Papers

This paper exposits relationships between the geometry of modular curves and the arithmetic of cyclotomic fields through the Eisenstein ideal. For an integer $N \ge 5$ and an odd prime $p$, we define two conjecturally inverse maps between the real part $P$ of the Eisenstein reduction of the homology $P$ of $X_1(N)$ and the real part of the $p$-part $Y$ of the second $K$-group of the $N$th cyclotomic integer ring. We recall a motivic construction of the map from $P$ to $Y$ and describe the recent proof of the Eisenstein property of the underlying map on modular symbols. We provide a fully equivariant construction of the second map from $Y$ to $P$ using the Eisenstein reduction of the first étale cohomology of $X_1(N)$. We then explain an equivariant construction of the Euler system of cyclotomic units using extension classes in relative cohomology groups of modular curves and describe its place in our program.

In this article, we study the spatial fluctuations of the Skorohod solution to a stochastic heat equation on $\mathbb R^d$ for $d<4$, driven by multiplicative Gaussian noise with the non-separable covariance kernel $p_{|t-s|}(x-y)$. We prove that the solution is strictly stationary and spatially ergodic at every fixed time. For the centered spatial integral $F_R(t)=\int_{\{|x|<R\}}(u(t,x)-1)dx,$ we show that $\mathbb E[F_R(t)F_R(s)]\sim K(t,s)R^d$ as $R\to\infty$. Using moment estimates for the first two Malliavin derivatives and a second-order Gaussian Poincaré inequality, we establish a quantitative central limit theorem in total variation distance with rate $R^{-d/2}$. We also prove a functional central limit theorem for the process $\{R^{-d/2}F_R(t)\}_{t\geq0}$.

In this paper, we study radial normalized solutions to nonlinear Schrödinger equations in the unit ball \[ -\Delta u+\lambda u=|u|^{p-1}u, \qquad \int_{B_1}|u|^2\,dx=\rho, \] where $\lambda$ is unknown and $p\ge p_c:=\frac{N+2}{N-2}$. Soave \cite{Soave} proved that for every $1<p<p_c$ every positive mass is attained by infinitely many radial solutions and explicitly raised the question of whether every mass is attained in the Sobolev critical and supercritical regimes. We answer this question negatively in the radial class: for every $p\ge p_c$, the set of masses attained by nontrivial radial solutions is an interval with finite positive supremum. In particular, at $p=p_c$ the set of attainable masses is $(0,\rho^*_{\rm rad})$ or $(0,\rho^*_{\rm rad}]$, with $0<\rho^*_{\rm rad}<\infty$, and the same finite threshold alternative holds for every $p>p_c$. Thus, in the radial class, the subcritical all masses phenomenon fails throughout the full Sobolev critical and supercritical range.

In this paper, we study multidimensional perturbations of radially symmetric rarefactive flows for the compressible Euler equations. No assumptions on the smallness of the background are imposed. We derive lower bounds on both the classical lifespan and stability timespan of the perturbed solution. These lower bounds are demonstrated to tend to infinity as the size of the perturbation approaches zero. Furthermore, owing to the rarefactive nature of the background, the resulting estimate is shown to be substantially longer than the corresponding estimate for a general classical background with bounded first derivatives.

We study the stochastic heat equation on a bounded interval with zero Dirichlet boundary conditions, driven by symmetric $\alpha$-stable space–time Lévy white noise, where $1<\alpha<2$. Both the drift and noise coefficients are continuous and have at most linear growth. For bounded deterministic initial data and every $r>1/2$, we construct a weak solution with càdlàg paths in the negative Sobolev space $\mathbb{H}_{-r}$. We also construct a predictable mild solution on the same stochastic basis and with the same noise. At each fixed deterministic time, this mild solution is a density of the distribution-valued weak solution.

We study the initial–boundary value problem for the one-dimensional full compressible Navier–Stokes system whose viscosity and heat conductivity are given by $\mu(\theta)=\mu\theta^\alpha$ and $\kappa(\theta)=\kappa\theta^\beta$, respectively. The initial density is nonnegative and is allowed to vanish. For $\alpha>2$ and $\beta>2\alpha+4$, we prove the existence of a global strong solution for arbitrarily large compatible $H^2$ data, provided that the prescribed boundary temperature is sufficiently high relative to their size. No velocity or temperature perturbation is required to shrink as the boundary temperature tends to infinity. A shifted initial-layer weight and an exact symmetrization of the two material-derivative equations separate the fast thermal mode from the momentum mode. The resulting estimates are uniform with respect to the positive lower bound used in the approximation and therefore survive the limit to genuine vacuum. The proof further combines a relative-entropy identity centered at the conserved mean density, the physical effective viscous flux, a Zlotnik-type estimate along particle trajectories, and finite-time transport estimates for the density derivatives. The density remains uniformly bounded and the temperature stays uniformly comparable with the prescribed boundary temperature. After the initial layer we establish vacuum-compatible exponential relaxation: the density converges to its conserved mean in every finite $L^p$, the velocity and temperature perturbation decay in $H^1$, and the weighted material-derivative energy decays at rate $c\bar\theta^{1-\alpha}$.

Let $G$ be a graph. Under an injective edge coloring of $G$, any two edges at distance $2$ or belonging to a common triangle receive distinct colors. The injective chromatic index of $G$, denoted by $\chi'_{inj}(G)$, is the smallest integer $k$ such that $G$ admits an injective edge coloring with $k$ colors. Let $\Delta$ be the maximum degree of $G$. Ferdjallah et al.~proved that $\chi'_{inj}(G) \le 2(\Delta-1)^2$. In this paper, we improve this bound by proving that $\chi'_{inj}(G) \le 2(\Delta - 1)^2 -\Delta + 3$ if $\Delta \ge 4$.

Let $r\geq 2$ be an integer. An $r$-partite graph $\Gamma$ with vertex partition $V_1,\ldots,V_r$ is $2$-balanced if there exists a positive integer $n$ such that $n\leq |V_i|\leq 2n$ for every $i\in[r]$. For an integer $\ell\geq3$, let $C_\ell$ denote the cycle of length $\ell$, and define $\hat{\delta}(\Gamma)=\min\{d_\Gamma(v,V_i)/|V_i|:i\in[r],\ v\in V(\Gamma)\setminus V_i\}$. Let $\hat\delta^r_{C_\ell}$ denote the $C_\ell$-decomposition threshold for $2$-balanced $r$-partite graphs, that is, the least nonnegative real number $\delta$ such that, for every $\varepsilon>0$, there exists $n_0$ such that every $C_\ell$-divisible $2$-balanced $r$-partite graph $G$ with $\min_i|V_i|>n_0$ and $\hat\delta(G)\geq\delta+\varepsilon$ admits a $C_\ell$-decomposition. We prove that $\hat\delta^{r}_{C_4}=\frac{2}{3}$ and $\hat\delta^{r}_{C_{2k}}=\frac{1}{2}$ for every $r\geq2$ and every $k\geq3$.

We investigate the inhomogeneous incompressible Navier-Stokes system in the 2D global domain with zero initial total momentum, allowing vacuum and densities with finite moments but not necessarily compact support. Exploiting the cancellation structure inherent in the 2D equations along with spatial moment bounds on the density, we derive that the inertial force belongs to the Hardy space and establish weighted Stokes estimates through endpoint Hardy-Littlewood-Sobolev inequalities. These estimates identify the far-field velocity and control the remainder, leading to uniform exponential decay of the velocity throughout global domains. Combined with higher order regularity bounds, we establish the global well-posedness of strong solutions for the density-dependent viscosity case, and prove the uniform regularity of flow maps for measurable density patches in the constant viscosity model.

Data centers consume a large amount of energy, much of which is wasted due to idle servers. Turning off idle servers might be an effective power-saving solution; however, there is a trade-off between energy savings and system performance. Hence, we propose a setup queueing model with a batching policy that allows servers to process a set of jobs simultaneously to minimize power consumption while maintaining acceptable performance. We consider an M/M/c/SET–BATCH queue, a multi-server batch service queue with a fixed batch size and setup times, and some variants, including systems in which idle servers delay before turning off or systems in which the batch size is dynamic. We analyze the steady-state probabilities and system performance of the M/M/c/SET–BATCH system and its variants. Our analysis of the M/M/c/SET–BATCH system with lower computational complexity is made possible by utilizing the special structure of the model. In addition, we use simulations to compare the M/M/c/SET–BATCH model with some other variants with different setup time distributions. The results suggest that the model performs better when the setup time has a larger coefficient of variation. Our results indicate that the batching policy enhances the system performance, especially when we allow servers to be idle before turning them off.

We develop a high-order discontinuous Galerkin formulation of the general synthetic iterative scheme (GSIS) for steady rarefied gas mixture flows. The resulting GSIS-DG discretizes the mesoscopic kinetic equations for monatomic gas mixtures and the macroscopic synthetic equations with the same polynomial degree and solves them alternately. The kinetic solution supplies nonequilibrium corrections to the stress and heat flux, while the synthetic equations accelerate the convergence toward the steady state. Asymptotic analysis shows that GSIS-DG recovers the continuum limit without requiring the spatial mesh to resolve the molecular mean free path. Numerical results over a wide range of Knudsen numbers demonstrate high-order spatial accuracy and rapid steady-state convergence, with more than an order-of-magnitude reduction in computational cost relative to conventional kinetic iteration in the near-continuum regime. Comparisons with second-order finite-volume GSIS further demonstrate that GSIS-DG achieves comparable or better accuracy on coarser physical meshes with fewer total spatial degrees of freedom.

For an odd prime power $q$, a BLT-set $\mathcal B$ is a set of $(q+1)$ points of $\mathcal Q(4,q)$ such that every point of $\mathcal Q(4,q)$ is collinear with at most two points of $\mathcal B$. We complete the classification of BLT-sets of $\mathcal Q(4,q)$ admitting a non-solvable transitive automorphism group. Nelson and Penttila completed the classification of BLT-sets $\mathcal B$ with a non-solvable transitive automorphism group acting irreducibly on the underlying $5$-dimensional vector space. In this paper, we prove that a BLT-set with a non-solvable transitive full automorphism group acting reducibly on the underlying $5$-dimensional vector space is either a classical BLT-set or a Kantor semifield BLT-set.

In this note, we prove that the noncommutative AM-GM inequality holds if $n\ge 2\lceil m/2 \rceil^2$. The motivation comes from counterexamples constructed in [De Sa, Random reshuffling is not always better, NeurIPS2020]. The proof constructs a vertex measure based on the Chebyshev nodes on the Boolean cube to extract the distinct indices. The main difficulty is that the measure is not positive on non-integer nodes. The key technique comes from [Grigoriev, Complexity of Positivstellensatz proofs for the knapsack, Computational Complexity (2001)] and eventually transforms the problem into a quadrature estimate.

The Hamilton-Jacobi-Bellman (HJB) equation of a nonlinear system is a nonlinear partial differential equation, and in that form it gives little away. This paper shows that it hides a Riccati equation. Carleman linearization reveals this structure. For scalar input-affine systems with analytic data, the lifted HJB equation is a single infinite-dimensional Riccati equation in a lower-triangular Toeplitz matrix. Its first entry is the Riccati equation of the standard linearization at equilibrium. Its remaining entries hold the nonlinear terms. Nothing about this is an approximation. Under controllability of the standard linearization and detectability of the cost, the equation has exactly one solution with positive diagonal entries, and it can be computed one entry at a time. This solution is the derivative of the value function. We prove that finite-sections of the solution are exact and converge exponentially to the derivative of the value function near the origin. The scalar case has a closed-form solution. This closed form is our benchmark: every result on the lifted equation is checked against it. Nothing in this structure depends on the state dimension. Our methodology points to a general theory of optimal control in which the HJB equation of analytic nonlinear systems becomes an infinite-dimensional Riccati equation.

We prove strong ill-posedness for the two-dimensional incompressible Euler equation in the endpoint velocity space $W^{3,1}_\sigma(\mathbb R^2)$. Given any compactly supported smooth divergence-free background and any $\delta>0$, we construct a single compactly supported divergence-free datum within $\delta$ of the background in $W^{3,1}$. The perturbation can be localized inside any prescribed nonempty open neighborhood of the background support while remaining disjoint from that support. The associated unique global finite-energy solution $u$ satisfies ${\rm curl} u\in C([0,\infty);W^{2,1}(\mathbb R^2))$, but \[ \operatorname*{ess\,sup}_{0<t<T}\|u(t)\|_{W^{3,1}}=\infty \qquad\text{for every }T>0. \] The proof combines localized hyperbolic insertion, the Bonami–Poornima nonmultiplier mechanism, and many-packet dilution. Moment cancellation and positive material separation control the nonlocal Biot–Savart interactions, allowing a nested construction inside one fixed compact region. Without the compact-support requirement on the limiting datum, a complementary far-field argument yields a dense $G_\delta$ set of data whose solutions leave $W^{3,1}$ at every positive rational time and on a residual set of times, with unbounded essential supremum on every nonempty open time interval.

Motivated by the EPPA problem for finite tournaments,we consider various extension properties for partial permutations. We show that for any set $\Pi$ of prime numbers, the $\Pi$-extension property is equivalent to the $\Pi$-LERF. As a consequence of known results, the odd-extension property is then equivalent to the EPPA for finite tournaments. To study the $\Pi$-extension property, we reformate the property using the concepts of EP problems and $\Pi$-solutions. We then show that the existence of a $\Pi$-solution for an EP problem depends entirely on its fundamental group. In particular, when the fundamental group is trivial or cyclic, the EP problem has a $\Pi$-solution for any $\Pi$. These extend some known results of Huang, Pawliuk, Sabok and Wise [HPSW19]. We give examples of EP problems without nilpotent-solutions; they witness that the $p$-extension property and the $p$-LERF fail for any prime $p$. Then we consider a special kind of EP problems whose fundamental groups have two generators. We show that any 1-weakly wandering problem has a $\Pi$-solution for any $\Pi$. For 2-weakly wandering problems, we show that they all have odd-solutions and we completely characterize those without nilpotent-solutions.

We study the refined de Sitter conjecture (RdSC), the Trans-Planckian Censorship Conjecture (TCC) and the species scale $\Lambda_s$ in the multifield flux potential of type IIB string theory compactified on one-parameter Calabi–Yau threefolds. The scalar potential depends on the axio-dilaton and on the single complex-structure modulus, and is no-scale along the Kähler directions. Restricting to $h^{2,1}=1$ allows us to work with the exact periods obtained from the Picard–Fuchs equations rather than with their large-complex-structure expansion, and therefore to probe the conifold region. For the mirror quintic, $(h^{2,1},h^{1,1})=(1,101)$, we scan 200 flux configurations and find unstable extrema in $96\%$ of the cases, with $c'=0.18$. Once the Kähler runaway is taken into account, the no-scale structure fixes the logarithmic slope of the potential at every critical point to the universal value $\sqrt6$, so that the first RdSC condition is satisfied throughout. Statistically, the complex structure is the heavier sector and the axio-dilaton dominates the unstable directions. For the TCC we obtain analytic results valid for the whole class of one-parameter models: the slope approaches the universal value $2$ at large complex structure, diverges at the conifold, and equals exactly $\sqrt2$ in the axio-dilaton directions, where the leading potential takes a single $SL(2)$-covariant form, as the multifield evolutions confirm numerically. The TCC further translates into a lower bound on the compactification volume, $v>v_0\sim e^{N}$, controlled by the number of e-folds. Finally, the field displacements along our trajectories are compatible with the species-scale field-range bound, and restoring the volume dependence of $\Lambda_s$ shows that $V\lesssim3\Lambda_s^2$ is implied by the TCC volume bound, which is thus the genuine constraint on these models.

We study relative completions of fundamental groups of moduli stacks of curves of compact type and of partial compactifications of the smooth moduli stack. For $g\geq3$, the compact-type completion has commutative prounipotent radical with Lie algebra $\Lambda^3_0H\oplus H^{\oplus n}$. Fix $1\leq h\leq g-1$. We show that adjoining the one-node loci of the separating divisors whose genus-$h$ side carries at most two markings gives the same relative completion. The proof uses explicit separating-twist calculations in weight $-2$, pointed Johnson homomorphisms, and the lantern relation. Functorial specialization gives the corresponding results in positive characteristic for odd $\ell$ different from the characteristic. As a consequence, extension to compact type of iterated extensions of algebraic symplectic local systems can be tested on these selected divisors. We also prove that any collection of separating divisors giving the same relative completion contains at least $\binom n2$ divisors.

We study permutation and monomial equivalence for one-point codes on weighted superelliptic curves. On strata where $H_m\simeq \Z/n\Z\times\Z/m\Z$ acts regularly on the evaluation set, we prove that these codes are Cartesian character codes and determine their primal and dual minimum distances. We compute the projective monomial normalizer of the regular action and classify the compatible permutation and monomial equivalences through the induced actions on the defining sets. When $d > n$ and $m \geq 2$, the common permutation automorphism group of the full filtration is exactly $H_m$, and the regular action can be recovered from its two lowest nonconstant members. We determine the full permutation automorphism group in closed form at the top of the filtration and give a field-independent finite certificate for $\PAut ( \CC_s ) = H_m$ in the remaining primitive cases treated here. The hull dimension is bounded by the genus on every regular stratum, apart from the type $(3,3,3)$. Explicit examples show that length, dimension, and hull dimension do not determine monomial equivalence, and that the hull profile together with the regular automorphism subgroup does not determine the curve up to isomorphism. As an application, we determine the corresponding hull-based invariants and punctured signatures used in support-splitting methods.

This paper studies privacy-preserving distributed optimization using gradient tracking with one communication-free local update between consecutive communication rounds. Since local computation alone does not protect gradient information, we perturb both the primal state and the gradient-tracking direction at communication iterations while keeping local updates noise-free. We establish an aggregate variable for tracking, characterize the limiting consensus point in the presence of added random noise, and prove convergence in mean under certain parameter conditions. We also establish infinite-horizon pure differential privacy for the complete communication transcript under an affine objective adjacency relation and geometrically decaying Laplace perturbations. Simulation results illustrate the advantages of the proposed method in terms of the convergence and optimization accuracy given a privacy budget.

Let $Q_0:=(0,1)^n$. We define $\mathcal{B}_q(Q_0)$ by replacing the local $L^1$ mean oscillation in the Bourgain–Brezis–Mironescu space $\mathcal{B}(Q_0)$ with its $L^q$ counterpart, which coincides with $\mathrm{BMO}(Q_0)$ when $n=1$. For $n\geq2$, our results reveal different endpoint behaviors for general functions, characteristic functions and the exact perimeter formula. (i) For general functions, $\mathcal{B}_q(Q_0)=\mathcal{B}(Q_0)$ precisely when $q\in(0,\frac{n}{n-1})$. We also establish the corresponding sharp self-improvement and duality results for Hardy-type preduals. (ii) For characteristic functions, the endpoint is attained: the equivalence $\|\mathbf{1}_E\|_{\mathcal{B}_q(Q_0)}\sim\|\mathbf{1}_E\|_{\mathcal{B}(Q_0)}\sim\min\{1,P(E,Q_0)\}$ holds uniformly over measurable $E\subset Q_0$ precisely when $q\in(0,\frac{n}{n-1}]$, where $P(E,Q_0)$ denotes relative perimeter. (iii) The isotropic functional $I_{\varepsilon,q}^{*}(E)$, defined using arbitrarily oriented cubes, converges to $\frac{1}{2}\min\{1,P(E,Q_0)\}$ as $\varepsilon\to0^+$ throughout $q\in(0,\frac{n}{n-1}]$. In dimension two, the formula extends to $q\in(0,q_*]$ for some universal $q_*>\frac{19}{9}$, whereas it fails in every dimension for every $q>q_0$, where $q_0>2$ is defined variationally and numerical computation suggests that $q_0\approx2.15391$. The proofs of the three sharp self-improvements use sparse median domination, atomic block decompositions, and relative isoperimetry, respectively. For the perimeter formula, a uniform De Giorgi blow-up reduces the problem to a geometric cube-section inequality; geometric averaging and an explicit packing yield the planar extension and the obstruction, respectively.

We consider inference for network stochastic differential equations in a small-noise, high-dimensional regime, where the diffusion coefficient vanishes with $\ep\to0$ while the network size and parameter dimension may diverge. We develop a minimum-distance estimation framework based on a deterministic measurement model that may omit nuisance terms in the drift, and derive a non-asymptotic error bound decomposing the estimation error into stochastic fluctuation, nuisance discrepancy, and identifiability terms. For a linearly parametrized interaction model under repeated measurements, we obtain explicit non-asymptotic bounds on the nuisance discrepancy and the identifiability gap in terms of graph connectivity and noise level. Under suitable mixing and balance conditions, this yields consistency and convergence rates for the estimator. We further study graph recovery via adaptive Lasso. The penalized estimator is shown to inherit the rate of the preliminary estimator and to recover the true interaction graph consistently. These results provide a theoretical basis for parameter estimation and edge selection in high-dimensional network SDEs with nuisance drift components.

Let $(M^n,g)$ be a compact Riemannian manifold of dimension $n\ge 2$, possibly with nonempty convex $C^2$ boundary. We study the first nonzero eigenvalue $\lambda_1(M)$ of the Laplace–Beltrami operator, subject to the Neumann boundary condition when $\partial M\neq\emptyset$. Given a reference scalar curvature $R_E>0$, set $K_E:=R_E/(n(n-1))$, and define the traceless Ricci tensor $\mathring{\Ric}:=\Ric-\frac{R}{n}g$ and the Einstein deviation \[ \Theta:=|\mathring{\Ric}|+\frac{|R-R_E|}{n},\qquad \kappa_{\mathrm{Ein}}(p):=\left(\frac{1}{\vol(M)}\int_M\Theta^p\,dV\right)^{1/p}, \quad p>n/2. \] We say that $M$ is $\varepsilon$-nearly Einstein if $\kappa_{\mathrm{Ein}}(p)<\varepsilon$. The main result states that for every $\alpha\in(0,1)$ there exists $\varepsilon_0=\varepsilon_0(n,p,\alpha,D,R_E)>0$ such that if $\diam(M)\le D$ and $\kappa_{\mathrm{Ein}}(p)<\varepsilon_0$, then \[ \lambda_1(M)\ge \alpha\,\lambda_1(n,K_E,D), \] where $\lambda_1(n,K,D)$ is the first nonzero eigenvalue of the Bakry–Qian one-dimensional model problem with curvature parameter $K$, with the convention $\lambda_1(n,K,D)=nK$ when $K>0$ and $D\ge\pi/\sqrt{K}$. The proof combines the Bochner formula for the decomposition $\Ric=(n-1)K_Eg+\theta g+\mathring{\Ric}$ with an auxiliary function $J$ satisfying \[ \Delta J-\tau\frac{|\nabla J|^2}{J}-2J\Theta=-\sigma J \] and Neumann boundary conditions, together with Moser iteration and a maximum principle. As $\kappa_{\mathrm{Ein}}(p)\to0$, the estimate recovers the classical Bakry–Qian theorem.

Let \(\mathcal{H}\) be a complex Hilbert space, and let \(T=H+iK\in\mathcal{B}(\mathcal{H})\) be the Cartesian decomposition of \(T\), where \(H=H^{*}\) and \(K=K^{*}\). We introduce the \(L^2\)-Cartesian numerical radius \[ \omega_2(T)=\left(\frac{1}{2\pi}\int_0^{2\pi}\|\cos\theta H+\sin\theta K\|^2\,d\theta\right)^{1/2}. \] We study its basic properties and compare it with the classical numerical radius \(\omega(T)\). In particular, we establish the sharp estimate \[ \frac{1}{\sqrt{2}}\omega(T)\le \omega_2(T)\le \omega(T), \] and establish lower and upper bounds in terms of the Cartesian parts \(H,K\). As a product, we obtain \[ \frac{1}{4}\left\|T^{*}T+TT^{*}\right\|\le \omega_2(T)^2\le \omega(T)^2. \]

We prove several conjectures from the On-Line Encyclopedia of Integer Sequences about the lower and upper Wythoff sequences and the Fibonacci word. Among them are two of Kimberling's three conjectures on the number of ways to write $n=\lfloor h\varphi\rfloor+\lfloor k\varphi^2\rfloor$ with $h,k\ge1$ (A259598): exactly one way if and only if $n+1=2F$ for a Fibonacci number $F\ge2$, and exactly two ways if and only if $n+1\ge7$ is a Lucas number. The third conjecture, that no way exists if and only if $n+1$ is a Fibonacci number, was proved earlier by Kawsumarng et al. We also observe that Kimberling's conjecture on the gaps of the sums of two distinct terms of A003622 and of their complement (A333308, A333309) follows, after a shift by $2$, from earlier Walnut results of Shallit and of Bosma et al. on A260317, and we re-verify it. Next, we prove Kimberling's five 2025 conjectures on the gaps between positions where the Fibonacci word and the "Lucas word" take prescribed values (A383423-A383427). Finally, we prove a conjecture of Mathar on A285383, and we point out that a conjecture of Schmidt on A003250 follows from theorems of Carlitz, Scoville and Vaughan (1973); we also confirm it with Walnut. Most proofs are decision procedures run in the free prover Walnut, and we supply the complete command file.

We establish global well-posedness of classical solutions to the two-dimensional primitive equations with fractional horizontal dissipation for arbitrarily large initial data in the full subcritical range $1<\alpha\leq2$. Together with the known ill-posedness results for $0\leq\alpha<1$, this establishes the sharp dissipation threshold for large-data global well-posedness in the corresponding solution framework. The key ingredient is a hydrostatic energy estimate obtained by splitting the nonlinear energy into symmetric and antisymmetric parts and exploiting the commutator structure of the latter. Using anisotropy and incompressibility, we bound the nonlinear energy by the $L^\infty$ norm of the hydrostatic vorticity times a quadratic velocity norm with only one-half additional horizontal derivative. The vorticity maximum principle then yields enhanced velocity bounds, which close the vorticity estimates and verify the continuation criterion throughout the subcritical regime.

We classify nontrivial simple super-Virasoro modules finitely generated over \(\C[L_0]\) using simple Laurent Weyl algebra modules finite free over \(\C[D]\). The classified modules are finite Cartan-free with equal parity ranks. We also prove finite length without simplicity and construct explicit nonsplit self-extensions.

We introduce a quantum coding framework for discrete small-shift noise, in which errors on qudits are modeled as low-weight $X$- and $Z$-type Pauli shifts, analogous to small phase-space displacements in continuous-variable systems. This structure is approximately respected by nuclear-spin noise and captured by the Lee metric, motivating a quantum extension of classical Lee-metric coding theory. We develop such a formalism for stabilizer codes over $\mathbb{Z}_q$ for arbitrary $q$, using joint and separate Lee metrics for the $X$- and $Z$-components of errors. For qubits, the joint metric counts $Y$ errors twice and can yield codes that detect and correct $X$ and $Z$ errors with fewer physical qubits than codes designed for the conventional Hamming metric. We discuss Lee-weight spreading under Clifford gates and qubitize codes over $\mathbb{Z}_4$ via the Gray map, finding codes with two-fold transversal non-qubit-Clifford gates. For quantum CSS Lee-LDPC codes on $n$ qudits, we prove that the Lee distance cannot exceed $O(n)$, uniformly in $q$, demonstrating an unexpected obstruction to using the large internal Hilbert space of a large-$q$ qudit to make high-Lee-distance codes. Under local Metropolis dynamics, certain classical $q$-ary “helical repetition codes” have exponentially long memory times at fixed temperature for sufficiently large $q$ (that grows with system length), which can be understood as spontaneous symmetry breaking at finite temperature, even in local one-dimensional models. Hypergraph products of these helical repetition codes provide local two-dimensional quantum codes that inherit self-correction for $Z$ errors, but self-correction for $X$ errors remains an open question.

This is the second paper in our series of works to develop the min-max theory for free boundary minimal hypersurfaces in Riemannian manifolds with boundary via the Allen-Cahn equation. Building on our previous work for general critical points, we study the boundary behaviour of limit-interface integral varifolds arising from stable (or finite Morse index) solutions to the Allen-Cahn equation with Neumann boundary condition. More precisely, we prove that the limit-interface is a curvature varifold with boundary in the sense of Mantegazza satisfying a generalized stability inequality. Furthermore, we rule out classical tangent cones at the boundary and show that the first variation of the limit-interface varifold is singular with respect to its weight measure. In particular, our result confirms a folklore conjecture that the boundary concentration examples of Malchiodi, Ni and Wei have unbounded Morse index. As will be evident in our forthcoming work, the results obtained in this paper are crucial ingredients towards the optimal boundary regularity in the Allen-Cahn free boundary min-max theory.

We develop entanglement-of-formation potentials for feedback-assisted classical communication and one-way randomness distillation. For a finite-dimensional quantum channel without initial shared entanglement, an exact amortization theorem bounds the increase of message information plus entanglement remaining across the communicating laboratories. This proves a classical-feedback capacity bound using a mixed-convex-roof expression previously studied by Winter and Yang in the context of potential capacities. We determine the capacity of flagged mixtures of noiseless and entanglement-breaking channels, improve a published depolarizing-channel converse, and evaluate the new bound exactly for the amplitude damping channel. A state analogue gives a finite-blocklength upper bound on the net shared randomness obtainable with one-way classical communication. Amplitude-damping Choi states provide exact benchmarks, while a full-rank generalized amplitude-damping Choi state demonstrates a useful improvement where we do not evaluate the operational rate exactly. We also distinguish one-way from two-way randomness distillation: a two-round weak-measurement protocol strictly exceeds the exact one-way rate of every nontrivial qubit isotropic state. The protocol preserves quantum side information during its first decoding and extracts additional randomness in the reverse direction. This shows why the one-way state potential is not a general two-way converse. Together, these results establish a common entanglement-based approach to sharper communication and randomness bounds, while revealing how retained quantum correlations enable two-way distillation to surpass one-way limits.