Papers

We consider initial and initial boundary value problems for the one-dimensional compressible Navier–Stokes equations of an ideal polytropic gas in unbounded domains. The viscosity and heat conductivity are proportional to $\theta^\alpha$ and $\theta^\beta$, respectively. For every fixed $\beta\geq0$ and sufficiently small $\alpha\geq0$, depending on the initial data, we prove the global existence and asymptotic stability of a strong solution for arbitrarily large $H^1$ initial data. Dividing the momentum equation by the viscosity allows us to estimate the specific-volume derivative without an initial $H^2$ assumption.

In this paper, we study the anisotropic-length-preserving weighted anisotropic curvature flow for smooth convex closed plane curves. For any smooth, embedded, closed, convex initial curve, the flow exists smoothly for all time, and the anisotropic curvature converges smoothly to that of the boundary of the corresponding homothetic Wulff shape as time tends to infinity.

We consider $N \times N$ random band matrices $H = (H_{xy})$ with centered complex Gaussian entries, indexed by points $x,y$ on the two-dimensional discrete torus $(\mathbb{Z} / \sqrt{N} \mathbb{Z})^2$. The matrix entries $H_{xy}$ vanish whenever the distance between $x$ and $y$ exceeds a constant multiple of the bandwidth parameter $W$. We prove that, for any fixed $\varepsilon>0$, if $W\geq(\log N)^{2+\varepsilon}$, then all bulk eigenvectors are delocalized with high probability. Equivalently, this yields a lower bound of order $\exp(W^{1/2-\varepsilon})$ on the localization length of two-dimensional random band matrices for arbitrarily small constant $\varepsilon>0$, improving the superpolynomial lower bound $W^C$ for every fixed constant $C>0$ established in arXiv:2503.07606. A key ingredient of our proof is a new strategy that closes the resolvent flow analysis using only loop observables of order at most two, without relying on the higher-order loop hierarchy.

We consider a standard convex composite optimization problem with either smooth or nonsmooth objective function, and under quadratic growth. In recent years, several works gave algorithms based on a weak proximal oracle (WPO) that essentially match in oracle complexities proximal (sub)gradient methods relying on exact prox operations. Importantly, such WPOs, which relax the strong optimality condition of the standard prox operator, may admit much more efficient implementation in terms of runtime when optimal solutions have some sparse structure. A question remained if such WPO-based methods can be accelerated (in the sense of Nesterov's accelerated gradient). In this work we provide a negative answer by establishing lower bounds against both deterministic and randomized methods. Thus, while WPOs can substantially reduce the cost of individual oracle calls, this comes with an inherent loss in oracle complexity. We also provide a new upper-bound for WPO-based nonsmooth convex composite optimization, nearly matching the proximal subgradient method.

In this paper, we study standard and distributionally robust $\mathcal{H}_2$ synthesis problem of a stabilizing state-feedback controller for discrete-time linear time-invariant systems. Without requiring a nonsingular disturbance controllability Gramian or a positive-definite control penalty, we establish that stationarity of a stabilizing gain in standard $\mathcal{H}_2$ synthesis is equivalent to global optimality and derive an exact Pythagorean identity for the performance difference. We then generalize the $\mathcal{H}_2$ synthesis problem by considering i.i.d. disturbances with zero mean and uniformly bounded second moments, and extend the stationarity-optimality equivalence to the corresponding distributionally robust $\mathcal{H}_2$ synthesis. Moreover, we show that every standard $\mathcal{H}_2$-optimal gain is simultaneously optimal for every such distributionally robust problem. Finally, we specialize the framework to Frobenius, Kullback–Leibler, and Wasserstein-2 ambiguity sets and obtain explicit characterizations of their worst-case covariances, which, in turn, provide performance certificates for a common optimal controller.

Let $f\colon X \to Y$ be a proper and tame morphism of algebraic stacks, where $X$ and $Y$ are locally of finite type over an algebraic stack $S$. We prove that $R f_*$ sends complexes that are pseudo-coherent relative to $S$ to pseudo-coherent complexes relative to $S$. In the scheme case, this resolves a conjecture of Illusie from SGA6. We also prove related and new results in the non-tame setting (e.g., infinite stabilizers). Our methods use derived algebraic geometry and also give new proofs of classical statements for schemes due to Kiehl. Along the way, we extend some foundational results for quasi-coherent sheaves on algebraic stacks to the derived setting.

We prove a recursive formula for construction of generalized spectral basis. Example of application is presented for symbolic computation with non-diagonalizable matrix that describes electromagnetic wave propagation in anisotropic media. The method is general enough and can be applied, for example, to multivectors of the Clifford algebra.

We study global strong solutions and their large-time behavior for the one-dimensional outer pressure problem of a viscous, heat-conducting ideal polytropic gas. The viscosity and heat conductivity satisfy $\mu=\tilde\mu\theta^\alpha$ and $\kappa=\tilde\kappa\theta^\beta$. For each prescribed nonnegative pressure and each fixed $\beta\geq0$, we allow large $H^1$ initial data with positive specific volume and temperature, provided that $\alpha\geq0$ is sufficiently small. When the limiting outer pressure is positive, the solution has uniform bounds and converges to a stationary state. When the limiting outer pressure is zero and the stated weighted pressure conditions hold, the normalized solution converges to an expanding state at an algebraic rate in physical time.

We study necessary and sufficient conditions on a Banach space $X$ such that $L(Y,X)$ is transfinite (rigid) octahedral for any Banach space $Y$. We present several examples of such $X$. As application of our results, we find the first example in the literature of Banach space which is rigid octahedral but fails any transfinite version of octahedrality. We also improve some previous results about octahedrality in ultrapower spaces.

For a positive integer $n$, let $$S_0(n)=\{1,2,\ldots,\lfloor n/2\rfloor\},\qquad S_{j+1}(n)=\{n\bmod k:k\in S_j(n)\setminus\{0\}\},$$ and put $s_j(n) := |S_j(n)|$. The sets defined above arise naturally in the study of the length of the Pierce series expansion of a rational number. In \cite{Ba-Vu}, Baraskar and Vukusic conjectured that for every fixed $j\geq 2$, the limit $$\lim_{n\to\infty} s_j(n)/n$$ does not exist. In this paper, we prove this conjecture in the affirmative. Moreover, we define a new class of iterated remainder sets $T_j(n)$ that naturally arises from the study of the length of the Engel series expansion of a rational number. We analogously study the asymptotic behavior of $t_j(n) := |T_j(n)|$. We show that $$\lim_{n\to\infty}\frac{t_j(n)}n$$ exists precisely for $j\in\{0,1\}$ and fails to exist for every fixed integer $j\geq2$.

The exact behaviour of the enveloppes of the single-time and two-time correlators is found for the voter model on a fractal substrate, with nearest-neighbour interactions. Herein the geometry of the fractal substrate is described by its non-integer geometric fractal dimension $d_f$, and its topology and diffusive transport by the distinct spectral dimension $d_s$. On the level of the equations of motion of the correlators this can be modelled by considering a space-dependent diffusion constant ${\cal D}(r)\sim r^{-\theta}$ which implies the spectral index $\theta$, itself a function of $d_f$ and $d_s$. With a scaling ansatz, the generic phenomenology of ageing is confirmed and the dynamic exponent ${z}=2+\theta$ and the autocorrelation exponent $\lambda=d_f$ are derived. The explicitly found dynamic scaling functions are shown to depend only on the spectral dimension $d_s$. The decay of the enveloppe of the density of active interfaces with time is described by the exponent $\alpha=1-d_s/2$ for $d_s<2$, confirming the results of preexisting numerical simulations.

In this paper, we establish an upper bound for the second eigenvalue of the scalar Jacobi operator of an arbitrary closed two-sided surface immersed in a contracted Berger sphere. The estimate involves the total squared mean curvature, the Euler characteristic, and an explicit nonpositive term determined by the angle between the surface normal and the Hopf direction. The proof combines the realization of a Berger sphere as a geodesic hypersurface of a complex projective plane, the first standard embedding of the latter into a Euclidean sphere, a weighted Hersch–Li–Yau balancing argument, and the conformal invariance of the Willmore functional. No minimality or constant mean curvature assumption is imposed. As an application, if $1/3\leq\alpha\leq1$ and the surface has nonpositive Euler characteristic, then its second Jacobi eigenvalue is nonpositive; it is strictly negative for $\alpha>1/3$. At the critical value $\alpha=1/3$, equality forces the immersed image to be congruent to the minimal Clifford torus; in the embedded category, this yields a complete characterization of the equality case.

Heffter arrays, introduced by Archdeacon in 2015, are combinatorial structures with applications to cyclic cycle systems and biembeddings of graphs on surfaces. Costa, Morini, Pasotti and Pellegrini proposed in 2020 the notion of relative Heffter arrays as a generalization of classical Heffter arrays, inspired by the concept of relative difference families. In their work, the existence problem of integer relative Heffter arrays $\mathrm{H}_k(n;k)$ was solved for all $k\neq 5$, while the case $k=5$ and $n \equiv 0 \pmod{4}$ remained open, apart from two sporadic examples with $n=8,16$. In this article, we consider this open problem and we construct an $\mathrm{H}_5(n;5)$ for every $n\equiv 0 \pmod{4}$, $n\geq 12$. As a consequence, the existence problem of integer relative $\mathrm{H}_k(n;k)$ is completely settled.

We study a stochastic Navier–Stokes system in which the spatial modes of the transport noise evolve dynamically and are coupled to the resolved velocity. The model couples a Navier–Stokes stochastic PDE to an infinite family of linearized Navier-Stokes equations for the noise correlation modes. These latter equations may contain a hyperviscosity. We prove existence of martingale solutions in two dimensions for hyperviscosity exponent $s\geq1$, and pathwise uniqueness for $s>1$ under additional regularity of the initial noise modes. In three dimensions, we establish existence of martingale solutions for $s>3/2$. The proof combines Galerkin approximations and compactness with estimates adapted to the coupled drift and stochastic terms.

We prove that the Lipschitz-free space $\mathcal{F}(M)$ is the strongly unique isometric predual of $\mathrm{Lip}_0(M)$ for every metric space $M$, solving a longstanding open problem of Weaver. The result is proved first for length metric spaces by analyzing the behavior of Lipschitz functions on paths in $M$ and the topological properties of subspaces of functions vanishing on certain barrier-sets of $M$. Then, the result is extended to general metric spaces by embedding them into length spaces in such a way that the restriction operator for Lipschitz functions is weak$^*$ continuous with respect to their selected preduals. Finally, we also give a counterexample to the codimension-one inheritance assertion used in Weaver's original proof for bounded and geodesic metric spaces.

We prove that no nontrivial finite group $G$ satisfies the equivariant generating hypothesis of Bohmann, even when all $RO(H)$-graded homotopy groups are tested for every subgroup $H\leq G$. For each $G$, we construct ghosts on finite $G$-spectra with arbitrarily long nonzero composition powers. We also prove that the homotopy-module functors are nonfull and construct non-equivalent finite $G$-spectra with isomorphic full homotopy modules. A central ingredient of our proof is the natural ghost operation $\Psi=(p-[C_p/e])(1-g)$ on $C_p$-spectra. Motivated by Ma–Xu's examples in the motivic settings, we apply this operation to both a two-cell complex and products of complex projective spaces with cyclic permutation action, which produces arbitrarily long nonzero ghost powers. We also construct a ghost on an explicit four-cell complex that is detected in the derived category over the group ring.

We study latent-space watermarking through a pretrained generator using a prescribed latent-to-output stochastic mapping, called the renderer. A watermark encoder selects the latent input using a message and secret key. For every message and semantic context, the released output must have exactly the desired conditional output distribution. For finite alphabets, we derive rate–key inner and outer bounds and characterize the coding and coordination requirements for realizing watermark communication through the prescribed latent interface. When the target output distribution of the generator uniquely determines the corresponding latent input distribution through the renderer, a strengthened converse yields the capacity region; the same region governs explicit preservation of the pretrained latent distribution. We extend the analysis to general jointly Gaussian models and identify a sufficient statistic of the latent that captures both the watermark-bearing information available at the generated output and the latent coordination required to preserve its target distribution. For the vector Gaussian model, we further characterize the optimal allocation of the secret-key resource across the resulting modes. Finally, we turn to an emerging robustness threat that is particularly natural in generative watermarking: an adversary can regenerate the released sample to obtain a fresh realization of the same underlying content while attenuating or destroying the embedded watermark. We incorporate this robustness axis into our framework and characterize the one-pass compound capacity of the scalar Gaussian model when the semantic context is known to the encoder but hidden from the detector, while the regeneration attack may depend on that context. Extending the analysis to multiple rounds of repeated canonical regeneration, we characterize the resulting watermark-capacity decay.

We study the problem of nonparametric estimation of the interaction function in particle systems whose interaction enters through the diffusion coefficient. The particle system is observed discretely over a fixed time interval, and the interaction function is assumed to belong to a suitable nonparametric function class. We propose an estimation procedure based on empirical risk minimization over finite-dimensional sieve spaces, introducing a computable contrast function that exploits the local volatility of the observed trajectories. We derive a general non-asymptotic error bound under the intrinsic norm associated with the statistical problem and, focusing the analysis on multiresolution wavelet approximation spaces, we obtain explicit polynomial convergence rates in two distinct regimes, depending on whether the particle or discretization error dominates. Finally, we establish convergence rates for the classical $L^2$-norm.

Given two positive integers height $h$ and order $n$, the barrycade construction problem asks for a set of $h$ permutations of the integers from $1$ to $n$ such that all the proper partial sums given by these permutations are pairwise distinct. The name barrycade was coined by Richard K. Guy and refers to Barry Cipra, who introduced this kind of arrangement problem. A simple calculation shows that a solution can only exist for $n \ge 2h-2$ and it is conjectured that there always exists a solution for every height $h \ge 2$ and order $n \ge 2h-2$. In this work, for every height $h \ge 1$, we present a construction of a barrycade of height $h$ and order $n = 2h+3$. We also present a randomized heuristic that allows us to find a barrycade of height $h$ and conjectured optimal order $n=2h-2$, for every height $2 \leq h \leq 50$. Thus, we confirm the conjectured optimal order for all heights up to $50$. We also consider a related corral construction problem, where the permutations define a cyclic arrangement. In this setting, for every height $h \ge 1$, we present a construction of a corral of height $h$ and order $n=2h$. A heuristic approach, similar to the one used for barrycades, allows us to find a corral of height $h$ and conjectured optimal order $n=2h-1$, for every height $1 \leq h \leq 50$. We confirm Tomoki Nakamigawa's conjecture on well-dispersed partitions of cyclic groups for the number of parts up to $20$.

We present a method for determining whether there exists a degree $D$ rational map from $X_0(N)$ to some elliptic curve $E/\mathbb{Q}$. Moreover, we explain how to obtain a quadratic form that represents all possible degrees of such maps using the degree pairing method. This method previously required odd analytic rank and some additional technical conditions to be satisfied. In this paper, we extended the method to even rank curves as well and remove all technical conditions. We also show how to prove or disprove the existence of degree $D$ maps to elliptic curves by a lifting criterion on the lattices $H_1(-,\mathbb{Z})$ As an application of this method, we determine all $D$-elliptic curves $X_0(N)$ for all $D\leq 100$. Interestingly, in some cases we found degree $D$ rational maps that we could not explain by degeneracy maps, quotient maps, and modular parameterisation.

In this paper we propose a universal way to define multivariate orthogonal polynomials for the positive branch of WLZZ matrix models. This allows us to generalize from the Gaussian case, previously considered in the literature, the strict superintegrability property and the Miwa deformation formulas.

We study rolls and homoclinic snaking in a variation of the one-dimensional Swift-Hohenberg equation, whose standard forms are prototypical order-parameter models for pattern formation in the sciences. Motivated by classes of differential equation models that involve continuous non-smooth low order nonlinear terms, we replace the standard quadratic-cubic nonlinearity by $\nu |u|^\alpha-u^3$, $\alpha \in [1,2]$ with $\nu > 0$. In the vicinity of zero, for $\alpha<2$ this nonlinearity falls outside the scope of classical Taylor expansion and bifurcation analysis. Our partially analytical and partially numerical results highlight that the non-smooth term modifies the criticality of pattern-forming bifurcations and alters the associated branches of solutions. In particular, $\alpha\in(1,2)$ implies subcriticality of roll bifurcations for any $\nu>0$. At $\alpha=1$ differentiability is lost, which has a strong impact on the bifurcations of sign-changing rolls including the disappearance of homoclinic snaking. Homoclinic snaking thus emerges non-smoothly as $\alpha$ increases from $\alpha=1$, and persists when retaining an additional, e.g., quadratic term.

In this paper, we further develop the real link Floer homology defined by the first author. We introduce a new base-pointing convention that leads to a different version of real link Floer homology and show that this new theory is related to the old one by an exact triangle. We also define a real link Floer theory for multi-based strongly invertible links, which is a strong real Heegaard invariant, and take a first step toward a real link Floer TQFT. A computer implementation for the new theory via grid diagrams was written by Zhenkun Li. We also include an appendix containing real grid homology of more than 50 small knots.

For a family $\mathcal{F}$ of $k$-graphs, $ex_k(n,\mathcal{F})$ denotes the maximum number of edges in an $n$-vertex $\mathcal{F}$-free $k$-graph. Let $M_{s+1}^k$ denote a matching of size $s+1$ in $k$-uniform hypergraphs. Recently, Alon and Frankl (JCTB, 2024) determined $ex_2(n,\{M_{s+1}^2,K_{\ell+1}\})$ for all $n\geq 2s+1$ and $\ell\geq 2$. For every non-bipartite graph $F$, Gerbner (JGT, 2024) determined $ex_2(n,\{M_{s+1}^2,F\})$ for sufficiently large $n$. In this paper, we investigate this problem for different ranges of the matching parameter. First we prove that for every graph $F$ with $\chi(F)>3$, there exist constants $\beta>0$ and $s_0$ such that $ex_2(n,\{M^2_{s+1}, F\})=ex_2(2s+1,F)$ for $\max\{s_0,n/2-\beta n\}<s<n/2$. For integers $\ell\ge k\ge3$, let $\mathcal{K}_{\ell+1}^k$ be the family of all $k$-graphs $F$ with at most $\binom{\ell+1}{2}$ edges for which there is an $(\ell+1)$-set $L$ such that every pair of vertices of $L$ is covered by an edge of $F$, and let $H_{\ell+1}^k$ be the $k$-uniform hypergraph obtained from the complete graph $K_{\ell+1}$ by enlarging each edge with a set of $k-2$ new vertices, which is a member of $\mathcal{K}_{\ell+1}^k$. We determine $ex_k\bigl(n,\mathcal{K}_{\ell+1}^k\cup\{M_{s+1}^k\}\bigr)$ for $s< \frac{n}{8(k-1)^{k-2}(\ell-1)^3}$. For sufficiently large $s$, we also determine $ex_k\bigl(n,\{M_{s+1}^k,H_{\ell+1}^k\}\bigr)$ for $\frac{n}{k}-\beta n<s\leq \frac{n-k}{k}$ and $s< \frac{n}{8(k-1)^{k-2}(\ell-1)^3}$.

In this paper, we consider the asymptotic density of $T_c(x):=\#\{p\leq x:P^+(p-1)\geq p^c\}$, where $P^+(n)$ denote the largest prime factor of $n$. We show that for $x\rightarrow\infty$, for any $0<c\leq 1/2$, one has \begin{align*} \mathop{\lim\inf}_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}&\geq\max\left(1-\frac{16}{5}\rho\left(\frac{1}{c}\right),1-c+g(c)\right), \end{align*} where $g(c)>0$ for $11/32<c\leq1/2$, and $g(c)=0$ for $0<c\leq11/32$. In particular, we have $g(1/2)>0.0436$. This improves a previous result by Liu-Wu-Xi (2020), who showed \begin{align*} \mathop{\lim\inf}_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}&\geq\max\left(1-4\rho\left(\frac{1}{c}\right),1-c\right), \end{align*} for $0<c\leq 1/2$.

We determine the maximum stable volume of bubbles attached to flat surfaces as a function of contact radius, contact angle, and capillary length. By solving the Young-Laplace equation using a shooting method, we calculate equilibrium bubble shapes and determine when bubbles detach through three distinct modes: necking at the cavity, sideways instability, and spreading followed by necking on the surrounding surface. Our predicted detachment volumes agree well with experimental measurements of electrolytic, boiling, and carbonated bubbles, as well as pendant drops. These results bring together and extend previous analytical estimates and, by symmetry, also apply directly to pendant drops.

We establish a discrete $H^1$-Korn inequality for finite element spaces satisfying a second order weak continuity condition. This implies that classical $H^2$-nonconforming elements satisfy the discrete Korn inequality. Based on this result, we develop a stabilized Morley finite element method for linear strain gradient elasticity using only piecewise quadratic polynomials. Thanks to the discrete $H^1$-Korn inequality, this method does not require any penalty parameter. We prove that the method is robust with respect to both $\lambda$ and $\iota$. Numerical experiments confirm the theoretical results.

We prove that the first-order theory of nonabelian free groups does not admit a model companion.

Autoencoders (AEs) have become a dominant approach to nonlinear latent-space construction in data-driven reduced-order modelling (ROM), with their decoders lifting latent representations back to the ambient state space. Their prominence, however, has overshadowed an established alternative: manifold-learning methods grounded in classical numerical analysis. We revisit this alternative using Parsimonious Diffusion Maps (PDMs), benchmarking them against Proper Orthogonal Decomposition (POD)-based ROMs and several convolutional AE architectures for the two-dimensional incompressible flow past a rotating cylinder —a bifurcating Navier-Stokes (NS) system organized by a codimension-2 Bogdanov-Takens point and its associated Hopf, saddle-node, and homoclinic bifurcations. Our approach uses PDMs to identify a parsimonious and interpretable set of intrinsic latent coordinates and to estimate their dimension directly from data. Gaussian process regression then learns the latent dynamics, while convex K-nearest-neighbor (K-NN) interpolation in PDMs space constructs the pre-image map, for which we establish pointwise consistency. The resulting nonlinear ROM substantially outperforms POD-based ROMs and achieves reconstruction and prediction accuracy comparable to —and, in some bifurcating regimes, better than— that of AE-based ROMs. At the same time, latent-variable learning with PDMs requires orders of magnitude less computational time than AE training.

In previous work, we analysed a number of categorical properties, of interest in the context of Janelidze-Márki-Tholen semi-abelian categories, for the variety $\mathsf{HSLat}$ of Heyting semilattices. In this paper, we focus on the subvariety $\mathsf{ELHSLat}$ of equationally linear Heyting semilattices. Our main objective is to show that, unlike $\mathsf{HSLat}$, this category is algebraically coherent. We furthermore prove that it is neither locally algebraically cartesian closed nor cosmash associative.