Papers

With the advance of measurement technologies and increasing computing power, large spatial data with heterogeneous structures are often collected over high-dimensional domains. Existing Gaussian process (GP) models and computational strategies are often inadequate for analyzing such datasets in multi-dimensional domains. To address these challenges, we develop a Bayesian residual tree GP methodology called ResTGP for large spatial data with potentially heterogeneous structures in multi-dimensional domains. The key idea is to decompose a Gaussian process at a cascade of resolutions along a dyadic tree through iteratively computing predictive and residual processes so that the residual process on each tree node, both interior and leaf, becomes sufficient for the finer-level dependency within that node. This allows characterization of the underlying covariance structure in a flexible, multi-scale manner while achieving divide-and-conquer on the data domain, which leads to computational efficiency. To allow efficient tree inference, we introduce a computational strategy for Bayesian inference based on recursive message passing, which scales linearly with the sample size given the tree. This paper also proves posterior consistency of the model for estimating continuous functions in a nonparametric regression framework. Extensive numerical examples and the storm surge application confirm the advantages of the proposed method.

We study how spectral averaging interacts with the logarithm in large-dimensional rank-one Harish-Chandra–Itzykson–Zuber (HCIZ) integrals. For independent bi-unitarily invariant matrices $A,B$ with uniformly bounded deterministic singular values and weakly convergent empirical spectral measures of their Gram matrices, the integral for $(BA)^*(BA)\otimes\mathbf{1}_N$ has two limiting descriptions at speed $N^2$ for small real parameters: averaging its logarithm gives free multiplicative convolution, whereas taking the logarithm after averaging gives classical multiplicative convolution. We also prove an exact finite-dimensional equality between this annealed integral and the one for $(A^*A)\otimes(B^*B)$. For every fixed number of factors $k\geq3$, we construct counterexamples showing that the analogous equality can fail even asymptotically. For tensor products with any fixed number $k$ of factors, global Haar vectors at speed $N^k$ and product-state vectors at speed $N$ yield classical and free multiplicative convolution, respectively, for small real parameters under the corresponding hypotheses. We also derive the known multiplicativity of the $S$-transform.

In this paper, we propose a highly efficient and accurate method, called “lift-and-decoupling generalized transferable neural network" (LD-GTransNet), for numerical solution of elliptic and moving interface problems. Our method makes use of the multi-layer GTransNet approach and incorporates a novel {lift-and-decoupling} strategy, in which the hidden-layer neuron parameters are preset and the physical coordinates and the added auxiliary variable are processed by separate subnetworks. Such design enables a unified global approximation of piecewise smooth solutions without explicit domain decomposition, while preserving network expressivity and avoiding the redundancy inherent in conventional {lift} approaches. The resulting formulation reduces to a linear least-squares problem, thereby eliminating the need of nonlinear training and ensuring the high computational efficiency and accuracy. Extensive numerical experiments on benchmark interface problems in two and three dimensions demonstrate that, compared with existing neural network-based and traditional numerical methods, our proposed LD-GTransNet consistently delivers superior performance, particularly for problems with high-contrast coefficients and complex interface geometries.

Bhatt-Morrow-Scholze motivic filtration on topological Hochschild homology ($THH$) is extended to Azumaya algebras over quasisyntomic rings. We prove the Azumaya descent theorem: $THH$ of such an algebra is equivalent, as an underlying spectrum, to $THH$ of its center, normalized on $\pi_0$ by the reduced trace instead of the unit map. This comparison $\Phi_A$ is not shown to be étale-locally cyclotomic. But what is cyclotomic étale-locally is a separate Morita trace, agreeing with $\Phi_A$ only on $\pi_0$. The resulting transported filtration is exactly the center BMS filtration, well defined up to the automorphism ambiguity of $\Phi_A$. It is insensitive to the Brauer class, reflected instead by $K$-theoretic and module-categorical pairings $THH$ does not see. We establish the analogous statement for derived Azumaya algebras over connective $\mathbb{E}_\infty$-rings, under a connectivity hypothesis.

Let $C^1(I)$ denote the space of all continuously differentiable real-valued functions on the closed unit interval $I=[0,1]$. For $1\le p\le\infty$, we equip $C^1(I)$ with the norm \[ \Vert{f}\Vert_{(p)} := \begin{cases} \left(|f(0)|^p+\Vert{f'}\Vert_{I}^p\right)^{1/p}, & 1\le p<\infty, \\ \max\{|f(0)|,\Vert{f'}\Vert_{I}\}, & p=\infty, \end{cases} \qquad (f\in C^1(I)), \] where $\Vert{\cdot}\Vert_I$ denotes the supremum norm on $I$. Let $\mathcal{S}_p^+:=\{f\in C^1(I):f(0)\ge0,\,f'\ge0,\,\Vert{f}\Vert_{(p)}=1\}$ be the positive unit sphere of $C^1(I)$. A map $T:\mathcal{S}_p^+\to \mathcal{S}_p^+$ is called norm-additive if \[ \|T(f)+T(g)\|_{(p)}=\|f+g\|_{(p)} \qquad (f,g\in \mathcal{S}_p^+). \] For $1<p\le\infty$, we prove that every surjective norm-additive map $T\colon\mathcal{S}_p^+\to\mathcal{S}_p^+$ is of the form \[ T(f)(t) = f(0)+\int_0^t f'(\tau(s))\,ds \] for all $f\in\mathcal{S}_p^+$ and $t\in I$, where $\tau\colon I\to I$ is a homeomorphism. Conversely, every map of this form is a surjective norm-additive map. Consequently, every such map extends to a surjective linear isometry from $(C^1(I),\Vert{\cdot}\Vert_{(p)})$ onto itself. For $p=1$, however, we construct a surjective norm-additive map on $\mathcal{S}_1^+$ which is not of the above form and admits no linear extension to $C^1(I)$, showing that the assumption $p>1$ is essential.

We consider the existence of nonradial sign-changing solutions to the problem $\Delta u-\mu u+|u|^4u=0$ with zero Neumann boundary conditions in the unit ball for an arbitrary fixed parameter $\mu>0$. We prove that, for every $\mu>0$, this problem admits infinitely many distinct nonradial sign-changing solutions. More precisely, for every sufficiently large even integer $K$, we construct a solution by gluing $K$ symmetrically arranged, suitably transformed copies of a sign-changing entire solution of the critical equation in $\mathbb{R}^3$. The construction combines an inner–outer gluing scheme with a finite-dimensional reduction. A detailed expansion of the reduced energy allows us to locate an interior critical point in the admissible parameter region and thereby complete the construction.

An algebra is called binary Zinbiel if every subalgebra generated by at most two elements is a Zinbiel algebra. Over a field of characteristic zero, we prove that every finite-dimensional binary Zinbiel algebra is nilpotent. We also show that a subvariety of binary Zinbiel algebras is nil if and only if it does not contain the variety of all Zinbiel algebras. We further prove an analogue of Itô's theorem for binary Zinbiel algebras.

The filtering problem asks to estimate a signal process, the initial state of which is unknown, given its partial and noisy observations. From a probabilistic perspective, the best estimate at each time is the optimal filter, which is the conditional law of the signal given all observations up to the time. The problem of asymptotic stability of the optimal filter, which asks whether the filter initialized with the incorrect initial law converges to the correctly initialized filter, is of fundamental practical and theoretical importance. This paper studies the quantitative asymptotic stability of the optimal filter for signal-observation models with linear drift and additive Brownian noise. When the initial law is additionally assumed to be Gaussian, the optimal filter reduces to the celebrated Kalman-Bucy filter, for which the detectability condition on the signal and observation matrices is the sharp condition ensuring asymptotic stability. In the linear setting where the detectability condition holds but the initial law need not be Gaussian, we make the following contributions by studying the Zakai equation satisfied by the unnormalized filter. First, we study the asymptotic behavior of the deterministic part of the Zakai equation in the general case with nonlinear drift, and place the detectability condition in this context. Second, in the case of linear drift, we characterize the solution of the Zakai equation (hence the optimal filter) by a gauge transformation that separates the finite-dimensional stochastic dynamics from the infinite-dimensional deterministic dynamics. Third, we use this representation to establish asymptotic exponential filter stability in the $\chi^p$-divergence (with respect to a reference optimal filter), the total variation norm, and the $2$-Wasserstein distance.

We study the expected aperiodic autocorrelation sidelobes of random communication symbols transmitted through a unitary modulation matrix. We consider proper sub-Gaussian constellations, namely those with fourth absolute moment $\mu_4<2$ and zero pseudo-variance, including the commonly used uniform square quadrature amplitude modulation (QAM) and uniform phase-shift keying (PSK) constellations. For every prescribed lag, we determine the exact minimum expected sidelobe level (ESL) and a unitary modulation attaining it by applying a normalized inverse discrete Fourier transform (IDFT) separately on each chain of sample indices spaced by that lag. The full orthogonal frequency division modulation (OFDM) modulation attains this minimum if and only if the lag divides the block length $N$. Combining this criterion with the complete characterization of the minimizers at lag one, we show that a unitary modulation minimizes the ESL simultaneously over the first $r$ positive lags, called the prefix $\{1,\ldots,r\}$ with $1\le r<N$, if and only if every lag in that prefix divides $N$. In particular, simultaneous optimality over all positive lags is impossible for $N\ge 3$. We also prove that OFDM globally minimizes the partial expected integrated sidelobe level (EISL), defined as the sum of the ESLs over a selected subset of positive lags, for every such prefix. This also resolves the full EISL conjecture when $r=N-1$.

Dynamic programming for nonstandard infinite-horizon criteria requires more than finite stream values: one-step control substitutions must remain evaluable, and continuation information must propagate recursively. For expected stage-reward streams, we separate domain admissibility, finite-dimensional continuous autonomous realizability, scalar sufficiency, and contextual order consistency. For absolutely summable weighted criteria, we prove that the minimum Euclidean dimension of an exact continuous autonomous continuation state equals the temporal Hankel rank. Thus finite Hankel rank exactly characterizes when temporal nonstationarity can be replaced by finite endogenous state augmentation; the lower bound applies even to nonlinear continuous encoders and already holds in a one-state MDP. For literal Candelpergher Ramanujan summation, the same common-continuation test fails earlier: changing only the current action can leave the evaluation domain. Abel summation provides a contrasting prefix-closed benchmark.

We prove both Erdős-Ulam conjectures on monochromatic union-closed families. Every two-colouring of the subsets of a finite set contains a monochromatic union-closed family whose size grows faster than any fixed power of the size of the set, while suitable colourings admit no monochromatic union-closed family of exponential size. Both results hold for any number of colours and are verified in Lean.

Inverse design has been transformative in nanophotonics, providing an automated means of realizing high-performance, non-intuitive devices. However, the primary bottleneck of existing inverse design approaches is their reliance on time-consuming and computationally expensive full-wave electromagnetic simulations. In this work, we extend the Precomputed Numerical Green Function (PNGF) method for the first time to nanophotonic inverse design, accelerating evaluation of the forward problem by multiple orders of magnitude with no loss in accuracy compared to the full-wave solution. After a single, fully parallelizable precomputation step, both the objective function and its gradient can be evaluated through simple calculations whose cost scales linearly with the size of the design region. A low-rank matrix update technique further reduces the cost of objective function evaluation, yielding sub-millisecond computation times per iteration. Design examples using both gradient-based level-set optimization and tile-flipping direct binary search often achieve more than three orders of magnitude speedup in design time, resulting in an ultrafast photonic inverse design platform that can design new devices from scratch in seconds.

Under mild mass or energy assumptions, we prove uniform $L^\infty$ bounds for all distributional solutions of a multi-dimensional system of nonlocal semilinear equations. The setting comprises Ginzburg-Landau systems and Allen-Cahn equations, and the results obtained are new even for scalar equations.

The Beardwood-Halton-Hammersley theorem characterizes the asymptotic length of the optimal Euclidean traveling salesman tour through independent and uniformly distributed random points $X_1,\ldots,X_n$ in the unit square. It states that there exists a universal constant $\beta$ such that the length of the shortest tour is asymptotic to $\beta \sqrt{n}$ almost surely. The best bounds established to date are $0.6277\leq \beta \leq 0.90367$. In this paper, we improve these bounds to $0.6421\leq\beta\leq0.8810$. Using importance sampling, we further show that $0.6536\leq \beta\leq 0.8749$ holds with probability at least $1-2\times 10^{-4}$. Here the probability is taken with respect to the randomness of the sampling procedure.

We establish several general results on the categorical dynamics of abelian varieties. (1) We prove the exponential and polynomial Gromov–Yomdin equalities for every autoequivalence of the derived category of an arbitrary abelian variety. The exponential equality extends earlier results of Kikuta for elliptic curves and of Yoshioka for abelian surfaces and simple abelian varieties. (2) We prove the categorical polynomial entropy of every autoequivalence is bounded above by $g^2$, where $g$ is the dimension of the abelian variety. This gives a categorical generalization of the polynomial log-volume growth of Lin–Oguiso–Zhang and recovers their upper bound as a direct consequence. (3) We identify the shifting number of every autoequivalence of an abelian variety with the pullback of the normalized Barge–Ghys symplectic translation number on the universal cover of $\text{Sp}(4g,\mathbb{R})$. (4) We prove that the set of reduced shifting numbers is finite if and only if the nef cone is rational polyhedral. Moreover, when this set is infinite, there exists an autoequivalence with transcendental shifting number. In particular, this gives a negative answer to a question of Dimitrov–Haiden–Katzarkov–Kontsevich concerning the algebraicity of categorical entropy functions.

Let $n$ be even, let $\zeta$ be a primitive $n$-th root of unity, and let $\Delta_n$ be the set of derangements in the affine group $\overrightarrow{GL}(\mathbb{Z}/n\mathbb{Z})$. We evaluate the Wang–Sun type sum \[ S_n=\sum_{\pi\in\Delta_n}\mathrm{sign}(\pi) \prod_{x\in\mathbb{Z}/n\mathbb{Z}} \frac{1+\zeta^{x-\pi(x)}}{1-\zeta^{x-\pi(x)}}. \] The main step is an arithmetic grouping: for an affine map $\pi(x)=vx+u$, the product depends only on $\sigma=\gcd(1-v,n)$ and on the residue class of $u$ modulo $\sigma$. After summing the translation signs, the full sum becomes a sum over even divisors with explicit local multiplicities. The only remaining analytic blocks are alternating even cotangent power sums; a recent formula of Liu and Xin then gives a closed expression in Bernoulli polynomials and universal coefficients. Moreover, using results by Cvijović and Klinowski we can evaluate the cotangent power sums using exclusively rational arithmetic. This supplies a general formula for the full affine sum whose initial values were computed using the definition only in an earlier work. The formula also lets us prove some general patterns of the sum: it has integral value for powers of $2$, prime denominator for $2p$ for primes $p>3$ and it has alternating signs.

The Muon optimizer applies spectral orthogonalization to matrix-valued updates and has shown strong performance in large-scale neural network training, yet the mechanisms of this transformation in feature learning remain poorly understood. In this work, we investigate this question through a tractable factual-recall model, where a fact maps each subject-relation pair to an answer, and a linear transformer learns the subject- and relation-dependent information required to recover this mapping. The transformer is optimized with gradient flow (GF), spectral GF, or Sign GF, which are continuous-time limits of gradient descent, Muon, and Adam, respectively. Prior studies (Nichani et al., 2025) have shown that when the number of subjects exceeds the number of relations, GF learns relation-dependent information before subject-dependent information, producing a feature-separation phase during training. We characterize this separation with the learning times when the subject- and relation-dependent components of the prediction reach a target accuracy. With $S$ subjects and $R$ relations, GF has a learning-time ratio of $\widetilde{\Theta}(\sqrt{S/R})$, whereas Spectral GF reduces this ratio to $\widetilde{\Theta}(1)$. In addition, for fixed $S$ and $R$, the subject- and relation-dependent errors decay as $1/(T\log T)$ in training time $T$ under GF, but as $\exp(-\mathrm{poly}(T))$ under spectral GF. Finally, we show that GF and spectral GF are equivariant under orthogonal transformations of the token embeddings, whereas Sign GF is not: Different orthonormal embeddings can potentially produce no feature separation, a large feature-separation phase, or even a reversed learning order. These results provide a mechanistic view of how spectral orthogonalization can fundamentally reshape feature-learning dynamics.

We study the decomposed Möbius energies $E_1$ and $E_2$ at the round circle and determine their complete normal Hessians: $H_1(C)=2\pi(|D_s|^3-|D_s|)I_2$ and $H_2(C)=-(4\pi/3)(|D_s|^3-|D_s|)I_2$. Thus $H_1(C)=3H_E(C)$ and $H_2(C)=-2H_E(C)$, where $H_E(C)$ is the Hessian of the full Möbius energy. This exact proportionality singles out the Möbius-invariant combination $F=2E_1+3E_2+2\pi^2$. We prove that $F(C)=0$ and that its first three derivatives vanish on the full space of parametrized variations. For scalar binormal variations, the fourth variation has a Fourier tensor with complete $3+1$ resonance cancellation and finite $2+2$ fixed-sum blocks. Their positivity follows from an explicit weighted-difference tridiagonalization and a closed determinant formula. For arbitrary normal variations in $\mathbb{R}^3$, Möbius invariance in $\mathbb{R}^4$ reduces the radial component to a second binormal direction. The resulting two-component quartic form splits into trace, symmetric-traceless, and antisymmetric blocks, which are positive on the active coordinates. Consequently, $D^4F(C)[u,u,u,u]\geq 0$ for every smooth real normal field $u$, with equality exactly on the normal tangent space of the Möbius family of round circles. This yields a fourth-order nondegeneracy phenomenon that is not determined by the total-energy Hessian alone.

We study the query complexity of finding approximate stationary points under higher-order smoothness using only function values or only gradients. Our results cover different dimensions, oracle models, randomization, and constraints. For constant dimension $d$ and $C^{p,\beta_p}$ objectives, with $\nu:=p+\beta_p$, we give deterministic function-value gradient trapping algorithms with query complexities $O(\varepsilon^{-(d-1)})$ on $\mathbb R^d$ and $O(\varepsilon^{-(d-1)/\nu})$ on $[0,1]^d$. High-order line integration enables gradient-only trapping. For two-dimensional $C^{1,1}$ objectives, randomized integration yields $O(\varepsilon^{-4/3})$ gradient queries on $\mathbb R^2$, breaking the deterministic gradient-only $\Theta(\varepsilon^{-2})$ barrier. For general dimension, we introduce a trapping-or-descent framework that lazily reuses boundary models. For $C^{p,\beta_p}$ objectives with $p\ge2$, it achieves deterministic query complexities $O(d^p+d^{p-1+\beta_p/(\nu-1)} \varepsilon^{-\nu/(\nu-1)})$ using function values and $O(d^{p-1}+d^{p-2+\beta_p/(\nu-1)} \varepsilon^{-\nu/(\nu-1)})$ using gradients. These bounds apply to approximate stationary points on $\mathbb R^d$ and approximate KKT points on $[0,1]^d$. Their dimension-independent accuracy exponent matches the classical AR$p$ exponent $(p+1)/p$ when $\beta_p=1$. For sufficiently high dimension, we prove randomized lower bounds for gradient-only methods. For $C^{1,1}\cap C^{p,1}$ objectives, our lower bound is $\Omega(\varepsilon^{-(3/2+1/(2p))})$, matching known upper bounds for $p=2$ and, up to logarithmic factors, for $p=3$. Without the $C^{1,1}$ assumption, we obtain an $\Omega(\varepsilon^{-5/3})$ lower bound for every fixed $p\ge3$.

Let $w_0$ be the positive homoclinic solution of $-w''+w=w^p$ and set $c_p=w_0(0)$. We study bounded positive solutions of \[ -\Delta v+v=v^p\quad\text{in }\mathbb R^N_+, \qquad v=c_p\quad\text{on }\partial\mathbb R^N_+, \qquad v(x',x_N)\to0 \] uniformly as $x_N\to\infty$. This is the threshold boundary value left open by Fernández and Weth [Math.\ Ann.\ 383 (2022), 361–397]. We prove a dimension-dependent rigidity/nonuniqueness picture. For every $p>1$, the profile $w_0(x_N)$ is the unique bounded positive solution when $2\le N\le5$, whereas for $N\ge8$ there is a one-parameter family of bounded positive non-one-dimensional solutions. In the critical dimension $N=7$, nonuniqueness holds whenever a cubic coefficient $\kappa(p)$ of the exact reduced nonlinearity is positive; in particular $\kappa(p)>0$ for all $p\ge2$, and continuity extends this range below $2$. The corresponding amplitudes have Fowler ends, with their limiting energy, neck size, bulge size, and logarithmic period determined to leading order. The case $N=6$, as well as the complementary range in $N=7$, remain open. The proof combines a boundary-adapted Modica estimate, projection onto the kernel of the one-dimensional linearized operator, an automatic tail non-concentration estimate, and a nonlinear cell problem yielding the exact reduced nonlinearity. For $N\ge8$ the reduced linearization is invertible in weighted spaces. In $N=7$ the critical resonance is bypassed by solving the projected radial equation nonlinearly and closing the transverse equation by a Schauder–Tychonoff fixed point.

In their seminal work, F. Ricci and E. M. Stein [Proc. Nat. Acad. Sci. USA 83, 1–3 (1986)] introduced the oscillatory singular integral \[ T_Pf(x)=\mathrm{p.v.}\int_{\mathbb{R}^n} e^{iP(x,y)}\frac{\Omega(x-y)}{|x-y|^n}f(y)\,dy, \] and S. Chanillo and M. Christ [Duke Math. J. 55, 141–155 (1987)] established its weak-type $(1,1)$ inequality for smooth $\Omega$. For the rough variant, however, the weak-type $(1,1)$ inequality remained a long-standing open question. In the present paper, we settle this problem: specifically, we prove that the rough oscillatory singular integral is of weak type $(1,1)$ with $\Omega\in L\log^+L(\mathbb{S}^{n-1})$. Moreover, we establish weak-type $(1,1)$ bounds for the rough maximal oscillatory operator associated with $T_P$, a result of independent interest.

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let $\mathcal{C}_N\mathfrak{g}=\mathfrak{g}\otimes (\mathbb{C}[t]/t^{N+1}\mathbb{C}[t])$ denote the $N$-th Takiff algebra of $\mathfrak{g}$. We introduce a notion of parabolic subalgebras of $\mathcal{C}_N\mathfrak{g}$ and classify them in terms of $(N+2)$-partitions of a simple root system of $\mathfrak{g}$. We then study the corresponding parabolically induced $\mathcal{C}_N\mathfrak{g}$-modules and establish a necessary and sufficient criterion for their irreducibility, extending Jantzen's criterion for parabolically induced $\mathfrak{g}$-modules. As an application, we derive character formulas for almost all irreducible Harish-Chandra modules in Chari's category $\widetilde{\mathcal{O}}$ over the (centerless) affine Lie algebra associated with $\mathfrak{g}$. In particular, when $\mathfrak{g}=\mathfrak{sl}_3(\mathbb{C})$, we obtain explicit character formulas for all irreducible Harish-Chandra modules in this category.

We develop a method for lossy compression of incompressible (i.e., divergence-free) velocity fields that preserves zero divergence and, on periodic domains, periodicity. We fit the nodal velocities with property-preserving matrix-valued kernels and compress their coefficients using standard lossy compressors. Every decompressed coefficient vector defines a field with the same constraints, independently of the compression tolerance. We present several fitting schemes designed around either low-rank representation of or sparsity in the interpolation matrix. More specifically, we show results for global Matérn interpolation, a two-level Nyström approximation, Nyström–Wendland residual interpolation, multilevel Wendland interpolation and a property-preserving (redundant) kernel frame. We derive nodal error bounds that relate the compressor tolerance to the additional velocity error. Experiments with four compressors on cavity, Taylor–Green, and species-transport flows verify the bounds under their coefficient-error assumptions and compare storage at matched velocity accuracy. The low-rank methods give compact representations when they resolve the velocity field sufficiently accurately; residual, multilevel, and frame interpolation extend the attainable accuracy when they do not. We also measure changes in turbulent velocity statistics. The resulting representations support an accuracy–compression tradeoff while preserving incompressibility and periodicity analytically.

The space of $n\times n$ density matrices is the full state space of an $n$-level quantum system. While it has been asserted in the literature on quantum information theory that decomposing this state space into equirank pieces yields a Whitney stratification, rigorous proofs have remained absent. Using methods from invariant theory, this work provides a proof that the rank decomposition of the matrix algebra constitutes a (b)-regular stratification in the sense of Whitney. Furthermore, we demonstrate how the Uhlmann connection - which describes the adiabatic holonomy of mixed states and reduces to the Berry connection on pure states - can be geometrically treated as a stratified connection on a stratified vector bundle. Finally, our appendix provides a self-contained exposition of stratified fiber bundles and the new concept of stratified connections, which may be of independent interest.

We prove that a Cohen-Macaulay standard graded integral domain over a field is almost Gorenstein as a graded ring if and only if the completion of its localization at the graded maximal ideal is almost Gorenstein as a local ring.

Let $f:S\to C$ be a relatively minimal semistable fibration of genus $g\ge2$ over an algebraically closed field of characteristic $p>0$ with smooth and geometrically connected generic fiber. Let $\Sigma\subset C$ denote the set of points over which the fibers are singular, with $s=|\Sigma|$. If the total surface $S$ is Hodge Witt and $s\ge2$, we prove the classical Arakelov inequality $$ \text{deg}\, f_*\omega_{S/C}\le \frac g2 \text{deg}\, \Omega_C^1(\log\Sigma), $$ where $\omega_{S/C}$ denotes the relative dualizing sheaf. For $C=\mathbb P^1$, we obtain the stronger estimate $$ \text{deg}\, f_*\omega_{S/\mathbb P^1}\le \frac g2(s-2)-\frac12 b_1(S), $$ where $b_1(S)$ is the first Betti number of $S$. We also construct a genus-$2$ semistable fibration over $\mathbb P^1$ with exactly $4$ singular fibers in characteristic $5$. Its total surface is Hodge Witt, and $$ \text{deg}\, f_*\omega_{S/\mathbb P^1}=2=\frac g2(s-2). $$ Thus Nguyen's lower bound $s\ge4$ and the Hodge–Witt Arakelov inequality are both sharp. As applications, we derive a canonical-class inequality and a Szpiro-type inequality with coefficients linear in $g$.

We prove the following conjecture of Zhang (arXiv:2604.10779, Conjecture 6.1): for $n \geq 0$, the number of $2$-stack-sortable permutations of $\{0,1,\dots,n\}$ ending in $0$ is the $n$th Motzkin number. By Zhang's result, there is a bijection between $2$-stack-sortable permutations ending in their least element and standard composition tableaux of width at most $2$. We then show bijectively that there are an equal number of these and standard Young tableaux of width at most $3$, which are known to be counted by the Motzkin numbers.

We study $-\Delta_p u = |u|^{q-1}u$ in $\mathbb{R}^N$ with $N>p\ge 2$, for solutions that are stable outside a compact set, with no assumption of sign, boundedness or symmetry. Damascelli, Farina, Sciunzi and Valdinoci settled the subcritical range for $p>2$, and treated the supercritical range $p^*-1<q<q_c(N,p)$ only for radial solutions, stating that without radial symmetry they were not able to conclude. The obstruction is that the arguments available for $p=2$ rest on a monotonicity formula with no monotone analogue when $p\ne2$. We remove the radial hypothesis: every $C^1$ weak solution stable outside a compact set is trivial when $p^*-1<q<q_c(N,p)$. The whole range $p-1<q<q_c(N,p)$ with $q\ne p^*-1$ is thereby settled, as it is for $p=2$ by Farina's theorem, and every finite Morse index solution in it is trivial. In its place we iterate the comparison principle against an explicit two-parameter supersolution, upgrading the decay forced by stability to that of the $p$-harmonic fundamental solution; this step uses no stability, only smallness of the self-similar decay, and is stated on its own. The upper threshold is sharp: for every $q>q_c(N,p)$ we exhibit positive bounded radial solutions stable outside a compact set. At the Sobolev endpoint, every $C^1$ solution stable outside a compact set has finite energy; for $p>2$ this removes the a priori $\mathcal D^{1,p}$ assumption from the low-Morse-index classification of Farina, Mercuri and Willem and answers their question on the Morse index of the Aubin–Talenti extremals: it is one. At the upper endpoint, $q_c(N,p)$ is the exact threshold for a nonzero homogeneous weak solution on $\mathbb R^N\setminus\{0\}$ to be stable outside a ball, and at the threshold there are exactly two.

In this paper, we consider the sixth-order Boussinesq equation with a cubic nonlinearity, \[ u_{tt}-u_{xx}+k u_{xxxx}-u_{xxxxxx}+\left( u^3 \right) _{xx}=0, \ \ \mbox{with }k=\pm 1,\] posed on both $\mathbb{R}^+$ and $\mathbb{R}$. The local wellposedness are improved in related Bourgain-type spaces, $X^{s,b}$, for $s>-\frac12$ with $b=\frac12$ and $b>\frac12$ accordingly (See. \cite{esfahani2025well,zhong2024local}). The key ingredient is to refine the trilinear estimate on Burgain type space by adapting ideas from Tao's multiplier method, interpolation theory and our early work on IBVPs of dispersive equations \cite{li2020low,tao2001multilinear}.

In this paper, we continue the study of Himonas and Yan\cite{himonas2024schrodinger,himonas2026higher,himonas2024higher} on the nonlinear Schrödinger equation with a dispersion of order $2m$ and cubic nonlinearities, \[ iu_t+\left( -1 \right) ^{m+1}\partial _{x}^{2m}u=N_k\left( u,u,u \right), \] where, $m\ge1$ being an integer, \[N_0\left( u,u,u \right) =uuu,\quad N_1\left( u,u,u \right) =\bar{u}uu,\quad N_2\left( u,u,u \right) =\bar{u}\bar{u}u,\quad N_3\left( u,u,u \right) =\bar{u}\bar{u}\bar{u},\] We derive trilinear estimates at lower regularity and thereby prove that the cNLS-2m on the half-line is well-posed at the optimal regularity $s=-\fr{m-1}{2}$. This improves the previous result, $s>-\fr{m-1}{2}$, and answers an open question left in \cite{himonas2026higher}. In addition, for $k=0,2$, we establish well-posedness at $s=-\fr{m-1}{2}$ as well. Moreover, for $k=3$, the nonlinearity exhibits a stronger resonance relation, which implies well-posedness for $s>-\fr{2m-1}{3}$. Our derivation of the trilinear estimates relies on varieties of Strichartz estimates, which differs from the $[k;Z]$-multiplier norm method employed in \cite{himonas2026higher}.