Papers

In this paper, we introduce a novel multi-fluid approximation scheme for the one-dimensional Vlasov-Poisson system, that is relevant for arbitrarily long time intervals. For low regularity solutions, we establish an error estimate in the Wasserstein-1 distance, obtaining a convergence rate of order $O(h^{\frac{2}{3}})$, where h denotes the velocity grid size. This establishes the consistency, as the grid size tends to zero, between the pressureless Euler-Poisson and the Vlasov-Poisson systems. Numerical illustrations are given to illustrate the efficiency of our approach.

We establish an integral Harnack inequality and expansion of positivity for solutions to fractional porous medium-type equations in the fast diffusion regime, under optimal tail assumptions.

We prove that strong solutions to the incompressible, homogeneous Navier-Stokes system on $\mathbb{R}^3$ are global in time for initial data in the Sobolev space $H^1_\sigma(\mathbb{R}^3)$. The central ingredient of the proof is a new energy estimate that rules out finite-time blow-up in $L^r(\mathbb{R}^3)$ for $r = 2 + 2 / \sqrt{3}$. This estimate is based crucially on properties of the material derivative in combination with the divergence-free constraint, together with an intricate estimate that harnesses the delicate interplay between the viscous coercivity of the new energy functional and the control of a perturbing pressure term. Notably, the estimate does not carry over to systems that merely share the standard energy structure, such as T. Tao's averaged Navier-Stokes system. As a consequence of the new energy estimate, every Leray-Hopf weak solution $u$ to the homogeneous Navier-Stokes system is globally unique whenever the initial value $u(0, \cdot)$ belongs to $L^2_\sigma(\mathbb{R}^3) \cap L^3(\mathbb{R}^3)$. Even for $u(0, \cdot) \in L^2_\sigma(\mathbb{R}^3)$, the only possible non-uniqueness in the class of Leray-Hopf weak solutions is initial branching, and every such solution is $C^\infty$-smooth on $\left( 0, \infty \right) \times \mathbb{R}^3$. Moreover, if $u(0, \cdot)$ is smooth with derivatives of all orders decaying rapidly at infinity, we show that $u$ is in fact $C^\infty$-smooth on all of $\left[ 0, \infty \right) \times \mathbb{R}^3$.

This work studies the placement of an entanglement source along a communication line formed by two noisy qubit channels. Recent work argued, on analytical and numerical grounds, that midpoint placement should be at least as favorable as endpoint placement. Here it is shown that, for arbitrary qubit channels, if a sequential composition can preserve entanglement, then the corresponding parallel action cannot annihilate all entanglement. The proof uses the transpose-factorization criterion introduced in that recent work. Quantum Sinkhorn scaling converts every strictly positive qubit channel into a unital representative, and the special normal form of unital qubit channels then yields the required factorization of the transposed map through the original channel. Depolarizing regularization and the closedness of the set of entanglement-breaking channels extend the result to arbitrary channels. Consequently, $\Lambda_1 \otimes \Lambda_2 \in \mathrm{EA}$ implies $\Lambda_2 \circ \Lambda_1 \in \mathrm{EB}$, and, by exchanging the two channels, the same holds for the opposite composition order. This proves the recent conjecture that midpoint placement is optimal for all qubit channels, in the feasibility sense in which that optimality was originally defined.

We study the geography of surfaces of general type with extremal cotangent dimension, in other words surfaces which have either no global holomorphic symmetric differentials at all or the maximal asymptotic growth of their number, i.e. big cotangent bundle. We are mainly interested in surfaces with low slope K^2 /$\chi$, which, as far as maximal cotangent dimension is concerned, were out of reach of previous methods. We prove vanishing theorems for symmetric logarithmic differentials on minimal rational surfaces and for differentials on their double covers and extend a bigness criterion of Sakai to fibrations of general type in the sense of Campana. As a consequence, we prove, on the one hand that generic Horikawa surfaces have no nontrivial symmetric differentials and on the other hand that there exist Horikawa surfaces with big cotangent bundle.

Applying the BBDJS minimal model to the derived symplectification of a $-1$-shifted contact derived Artin stack and descending algebraically along the structural free $\mathbb{G}_m$-action, we construct an $\ell$-adic perverse sheaf on any oriented such stack, and use Verdier's specialization equivalence for monodromic sheaves to equip it with a tame twisted monodromy operator $\theta$. We show that the local fundamental class of a Legendrian $L$ satisfies $\theta \circ \mu_L = (-1)^{\mathrm{vdim} L}\mu_L$, so that on Legendrians of odd virtual dimension, the class vanishes due to $\theta$-invariance. Moreover, both parities occur already on the $A_1$ chart while an odd Legendrian can carry a nonzero local class. We further formulate a contact analogue of Joyce's conjecture for a graded orientation, in which the orientation datum is twisted by the parity of the virtual dimension. Under the assumption of a monodromic refinement of the symplectic conjecture, we construct the categorified Legendrian 2-categories $\mathfrak{L}\mathcal{F}_c(X)$ and $LLeg_0$ via $\ell$-adic pull-push functors. Finally, we show that the contact Behrend function is identically $1$, so that the associated Donaldson-Thomas invariant is the compactly supported étale Euler characteristic of the classical truncation, and that the higher traces of $\theta$ recover the singularity type that the first trace discards. As an application, we show that the symplectic invariant of a derived intersection of conic Lagrangians in a cotangent bundle vanishes identically, while the contact invariant computes the Euler characteristic of the projectivized intersection, with an explicit formula for conormal bundles.

We consider solutions of space-periodic reaction-diffusion-advection equations in the whole space R^N, with general unbounded initial support. We show the existence of spreading sets for the solutions at large time, and variational formulas for the spreading speeds. The location of the upper-level sets of a solution with unbounded initial support is estimated in terms of that initial support and the spreading set for the solutions with bounded initial supports. The results hold for the standard classes of Fisher-KPP, ignition, or bistable reactions. The general results apply to the cases of V-shaped and Lambda-shaped initial supports. The proofs rely in particular on upper estimates of what we call retracting solutions.

Let \(G\) be an infinite residually finite countable discrete group with Kazhdan's property~\((T)\). Assume that its finite-dimensional irreducible unitary representations admit an exhaustive ordering with nondecreasing dimensions tending to infinity and bounded consecutive dimension ratios. From this data we construct a separable unital residually finite-dimensional \(C^*\)-algebra with a faithful quasidiagonal tracial state \(\tau=\frac14\mu_1+\frac34\mu_2\), where \(\mu_1\) is not quasidiagonal. For the resulting algebra, quasidiagonal traces do not form a face of the tracial state space. The construction has two copies of a weighted representation ladder joined at their top levels. We prove that every quasidiagonal trace gives equal weight to the two copies. The proof combines uniform Kazhdan spectral projections, exact ranks of rounded finite-rank compressions, and a weighted Hilbert–Schmidt comparison that controls all representation blocks.

We construct theories of Carrollian electromagnetism coupled to an axion field. Obtained by performing a $c \to 0$ expansion of axion electrodynamics, these are of two types: electric and magnetic. We derive the Lagrangians and work out the general classical solutions for both. While the electric theory has the axion effectively decoupled from the electromagnetic fields, the magnetic theory is endowed with a nontrivial axion coupling that leads to interesting phenomenology. In particular, a Carrollian analogue of the Witten effect, wherein a magnetic monopole in the presence of a constant non-vanishing axion acquires an electric charge and becomes a dyon, is present in the magnetic theory. At the same time, the linear magnetoelectric effect vanishes entirely, a conclusion we establish by performing a thorough Dirac-Bergmann analysis of the constraint structure of the magnetic theory and a proper identification of its physical phase space.

We present a fixed nondegenerate tetrahedron with a nested largest-dihedral-anglebisection branch that collapses onto a unit edge. The selected dihedral angles to bisect are uniquely largest ones at every bisection step. Some face angles of the branch elements tend to zero, the corresponding diameter-to-inradius ratio diverges, and the diameters of the elements produced do not tend to zero, although every dihedral angle stays uniformly away from both 0 and {\pi} and all face and dihedral angles satisfy a uniform maximum-angle bound. The proof is based on an exact invariant-range argument. This example shows that the largest-dihedral-angle bisection algorithm can produce degenerating tetrahedral partitions.

In this paper, we study the effect of an external random field on the localisation / delocalisation of the long-range Discrete Gaussian Chain (DGC), recently studied by Garban arXiv:2312.04536 and Coquille et al. arXiv:2412.15782. We prove that the model is delocalised at every inverse temperature for any polynomial decay $\alpha > 3/2$. This behaviour differs qualitatively from the one of the disorder-free long-range DGC, for which delocalisation at every temperature occurs only for $\alpha > 2$, similarly to what happens for long-range Ising models as already proved by Aizenman and Wehr. Additionally, we provide a more detailed description of the delocalised phase by identifying the growth exponent and scaling limit of the height at the origin for any decay exponent $\alpha > 3/2$.

Consider the random lacunary series $f(z) = \sum_{k=1}^\infty \frac{\xi_k}{\sqrt{k}} \, z^{2^k}$ on the unit disk, where $\{ \xi_k \}$ are independent standard complex Gaussian random variables. We show that almost surely, a.e. $\zeta \in \partial \mathbb{D}$ is a Plessner point of $f$, yet the image of every Stolz angle with vertex at $\zeta$ has asymptotic density zero. This gives a negative answer to questions of Collingwood and Baernstein concerning possible strengthenings of Plessner's theorem. In this example, for a.e. $\zeta \in \partial \mathbb{D}$, the non-tangential range of $f$ at $\zeta$ has zero area. The non-tangential range cannot be much smaller: we show that for an arbitrary holomorphic function on the unit disk, the non-tangential range has Hausdorff dimension 2 at almost every Plessner point.

The entropy rate of a uniformly quantized stationary Gaussian process is the fundamental limit on its lossless compression. It has no closed form, and the classical high-resolution approximation breaks down whenever the spectral density is small on part of the band, as routine smoothing and filtering produce. Here we present two methods for computing the rate. The first is an analytical approximation, obtained from an exact dithering identity and the Kolmogorov-Szegő formula, in which the quantization noise power acts as a floor on the spectral density. The second is a Monte Carlo estimate of the exact rate: a minimum-phase moving-average representation makes the quantized sequence a hidden Markov process whose optimal particle filter is available in closed form, and the filter's log-likelihood on one long sequence gives the rate. The approximation agrees with the estimate to within a few millibits per sample over most of the parameter range for quantization steps up to the signal standard deviation, including processes on which the classical formula fails; for coarser steps the estimate should be used instead. At a step of a quarter of the standard deviation, linear predictive coding comes within 2 to 4 percent of the rate, FLAC within 3 to 32 percent, and general-purpose compressors remain 12 to 89 percent above it.

We establish convergence bounds for deep $V$-learning with horizon $H$. The algorithm fits a scalar value function to targets from executed transitions and selects actions using a predictive model and the value function. For current observed-successor targets with fresh true-kernel outcomes, the conditional mean is $\mathcal{T}^\beta V$, which averages over behavior-policy actions. The Bellman optimality update is $\mathcal{T} V$. We decompose the update error into six residuals: fitting, transition reuse, target construction, replay, action selection, and exploration. Under $L^s$ concentrability, their $L^p$ norms ($p=s/(s-1)$) control expected $L^1$ policy loss. The bound explicitly weights residuals from only the last $H-1$ update blocks, plus an initialization term for shorter runs. We quantify the cost of a shared sampling distribution across horizon levels. For statistical error bounds of order $n^{-\nu}$, we derive optimal continuous allocations and an integer allocation whose objective is within a factor $2^\nu$ of the constrained optimum. A margin condition with exponent $\alpha$ gives action error of order $\Lambda^{1+\alpha/p}$, where $\Lambda$ combines network drift and score error; a one-step construction proves the exponent sharp. Bounds on the distance between frozen and optimal scores transfer an optimal-gap condition to frozen-iterate gap bounds while retaining the mass of optimal ties. Survival probabilities and coverage conditions at deployment yield bounds for policies selected with approximate scores. Separate spatial ReLU networks per horizon level give a conditional neural regression rate, and the finite-state case gives a log-free expected fit rate. These results give expected policy-loss consistency for the fixed-horizon generative-reset approximate-ERM procedure with exact action scores and provide an explicit residual-decay criterion for FIFO/interleaved SGD.

The flip symmetry on link diagrams induces an involution on Khovanov-Rozansky ${\frak{gl}}_{N}$ homology. We prove that this involution is diagonalizable with eigenvalues $\pm1$. On the one hand, it is the identity over $\Bbb{F}_2$, generalizing a previous result of Chen and the author. On the other hand, it is expected to be nontrivial over $\Bbb{Z}$ in general. The key ingredients of the proof are (1) a homotopy perturbation argument via a detailed study of the fork twist, which allows us to reduce the computation to planar ${\frak{gl}}_{N}$ webs, and (2) the computation of the flip map for planar ${\frak{gl}}_{N}$ webs via diagrammatics of Soergel bimodules. The latter computation can also be interpreted as a naturality result for the half twist action on type $A$ Soergel bimodules, which might be of independent interest.

In this paper, we study multiple Eisenstein series (MES) in positive characteristic. By computing and analyzing the $t$-expansions of MES, we determine the precise "weights" of their coefficients. This framework enables us to establish a graded algebra structure for MES, extending the direct sum result for Thakur's multiple zeta values proved in [Cha14]. Our result may also be viewed as a function field analogue of the corresponding direct sum result of Bachmann and Kanno [BK26].

We consider a class of neural field models describing the macroscopic activity of an infinite population of neurons organized into finitely many stacked layers, written as a system of semi-linear higher-order parabolic equations. For this class of biologically inspired equations, we show that the higher-order dissipation, modeling gap junction activity regulation, induces a spectral gap ensuring that the associated semigroup possesses an inertial manifold, that is, a finite-dimensional manifold attracting all orbits of the infinite-dimensional system of neural activity. Our main result is an upper bound on the dimension of this manifold, giving its explicit scaling in the dissipation order and strength, the leaking rate, and the norm of the connection operator. We interpret this as a step toward the mathematical modeling of the empirically motivated concept of neural manifold, namely the idea that the collective activity of a large neural population lies on a low-dimensional manifold. For this class of equations, we further prove universal approximation properties, namely stationary pattern expressivity in terms of the input and input-dependent dynamical expressivity over finite time horizons. Finally, we provide sufficient conditions for the existence of traveling waves and nontrivial stationary solutions.

This paper studies the regularity of the value function arising from a multidimensional continuous-time principal-agent model with separable, nonquadratic effort costs. The associated stochastic control problem has the output and the agent's continuation utility as state variables, and its Hamilton-Jacobi-Bellman equation is fully nonlinear and degenerate, with potentially unbounded coefficients. We address these difficulties by adding an independent regularization noise and bounding the effort. For the resulting problem, we establish classical regularity of the value function and show that the optimal effort is unique, positive and remains in a fixed compact subset, uniformly with respect to both the control restriction and the regularization parameter. These estimates allow us first to remove the control restriction and then to let the additional noise vanish. Consequently, we prove that the regularized value function converges to the original value function and conclude that the latter belongs locally to the Sobolev space $W^{2,1}_{\infty, \mathrm{loc}}$, thereby extending the regularity analysis to separable nonquadratic effort costs, for which the arguments yielding classical solutions in the quadratic-cost setting no longer apply.

In this paper, we study the Liouville equation with a nonlinear Neumann boundary condition \begin{equation*} \begin{cases} -\Delta u = Ke^{2u} & \text{in } \mathbb{B}^{2},\\[2mm] \dfrac{\partial u}{\partial \nu}+\lambda = ke^{u} & \text{on } \partial\mathbb{B}^{2}, \end{cases} \end{equation*} where $K,k\in\mathbb{R}$, $\lambda\in(0,1]$ are constants, and $\nu$ denotes the outward unit normal on $ \partial\mathbb{B}^{2}$. We establish a classification of all smooth solutions to the equation. Our approach is a boundary-adapted P-function method which treats all signs of $K$ and $k$ within a unified framework. As applications of the classification result, we establish a family of sharp Sobolev-trace-type inequalities encompassing the classical Lebedev–Milin inequality, together with a corresponding deficit estimate.

We prove that the very-high temperature long-range Ising model on $\mathbf{Z}^d$ is a factor of an i.i.d. process, and provide quantitative information on the coding radius. We do not assume that the model is ferromagnetic, but require the interactions to decay faster than $1/n^{2d}$. While previous constructions were known, the novelty of our approach is that we obtain an explicit polynomial upper bound on the tail of the coding radius. In particular, our construction yields a factor map with finite expected coding volume. This answers a question of a recent preprint by Chazottes, Gallo and Takahashi, see arXiv:2602.14618.

We construct a bounded Hochschild one-cocycle which detects amenability of tracial states on unital C*-algebras. For an arbitrary trace the construction uses a faithful representation containing the GNS representation as a direct summand. When \(\tau\) is faithful, \(H_\tau=L^2(A,\tau)\), and \(J\) is the canonical conjugation, the cocycle has the particularly simple form \[ \delta_\tau(a)(x) =J\pi_\tau(a^*)Jx-xJ\pi_\tau(a^*)J. \] It is inner in a natural Banach \(A\)-bimodule if and only if \(\tau\) is amenable. We also give the corresponding reformulation of embeddability into an ultrapower of the hyperfinite \(\mathrm{II}_1\) factor. Finally, for faithful traces we study a C*-algebraic variant of the Christensen–Sinclair two-cocycle and prove, with all admissibility details, that amenability makes this cocycle a coboundary.

In this letter, a novel 3-D Finite-Difference Time-Domain (FDTD) framework is proposed that circumvents the costly volumetric scaling of conventional FDTD methods. By representing the electromagnetic fields as low-rank Quantized Tensor Trains (QTT), the memory requirements scale logarithmically with system size. Moreover, the various numerical operations that constitute the FDTD scheme can be efficiently implemented in this format, with their computational cost also exhibiting a logarithmic complexity. The simulation of systems in open space is enabled by the inclusion of a uniaxial PML. A validation example demonstrates that the method achieves excellent accuracy compared to the traditional full-grid (FG) FDTD method, while significantly reducing the required computational resources. Memory savings of several orders of magnitude are obtained, highlighting the potential of the proposed framework for the simulation of large multiscale electromagnetic systems.

We prove that any embedded minimal Lagrangian torus in the complex projective plane is congruent to the Clifford torus. The proof uses a two-point function constructed from normalized horizontal lifts to the five-sphere.

We prove that minimum-output Rényi-entropy additivity can fail by almost one bit simultaneously at every nonnegative order. For every $\varepsilon\in(0,\log2)$, there exists a finite-dimensional quantum channel with a real Stinespring isometry such that the same maximally entangled input witnesses a tensor-square entropy gap of at least $\log2-\varepsilon$ for all $p\in[0,\infty]$. The output dimension can be chosen to be $O(\varepsilon^{-3})$ as $\varepsilon\downarrow0$. The construction uses direct products of free groups: tensorized Haagerup estimates control the one-copy outputs, while commutation between distinct factors forces exact Bell-branch collisions at two copies. Strong convergence gives both an existential realization through finite-dimensional representations of right-angled Artin groups followed by realification, and a Haar-orthogonal model whose success probability tends to one as the matrix dimension grows. We also determine the exact Bell quotient, prove asymptotically sharp regular-radius bounds, and show that the cubic output-dimension scale is optimal within the present purity–rank certificate.

Let $k$ be a field and let $\Lambda=\left(\begin{smallmatrix}A&N\\M&B\end{smallmatrix}\right)_{\phi,\psi}$ be a finite-dimensional Morita context algebra. We introduce a bilateral approximation construction which glues support $\tau$-tilting modules over $A$ and $B$ by alternately correcting the two corner components through minimal approximations and pushouts. When this process terminates, it yields a support $\tau$-tilting $\Lambda$-module with the prescribed componentwise torsion class. The one-sided case recovers Zhang's triangular-matrix construction, while the two-sided compatibility conditions give direct corner induction and, for radical-valued connecting maps, are also necessary, extending the Gao–Huang criterion. Examples show that the bilateral correction process can terminate even when neither one-sided compatibility condition is satisfied, while in other examples the process never terminates.

Rapid algorithmic developments have accelerated the sampling of redistricting ensembles (balanced graph partitions), yet evaluating rare events and sampling complex target measures remains a core challenge due to the high-dimensional and combinatorial nature of the phase space. We address a prerequisite for stratified sampling on this space: constructing and diagnosing candidate strata with suitable coverage and overlap. We build a grammar on observed plans by clustering districts into representative “letters” and using them to form plan-level “words.” A partition of unity over these words gives a soft assignment of plans to strata and allows us to estimate stratum masses and an overlap-induced flux matrix. We demonstrate this computational pipeline using real-world congressional redistricting data from Connecticut and examine how strata learned from one target distribution behave under related distributions. The resulting construction provides a foundation for future stratified sampling on spaces of redistricting plans or balanced graph partitions. We do not implement a complete stratified sampler here; evaluating whether the proposed strata improve sampling efficiency or reduce estimator variance is left for future work.

A path in a properly edge-colored graph is rainbow if its edges have pairwise distinct colors. For a proper edge-coloring $c$ of a graph $G$, let $\operatorname{rpc}(G,c)$ be the minimum number of rainbow paths needed to cover $E(G)$, and let $\operatorname{rpc}(G)$ be the maximum of $\operatorname{rpc}(G,c)$ over all proper edge-colorings of $G$. We prove that, for every fixed $0<\alpha<1$, every properly edge-colored $n$-vertex graph with minimum degree at least $\alpha n$ satisfies $\operatorname{rpc}(G,c)\leq(1+o(1))n/2$, where the coefficient $1/2$ is best possible. We also determine $\operatorname{rpc}(G)$ asymptotically for every complete multipartite graph. If $G=K_{n_1,\ldots,n_r}$ has order $n$ and largest and smallest part sizes $M$ and $s$, respectively, then, uniformly over all choices of the number and sizes of the parts, $\operatorname{rpc}(G)=(1+o(1))\max\{\min\{\lfloor n/2\rfloor,n-M\},(n-s)/2\}$. The proof combines pseudorandom packings of globally rainbow linear forests with a decomposition into dense parts and prescribed avoidance for arbitrary dense graphs, and with reserved connectors and a direct dominant-part argument for complete multipartite graphs.

We consider the Kolchin-generic complete $1$-type of the inviscid Burgers equation $Dx=x\partial x$ in a differentially closed field of characteristic zero with two commuting derivations. We prove that this type has Lascar rank $1$ and Morley rank $\omega$.

We study deterministic adaptive optimization of globally $\beta$-smooth, $\mu$-strongly convex functions using exact scalar function values. Queries and outputs lie in $B_2^d(R)$, and the minimizer lies in $B_2^d(R/2)$. Set $\kappa=\beta/\mu$, $Q=\beta R^2/\epsilon$, and $D_d=(d/\log(ed))^{1/3}$. For sufficiently large $d$ and $0<\epsilon\le c_\epsilon\beta R^2$, the minimax value complexity $N_\epsilon$ satisfies \[ \begin{aligned} N_\epsilon &\ge c d\min\{\sqrt Q,\sqrt\kappa,D_d\},\\ N_\epsilon &\le C d\min\left\{ \sqrt Q,\sqrt\kappa[1+\log_+(Q/\kappa)] \right\}, \end{aligned} \] where $c,C,c_\epsilon>0$ are universal constants and $\log_+(t)=\max\{0,\log t\}$. The lower bound uses an exactly shielded smooth chain and batched delayed rotations; the upper bound combines finite differences, acceleration, and restart. When $\min\{Q,\kappa\}\le D_d^2$, these bounds match up to constants in the accuracy-dominated regime $Q\le\kappa$ and at constant relative accuracy $\epsilon=\Theta(\mu R^2)$. For arbitrarily higher accuracy in the range $\kappa\le D_d^2$, the bounds differ by at most $1+\log(\mu R^2/\epsilon)$; the optimal accuracy dependence remains unresolved in general.

We present a counterexample to the relay channel capacity characterization proposed in a recent work, which is based on properties of typical sequences. The counterexample uses a binary relay channel with a noiseless source-destination component and binary symmetric relay links. An achievable scheme exceeds the proposed capacity characterization, thereby disproving the characterization of the capacity for general relay channels. We highlight that the counterexample is obtained with the assistance of ChatGPT-6 Astra and the proofs are simplified with human effort.