Papers

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 $A_\Gamma$ 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 $A_\Gamma$ to the dual Artin group is an isomorphism if and only if the projection from $A_\Gamma$ 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 $\omega_1<\mathfrak{c}$ and there exists a compact space $K$ whose Banach space $C(K)$ of continuous real-valued functions is Grothendieck, has density $\omega_1$, and admits no renorming which is strictly convex or sequentially Kadets–Klee.

In modern machine learning, parallelization of training is an important strategy for increasing scale. Asynchronous stochastic gradient descent (ASGD), which maximizes the utilization of available hardware by avoiding waiting for slow workers. However, with constant step sizes, the convergence of ASGD is nonetheless affected negatively by slow workers due to large delays in updates. At the same time, it has been empirically observed in asynchronous training of deep learning models that gradient clipping "stabilizes" training. In this work, we provide a theoretical justification for this behavior, as we show that clipping removes the dependence of the maximum delay in the oracle complexity. We employ a sub-Weibull model of gradient noise which generalizes sub-Gaussian and sub-exponential distributions to more heavy-tailed distributions, motivated by empirical observations in deep learning. We show convergence in expectation, and the first time in asynchronous optimization, convergence with high probability.

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.

We prove that two fixed sources in the Euclidean plane can be realized by a bounded-degree planar unit-edge graph on a 10-net, with graph distance from each source agreeing with Euclidean distance up to a universal additive constant. We ask whether the analogous statement holds for three non-collinear sources, and prove a logarithmic obstruction for large ordered source sets in the coordinate-planar setting.

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 propose a model for evaluating the performance of wireless communication networks beyond the ubiquitous full-buffer assumption, under which every transmitter is always active. The network is represented by N interacting queues arranged on a torus, with homogeneous arrival rate and service rates depending on the activity of neighboring interferers. More precisely, each queue is associated with a transmitter-receiver pair, and its service rate is given by the Shannon capacity, which depends on the corresponding Signal-to-Interference-plus-Noise Ratio (SINR). Since interfering transmitters only emit when their queue is non-empty, the SINR, and hence the service rate, improves when neighboring queues are empty. We first derive the stability region of the system. To investigate the stationary regime, we introduce two mean-field approximations. The first one is obtained from a finite K-replica system, for which we prove propagation of chaos as K goes to infinity. The second one describes a queueing system evolving in a suitably chosen environment. We derive quantitative estimates comparing these two approximations. Finally, we prove that the original interacting queueing system converges exponentially fast to stationarity.

The solution of large, sparse linear systems often dominates the computational effort of scientific applications and is a frequent optimization target. Modern libraries provide numerous solver and preconditioner configurations, but their performance varies significantly across problem instances. Previous works have addressed the selection of an optimal solver, but are typically limited by the problem set addressed (e.g., only symmetric positive definite matrices), the use of expensive matrix features, or the complexity of the approach. This work proposes a modular, low-cost embedding-based framework for solver selection that decouples performance modeling from feature representation and downstream prediction. Solver-problem relationships are learned directly from observed performance data, while inexpensive numerical features are used to project unseen problems into the learned embedding space. The framework focuses on multilabel prediction and evaluation using user-centric metrics, such as MAPE and nDCG, which better reflect relative performance. Experiments on 621 matrices from the SuiteSparse matrix collection across 101 PETSc solver configurations demonstrate a 17% increase in top-prediction accuracy over classical feature-based models when expensive numerical features are included, along with reductions of 37% in mean average percentage error (MAPE) and 46% in top-prediction error (1-error). When restricted to a reduced feature set, the embedding approach remains competitive, while still consistently achieving ca. 24% lower MAPE and 1-error across a broad range of problems.

We investigate the extension of the Kubo–Martin–Schwinger (KMS) thermal equilibrium condition to bounded non-Hermitian Hamiltonians with real spectra and biorthogonal eigensystems, providing a unified framework through three complementary constructions: a complete KMS theorem under quasi-Hermiticity, a biorthogonal KMS-type identity whose positivity characterises quasi-Hermiticity, and a quantum-detailed-balance condition for the associated open-system dynamics. Our main result is a thermodynamic characterisation of quasi-Hermiticity: for any diagonalisable $H\in M_d(\mathbb C)$ with real spectrum, the biorthogonal Gibbs functional $\omega_{\rm bi}(A)=Z_{\rm bi}^{-1}\sum_n e^{-\beta E_n}\langle\phi_n|A|\psi_n\rangle$ satisfies $\omega_{\rm bi}(A^\dagger A)\ge0$ for all $A$ if and only if $H$ is quasi-Hermitian. The proof reconstructs the metric $\eta$ directly from the eigenprojectors of $\omega_{\rm bi}$ via the Riesz representation theorem, yielding a metric-free criterion for quasi-Hermiticity. Under the quasi-Hermitian hypothesis, we prove that the $\eta$-Gibbs state $\omega_\eta(A)=Z_\eta^{-1}{\rm Tr}[\eta e^{-\beta H}A]$ satisfies the full analytic KMS condition using the Hadamard three-line theorem and Bari's theorem on Riesz bases. The transported state generally differs from the Gibbs state of the isospectral Hermitian partner whenever $[\eta,h]\neq0$, so the KMS property cannot be obtained by similarity transformation alone. Finally, within the Haag–Hugenholtz–Winnink programme, we establish the Tomita–Takesaki modular structure of the $\eta$-Gibbs state in finite dimensions, while the construction of a compatible $C^*$-norm and the proof of $\sigma$-weak continuity remain open.

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.

The relationship between linear codes in the Hamming metric and projective algebraic varieties has led to deep interactions between coding theory and algebraic geometry, with classical examples such as Reed-Solomon codes and the rational normal curve. On the other hand, the sum-rank metric has recently gained attention due to applications in network coding, distributed storage, and post-quantum cryptography, with linearized Reed-Solomon codes emerging as optimal constructions. Despite recent advances, their structural and geometric properties are still not fully understood, and existing distinguishers remain limited. In this paper, we develop a geometric framework for linearized Reed-Solomon codes by considering a $q$-analogue of the rational normal curve. This yields a geometric characterization for certain parameter choices and reveals that the corresponding sets of points satisfy unexpectedly many $(q+1)$-degree hypersurface conditions. Our approach extends Schur-product-based techniques from the Hamming and rank-metric settings to the sum-rank metric case. Finally, we study the Hilbert function of the associated coordinate ring, providing a detailed description of its behavior and identifying its regularity, which also sheds new light on Gabidulin codes.

We prove the global-in-time existence of weak solutions for the Maxwell–Stefan–Fourier system with physically motivated degenerate thermal conductivity $\kappa(\theta)=\theta^\alpha$, for $0<\alpha\leq2$. In contrast to existing nonisothermal compressible theories based on nondegenerate conductivities, the entropy inequality no longer gives control of the quantities $\nabla\theta$ and $\nabla \log\theta$. The main new ingredient is a renormalized energy estimate, which yields compactness of the temperature and allows the identification of the degenerate heat flux.

We introduce a switching Hamiltonian Monte Carlo method for sampling from finite mixture Boltzmann-Gibbs distributions. We propose symmetric numerical integrators to approximate switching Hamiltonian dynamics interlaced with Poisson jumps, where the regime-switching chain is simulated using the uniformization technique or the stochastic simulation algorithm. We prove geometric ergodicity of the resulting Markov chain. We develop an approach based on the discrete Poisson equation associated with numerical schemes to estimate the error in computing ergodic averages. Using this approach we prove that the proposed numerical integrators have second-order bias. This approach is simple and can be generalized to other settings, for example, kinetic Langevin equations. Finally, we verify the convergence result via numerical experiment.

Variational methods are widely used to approximate posterior distributions in Bayesian inference when exact computation is infeasible. We study when such approximations inherit posterior consistency. Our first result shows that, on a general metric space, a uniform bound on the Kullback–Leibler divergence from the approximating measures to a tight sequence of target measures forces the approximating sequence to be tight. It follows that if the target posteriors converge weakly to a Dirac mass at the true parameter, then any variational sequence with bounded Kullback–Leibler divergence to the targets is also consistent. We also give simple logarithmic-moment conditions that verify this boundedness condition, and illustrate them for smooth generalised posterior distributions.

We develop an embedded Trefftz-DG method for the Oseen problem and prove a complete stability and quasi-optimality theory in standard DG norms. The key ingredient is a construction of a suitable local complement space to the Trefftz space, on which the Oseen operator is stably invertible. We also derive a reduced formulation of the method, the resulting system is posed in terms of the velocity unknown only, a crucial step in the analysis especially for the nonlinear Navier-Stokes problem in Part II.

We show that the QR factorization of a banded-plus-semiseparable (BPS) matrix is computable in optimal linear complexity with respect to the discretization size by showing that the intermediate stages of a QR factorization as computed using Householder reflection maintain a specific structure which has optimal storage. While optimal complexity QR factorizations of BPS matrices are known via Given's rotations, our Householder-based framework enables adaptive, partial QR factorizations whose computational cost depends solely on the number of upper-triangularized columns rather than the global matrix size and represents the factors in a format consistent with LAPACK. This allows users to switch to highly optimized dense matrix operations for the final solve, creating a fast hybrid approach. We further deduce that the Compact WY format also has BPS structure and can be computed in optimal complexity. Finally, for symmetric BPS matrices, we show that the reverse $RQ$ product preserves the BPS structure, establishing the algebraic closure of this matrix class under orthogonal similarity transformations and facilitating efficient chain-structured matrix decompositions. Numerical experiments validate the optimal linear complexity, confirm high numerical accuracy, demonstrate excellent computational efficiency via LAPACK integration, and show substantial speedups compared with existing hierarchical approaches. The algorithms have been implemented in an open-source Julia package, providing an efficient and accessible platform for practical use.

This paper introduces Bott-Chern and Aeppli homotopy sets for a fibrant class of bisimplicial sets and establishes their basic properties. In positive bidegrees, Bott-Chern homotopy sets carry natural monoid structures, while Aeppli homotopy sets carry natural group structures. They are related by a loop-space comparison: after a bidegree shift, the Aeppli homotopy groups of X are naturally identified with the Bott-Chern homotopy monoids of the loop space of X. In particular, the Bott-Chern homotopy monoids of loop spaces are groups. To justify our definitions, we show that the Bott-Chern homotopy monoids of a bisimplicial abelian group are naturally isomorphic to the Bott-Chern homology groups of its associated normalized Moore bicomplex. An analogous statement holds for Aeppli homotopy.

We develop and analyze an embedded Trefftz-DG method for the steady incompressible Navier-Stokes equations, based on the reduced Oseen discretization from Part I. The main difficulty is that the reduced Trefftz space depends on the convection field, so successive Picard iterates live in different discrete spaces. We address this by constructing projections between convection-dependent Trefftz spaces and using them to control the reduced Oseen solution map. Under suitable resolution and small-data assumptions, we prove existence of discrete solutions, uniqueness, and convergence of the Picard iteration. We also derive an a priori error analysis by relating the method to the underlying DG discretization, thereby inheriting convergence properties from compatible DG Navier-Stokes analyses. Numerical experiments on standard incompressible-flow benchmarks illustrate the theory.

Although the successive cancellation list (SCL) decoding of polar codes exhibits excellent performance, it retains many decoding paths in the list with negligible contribution to the final output, resulting in high sorting and computational complexity. In this letter, we propose a novel pruning strategy to mitigate the decoding complexity. By leveraging the blockwise soft output extraction process of soft-output SCL and soft-output fast SCL decoding, we provide an accurate approximation of the probability that a decoding path is correct, and thus accordingly prune the paths failing to meet a pre-defined reliability threshold. The complexity reduction achieved by the proposed soft-output-based pruned SCL (SOP-SCL) decoder and its fast version, SOP-FSCL decoder, is significant, without any compromise in error-correction performance. Meanwhile, they also prove to be more efficient than state-of-the-art pruned polar decoders.

In this paper, we develop a general theory of twist deformations for Hopf coquasigroups in a symmetric monoidal category. To this end, we first introduce and study non-coassociative bimonoids endowed with left and right codivisions, and establish their connection with left and right Hopf coquasigroups. Next, motivated by the classical theory of Drinfeld twists for Hopf algebras, we define twists for non-coassociative bimonoids and prove that they induce deformations of Hopf coquasigroup structures through suitable modifications of the coproduct. In particular, we obtain explicit deformation procedures for right and left Hopf coquasigroups and analyze the corresponding antipodes. Finally, we apply the general theory to construct nontrivial examples arising from Hopf coquasigroups associated with the sphere ${\sf S}^7$, obtaining new examples of twisted Hopf coquasigroups that are neither commutative nor cocommutative.

Assume that \(M_n\), the canonical inner model with \(n\) Woodin cardinals, exists. We force a model with continuum \(\aleph_2\) in which every \(\boldsymbol{\Sigma}^1_{n+2}\) set of reals is Lebesgue measurable and has the Baire property, the \(\Sigma^1_{n+2}\)- and \(\Pi^1_{n+3}\)-uniformization properties hold, and the reals admit a \(\Delta^1_{n+3}\)-definable well-order. Thus regularity up to a fixed finite projective level, together with a definable well-order of the reals at the adjacent level, does not force the determinacy strength which would normally explain that regularity, even when this package is strengthened by adjacent \(\Sigma\)- and \(\Pi\)-uniformization. In particular, this gives a negative answer to a local form of Woodin's twelfth Delfino problem asked by Friedman-Schindler.

Assuming that $M_n$, the canonical inner model with $n$ Woodin cardinals, exists, we force a model in which every $\boldsymbol{\Sigma}^1_{n+2}$ set is Lebesgue measurable and has the Baire property, and in which $\Sigma^1_{n+2+m}$-uniformization holds for every $m\in\omega$. Additionally, this universe has a $\Delta^1_{n+3}$-definable wellorder of the reals. This answers a question of S. D. Friedman and R. Schindler from 1999. In the case $n=1$, the construction also gives a model with one Woodin cardinal in which all $\Sigma^1_3$ sets are measurable with respect to the random, Cohen, Sacks and Miller notions of measurability, while a $\Delta^1_4$-definable wellorder of the reals exists answering an instance of a question of S. D. Friedman and D. Schrittesser.

In this paper, we study a class of monotone generalized Nash equilibrium problems (GNEPs) with jointly linear constraints. The players' strategy spaces are real Hilbert spaces, while the joint constraint is formulated in a Banach space. To solve such problems, we propose a novel symmetric Gauss-Seidel (sGS) based alternating proximal augmented Lagrangian method (sGS-APALM) which incorporates newly designed quadratic surrogates. In contrast to existing regularization and ALM-type methods, the proposed method avoids solving coupled Nash equilibrium subproblems at each iteration and instead updates the players' strategies alternately by solving a sequence of unconstrained quadratic programs. Moreover, unlike many existing splitting-based methods, our global convergence analysis and convergence rate estimation require only monotonicity and Lipschitz continuity of the pseudo-gradient mapping, without imposing stronger assumptions such as strong monotonicity or co-coercivity. Finally, we apply the method to a class of risk-neutral PDE-constrained GNEPs with joint state constraints, and preliminary numerical results demonstrate its efficiency and effectiveness.

We investigate two problems concerning spaceability in function spaces. First, we study the spaceability of sets of nowhere regular periodic functions, for various notions of regularity, including Hölder regularity and real analyticity. Second, we show that the set of real analytic functions is spaceable in $C((0,1))$. Moreover, we provide a complete characterization of the closed subspaces of $C((0,1))$ consisting of real analytic functions. Our work solves several open questions posed by Bernal-González et al. [4, 5, 6].

We develop and analyze a fully discrete scheme for the weakly damped semilinear wave equation that combines a Symmetric Interior Penalty Discontinuous Galerkin (SIPG) spatial discretization with a hybrid Crank–Nicolson/second-order Backward Differentiation Formula (CN–BDF2) time integrator. A chord-slope linearization of the nonlinear reaction term is employed, which preserves an exact discrete gradient structure and, crucially, requires {no global Lipschitz continuity assumption} on the nonlinearity. Stability of the fully discrete solution is established through a Lyapunov-based analysis-rather than spectral arguments-by constructing a discrete Lyapunov functional that yields existence, uniqueness, and uniform boundedness of the numerical solution. Under standard regularity assumptions, optimal a~priori error estimates of order $\mathcal{O}(h^{k}+\tau^{2})$ in the DG energy norm and $\mathcal{O}(h^{k+1}+\tau^{2})$ in the $L^{2}$-norm are proved, where $h$ is the mesh size, $\tau$ the time step, and $k$ the polynomial degree. Numerical experiments on two-dimensional problems with linear, cubic, and trigonometric nonlinearities confirm the theoretical convergence rates and illustrate the long-time energy-dissipation properties guaranteed by the Lyapunov structure.