Papers

We present a detailed study of closed smooth manifolds having Kähler forms that pullback to exact forms on the universal cover. We show that these manifolds, which we call symplectically aspherical Kähler manifolds, exist in abundance, even outside the aspherical setting, and have interesting topological and geometric features, such as large fundamental group á la Kollár and the absence of Kähler metrics of positive scalar curvature. Motivated by the latter, we extend the Gromov–Lawson Conjecture on aspherical manifolds to symplectically aspherical manifolds and prove it in the spin case. We also study Kähler cones on symplectically aspherical Kähler manifolds and the realizability problem of their fundamental group, and explore their other complex geometric properties.

This paper intends to understand the regularity and stability problem on the 3D incompressible magneto-micropolar equations with zero magnetic and angular viscosities in a strip domain. The magneto-micropolar system models the electrically conducting micropolar fluid in the presence of a magnetic field. The lack of magnetic diffusion and angular dissipation makes it impossible to prove even small data global well-posedness result, let alone general large data global regularity. This paper presents a steady-state setup around which any perturbations can be shown to be globally regular and stable. More precisely, any small perturbation near a steady magnetic field perpendicular to the horizontal boundary leads to a unique global classical solution. In addition, the solution is shown to converge to the steady state at an almost exponential rate as time goes to infinity. These appear to be the very first rigorous global results on the magneto-micropolar equations concerned here.

In analogy with ergodic theory, we introduce the notions of joint topological transitivity and joint minimality for commuting minimal homeomorphisms on a compact metric space $X$. We prove an analogue of the Berend–Bergelson theorem for the first notion and investigate to what extent this theorem holds for the second.

We present a new algebraic modeling of the Supersingular Isogeny Problem as a system of multivariate polynomial equations, in the case where the elliptic curves are connected by an isogeny whose degree is a power of $2$ or $3$. This modeling relies on Renes formulas for elliptic curves in Montgomery form (degree $2$) or triangular form (degree $3$). We investigate several algebraic properties of these systems: we prove that they are zero-dimensional, compute the dimension of their highest degree part, and show that they are not in generic coordinates. Experimental results show that solving these systems via Gröbner basis techniques is significantly faster than solving the algebraic modeling with modular polynomials.

We consider the optimal decay rate of global solutions to the convection-diffusion equation with zero-mass initial data. We establish an improved decay estimate of the global solutions under the zero-mass condition. Moreover, we derive a self-similar asymptotic profile of the global solutions. This result shows that the improved decay rate is optimal and provides a necessary and sufficient condition for attaining this decay rate.

Trace estimation is central in many lattice QCD computations, but the accuracy of the standard, stochastic Hutchinson method improves only with the square root of the sample size, making precise results expensive. We investigate two complementary variance reduction strategies. First, multigrid multilevel Monte Carlo uses a multigrid hierarchy to construct an unbiased multilevel estimator via recursive coarse grid corrections available from the multigrid hierarchy of the solver. Second, stochastic probing uses distance-$d$ graph colorings; we propose a torus based coloring that requires substantially fewer colors than hierarchical probing at the same distance. We test these approaches on two representative problems: the connected pseudoscalar correlator and disconnected fermion loops. For the connected pseudoscalar two-point function, the multilevel decomposition yields a variance reduction of up to $\mathcal{O}(10^5)$ at large time separations and translates into a clear cost reduction at fixed accuracy, thus confirming earlier results of arXiv:2412.06347. For the disconnected loops, in contrast, the multilevel decomposition provides only moderate gains, whereas probing combined with dilution delivers a substantial cost reduction that improves as the number of probing vectors is increased. Overall, the results highlight a pronounced complementarity: deflation schemes are most effective for observables dominated by long distance propagation, while probing is most effective for localized quantities.

We investigate metric dimension and the localization game for several families of directed analogues of strongly regular graphs and their generalizations, adapting a probabilistic method of Babai (1980) for bounding the size of resolving sets in undirected strongly regular graphs. We derive upper bounds on the localization number and metric dimension depending on the order of the graph and the maximum number of common out-neighbours for a pair of vertices. We consider normally regular digraphs, so-called "ordinary graphs", classes of Deza digraphs, divisible design digraphs, nearly doubly regular tournaments, and certain doubly regular team tournaments. In particular, for asymmetric normally regular digraphs on $n$ vertices, we show that these invariants are bounded above by $O(\sqrt{n} \log n)$, and improve this to $O(\log n)$ for a class of doubly regular team tournaments.

For the Potts model on the Cayley tree $\mathbb{T}^d$ with branching factor $d\geq 2$, we consider the free state which is obtained as the limiting Gibbs measure with free boundary conditions. We prove that the free state is either extremal or glassy (i.e., whose decomposition into extremal Gibbs measures contains uncountably many components). As a corollary, the free state for the Ising model on $\mathbb{T}^d$ is glassy if and only if the inverse temperature $\beta>\operatorname{arctanh}(1/\sqrt{d})$; this generalizes a previous result by Gandolfo, Maes, Ruiz and Shlosman (2020) from a very low temperature regime to the entire spin-glass regime.

We provide a classification result of positive solutions with a non-removable singularity at the origin to the subcritical Lane-Emden equation $-\Delta_p u = u^q$ in $\mathbb{R}^N \setminus \{0\}$, where $p\in(1,N)$, $q\in (p-1,p^*-1)$ and $u^{q-p+1} \in L^{\frac{N}{p}}(\mathbb{R}^N \setminus B_R)$.

We prove a quantitative Roth theorem in the plane for the two-dimensional polynomial pattern $(x_1,x_2), (x_1,x_2)+(t_1,t_2), (x_1,x_2)+(t_1^2+t_2^2,t_1^3+t_2^3)$. A pointwise convergence result for the associated polynomial ergodic average is also obtained. A new bilinear Sobolev improving estimate serves as the primary analytic tool, derived from a new sublevel set estimate.

We study the $L^p(\mu)$-norm of the metric projection onto a closed, convex set $C \subset \mathbf{R}^n$ when $\mu$ is the uniform measure on the sphere or the standard Gaussian measure on $\mathbf{R}^n$. Up to universal constants, we determine the optimal reverse Hölder inequalities (i.e., $L^q-L^p$ estimates for $q > p$) for both settings and for all $1 \leq p < q \leq \infty$. The optimal constants in these inequalities depend polynomially on the dimension $n$. We establish upper bounds for the expected norm of the metric projection for a wide class of probability measures. Our inequalities improve and extend previous results of S. Chatterjee.

We study a Marker-and-Cell (MAC) scheme for the barotropic Euler system. First, we apply the recently developed Lax-type convergence theorem to show the convergence of the MAC scheme to i) a dissipative weak solution unconditionally and ii) a strong solution as long as it exists. Second, We derive relative energy error estimates up to the lifespan of a strong solution, without assuming uniform boundedness of the numerical sequence. Additionally, assuming the boundedness of the numerical solutions, we obtain the optimal relative energy rate of 1, corresponding to a convergence rate of 1/2 for the numerical solutions. Finally, we corroborate our theoretical results by numerical experiments.

A measurable set $A\subseteq \mathbb S^{d-1}$ is called zero-sum-free if there are no $\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in A$ with $\boldsymbol{x}+\boldsymbol{y}+\boldsymbol{z}=\boldsymbol{0}$. Bukh asked whether every zero-sum-free measurable subset of $\mathbb S^{d-1}$, for $d\ge3$, has normalized surface measure at most $\frac{1}{2}$. He also pointed out that even the asymptotic behavior as $d\to\infty$ was unknown. We answer Bukh's asymptotic question by proving that every such set has normalized surface measure at most $\frac{\lfloor (d+1)^2/2\rfloor}{d(d+1)}=\frac{1}{2}+O\left(\frac{1}{d}\right).$ Since the lower bound $\frac{1}{2}$ comes from open hemispheres, this determines the asymptotic extremal density. By monotonicity, upper bounds in low-dimensional cases are especially important. We use a stability argument to improve the bound from $\frac{3}{5}$ to $\frac{71}{120}$ in dimensions $4$ and $5$.

In this paper we analyze the Lyapunov exponents of a slow-fast system where the slow component is an Ornstein-Uhlenbeck process which perturbs the linear evolution of a fast variable through a bilinear form. These naturally arise in many finite-dimensional models for turbulence such as Galerkin truncation of 2D Navier-Stokes, the Lorenz 96 system, and the Lorenz 63 system. Using our general results about the slow-fast system, we are able to prove phase transitions in the ergodicity of each of these models when degenerate stochastic forcing is applied: as a parameter (e.g. noise strength or viscosity) varies, the number of invariant measures of the system switches from one to several. We are also able to obtain precise asymptotics for the top Lyapunov exponent associated to the unstable invariant measure. The crux of our proof is using a Wiener chaos expansion to show that mass quickly transfers from stable modes to unstable ones.

Matrix Brownian motion provides a powerful framework for studying crossover ensembles in quantum chaos and quantum transport, as well as thermalization and information scrambling in many-body dynamics. Here, we develop a unified diagrammatic framework to characterize Brownian ensembles for orthogonal and symplectic random matrices, which describe systems with particle-hole symmetry. We compute polynomial averages up to fourth order and construct an orthogonally invariant interpolation for the disconnected $\mathrm{SO}^-(q)$ sector of the orthogonal group. We consider applications relating to the fields of quantum information, quantum chaos, and quantum transport.

We prove that the maximal normalized surface measure of a spherical set in d dimensions avoiding solutions to x + y + z = 0 approaches 1/2 as d goes to infinity. This gives a partial answer to a question of Bukh, who conjectured 1/2 to be the optimal bound for all d >= 3.

We give a complete classification, up to $k$-Möbius equivalence, of exceptional rational functions of degree five over every finite field. The classification gives explicit normal forms in every characteristic, exact parameter identifications, and the resulting class counts. The main structural input is an arithmetic theory of shifted dihedral isogeny quotients valid in every odd degree. For each odd $n\geqslant3$, the separable degree-$n$ rational maps whose geometric monodromy group is isomorphic to $D_n$ and whose nontrivial inertia groups are generated by reflections are precisely the maps induced by cyclic $n$-isogenies on shifted Kummer quotients. We determine the exact ambiguity of the isogeny data under two-sided $k$-Möbius equivalence by means of a signed Frobenius-descent invariant. The theory allows arbitrary odd $n$, reduced kernels when the characteristic divides $n$, and wild reflection inertia. If Frobenius acts on the cyclic kernel by $\lambda\in(\mathbb Z/n\mathbb Z)^\times$, the induced map is exceptional exactly when both $\lambda-1$ and $\lambda+1$ are units modulo $n$.

This paper studies a non-separable composite $\ell_0$-$\ell_2$ regularization model that simultaneously enforces sparsity and smoothness for inverse problems. The $\ell_0$ norm induces inherent nonconvexity and nonsmoothness, while linear transformations further introduce nonseparability, making the problem computationally challenging to solve. The existing inexact augmented Lagrangian method suffers from high computational complexity and unstable convergence. To overcome these difficulties, we develop two novel augmented Lagrangian algorithms with exact multipliers, designed respectively for the full row-rank case and the general matrix case, where all subproblems are globally optimized via closed-form solutions. Furthermore, we prove linear convergence of the proposed method when the transformation matrix is full row rank. In the general setting, all accumulation points of the generated sequence are KKT points for the original problem. Numerical experiments on synthetic data, trend filtering, and image smoothing demonstrate the superior efficiency and accuracy of the proposed methods over the existing method, confirming our theoretical analysis.

In this paper, we study the behavior of integral kernel operators acting on functions of a single real variable. In particular, we attempt to find possible candidates for their optimal endpoint spaces in the class of Banach spaces endowed with a rearrangement-invariant quasinorm. To achieve this, we characterize the existence of a concrete pointwise estimate for their nonincreasing rearrangement. We also build a theory of optimal endpoint spaces for the Calderón operator given by this pointwise estimate. We use these results to propose optimal endpoint spaces for some integral kernel operators.

$2$-designs admitting a flag-transitive automorphism group $G$ with socle $PSL(2,q)$, where $q=p^{f}\geq 4$, are investigated in both the point-primitive and point-imprimitive cases. In the latter case, a complete classification is achieved, and three known examples occur, namely: the complementary designs of $PG(3,2)$ and $PG(3,4)$, and the $2$-$(36,8,4)$ design constructed by Devillers and Praeger in [14]. In the point-primitive case, apart from the Witt-Bose-Shrikhande linear spaces of even order $q$, $48$ sporadic examples are classified. Surprisingly, one of these numerical examples is the linear space with $v=496$ and $k=4$ admitting $P\Gamma L(2,2^{5})$ as a flag-transitive automorphism group, which was missing in the 1990 classification by Buekenhout et al. [7,36,12].

Over an algebraically closed field of characteristic zero, let $H$ be a coquasi-Hopf algebra with bijective antipode, and let $M$ be a tuple of finite-dimensional simple Yetter–Drinfeld modules over $H$. We prove that, if $M$ admits all reflections, then its associated semi-Cartan graph is a Cartan graph. We characterize the finiteness of this Cartan graph by tensor decomposability of $\mathcal B(M)$ and obtain a finite-dimensionality criterion of Nichols algebras. We also show that braided monoidal equivalences preserve reflections and the associated Cartan graphs. As applications, we prove that every Cartan graph associated with a diagonal type tuple over a finite abelian group equipped with an abelian \(3\)-cocycle is covered by one arising from a diagonal type tuple over some finite abelian group $G$ with trivial associator, and that their real-root sets agree at corresponding objects. We also construct a Cartan graph of the former kind that cannot be obtained from any diagonal tuple in \({}_G^G\mathcal{YD}\).

This work sets out to apply primer vector theory to optimal impulse maneuvers in Low Earth Orbit (LEO) and adapt it to the types of perturbations encountered in this environment, which is not readily available in the literature. A review of the theory of optimal control and orbital maneuvering is made and in particular, primer vector theory is laid out in detail based on the extension of the thrust control to a measure, including its generalization to conservative and non-conservative perturbation models. An impulsive multiple shooting optimization scheme in Cartesian coordinates is presented, through a piecewise approach to the problem. Then, some maneuver scenarios with known solutions under the Keplerian model are solved under perturbed orbital dynamics with the help of the primer vector, and the resulting trajectories are compared. The perturbation models include a J2 model, representing the class of conservative perturbations by modeling Earth's nonspherical gravity field, and a J2+Drag model, representing the class of non-conservative perturbations by including the effects of atmospheric drag in LEO\@. For each model, some valid methods of primer vector calculation have been tried and validated between each other, and the primer vector is proven to be a useful tool in reducing the cost of orbital maneuvers.

The dynamics of polynomial roots under repeated differentiation has recently been conjectured to converge to a limiting measure governed by specific nonlinear PDEs, the conjectures being shown in some particular settings. For rotationally invariant initial distributions, a deterministic structured sampling model placing roots on concentric circles was recently introduced by Galligo, Najnudel, and Vu. In this paper, the authors proved convergence under the technical growth condition $m_n / (n \log n) \to \infty$, where $n$ is the number of circles and $m_n$ is the number of points per circle. In this paper, we significantly improve this result by relaxing the growth condition to $m_n / \log n \to \infty$, thus allowing for regimes where the number of points per circle grows proportionally to the number of circles. The key innovation is a refined upper bound on the root magnitudes after differentiation. This sharper estimate prevents the rapid accumulation of errors over multiple differentiations, fully validating a recent conjecture regarding the robustness of the sampling scheme.

This paper investigates the problem of statistical inference for a mixture distribution consisting of a discrete and a continuous component, with a particular focus on the class of rational-infinitely divisible distributions. We consider non-parametric estimation of both components of the mixture as well as the quasi-Lévy measure, assuming that the mixture belongs to the class of rational-infinitely divisible distributions. We propose an estimation framework based on band-limited kernels, which are the functions characterized by compactly supported Fourier transform. Under mild assumptions, the proposed estimators are theoretically shown to achieve polynomial (and in some cases even almost parametric) convergence rates. Finally, we demonstrate the numerical performance of the algorithm on simulated examples.

In this paper, we present a new construction of divisible design graphs with new parameters, obtained by plugging a difference set of a quotient group into a known construction of antipodal distance-regular graphs of diameter 3. Also, we show that in characteristic 2 the new divisible design graphs are Cayley graphs over an elementary abelian 2-group.

In physical layer authentication, verification of a user's identity is based on the characteristics of the transmission channel through which signals are delivered to the authenticator (Bob). In this paper, we assume that the signals received by Bob pass through a \ac{RIS} (controlled by Bob) and that the legitimate transmitter (Alice) is equipped with one antenna. Conversely, the attacker (Trudy) has multiple antennas and uses precoding to deceive Bob's verification. Assuming that Trudy knows all the channel matrices, we first derive her optimal attack strategy. Then, we analyse the conditions under which the channel estimated by Bob is indistinguishable when either Alice or Trudy is transmitting. When Trudy has a single antenna, we show that the indistinguishability condition cannot be met when the channels to the RIS are the result of propagation over multiple paths. For single-path line-of-sight (LOS) conditions, instead, Trudy can impersonate Alice although transmitting from a different position. We verify these results numerically and assess the security of the considered scenario, even when the indistinguishability conditions cannot be met.

We introduce a notion of coarse $r$-median spaces that is a higher-rank analog of Bowditch's coarse median spaces. Our notion is stable under quasi-isometries and recovers Bowditch's coarse medians when $r$ equals 1. We prove that several families of higher rank-symmetric spaces admit coarse $r$-medians with $r$ being the rank of the symmetric space. In particular, our list includes plenty of examples that are known to not admit Bowditch's coarse medians. Our main tools come from convex projective geometry, and we prove the existence of coarse higher medians on all divisible as well as quasi-homogeneous properly convex domains.

A structural classification-and-bridge result is proved for RS series on \(\Gamma_0(2)^+\). Among the fundamental Heegner discriminants with \(h(D)=1\), the irreducibly cubic Fricke-cusp companions occur exactly for \[ D=-11,-19,-43,-67,-163. \] The proof gives an explicit bridge between \(z=1728/j\) and \(X=256t/(t+64)^2\), a hypergeometric gauge identity, and a CM interpretation of the fibre. For \(D=-163\), the cubic Fricke RS series is the exact level-two transport of the Chudnovsky linear form.

In 2017, McDiarmid and Yolov introduced the bipartite-hole-number $\widetilde{\alpha}(G)$ and proved that $\delta(G)\ge \widetilde{\alpha}(G)$ forces a Hamilton cycle. They also gave a sufficient condition for packing edge-disjoint Hamilton cycles, and asked whether this condition is sharp or can be relaxed. For integers $a,k\ge 2$, let $f(a,k)$ be the least integer $d$ such that every graph $G$ on at least three vertices with $\widetilde{\alpha}(G)\le a$ and $\delta(G)\ge d$ contains $k$ pairwise edge-disjoint Hamilton cycles. We prove that $f(a,k)=\Theta\left(a+k+\frac{ak}{\log(k+2)}\right).$ The upper bound uses a deletion lemma for the bipartite-hole-number together with the McDiarmid–Yolov Hamiltonicity theorem and a greedy packing argument. The lower bound is obtained from three extremal constructions, the logarithmic one using a sparse random auxiliary graph with no prescribed bipartite hole.

This paper investigates the existence, construction and classification of limited magnitude error-correcting codes, with a focus on splitter sets and their connections to group splittings. We establish new nonexistence results for quasi-perfect splitter sets and provide a complete classification of quasi-perfect $B[0,3](n)$ splitter sets in both singular and nonsingular cases. Furthermore, we derive improved lower bounds for the size of maximal $B[0,3](q)$ sets by investigating Cayley graphs, where $q$ is a prime. We also provide existence criteria for perfect $B[0,6](q)$ splitter sets and quasi-perfect $B[-4,4](2p)$ sets for prime $p$. For perfect burst-correcting codes, we develop a general construction framework, and prove the existence of infinite families of $(k_2,k_1)$-limited-magnitude cyclic $b$-burst-correcting codes for $k_1+k_2\le 4$ and arbitrary burst length $b$. We further provide sufficient existence conditions for general parameters $k_1$ and $k_2$. Our results combine algebraic, combinatorial, and number-theoretic methods to advance the understanding of codes tailored for flash memory and related storage systems.