Papers

Embedding subsets of Wasserstein spaces into linear spaces is a problem of both theoretical and computational interest. In this work, we prove that three classical and natural embeddings of Wasserstein spaces into Banach spaces of non-trivial type are bi-Hölder on subsets of Wasserstein spaces given by moment bounds. These embeddings, all considered in multiple applications, are the linearized optimal transport embedding, the sliced-Wasserstein embedding, and a particular kernel mean embedding. Our results contrast with known obstructions to embeddin the full Wasserstein space into Banach spaces of non-trivial type. We also show the sharpness of our results regarding the moments required for our conclusions to hold.

The oscillator groups are four-dimensional solvable Lie groups, extensions of the Heisenberg Lie group. We present them as circle bundles of the standard contact sub-Riemannian structure on $\mathbb{R}^{3}$. We define sub-Riemannian and sub-Lorentzian structures on them, describe their geodesics explicitly and determine which of them are periodic. We also study a generalization: Let $G$ be a semisimple Lie group and $K$ a compact subgroup such that $\left( G,K\right) $ is a symmetric pair of the compact or the noncompact type. Let $\mathfrak{g}$ and $% \mathfrak{k}$ be the Lie algebras of $G$ and $K$, respectively. We consider on $\mathfrak{g}$ the usual nilpotent Lie group structure with center $\mathfrak{k}$ and call it $N\left( \mathfrak{g},\mathfrak{k}\right)$. A suitable semidirect product with $\operatorname{Ad}\left( K\right) $ yields a solvable Lie group $\operatorname{Osc}\left( G,K\right) $, which generalizes the oscillator groups and is also quadratic (that is, it possesses bi-invariant metrics; in particular, a canonical one). We define nonholonomic pseudo-Riemannian structures on it and find their geodesics explicitly. In doing so, we obtain a result that may not be merely auxiliary: formulas for the monoparametric subgroups of $\operatorname{Osc}\left( G,K\right) $ and for the sub-Riemannian geodesics of $N\left( \mathfrak{g},\mathfrak{k}\right) $ with the standard left-invariant distribution $\mathcal{D}$. Moreover, when $G/K$ is compact with rank one, we realize $\operatorname{Osc}\left( G,K\right) $ as a Stiefel bundle of $\left( N\left( \mathfrak{g},\mathfrak{k}\right) ,\mathcal{D}\right) $, up to finite coverings.

We establish first-order variational formulas for the driving function and the half-plane capacity of a chordal Loewner chain under quasiconformal deformations. Under the standard normalization fixing $0,1$ and $\infty$, the formulas hold for every bounded Beltrami differential. If the curve has positive area, the formulas contain additional integrals over the curve; these terms vanish when the curve has zero area or when the differential is extended by zero on the curve. The proof uses the normalized Ahlfors–Bers variation together with a Stoilow factorization adapted to the slit domain. We then consider the asymptotic normalization for differentials whose symmetric extensions to $\mathbb{C}$ have the form $\nu(z)=a(\bar z/z)^{n-1}+\nu_0(z)$, where $a\in(-1,1)$, $n\ge1$ is an integer, and $\nu_0\in L^r(\mathbb{C})$ for $0<r<2$. The asymptotically normalized map differs from the standardly normalized map by a positive dilation. Using this relation, the variational formulas in the second normalization follow directly from the standardly normalized case.

Local coordinate-wise linear (LCL) properties provide a unified framework for studying list decoding, list recovery and other local properties of linear codes. We prove that random one-point algebraic geometry (AG) codes have the same LCL threshold rates as random linear codes under explicit alphabet and sampling conditions. More precisely, for a family of local profiles with random linear code threshold $R_{\mathcal P}$, a random one-point AG code avoids all such profiles below $R_{\mathcal P}-\varepsilon$ and contains one above $R_{\mathcal P}+\varepsilon$, with explicit failure bounds on both sides. Our proof also removes an auxiliary lower bound on the number of available rational places that arises in a direct adaptation of the Reed–Solomon argument and requires no additional genus-dependent rate gap. We also observe that the previously claimed above threshold proof for random Reed–Solomon codes does not establish the required probability lower bound. The containment relation used in that argument gives the probability comparison in the opposite direction from what is needed. We give a different above threshold argument, which in particular also yields an alternative proof in the genus zero case. Combining our threshold theorem with recent results for random linear codes gives a collection of new list decoding, average weight list decoding and list recovery guarantees for random one-point AG codes. We further obtain new correlated agreement and proximity gap results for polynomial curves. These results extend the LCL threshold framework from random linear and Reed–Solomon codes to random one-point AG codes.

We consider reduced basis approximations of non-equilibrium steady states of parameter-dependent Lindbladians, whose ambient matrix space grows exponentially with system size. We give a unified structural view of the problem through its characterizations as an eigenvalue problem or as a trace-constrained source problem. Building on these characterizations, we propose a residual-driven reduced basis framework for both formulations. Under the assumption of a unique steady state, we derive residual-based state-error estimates, and establish (sub-)exponential convergence of the greedy algorithms under real-analytic parameter dependence of the problem data. Numerical experiments on boundary-driven Fermi–Hubbard and dissipative XYZ chains support the analysis and suggest that the trace-constrained greedy method offers the best overall balance of accuracy and computational cost.

We prove that Brandt semirings over nontrivial groups with at least two indices admit no identity basis with a bounded number of variables. A uniform family of finite witnesses has size linear in the variable bound and applies to groups of arbitrary cardinality. A commutative construction yields the corresponding result for flat extensions of groups of unbounded exponent and gives an exact finite basis criterion for arbitrary flat group extensions. Structural representations lead to a classification of finite natural inverse semirings and a single identity characterizing locally Clifford natural inverse structure among regular semirings. We also describe products, orthogonal sums, minimum generators, colored graph identities and linear variable complexity. Further results concern infinite generating algebras and inherent nonfinite basability.

Let $X/\mathbb{Q}$ be a smooth, projective, geometrically integral curve of genus $g \geq 2$ with a point $ b \in X(\mathbb{Q})$ and geometrically CM Jacobian. Fix an odd prime $p$ of good reduction, and let $U$ be the étale pro-unipotent $\mathbb{Q}_p$-fundamental group of $X$. We prove, for all $n \gg 0,$ the inequality $\dim_{\mathbb{Q}_p} H^1_f(G_T, U_n) < \dim_{\mathbb{Q}_p}H^1_f(G_p, U_n)$ of Bloch–Kato Selmer schemes, with their ratio having limit superior at most $1/2$. Here $U_n$ is the lower central series quotient, and $T$ a finite set containing $p$ and the primes of bad reduction. This answers a particular case of a conjecture by Kim. Following the Iwasawa-theoretic method of Coates–Kim, we reduce the problem to bounding certain character multiplicities at degree $n$. We then define a twisted character lattice $B$ and its $p$-adic completion $B_p.$ Our main idea is to regard normalized multiplicities as $\mu_n(\lbrace x \in B_p: f(x,n)=0\rbrace)$ for a $p$-adic analytic function $f$ coming from Iwasawa theory and a sequence of measures $(\mu_n)_n$ weakly converging to the Haar probability measure $\mu$ on $B_p,$ which is the $p$-adic analogue of the uniform measure. We prove convergence by considering characters $\eta$ of $B_p$ and comparing $\int_{B_p} \eta \ d\mu_n$ to traces of operators induced by symplectic operators $h_\eta$, followed by a character formula. A compactness argument for sequences of $p$-adic loci gives our main Iwasawa estimate. Poitou-Tate duality, the global Euler characteristic formula, and Hodge filtration estimates then give the dimension inequality.

We prove interior $C^{1,\alpha}$ estimates for weak solutions of the fractional $p$-Poisson equation, with $p\geq2$. For right-hand sides in $L^q$, we obtain the estimate when $p<(1-d/q)/(1-s)$. For $\gamma$-Hölder continuous right-hand sides, the range is $p<(1+\gamma)/(1-s)$. Both ranges of parameters are optimal.

We prove instantaneous loss of Sobolev regularity across a fixed gap for the inviscid surface quasi-geostrophic equation on $\mathbb R^2$. More precisely, for every $1<s<2$, $T>0$ and $\varepsilon>0$, we construct initial data $\theta_0\in H^s(\mathbb R^2)$ with $\|\theta_0\|_{H^s}\leq\varepsilon$ that generate a solution $\theta$ on $[0,T]$ such that \[ \theta\in L^\infty([0,T];H^{1+\gamma}(\mathbb R^2)), \quad \theta(t)\notin H^\sigma(\mathbb R^2) \quad\text{for every }t\in(0,T],\ \sigma>\sigma_*(s), \] where $\gamma>0$ and $\sigma_*(s)$ satisfy \[ 1+\gamma<\sigma_*(s):= \frac{s(3-s)}{1+2s-s^2}<s. \] The solution is unique in a determined family of classical solutions with initial datum $\theta_0$.

Let $\varphi$ and $\psi$ be univalent functions in the unit disk $\mathbb D$ satisfying $\varphi(0)=\psi(0)=1$ and $\varphi\not\prec\psi$, and let $\mathcal P_{\varphi}$ denote the class of analytic functions $p$ in $\mathbb D$ such that $p\prec\varphi$. Under suitable analytic continuation, univalence, and boundary-behavior hypotheses on the branch of $\varphi^{-1}$ satisfying $\varphi^{-1}(1)=0$, we determine the sharp $\mathcal P_{\psi}$-radius of $\mathcal P_{\varphi}$ as \[ \mathcal R(\varphi,\psi) = \min_{|\zeta|=1} \left| \varphi^{-1}\bigl(\psi(\zeta)\bigr) \right|. \] For Ma–Minda functions satisfying these hypotheses, the same radius is sharp for the associated classes $\mathcal{ST}(\varphi)$ and $\mathcal{CV}(\varphi)$ with respect to $\mathcal{ST}(\psi)$ and $\mathcal{CV}(\psi)$, respectively. We apply the general result to $ \varphi_{\mathrm L}(z)=\sqrt{1+z}, \varphi_{\mathrm{\leftmoon}}(z)=z+\sqrt{1+z^2}, \varphi_{\mathrm{Lim}}(z)=\left(1+{z}/{\sqrt2}\right)^2, $ for several choices of the Ma–Minda function $\psi$, and obtain parameter-dependent extensions for the corresponding generalized families. The resulting boundary minima are evaluated analytically.

We revisit the discrete Wyner–Ziv problem and establish exact second-order asymptotics. Our main contribution is a second-order converse bound, which is derived using posterior decomposition, concentration inequalities, and a conditional Gaussian approximation for martingales. Furthermore, we extend the previous best known achievability result of Li and Li (arXiv 2025) by allowing the test channel to depend on the type of the observed source sequence. Exact second-order asymptotics are established by combining our achievability and converse bounds. In particular, we show that the achievability bound of Li and Li (arXiv 2025) is not optimal in general. Specifically, we provide two numerical examples to illustrate our results: a binary asymmetric source and a quaternary source. For the first example, the achievability bound of Li and Li (arXiv 2025) achieves the optimal second-order asymptotics. However, in the second example, we show that our achievability result is optimal, which reduces the second-order coding rate of Li and Li (arXiv 2025) by $10.77\%$ at the excess-distortion probability of $0.1$ and by $21.35\%$ at the excess-distortion probability of $0.2$.

A set of vertices in a digraph is majority-stable if each of its vertices has at most half of its outneighbors in the set. A fractional majority coloring assigns nonnegative weights to such sets, covering each vertex to total weight at least one; its minimum total weight is the fractional majority coloring number. We prove that every finite loopless digraph has fractional majority coloring number at most \(523/140<3.736\), improving the bound \(3.9602\) of Anastos, Lamaison, Steiner and Szabó. Our construction uses exclusive sampling from pairs in random cycle matchings, followed by deletion and a correction on the acyclic remainder. The same approach yields bounds below \(3.430\) for digraphs with a directed cycle factor, \(3.287\) for those with an even cycle factor, and \(3.324\) for tournaments. As a consequence, for every fixed \(\varepsilon>0\), we obtain a randomized \((2.491+\varepsilon)\)-approximation algorithm producing an explicit fractional majority coloring in expected polynomial time. On the complexity side, we prove NP-completeness of deciding whether the fractional majority coloring number equals $3/2$, and establish a multiplicative inapproximability threshold of $72/71$ and an additive upper-estimation threshold of $3/142$. All three results hold for acyclic oriented digraphs with outdegrees zero or two in which every directed path has length at most two.

We adapt the 'induction over scales' scheme of Duminil-Copin, Manolescu, and Tassion to the FK-Ising model on s-embeddings. Namely, we prove the existence of crossings of rough topological quadrilaterals with unfavorable boundary conditions, known as 'strong' RSW, under the 'uniformly bounded geometry' assumption (all edge lengths are comparable to 1 and all angles are bounded from below) on the tangential quads that form an s-embedding. This result has a number of standard consequences like quasi-multiplicativity of arm events and fractal properties of the clusters. It can be applied in a variety of setups including near-critical Ising models on regular grids below the correlation length scale by considering appropriate s-embedding thereof.

In this paper, we study the tensor triangular geometry of band algebras $RB$, where $B$ is a finite left regular band and $R$ is a commutative Noetherian ring. We show that the Balmer spectrum of $\Perf(RB)$ is homeomorphic to $\Spec R\times L_B$, where $L_B$ is the support lattice of $B$. We also show that there is a bijection between the localizing ideals of $D(RB)$ and the stable subsets of $\Spec R\times L_B$. As a consequence, the telescope conjecture holds for $D(RB)$. Finally, we construct a band $B$ such that the generalized Nerves of Steel Conjecture fails for the non-rigid tensor triangulated category $\Perf(RB)$.

We detail scalable methods for approximating the quadratic entropy $p^T d p$ for arbitrary distributions $p$ and common distances $d$ of negative type. We focus on the Euclidean and spherical geodesic cases, which both use random feature embeddings and projections to dramatically improve computational complexity within a simple framework. Amortization of a single large matrix multiplication and control variates further enable computation at large scale with low memory and runtime in situations where $d$ is held constant while $p$ varies. We demonstrate this with a comparison against direct pair sampling and bibliometric/scientometric examples on Open Graph Benchmark datasets, revealing papers, fields, and institutions with both particularly narrow and broad interdisciplinary reach from their citations and text features alone.

Let $S_k$ be the space of cusp forms of weight $k$ for ${\rm SL}_2(\mathbb Z)$, with $\dim S_k\geq1$, and let $\chi_p=\left(\frac{\cdot}{p}\right)$ for odd primes $p$. For fixed $s$ with ${\rm Re}(s)>(k+1)/2$, we prove that the proportion of primes $p\leq X$ satisfying $L(h,\chi_p,s)=0$ tends to zero as $X\to\infty$, uniformly over nonzero $h\in S_k$. We also prove an analogous statement for nontrivial linear combinations of twisted symmetric square $L$-values of the normalized Hecke eigenforms when ${\rm Re}(s)>k$. The main ingredient of the proof is a non-concentration theorem for random Euler products under a pairwise local separation condition. As applications, we obtain spanning and density-one basis results in $S_k$ for the kernels associated with twisted standard and symmetric square $L$-functions, with explicit basis criteria in specified right half-planes when $\dim S_k=2$ or $3$. We show that almost every tuple of $\dim S_k$ primes gives a basis of $S_k^{\ast}$ from twisted periods at every noncentral critical index. We also prove a density-one result for rational bases of $S_k$ obtained from traces of Rankin-Cohen brackets of Eisenstein series at prime levels.

In this work, we study the fast Fourier transform (FFT) of discrete data with crystallographic symmetry, for which standard FFT algorithms incur substantial redundant computation. We establish a rigorous mathematical theory of the symmetric fast Fourier transform (SFFT) for $n$-dimensional crystallographic groups and propose the SFFT and symmetric inverse fast Fourier transform (SIFFT) algorithms. We derive symmetry-reduction formulas for both linear and translational symmetries and unify them into a recursive framework. We analyze the arithmetic complexity to obtain theoretical speedup factors relative to the standard FFT, and we confirm the accuracy and efficiency of the proposed methods by numerical experiments on representative three-dimensional space groups.

We study the exact controllability of the cubic nonlinear Schrödinger equation in the exterior $\Omega=\mathbb{R}^n\setminus\Theta$ of a non-trapping obstacle $\Theta$, with internal controls supported outside a large ball, and the corresponding boundary control problem on $\Omega_0=B_{R_0}\setminus\Theta$ with Dirichlet controls acting only on the outer sphere $\partial B_{R_0}$. Using the local smoothing effect of Burq, Gérard and Tzvetkov [9], we prove observability inequalities for the linear Schrödinger equation in the whole scale of Sobolev spaces $H^\sigma_D(\Omega)$, $\sigma\in[-2,2]$, associated with the Dirichlet Laplacian; they require only the non-trapping assumption (no star-shapedness of $\Theta$), and their proofs use no normal boundary traces. Combined with Strichartz-type estimates and a perturbation argument, this yields local exact controllability in $H^1_0(\Omega)$ and in $H^2(\Omega)\cap H^1_0(\Omega)$, in dimensions two and three, around the zero solution and, for internal controls and under a unique continuation assumption, around nontrivial trajectories. The same method applies to the quintic equation, which is energy-critical in dimension three. Several open problems are discussed.

Let $p_i$ be the $i$th prime, and let $f(N)$ be the largest cardinality of a subset of \[ \left\{\prod_{i=1}^N p_i^{a_i}:0\le a_i<N,\ a_i\in\mathbb{Z}\right\} \] containing no two distinct legs of an integer right triangle. We prove \[ f(N) \ll N^N (\log\log N)^{-1/1296}. \]

This paper considers a nonconvex fractional optimization problem with composite structure and nonlinear equality constraints. We reformulate the problem as a non-fractional min-max problem with corresponding optimal solutions. By linearizing the nonlinear constraint terms in the augmented Lagrangian and incorporating Bregman proximal linearization and a relaxation factor in $(0, 2)$ into the alternating direction method of multipliers (ADMM), we develop a Bregman proximal linearized ADMM with guaranteed subsequential convergence to a lifted critical point. Convergence of the entire sequence is established under the Kurdyka–Łojasiewicz (KL) property. We further derive convergence rates under either a Hölderian value proximity error bound condition or the KL property, with the corresponding exponent in $[0, 1)$. In particular, we establish superlinear convergence for exponents in $(0, 1/2)$, together with finite, linear, and sublinear convergence in the remaining cases. Numerical experiments demonstrate the effectiveness of the proposed algorithm.

This paper presents upper and lower bounds on the minimax risk of a parameter estimation problem where the estimator not only observes independent and identically distributed observations but also a quantized version of the parameter, where the quantization has a limitation on the number of levels, but can be optimized otherwise. Our upper and lower bounds have similar decay rates for large number of observation samples and quantization levels.

Finding the extension complexity (xc) of a polytope is equivalent to finding the non-negative rank of its slack matrix. We provide a formulation for the rectangle and refined rectangle covering, bounding the non-negative rank of diverse non-negative matrices. While the bounds are known, our formulation enables us to use boolean satisfiability (SAT) solvers, which proves to be a strong tool. We obtain improved values for lower bounds on the non-negative rank of multiple matrices. In particular, we determined the xc of some regular polygons.

From previous results it follows that probability distributions on metric spaces featuring unique Fréchet means are dense among measures admitting means in both the quadratic Wasserstein metric and the total variation metric. Additionally, we show that the converse also holds on complete finite-dimensional noncontractible Alexandrov spaces with curvatures bounded from below, which encompass compact manifolds without boundaries and nonmanifold shape spaces: Probability distributions featuring nonunique Fréchet means are also dense in both the Wasserstein metric and the total variation metric. Moreover, in either metric, no nonempty open set of probability measures admits a continuous selection of means. This sharpens a previous result on zero reach of the Dirac embedding. Our argument combines cut-locus avoidance for atoms with a topological obstruction. For Riemannian manifolds, all of our conclusions hold for all exponents $1 < p < \infty$, also.

It is shown that, as $n\to\infty$, the logarithm of the number of decompositions into cycles of the $n$-dimensional Boolean cube $E^n$ is \[ 2^n(\ln n-1+o(1)), \] and the logarithm of the number of Hamiltonian cycles in $E^n$ is at least \[ 2^{n-1}(\ln n-1+o(1)). \] It is proved that, in $E^n$, every perfect matching whose edges belong to at most $k$ directions can be extended to a Hamiltonian cycle for every $n\geq n_0(k)$.

We study the transient behavior of two threshold-dependent stochastic models: a Cramér–Lundberg risk model and an M/G/1-type queueing model. In both models, the dynamics are driven by two different compound Poisson processes with drift, with the governing process depending on whether the current state is below or above a fixed threshold. For the risk model, we characterize the probability of ruin before an exponentially distributed epoch, while for the queueing model, we characterize the Laplace–Stieltjes transform of the workload at an exponentially distributed epoch. These quantities are characterized using fluctuation-theoretic results for spectrally positive Lévy processes, with Laplace transforms taken with respect to time for the risk model and with respect to both state and time for the queueing model. We consider general claim-size and job-size distributions and provide explicit representations of the relevant auxiliary quantities when these distributions are of phase-type.

We study minimizers of the Pekar functional with a transverse harmonic potential, which models a polaron harmonically trapped in the transverse plane but free along the longitudinal direction. For any confinement strength $\Omega>0$, minimizers exist and, up to a translation along the $x_3$-direction, are radially decreasing in $(x_1,x_2)$ and symmetric decreasing in $x_3$. For small $\Omega>0$ we prove uniqueness up to these symmetries. For sufficiently large $\Omega>0$ we derive a three-term asymptotic expansion of the ground state energy and show that, after a suitable rescaling, the minimizers converge in $H^1(\mathbb{R}^3)\cap L^\infty(\mathbb{R}^3)$ to the product of the normalized ground state of a two-dimensional linear harmonic oscillator and that of an effective one-dimensional nonlinear local problem, displaying asymptotic variable separation. This gives, for the transversely confined Pekar model, a rigorous ground-state counterpart of the three-dimensional-to-one-dimensional dimension-reduction phenomenon numerically observed for Coulomb-type Schrödinger equations with anisotropic confining potentials in [W. Z. Bao, H. Y. Jian, N. J. Mauser and Y. Zhang, SIAM J. Appl. Math., 2013]. Moreover, uniqueness up to the same symmetries is also established for all sufficiently large $\Omega>0$.

Let $(M^{2m},g,f,J)$ be a complete nonsteady gradient Kähler-Ricci soliton satisfying $\mathrm{Ric}+\nabla^2 f=\lambda g$, $\lambda\neq 0$. We prove that constant scalar curvature forces the soliton to be rigid. More precisely, the universal cover splits holomorphically and isometrically as $N^{2k}\times\mathbb{C}^{m-k}$, where $N^{2k}$ is Kähler-Einstein with $\mathrm{Ric}_{g_N}=\lambda g_N$. In the shrinking case the quotient is trivial; in the normalization $\lambda=1/2$ one has $R\equiv k$. The Kähler result rests on a Riemannian rigidity criterion. On a complete nonsteady gradient Ricci soliton, if $\mathcal{L}_{\nabla f}\mathrm{Ric}$ is nonnegative or nonpositive everywhere, then it vanishes and the soliton is rigid; no assumption on the scalar curvature is needed. Along the Ricci flow generated by the soliton, this means that a Ricci tensor that is monotone in time is constant in time, and that this forces rigidity. We also give a direct proof that the pinching $0\leq\mathrm{Ric}\leq\lambda g$ forces constant scalar curvature and radial flatness, yielding the rigidity conclusion through the Petersen-Wylie characterization. The proof combines a weighted cutoff argument with a partial Codazzi symmetry for the Ricci endomorphism.

Structure-preserving numerical methods for gradient flows have been extensively developed when the governing equations and free energies are known. However, accurately predicting dynamics from observational data while preserving intrinsic physical properties remains challenging. Although neural operators, such as Fourier neural operators (FNOs), provide powerful data-driven approximations, their predictions do not inherently guarantee physical structure preservation. To address this challenge, we integrate deep learning with structure-preserving exponential time differencing (ETD) methods. Instead of directly learning the evolution operator, we introduce a *trajectory-inferred free energy* (TIFE), which reconstructs the unknown energy density from observed trajectories through a Duhamel-based learning framework. The learned energy determines the variational force and stabilization parameter, enabling structure-preserving numerical evolution. We further extend the framework to jointly identify the free energy and diffusion coefficient. Under suitable assumptions, we establish the maximum-bound principle, unconditional dissipation of the original learned energy, and error estimates incorporating discretization and inference errors. Numerical experiments demonstrate accurate energy recovery and reliable long-time predictions.

For a non-empty bounded set $E\subset\R^d$, the upper and lower box dimensions of its orthogonal projections are constant onto almost all $m$-dimensional subspaces. These almost-sure values are called the upper and lower $m$-box dimension profiles of $E$, respectively. We study their relationship with intermediate dimensions, which interpolate between Hausdorff and box dimensions by restricting the relative sizes of covering sets. We establish several equivalent characterizations of both quantities and obtain upper and lower bounds for box dimension profiles in terms of intermediate dimensions, with the lower bounds also involving the upper Assouad spectrum. We further derive explicit formulas for the box dimension profiles in terms of the box dimension and the initial growth rate of the intermediate dimensions, under a suitable condition on the quasi-Assouad dimension.

We consider the pressureless Euler equations in one spatial dimension. These equations model the dynamics of particles on a line that interact only through perfectly inelastic collisions. The flow was first shown to be unique for given initial conditions by Huang and Wang, assuming the entropy and the strong initial continuity of energy conditions hold. In this paper, we prove an analogous uniqueness theorem using a Lagrangian interpretation. The key ingredients are showing that each solution satisfying the entropy condition and an initial kinetic energy bound has this particular type of Lagrangian interpretation and a monotonicity inequality involving conditional expectation.