, we obtain explicit bounding polynomials for the singularities of the intertwining period. Such bounds are a key ingredient of a new proof of the cuspidal case of a conjecture of D. Prasad on quaternionic symplectic models.

Replacing the circle by a manifold leads to current Lie algebras and their central and derivation extensions. We separate the coefficient algebra, harmonic analysis and geometric data. Differentials modulo exact forms describe central extensions; divergence-free vector fields give explicit cocycles, whose compatible derivations satisfy a stabilizer equation. Compact examples include the three-sphere and its current–Virasoro extension. We also explain the analytic qualifications for SL(2,R) and its symmetric space and the role of transported multiplication in soft geometries. The recent harmonic constructions of Campoamor-Stursberg, Marrani and Rausch de Traubenberg are compared with the cohomological work of Coquereaux, Frappat, Ragoucy and Sorba and Castellani's loop geometry and infinite-dimensional free differential algebras. A coefficient-algebra construction connects currents to higher brackets and clarifies the additional requirements of a dynamical theory. For Lie algebras of order F, central extensions obey coupled binary and symmetric F-ary conditions, with explicit examples and obstructions.

We determine explicit bulk and worst-case total variation cutoff profiles for the configuration law of a two-species Togashi–Kaneko model with a fixed population of $n$ particles and weak mutations. Boundary initial conditions introduce an additional logarithmic growth phase and change both the leading mixing-time coefficient and the limiting cutoff profile. Let $\kappa_1,\kappa_2>0$ be the autocatalytic rate constants and put $\lambda_*=\kappa_1\kappa_2/(\kappa_1+\kappa_2)$. For deterministic initial configurations whose density converges to $m\in(0,1)$, the total variation distance to equilibrium at time $t_n(s)=\frac{\log n}{2\lambda_*}+\frac{s}{\lambda_*}$ converges to $\mathcal G(r(m)e^{-s})$ for every fixed $s\in\mathbb R$, where $\mathcal G(r)=d_{\mathrm{TV}}(\mathcal N(r,1),\mathcal N(0,1))$, $r(m)>0$ is explicit, and $\mathcal N(a,b^2)$ denotes the Gaussian distribution with mean $a$ and variance $b^2$. For the worst-case distance $\mathscr D_n(t)$ over all initial configurations, we prove $\mathscr D_n(T_n+s/\lambda_*)\to\mathcal P(s)$, where $T_n=\left(\frac{1}{\min\{\kappa_1,\kappa_2\}}+\frac{1}{2\lambda_*}\right)\log n$.The profile $\mathcal P$ is the total variation distance between the equilibrium Gaussian and a Gaussian mixture whose random displacement is determined by a gamma-distributed amplitude arising from early growth at the boundary. Unlike the bulk profile, $\mathcal P$ depends on the mutation parameters. Both cutoffs have windows of constant order, and we identify the asymptotically worst boundary configurations.

We give two types of examples where two non isomorphic finite quandles have isomorphic enveloping groups. This implies that two non-isomorphic finite quandles can share isomorphic representation categories (as additive and tensor categories simultaneously).

The objective of this paper is to study the existence and uniqueness of positive ground state solutions of the following Schrödinger problem \[ -\Delta u=u\log u^2+\mu\abs{u}^{p-1}u\qquad\text{in }\mathbb {R}^N, \] where $N\ge2$, $1<p<2^*-1$ and $\mu>0.$ We prove that there exists a positive ground state solution for every $\mu>0$ and every such solution is smooth, radially symmetric with respect to some point, strictly decreasing in the radial variable. Moreover, as $\mu\downarrow0,$ the positive ground state solutions centered at the origin will converge, uniformly over the ground-state set, to $U_0(x)=\exp(\frac{(N-\abs{x}^2)}{2})$ in the logarithmic Orlicz–Sobolev space and in $C^{2,\alpha}(\R^N)$ for every $\alpha\in(0,1),$ respectively. Furthermore, for sufficiently small $\mu,$ we also prove that the positive ground state solutions are unique up to translations and the kernel space of the real scalar linearized operator can be spanned by the translation derivatives. The proof combines minimization on the Nehari manifold with uniform compactness and an exterior coefficient estimate that prevents normalized difference and kernel sequences from vanishing locally.

In this paper, we focus on an open problem whether every locally compact dcpo is sober for the Scott topology. Specifically, we obtain two sufficient conditions for answering this problem. The main results are (1) every meet continuous locally compact coherent dcpo is sober with respect to the Scott topology; (2) the Scott space of every locally compact second countable dcpo is a sober space.

Multi-continuum models provide an effective macroscopic description of transient flow in fractured and highly heterogeneous porous media by introducing several interacting state variables at each macroscale location. Although high-order multiscale (HOMS) methods reduce the cost of resolving microscopic heterogeneities while recovering local oscillatory features, repeated solution of the coupled homogenized system and repeated high-order reconstruction remain expensive in long-time and large-scale three-dimensional simulations. This work develops a proper orthogonal decomposition (POD) accelerated HOMS framework for transient multi-continuum problems. A vector-valued POD space is constructed from snapshots of the coupled homogenized solution, and the resulting block system is projected onto a low-dimensional space. Two recovery strategies are considered. The first reconstructs the reduced homogenized fields and performs full HOMS assembly at every output time. The second applies the spatial part of the multiscale recovery operator to retained POD modes offline and evaluates the online solution by direct linear combinations of corrected bases and time-derivative bases. An error estimate between the fully discrete homogenized solution and its POD-reduced approximation is established for the coupled system. A recovered POD-HOMS estimate is then obtained by combining the HOMS consistency error with POD truncation errors propagated through value, gradient, inter-continuum exchange, second-derivative, and time-derivative recovery terms. Numerical experiments for two- and three-dimensional heterogeneous configurations show that the reduced solutions retain the accuracy and local oscillatory behavior of the corresponding HOMS solution while substantially decreasing computational cost.

In answering the main criticism of the classical robust optimization that the size of the uncertainty set has to be determined beforehand, a recently proposed Joint Estimation and Robustness Optimization framework treats the size of the uncertainty set as a variable and maximizes it subject to some robust conditions over the uncertainty set. This paper studies its effect on portfolio selection when both the robust return and risk constraints are built into Joint Estimation and Robustness Optimization framework. Although this framework does not directly pursuit any optimality of portfolio return or risk, we report a surprising result that Joint Estimation and Robustness Optimization framework actually selects efficient portfolios. This is proved by reformulating the robust risk constraint as a tractable form through the S-procedure. Moreover, a closed-form formula was derived when there is only the budget constraint. An efficient algorithm is proposed when more constraints are added. Numerical experiments illustrate the validity of the reported results. This research paves the way for Joint Estimation and Robustness Optimization framework to be applied to the portfolio selection problem with return and risk related robust constraints.

In this paper, we establish vanishing viscosity limit of the 2D stationary Navier-Stokes equations in an expanding annulus with a fixed width. The flow is subjected to a large perturbation of a rotation of rigid body at one boundary and stays at rest at another boundary. Due to this different boundary condition setting, the boundary layer effects are different near the interior and exterior boundaries, which result in different thickness of boundary layer and different leading order boundary layer equations. We will construct an approximate solution to this 2D Navier-Stokes equations by using higher order asymptotic approximation and show the validity of the boundary layer expansion. The leading order of the Euler solution is the Taylor-Couette flow $\left(Ar+B\frac{1}{r}\right)e_\theta$ for some suitable constant $A$ and $B$, which is determined by using the principle of the Prandtl-Batchelor theory. The perturbation of the tangential velocity becomes large as the annulus expands, which causes heavy difficulties when we show the convergence of the approximate solution. Variously weighted energy estimates up to third-order derivatives of the error solution are constructed to close the energy of the nonlinear system.

We investigate a direct two-ends approach to restriction for the truncated cone in $\R^n$, which gives the diagonal restriction range \[ p>2+\frac{4(7n+15)}{11n^2-n-24},\qquad n\ge10. \] This improves the range obtained by applying the result of Wang–Wu and then slicing.

Let $\mathfrak p$ be the $(-1)$-eigenspace of a complex semisimple symmetric pair. For every $r\ge2$, we prove $\dim C_r(\mathcal N(\mathfrak p))\le \lfloor r\dim\mathfrak p/2\rfloor$ and show that every irreducible component is isotropic for a natural family of alternating forms. For a nonzero Hermitian pair with $\mathfrak g$ simple and $\mathfrak p=\mathfrak p^-\oplus\mathfrak p^+$, put $d=\dim\mathfrak p^+$ and $\ell=\operatorname{rk}(\mathfrak g,\mathfrak k)$. Then \[ \dim C_r(\mathcal N(\mathfrak p))=rd,\qquad \dim C_r(\mathfrak p)=\max\{rd,\,2d+(r-1)\ell\}. \] In this case we also determine the maximal-dimensional components of $C_r(\mathfrak p)$; for $r\ge3$, the only maximal-dimensional nilpotent components are $(\mathfrak p^+)^r$ and $(\mathfrak p^-)^r$. For arbitrary symmetric pairs, $\dim C_r(\mathcal N(\mathfrak p))=ra+m$ for all large $r$, where $a$ is the largest dimension of an abelian subspace of $\mathfrak p$ consisting of nilpotent elements and $m$ is the dimension of the variety of such subspaces. Applications give dimension and component formulas for equivariant punctual Quot schemes under explicit framing conditions, and bounds for punctual Gorenstein loci refined by socle parity.

We investigate whether Rényi divergence can sharpen McDiarmid's inequality and its distribution-dependent refinement induced by KL divergence. Starting from Marton's transportation proof, we show that its transport inequality fails in general when KL divergence is replaced by Rényi divergence of order $0<\alpha<1$. For sums of independent bounded random variables, however, the additivity of Rényi divergence allows us to adapt the argument and derive concentration bounds. For Bernoulli sums, these bounds strictly improve on the distribution-dependent bound induced by KL divergence over a range of deviation thresholds. We also establish a distribution-dependent refinement of Gilardoni's inequality relating total variation to Rényi divergence, providing explicit constants for our general concentration bounds.

The pentablock is a domain in $\mathbb C^3$ arising in the study of the $\mu$-synthesis problem and is defined by \[ \mathcal P = \left\{ (a_{21},\operatorname{tr}(A),\det(A)): A=[a_{ij}]_{i,j=1}^2\in M_2(\mathbb C),\ \|A\|<1 \right\}. \] We introduce the Hardy space $H^2(\mathcal P)$ associated with the canonical measure $\mu$ on the distinguished boundary $b\mathcal P$. We identify $H^2(\mathcal P)$ with a closed subspace of $L^2(b\mathcal P,\mu)$ and show that it is a reproducing kernel Hilbert space. We establish a new description of $\mathcal P$ and its closure and obtain a uniqueness set. In particular, we prove that $\mathcal P$ is the interior of the polynomial convex hull of $\Psi(\overline{\mathbb D}^3)$, where \[ \Psi(\xi,\lambda_1,\lambda_2) = \left( \frac{\xi(\lambda_1-\lambda_2)}{2}, \lambda_1+\lambda_2, \lambda_1\lambda_2 \right) \] is the parametrization map for $\mathcal P$. We also show that $H^2(\mathcal P)$ embeds isometrically as a closed subspace of $H^2(\mathbb D^3)$. We then obtain a Brown-Halmos type characterization of Toeplitz operators on $H^2(\mathcal P)$ and a commutant lifting theorem for $\mathcal P$-isometries and their minimal $\mathcal P$-unitary extensions. As consequences, we determine the commutant of the coordinate multipliers, characterize the compact Toeplitz operators and show that the coordinate multipliers have no nontrivial joint reducing subspaces. Finally, we characterize $\mathcal P$-unitaries as images of commuting triples of unitaries under $\Psi$ and present a class of $\mathcal P$-contractions from triples having $\overline{\mathbb D}^3$ as a spectral set.

We derive a recurrence for logarithmically weighted Möbius sums by combining Möbius inversion with the Euler–Maclaurin summation formula. Consequently, starting from estimates of orders zero, one and two, we obtain polynomial approximations for these sums for every nonnegative integer exponent of $\log$. We then extend these estimates to sums with coprimality conditions via a convolution identity and to sums weighted by the reciprocal of the sum-of-divisors function. We also establish the nonnegativity of the latter sums for every positive integer logarithmic order, proving that they never change sign.

We prove that the simulated annealing algorithm efficiently samples from the Sherrington-Kirkpatrick model at inverse temperature $\beta < 1$ to an arbitrary inverse polynomial accuracy in total variation.

We prove Yau's number-theoretic conjecture on positive integral points in real right simplices in every dimension $n\ge3$, and apply the resulting sharp polynomial bound to quantum harmonic oscillators. The lattice-point bound holds for every choice of positive real intercepts for which the count is nonzero, with equality precisely when all intercepts equal the same integer. For oscillators in dimensions $n\ge3$, we obtain explicit upper bounds for eigenvalue counts and spectral remainders for arbitrary positive frequencies at every threshold at or above the lowest eigenvalue. For frequencies with rational ratios in dimensions $n\ge2$, an additional lattice coordinate and interpolation give bounds for energy sums, or first Riesz means, at every threshold. When all frequencies are equal (the isotropic case), the oscillator attains these bounds: the counting bound is exact at its energy levels, and interpolation on its natural energy grid recovers the energy sum at every threshold. In three dimensions, the exact formula displays the periodic contribution associated with the resonance discussed by Fefferman and Seco. The proof builds on the slicing method of Yau and Zhang. Arithmetic and geometric consequences include explicit bounds for smooth numbers and lattice-point discrepancies.

We develop and analyze a fully discrete semi-Lagrangian method for vector master equations associated with mean field games (MFGs) with individual noise on finite graphs. The method discretizes both the characteristic evolution in time and the population variable on the probability simplex. A central challenge is the degeneracy of the transport coefficients near the boundary of the open simplex, where no boundary condition is prescribed. To overcome this difficulty in the analysis, we introduce an interior localization based on exact density characteristics, which keeps numerical arrivals within compact regions of uniform coefficient control. Combined with a closure argument for the coupled numerical recursion, this localization yields local solvability of the updates and quantitative error bounds on compact subsets of the open simplex. Under additional $C^2$-regularity of the exact solution and suitable space–time mesh coupling, these bounds imply first-order accuracy. Furthermore, the density and value trajectories generated by the numerical master feedback converge to the corresponding MFG characteristics at the same rate. In the potential setting, the associated reconstructed density–flux pair is admissible and asymptotically optimal for the planning problem, with an action gap quadratic in the feedback error. Numerical experiments illustrate the convergence rates, feedback responses to population redistribution, and the effects of graph topology.

Let $n,k$ be positive integers and $q$ be a prime power. We prove that there exists a real constant $C_k$ depending only on $k$ such that if $ \# M_n(\mathbb{F}_q) > C_k$ then every element in $M_n(\mathbb{F}_q)$ is a sum of two $k$th powers, proving Larsen's conjecture.

GPU-based first-order methods are emerging as an approach for solving large-scale linear programming (LP) problems. We present H-PDLP, a GPU first-order LP solver based on a generalized primal-dual hybrid gradient (PDHG) method for a homogeneous self-dual embedding of the LP, and study its numerical performance. The solver applies the restarted reflected Halpern iteration to the generalized PDHG operator, with the embedding encoding both primal-dual optimal solutions and certificates of primal or dual infeasibility. We derive a compact embedding for LPs with two-sided constraint and variable bounds in which the bounds enter through box cones, and we show that the resulting box-cone projections reduce to one-dimensional root-finding problems, so that each iteration remains matrix-free and well suited to GPUs. On the theoretical side, we establish conditions under which the iterates remain uniformly bounded away from the trivial zero solution of the embedding and derive complexity guarantees for the method. We also introduce MIPLIBInfeas, a reproducible benchmark of 229 infeasible and 105 unbounded LPs derived from MIPLIB 2017. Numerical experiments demonstrate the effectiveness of the proposed approach on both feasible LPs and infeasible or unbounded LPs. The source code for H-PDLP is publicly available at https://github.com/MIT-Lu-Lab/H-PDLP.

We prove two constant rank theorems for the Ricci tensor in two geometric settings: manifolds with harmonic curvature and gradient Ricci solitons. In both cases, if the $k$-intermediate Ricci curvature satisfies $\Ric_k\geq0$ for some $1\leq k\leq n-2$, then the Ricci tensor has constant rank and its nullity distribution is parallel. In the complete case, the universal cover splits isometrically along the Ricci-null directions. We also show that the range $1\leq k\leq n-2$ is sharp: in every dimension $n\geq4$, nonnegative Ricci curvature alone does not imply the constant rank property in either setting.

Weakly imposed boundary conditions in first-order system least-squares methods go back to Jespersen (1977), but two questions have remained open for decades: the right weights, especially for the Neumann residual, whose natural norm $H^{-1/2}$ suggests $h_F$, and norm and error equivalences, especially for indefinite equations. We answer both for general, possibly indefinite, second-order elliptic equations with non-homogeneous mixed boundary conditions. First, for the weights $h_F^{-1}$ on the Dirichlet and $1$ on the Neumann residual, we prove norm and error equivalences with mesh-independent constants for every discrete pair: the error in a mesh-dependent norm is equivalent to the square root of the functional plus a Dirichlet data term. The least-squares solution converges optimally in $H(\mathrm{div};\Omega)\times H^{1}(\Omega)$, and the functional is a reliable and efficient error estimator, with a recovered flux also for approximations from other methods, even ones violating the boundary conditions. The key idea is to apply the stability of the boundary value problem to pairs that are discrete only in their Dirichlet traces, whose $H^{1/2}$ norm an inverse inequality bounds by the weighted penalty. Second, these weights are the right ones: weaker powers, including $h_F$ on the Neumann boundary, destroy the norm equivalence, and stronger weights are admissible, since the penalty multiplies zero at a comparison function meeting the projected data. Third, for the fully discontinuous counterpart, a pure penalty method with both jumps weighted by $h_F^{-1}$, the error equivalence holds for every discrete pair, also for exact fluxes with normal traces not square integrable on interior faces.

We completely characterize the solvability profiles of tasks for two autonomous mobile robots in the plane under variable disorientation, chirality, and rigid movements. The four standard capability models and the schedulers considered here have a known quotient order Q of thirteen computational classes. A task's solvability profile is the set of classes in which it is solvable, and is necessarily an upset of Q. We prove that every upset is realizable and construct a minimum exact disjunctive basis of thirteen concrete tasks. The basis contains one task for each principal upset. For every antichain of Q, the disjunction of the corresponding tasks has profile exactly equal to the union of their profiles. This exactness is not implied by the individual profiles: an algorithm may satisfy different disjuncts in different executions. We establish it by constructing, for every algorithm in each maximal class outside the target union, a single admissible fair execution violating all selected predicates simultaneously. Consequently, the realizable profiles form the entire finite distributive lattice Up(Q): there are exactly 54 profiles, counted by the antichain polynomial 1 + 13z + 24z^2 + 16z^3, and each has a canonical disjunctive representation using at most three principal tasks. We also identify six tasks whose solvability signatures recover every inclusion and incomparability in Q, and prove that six is minimum even among arbitrary separating tasks. Their profiles are precisely the principal upsets generated by the join-irreducible elements of Q. In contrast, any task family generating every profile by exact disjunctions must contain all thirteen principal profiles. Thus our exact disjunctive basis has minimum size and is unique at the profile level.

We study certain simple affine vertex algebras $L_{-1-h/6}(\mathfrak{g})$ from rank-one Argyres-Douglas theories, where $\mathfrak{g}$ is a non-simply-laced simple Lie algebra and $h$ is its Coxeter number. For $L_{-3}(F_4)$, we determine the maximal ideal of $V^{-3}(F_4)$, Zhu's algebra, and the associated variety. We also classify the irreducible highest-weight $L_{-3}(F_4)$-modules and determine the characters of its irreducible ordinary modules. The associated variety is $\overline{\mathbb{O}}_{\widetilde A_1}$, and the level $-3$ is totally collapsing for the orbit $\mathbb{O}_{\widetilde A_1}$. The Poisson coinvariants of the $C_2$-algebra of $L_{-3}(F_4)$ are isomorphic to $\mathbb{C}[\ell]/(\ell^5)$, where $\ell$ is induced by the conformal vector. Consequently, the characters of ordinary $L_{-3}(F_4)$-modules satisfy a fifth-order modular linear differential equation. We also give modular linear differential equations of orders five and eleven satisfied by the characters of ordinary modules for $L_{-2}(B_3)$ and $L_{-2}(G_2)$ respectively.

We establish a matching lower bound for the sextic large sieve over the Eisenstein integers in an unbalanced range, with both indices restricted to squarefree ideals. Let $\Xi_6(A,B)$ denote the squared operator norm of the sextic residue-character matrix with row and column norms at most $A$ and $B$, respectively, outside a fixed finite set of primes containing those above $6$. For every fixed $0<\delta<3/11$ and every $\varepsilon>0$, we prove $\Xi_6(A,B)\gg_{\delta,\varepsilon}(AB)^{-\varepsilon}A^{5/6}B^{1/3}$ for $B^{16/11+\delta}\le A\le B^{2-\delta}$. Consequently, the full-cutoff norm satisfies $\Xi_6(X,X)\gg_\varepsilon X^{17/16-\varepsilon}$. The main difficulty is to preserve the large second moment supplied by Gauss sums when the column moduli are restricted to be squarefree. We prove a second-moment formula uniform in the prime exclusions and character twists arising from Möbius inversion, and use it to show that the squarefree projection retains a positive proportion of the original moment. A separate argument combining cubic and sextic large-sieve bounds then transfers the lower bound to squarefree rows.

In this paper, we study natural determinantal point processes on Stein manifolds defined by canonical sections and plurisubharmonic weight functions. We present a natural condition on the weights under which these processes satisfy a large deviation principle with a good rate function given by the Fenchel–Legendre transform of the pluripotential equilibrium energy. Along the way, we prove some pluripotential theoretic results on Stein manifolds, such as the $C^{1,1}$ regularity of appropriate envelopes. These results are noncompact analogues of results of Berman and Berman–Boucksom on compact Kähler manifolds.

We study the $s$-Schrödinger map equation \[ \begin{cases} \partial_t u = -u \wedge (-\Delta)^s u \quad \text{on } \mathbb{R}^n \times \mathbb{R}\\ u: \mathbb{R}^n \times \mathbb{R} \to \mathbb{S}^2\\ u(x,0) = u_0. \end{cases} \] For $s \in (1/2,1)$ and $n \geq 5$ we prove that for small initial data $u_0$ in the critical Besov space one can obtain a global solution.

A recent preprint constructs a finitely presented torsion-free group $G$ with a finite two-dimensional classifying space such that $\mathbb F_2[G]$ contains nonzero zero divisors. We extend this to every odd prime $p$: there is a finitely presented torsion-free group $G_p$, again with a finite two-dimensional classifying space, such that $\mathbb F_p[G_p]$ contains nonzero elements $\alpha,\beta$ with $\alpha\beta=0$. Together with the case $p=2$, Kaplansky's zero-divisor conjecture therefore fails over every field of positive characteristic. The new ingredients are a signed version of the parity cancellation, made possible by a bipartite random model on one side; a type design built from the projective plane over $\mathbb F_{p^3}$ with $p$ additional letters, which makes every cross-intersection congruent to $1$ modulo $p$; and a squared-word decay estimate for this design, proved with an explicit test vector. The geometric part of the argument is taken from the characteristic-two construction. The characteristic-zero case remains open.

Watermarking embeds an imperceptible signal in text from large language models that is statistically detectable with a secret key. The recent textGrain watermark (Li et. al. (2026b)) calibrates this signal against the model's sampling entropy through an explicit entropy budget, combining the stochasticity of Red-Green and the unbiasedness of Gumbel-max watermarks, and its accompanying report proposes a detector that sums pivotal statistics. We study when, and how efficiently, textGrain can be detected after human edits, modeled as a sparse mixture in which a fraction $\varepsilon_n\asymp n^{-p}$ of positions retain the watermark and token uncertainty decays as $n^{-q}$. First, entropy calibration comes at a cost: the minimax detection boundary of textGrain is $p+q=1/2$, so its detectable region is strictly smaller than that of Gumbel-max and maximal-coupling watermarks, whose boundary is $2p+q=1$. This cost stems from textGrain's construction rather than from entropy calibration itself: an unbiased scheme with the same entropy loss remains detectable where textGrain is not. Second, perhaps surprisingly, this loss makes the sum detector optimal at the boundary when the number of transport columns is fixed, unlike for Gumbel-max. When the number of columns grows and the entropy budget does not vanish, the detectable region expands while that of the sum detector does not. Third, for a fixed edit fraction, we characterize the optimal exponential rate of type II error decay and show that the sum detector falls strictly short of it at every fixed alternative. Finally, we propose a weighted multiscale binomial scan, valid in finite samples and needing no knowledge of the next-token distributions, entropy budget, or edit fraction; it attains every detection boundary above and the optimal exponent.