For the reversible speed-change exclusion process $(η_t)_{t \geq 0}$ in $\mathbb{Z}^d$, we study the scaling limit of additive functionals ${Γ_t(f) = \int_0^t f(η_s)\, \mathrm{d} s}$. Concerning the local centered function $f$, the previous work [Commun. Math. Phys. 104, 1-19, 1986] by Kipnis and Varadhan and [Comm. Pure Appl. Math., 66: 649-677, 2013] by Gon{ç}alves and Jara respectively covered the cases $d \geq 3$ and $d=1$. The present paper completes the missing part $d=2$, and also develops the theory for functions with higher degree. The novelty is a quantitative homogenization of the resolvent, which allows to overcome the obstacle of correlation function in non-gradient models.

Within an important class of discrete-time 2-valued dynamical systems on $\mathbb{C}$ defined by polynomials, we extract a family of systems induced by the action of a 2-valued group. We study this 2-valued group and its analogues obtained by replacing $\mathbb{C}$ with an arbitrary field $F$. We establish that all of them are double coset groups. For some of them we prove non-cosetness. In the case $F = \mathbb{C}$, this yields the first example of a topologically non-coset, non-discrete topological 2-valued group.

We propose a novel iterative solution framework for the unsteady Stokes equations in the pseudo-stress formulation. When solving this class of problems by using implicit time-integration schemes, standard solvers suffer from deteriorating convergence properties for small time steps, independently of the chosen space discretisation method. This is due to the singular modes of the dev-dev operator. For this reason, we introduce a computational framework obtained by combining a deflated Conjugate Gradient method with a W-cycle multigrid scheme that employs a Restricted Additive Schwarz smoother. The key point is to choose the deflation subspace so that the inner system to be solved within a deflated Conjugate Gradient scheme corresponds to a Laplace problem defined on the singular modes of the original dev-dev operator. This results to be independent of the spatial discretisation method and allows one to use efficient multigrid iterative solvers. Numerical experiments show that the proposed strategy significantly accelerates the Conjugate Gradient convergence and provides stable performance with respect to the time step, confirming its robustness for solving linear systems in the pseudo-stress framework.

The class-breadth conjecture of Leedham-Green, Neumann and Wiegold states that the nilpotency class of any $p$-group is at most $b(G) + 1$, where $\displaystyle{b(G) = \max_{g\in G}\log_p[G:Z_G(g)]}$ denotes the breadth of $G$. While several counter-examples to this conjecture have been found for $p = 2$, it is still open in general for $p>2$. This article is dedicated to the general case $p>2$ of the conjecture. We propose a generalization for the case $p>2$, which we prove under some additional conditions.

In this paper, we propose a coordinate-wise polyhedral method (CPM) for cutting polyhedra with theoretical guarantees of convergence. Unlike the existing polyhedral method, which designs pairwise comparison queries by solving coupled optimization problems and performs cuts subsequently, CPM specifies coordinate-wise cuts in advance and then designs corresponding pairwise comparison queries by solving a linear system of equations. Under this framework, we show that CPM reduces the diameter of the polyhedron at a linear convergence rate. Moreover, we extend CPM to a univariate piecewise-linear utility function by representing it with its increment vector over consecutive breakpoints. We show that the Kantorovich distance between the true utility function and the estimated one obtained by CPM decreases at a linear convergence rate. Further, we extend the CPM to general nondecreasing Lipschitz continuous utility functions by piecewise-linear approximation (PLA). We introduce an adaptive-breakpoint strategy to avoid direction errors caused by PLA. We prove that the set of piecewise-linear utility functions corresponding to the ambiguity set of increment vectors converges to the true utility function as the number of queries increases, and derive an explicit bound for the approximation. Finally, to evaluate the performance of CPM, we conduct a series of numerical experiments. The results demonstrate comparable convergence to the standard polyhedral method in the case of linear multivariate utility functions. For nonlinear univariate utility functions, CPM achieves stable convergence to the true utility function, in line with the theoretical findings.

In this paper, we partially extend the Ledrappier-Young entropy formula to invariant measures of $C^1$ diffeomorphisms with dominated splittings. For such measures, we show that whenever the $i$-th Lyapunov exponent has multiplicity one, the $i$-th transverse entropy equals the product of the $i$-th Lyapunov exponent and the corresponding transverse measure dimension. Furthermore, if all intermediate non-negative Lyapunov exponents have multiplicity one, then the Ledrappier-Young entropy formula holds. As applications, we derive $C^1$ versions of numerous results in measure dimension theory, including the famous works by Ledrappier-Young, Barreira-Pesin-Schmeling, and Ledrappier-Xie.

Let $\mathcal{W}=\mathcal{W}_{n}\boxtimes\mathcal{W}_{d-n}$ be a $d$-web on $(\mathbb{C}^2,0)$, where $\mathcal{W}_n$ is an $n$-web with a totally invariant irreducible curve~$C$, and $\mathcal{W}_{d-n}$ is a regular $(d-n)$-web transverse to $C$. We show that the curvature of $\mathcal{W}$ is holomorphic along $C$ if and only if the curvature of $\mathcal{W}_n$ is holomorphic along $C$. When $\mathcal{W}_n$ is non-degenerate along $C$, we prove that $K(\mathcal{W}_n)$, and hence $K(\mathcal{W})$, is holomorphic along $C.$ We deduce that, if $\mathcal{W}_n$ is irreducible and $\mathrm{mult}\left(Δ(\mathcal{W}_n),C\right)<3(n-1),$ then $K(\mathcal{W})$ is holomorphic along $C.$ This generalizes a result of \textsc{Mar\'ın} and \textsc{Pereira}, obtained in the case where $C$ has minimal multiplicity $n-1$ in the discriminant $Δ(\mathcal{W}_n).$ If $n$ is prime or $n=4$, the condition $\mathrm{mult}\left(Δ(\mathcal{W}_n),C\right)<3(n-1)$ can be weakened to $\mathrm{mult}\left(Δ(\mathcal{W}_n),C\right)<n(n-1).$ Moreover, we describe a natural decomposition of $\mathcal{W}_n$ as the product of two subwebs $\mathcal{W}_n=\mathcal{W}_{n}^{\rm{str}}\boxtimes\mathcal{W}_{n}^{\rm{wk}}.$ Under the assumption that $\mathcal{W}_{n}^{\rm{wk}}$ is non-degenerate along $C$, we show that the holomorphy of $K(\mathcal{W})$ on $C$ is equivalent to that of $K(\mathcal{W}_{n}^{\rm{str}}).$

We further explore connections between the symplectic homology persistence module and the properties of closed Reeb orbits for star-shaped domains in higher dimensions. Our first result is that the sequence of $S^1$-equivariant spectral invariants over a field of positive characteristic is bounded from above, in contrast with the case of characteristic zero. We also prove that the dimension of the filtered symplectic homology is bounded as a function of the action whenever the flow is a pseudo-rotation, i.e., it has finitely many prime closed orbits. Finally, we show that a non-degenerate Reeb flow has infinitely many prime closed orbits whenever it has one closed orbit with negative mean index.

Inspired by a statement about Extended Frege proof systems by Jain and Jin (FOCS 2022) we prove that: - there is a p-time binary relation $\approx$ between circuits that implies their logical equivalence, - the relation $\approx$ implies that each of the two circuits can be rewritten into the other one by possibly deleting some gates and adding at most seven new gates, - if the equivalence $C \equiv D$ has a size $s$ proof in an Extended Frege or a Circuit Frege proof system then there is a chain of circuits $E_i$ $$ C = E_0 \approx \dots \approx E_t = D $$ with $t \le s^{O(1)}$.

Solving nonlinear eigenvalue problems is an important and challenging task in scientific computing. Contour integral-based approaches are attractive for such eigenvalue problems because they reliably target all eigenvalues in a prescribed domain. However, unlike in the linear case, many traditional methods of this type, such as Beyn's method, lack an inherent iterative refinement mechanism. Consequently, achieving high accuracy requires high-quality quadrature rules for approximating the contour integral, which often leads to prohibitive computational costs. A notable exception is the so-called NLFEAST algorithm, which combines contour integral techniques with a nonlinear Rayleigh--Ritz extraction step. In this work, we propose a general framework of iterative contour integral-based methods for nonlinear eigenvalue problems that includes NLFEAST. This allows us to prove linear convergence of NLFEAST under mild assumptions and also explains why certain nonlinear eigensolvers do not combine well with iterative methods. Numerical experiments confirm our theoretical findings; in particular that NLFEAST can achieve high accuracy even with a limited number of quadrature nodes, significantly outperforming Beyn's method on challenging problems.

We provide two sufficient conditions for a Busemann G-space to admit a differentiable DC atlas with a continuous Finsler metric, from the viewpoint of comparison geometry. These results generalize previous work on G-spaces with Riemannian curvature bounds, namely the Alexandrov and CAT conditions, to the Finsler setting.

We consider the problem of minimizing a difference-of-convex objective over a convex composite inequality constraint and a compact convex set constraint. To solve this problem, we extend the ESQM in [1] via incorporating a variable smoothing scheme. In essence, in each iteration of our algorithm, we apply one proximal gradient step to a smoothed penalty function, constructed based on a smooth approximation of the convex composite constraint function; and we design explicit rules to update the smoothing and penalty parameters. Under suitable constraint qualifications, we establish an iteration complexity of $O(ε^{-3})$ for obtaining an $(ε,ε)$-KKT point. Moreover, in the convex setting, we show that the whole sequence generated by our algorithm is convergent and derive its local convergence rate under a standard Hölderian growth condition.

This paper is devoted to discussing the linear tensor product problems in the worst case setting. We consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We investigate algebraic $(s, t)$-weak tractability (ALG-$(s, t)$-WT) under the absolute error criterion in the case $λ_1 > 1$, where $λ_1$ is the square of the univariate maximal singular value. We solve the problem by giving the necessary and sufficient conditions for ALG-$(s, t)$-WT on univariate singular values and fill the gap left open.

We study the inverse scattering problem for the Kerr-nonlinear Helmholtz equation \[ Δu + k^2(1+q(x)|u|^2)u = 0 \quad \text{in }\mathbb{R}^n,\; n\geq 2, \] where the aim is to recover the unknown potential $q$ from the scattering amplitude. We obtain uniqueness for full data and partial data cases of backscattering, fixed angle scattering, and fixed energy scattering. For the linear Helmholtz equation, uniqueness in backscattering and fixed angle cases are classical and largely open problems. We are able to explicitly reconstruct individual Fourier modes of the potential, and if the measured directions and energies cover an open subset, we recover $q$. The simplicity of the approach leads to an efficient numerical method, and numerical experiments show accurate reconstructions, even in the presence of noise.

We prove a sharp quantitative stability result for Alexandrov's theorem in arbitrary dimension for bounded $C^1$ open sets in a small-excess regime. More precisely, if $E\subset \mathbb R^n$ is a bounded $C^1$ open set with the same volume as the unit ball $B$, small excess, and scalar distributional mean curvature $\mathcal H_{\partial E}\in L^2(\partial E)$, then, up to a translation, $$ \operatorname{Exc}(E)+|EΔB|^2+|μ-(n-1)|^2 \le C(n)\|\mathcal H_{\partial E}-μ\|_{L^2(\partial E)}^2 \qquad \forall\,μ\in \mathbb R. $$ In other words, both the excess and the symmetric difference from the ball are controlled by the optimal $L^2$-oscillation of the mean curvature. This yields a sharp stability estimate in a genuinely non-parametric regime. The proof combines a $BV$ version of Fuglede's spectral-gap argument, a star-shaped rearrangement for sets of finite perimeter, quantitative estimates for the part of the boundary contained in the tentacles, and a polyhedral approximation argument for the non-graphical region. We note that the $C^1$ regularity assumption enters only as a qualitative technical ingredient of the proof, but all constants in the final estimate depend only on the dimension.

We develop a pathwise integration theory based on Faber--Schauder energy spaces. The approach replaces the classical Hölder--Young and finite-variation Young conditions by dyadic summability conditions expressed in terms of Faber--Schauder coefficients. On the normalized interval $[0,1]$, these conditions define Banach spaces $\mathcal{E}^p$, which we call Faber--Schauder energy spaces. For $p,q>1$ satisfying $1/p+1/q\ge1$, we prove that every pair $f\in\mathcal{E}^p$ and $g\in\mathcal {E}^q$ admits a continuous pathwise integral $I_{f,g}$, constructed from dyadic left Riemann sums. We call $I_{f,g}$ the Faber--Schauder integral, and show that it depends boundedly and bilinearly on $(f,g)$ in the corresponding energy norms. The integral satisfies additivity, integration by parts, and a dyadic Young--Loève estimate. It is also the uniform limit of classical Riemann--Stieltjes integrals of finite Faber--Schauder approximations. The Faber--Schauder integral agrees with the classical Young integral whenever the latter is available, but also applies to deterministic and Gaussian examples for which neither the Hölder--Young condition nor the finite-variation Young condition can be verified. In this sense, it provides a Faber--Schauder coefficient-based extension of Young's framework.

We continue our investigation of roots of polynomials over semirings and hyperfields, via ``pairs'' with a surpassing relation $\preceq,$ employing a which we call $\preceq$-reversibility. The ensuing results include a fundamental theorem of algebra for pairs, that tangible polynomials with enough roots ``$\preceq$-split,'' at times uniquely, into linear factors over a suitable finite extension of pairs. We also see that polynomials that agree on ``almost'' all null roots are ``almost'' equal. Finally, we obtain roots of integral polynomials over extension pairs, providing a construction of integrally closed pairs over hyperfields and over zero sum free semirings.

Let $δ_n^L$ be the maximal density of a lattice sphere packing in the $n$-dimensional Euclidean space. We explain how Boaz Klartag proved the inequality $δ_n^L \geq c n^2 2^{-n}$ where $c>0$ is a universal constant. In higher dimension, even for non-lattice sphere packings, this new lower bound is a substantial improvement. Klartag's proof uses the probabilistic method in two different ways. The first, very standard, relies on the statistical properties of a uniformly chosen random lattice. The second, completely new, studies the stochastic evolution of an ellipsoid constrained to contain non nonzero lattice points in the interior.

Let $q$ be a prime power and $n$ a positive integer. A subspace \( U \subseteq \mathbb{F}_q^n \) is called cyclically covering if the union of all its cyclic shifts covers the whole space \( \mathbb{F}_q^n \). Let \( h_q(n) \) denote the maximum possible codimension of such a subspace. When \(\gcd(q,n)=1\), we derive necessary and sufficient conditions for \(h_q(n)=0\) via Discrete Fourier Transforms, and prove this equality is equivalent to the existence of full-weight codewords in cyclic codes of \(\mathbb{F}_q^n\). We also characterize codimension-$k$ cyclically covering subspaces. Based on these results, we give a unified characterization of \(h_q(n)\) in the case where $q$ and $n$ are primes with \(n>q\) and $q$ being a primitive root modulo $n$. Specifically, \(h_2(n) \geq 2\) and \(h_q(n) = 0\) for \(q \neq 2\). We prove that \(h_3(n) \ge 1\) for every prime \(n > 3\) with odd \(\operatorname{ord}_n(3)\). Moreover, for any prime \(q > 3\), the Generalized Riemann Hypothesis implies the existence of infinitely many primes \(n > q\) such that $q$ is not a primitive root modulo $n$ and \(h_q(n) = 0\). We provide algebraic interpretations for the inequalities \(h_q(mn)\ge\max\{h_q(m),h_q(n)\}\) and \(h_q(mn)\ge h_q(m)+h_q(n)\). Using Galois descent, we prove \(h_{q^m}(n)\le h_q(n)\). Furthermore, we generalize a class of constructions that achieve the upper bound \(\lfloor\log_q(n)\rfloor\). Finally, under the Generalized Riemann Hypothesis, we obtain average lower bounds of \(h_q(n)\) for $q=2,3$.

Starting from a closed, orientable three-dimensional Riemannian manifold, we consider the completion of the associated singular Ricci flow with respect to a natural spacetime distance. We show that this completion admits canonical intrinsic metrics on its time-slices, defined by conjugate heat kernel measures, and that these time-slices arise as metric limits of the regular part of the flow. More precisely, for any $t_0>0$ and any connected component $Z_{t_0}'$ of the time-slice $Z_{t_0}$ of the completion, we prove that \[ (\overline{\mathcal R_t'},d_{g_t}) \xrightarrow[t\nearrow t_0]{\mathrm{Gromov\text{-}Hausdorff}} (Z_{t_0}',d_{t_0}^Z), \] where $\mathcal R_t'$ is the corresponding connected component of the regular part and $\overline{\mathcal R_t'}$ denotes its metric completion. In particular, this yields the Gromov-Hausdorff convergence at the first singular time for closed three-dimensional Ricci flows. We also establish a refined structure theory for the singular set of the completion. In particular, the singular set is horizontally parabolic $1$-rectifiable, and its time image has vanishing $1/2$-dimensional Hausdorff measure. Moreover, on each time-slice, the singular set has Minkowski dimension at most $1$. The proof relies on heat kernel estimates for singular Ricci flows and the generalization of the structure theory of noncollapsed Ricci flow limit spaces.

Let $M$ be a two-dimensional compact manifold without boundary and let $N$ be a compact manifold without boundary. We study the $L^2$-gradient flow of an energy functional that interpolates between the harmonic map energy and the intrinsic biharmonic map energy. The critical points of this functional are called sesqui-harmonic maps. We investigate regularity properties of this flow, generalizing Struwe's classical regularity result for harmonic maps.

We show that an Artin group $A_Γ$ of XXL type (with all defining integers satisfying $m_{ij}\geq 5$) is isomorphic to the corresponding dual Artin group for any choice of Coxeter element. Our proof involves the set of Hurwitz words $Q$ which arise from the Hurwitz action on tuples of elements of the free group. We show that the canonical epimorphism from $A_Γ$ to the dual Artin group is an isomorphism if and only if the projection from $A_Γ$ to the Coxeter group is injective on the image of $Q$. Then, using geometric properties of $Q$ and the solution to the word problem for Coxeter groups, we show this to be the case when all $m_{ij} \geq 5$.

We prove that consistently $ω_1<\mathfrak{c}$ and there exists a compact space $K$ whose Banach space $C(K)$ of continuous real-valued functions is Grothendieck, has density $ω_1$, and admits no renorming which is strictly convex or sequentially Kadets--Klee.

In this paper, we investigate the symbol error probability (SEP) of phase-squeezed M-ary phase-shift keying (M-PSK). Since the relevant observable for M-PSK detection is the optical phase, we adopt the adaptive Mark-II receiver which is a physically realizable phase measurement. First, we develop a theoretical analysis based on the phase probability operator measure (POM) of the Mark-II scheme in the Fock basis. Then, we develop two SEP methods based on the statistics of the received PSK symbol and the error introduced by the Mark-II measurement. The first method derives the phase probability density induced by the squeezed state noise and incorporates the additional Mark-II phase uncertainty through an angular convolution. Since this convolution does not admit a simple closed form, we also introduce an effective tangential-variance model, which yields a closed form SEP expression in terms of the Owen's T-function. Numerical results show that phase squeezing substantially reduces the SEP of M-PSK compared to coherent state transmission, with greater gains for higher constellation orders. Notably, for the investigated scenario, squeezing can almost double the photon efficiency of M-PSK as the mean number of transmitted photons increases. Finally, the proposed approximations closely follow the Mark-II POM analysis, typically within an accuracy of 2-4 photons, and therefore provide accurate and computationally efficient tools for analyzing phase squeezed quantum M-PSK communication.

A refined statistical understanding of LLM pre-training requires the analysis of the transformer architecture for data distributions that encapsulate key characteristics of text data. To address this, we propose a text data distribution based on an extension of the log-bilinear language model from the natural language processing literature. For this data generating process, we derive generalization bounds for deep transformer architectures, highlighting the dependence on the network architecture, the vocabulary size, the number of documents and the document length.

A (strongly) concave composition of an integer $n$ is a sequence of positive integers that is (strictly) decreasing to a point and then (strictly) increasing thereafter, such that the sum of the entries equals $n$. The value at the low point is called the center part. The difference between the number of entries before and after the low point of the sequence is referred to as the rank of the (strongly) concave composition. The rank functions $V_d(m,n)$ and $V(m,n)$ are defined as the number of concave compositions and strongly concave compositions, respectively, of $n$ with rank $m$. By constructing the difference systems that characterize the rank generating functions, we establish monotonicity properties for the rank functions of both strongly concave compositions and concave compositions for all positive integers $n$. Moreover, we also study the monotonicity properties for the rank functions of (strongly) concave compositions with fixed center parts.

In this paper, we extend Tsukamoto's recent localization formula for metric mean dimension to actions of countable discrete amenable groups, which previously applied only to $\mathbb{R}^k$- and $\mathbb{Z}^k$-actions -- by proving that the global metric mean dimension is characterized by the asymptotic entropy of pointwise $\varepsilon$-stable sets (Theorem 2.3). To achieve this generalization, we introduce equivalent definitions of the invariant using topological entropy, packing topological entropy, and Bowen's dimensional entropy, respectively. A key technical contribution is our replacement of tiling arguments with Lindenstrauss's combinatorial covering lemma, which enables us to handle the general structure of amenable groups. Furthermore, we resolve all three questions regarding uniformity raised in Section 6 of a recent paper by Yang, Chen, and Zhou by constructing counterexamples (Theorem 2.6 and Proposition 5.8), which demonstrates that the supremum and limit superior in the localization formula cannot generally be interchanged, thereby highlighting the heterogeneous nature of the convergence. These results clarify the uniformity issue and offer insights into the link between local dynamics and global invariants, while our equivalent definitions provide flexible tools for computing metric mean dimension in concrete settings.

We establish valuative criteria for separatedness and (partial) properness for morphisms of adic spaces, which account for both vertical and horizontal specializations. The lifting test diagrams used in Huber's analytic valuative criteria are naturally adapted to vertical specializations. In the non-analytic setting, one must additionally include lifting conditions for horizontal specializations. Combining these horizontal criteria with Huber's vertical criteria yields unified valuative characterizations of separatedness, universal specialization, and (partial) properness for arbitrary adic spaces.

A 3d quantum trace map is a homomorphism from the skein module of an ideally triangulated 3-manifold to its quantum gluing module that quantizes the classical trace map. There are two constructions of such maps, one by Garoufalidis and Yu in [GY], and the other by Panitch and Park in [PP1]. However, the relationship between these two constructions was unknown. We propose a third construction of the 3d quantum trace map which agrees with the one given by Garoufalidis and Yu, and extends to certain types of manifolds with ideally triangulated boundaries. Our 3d quantum trace map can be compared with that of [PP1] relatively easily by subdividing the face suspensions of [PP1] and the ideal tetrahedra of our definition into a common subdivision based on face cones. This allows us to give an exact relation between the definitions, which partially addresses the equivalence between the constructions of [GY] and [PP1].

We study the exact-basis problem for Solvability Complexity Index (SCI) computational problem families through finite-query transports. A raw finite-query reduction permits arbitrary encodings and finite transcript reconstructions, with only a continuous output decoder. For the Colbrook-Hansen (CH23) singleton-window spectral/pseudospectral block, this raw preorder collapses the expected two-source structure: the diagonal exact spectral and fixed-$\varepsilon$ pseudospectral sources are raw- and continuous-finite-query equivalent, and, for computable $\varepsilon$ under the evaluation-name representations, TTE-finite-query equivalent, so the six-problem ambient is raw-principal. We then introduce modal finite-query preorders, whose admissibility conditions may restrict encodings, decoders, reconstructions, uniformity, and geometric naturality. We also characterize TTE finite-query transport as computable point transport with a uniform finite interface trace; after forgetting the trace this gives strong Weihrauch reducibility, and the implication is strict. Under a CH23 geometric modality generated by representation inclusions, unitary and graph relabelings, and neutral stabilizations, the same ambient has exactly two minimal exact sources. This gives a calibrated reformulation of the exact-basis problem: natural SCI families should be classified by modality-indexed exact bases and refinement maps, not by one raw preorder alone.