Papers

This article focuses specifically on the study of self-dual double cyclic codes over a finite field $\mathbb{F}_q$. A self-dual double cyclic code is a double cyclic code that is equal to its dual. Structurally, a double cyclic code of length $(r,s)$ over $\mathbb{F}_q$ is a $\mathbb{F}_q[x]$-submodule of $\mathbb{F}_{q,r,s}:=\mathbb{F}_q[x]/\langle x^r-1\rangle\times\mathbb{F}_q[x]/\langle x^s-1\rangle$. Moreover, any double cyclic code of length $(r,s)$ over $\mathbb{F}_q$ is generated by two pairs of polynomials in $\mathbb{F}_{q,r,s}$. From the properties of the generating elements, we provide the necessary and sufficient conditions such that two pairs of polynomials in $\mathbb{F}_{q,r,s}$ generate a self-dual code. Furthermore, we examine the existence of self-dual double cyclic codes for some specific lengths: $(r,r)$; $(r,2r)$ and $(2r,r)$; and $(r,s)$, where $\gcd(r,s)=1$. For each case, we provide a construction method with some explicit examples over various finite fields. We also observe some connections between self-dual double cyclic codes and other classes of self-dual codes.

We present a minimax adaptive control framework for discrete-time positive linear systems with parametric uncertainty and adversarial disturbances. The uncertainty in the system dynamics is assumed to lie in a finite set of possible plants. We formulate the problem as a dynamic game between the controller, which minimizes the cost, and an adversary, which selects both the disturbances and the plant dynamics to maximize the cost. An equivalent reformulation of the original game transforms the problem into a standard minimax two-player zero-sum dynamic game. This enables the problem to be addressed via minimax dynamic programming. We provide an explicit solution to the Bellman inequality, yielding stabilizing, positivity preserving policies without requiring an initially stabilizing controller. The resulting controller enjoys robustness guarantees in the form of bounded $\ell_1$-gain from disturbances to errors. Once the uncertain parameters have been sufficiently estimated, the controller behaves like a standard $\mathcal H_\infty$-type optimal controller for positive linear systems. The theoretical findings are supported by numerical experiments illustrating the resulting adaptive controller in action.

Let $\sigma_3:\mathbb{S}\to \mathbb{S}$ be the tripling map of the unit circle. For sequences $\{\mathcal{L}_i\}$ of $\sigma_3$-invariant dendritic laminations we study limits $(\overline{c}, \overline{d})$ of their critical portraits assuming that one such limit $\mathcal{P}=(\overline{c}_\circ, \overline{d}_\circ)$ is given. If the endpoints of $\overline{c}_\circ$ and $\overline{d}_\circ$ are non-periodic, then there is a unique lamination $\mathcal{L}$ with finite critical sets such that $\overline{c}$ and $\overline{d}$ can be any couple of critical chords compatible with $\mathcal{L}$. As the extreme opposite case we consider $\mathcal{P}=(\overline{0 \frac13}, \overline{0 \frac23})$ and describe the corresponding countable closed family of possible critical portraits $(\overline{c}, \overline{d})$ and the distinct laminations corresponding to them. These results can be useful for the construction of a model for the cubic connectedness locus.

We prove the equality of Hausdorff and Gromov-Hausdorff distance $d_{GH}(G, U) = d_{H}(G, U)$ for a metric graph $G$ and a subset $U \subseteq G$, given that the Hausdorff distance between $G$ and $U$ is attained not just near the leaves of $G$ and the Gromov-Hausdorff distance is not too big compared to the graph's systole.

We study a class of mean-field control (MFC) problems with singular controls over a finite horizon, allowing for general dependence of the cost functional on the measure argument. We derive an auxiliary mean-field game (MFG) with singular controls, which we refer to as a potential MFG, and show that, under suitable convexity assumptions, any solution to this potential MFG yields a solution to the original MFC problem. We apply this general result to a version of the classical Monotone Follower Problem by I. Karatzas and S. E. Shreve (SIAM Journal on Control and Optimization 22(6), pp. 856-877, 1984) with scalar mean-field interaction. The associated potential MFG with singular controls is solved by exploiting its connection with optimal stopping for the optimization step and by a suitable application of the Kakutani-Fan-Glicksberg fixed-point theorem. In the case of strategic complementarities, the mean-field equilibrium (and hence the optimal policy of the original MFC problem) is characterized by a continuous nonincreasing free boundary that uniquely solves a nonlinear integral equation. To the best of our knowledge, this is the first paper to provide a complete characterization of the optimal policy in a finite-horizon mean-field singular stochastic control problem.

We prove the P=W identity for isolated cluster varieties of dimension 3 without the full rank hypothesis. These cluster varieties are generally singular, and the associated Lagrangian fibrations are not complex algebraic. On the P side, we construct the perverse truncation by a detailed analysis of the explicit real-analytic geometry of the Lagrangian fibration. On the W side, we construct a natural non-proper algebraic morphism from the cluster varieties and investigate the decomposition of the derived push-forward of the constant sheaf along this morphism within the derived category of mixed Hodge modules. In the P=W phenomenon for non-full rank isolated cluster varieties of dimension 3, both the curious hard Lefschetz property on the W side and the relative hard Lefschetz property on the P side fail.

We establish sharp strong error estimates for a fully discrete approximation of semilinear stochastic convection–diffusion equations driven by additive noise subject to nonsmooth initial data. The problem is discretized in space using a standard Galerkin finite element method and in time via the linearly implicit Euler scheme. By decoupling the noise regularity index $\beta \in (0, 2]$ from the initial data regularity index $\mu \in (0, 2]$, we derive time-weighted spatial and temporal regularity bounds that isolate the initial singularity at $t = 0$ into an integrable temporal weight. By allowing the drift operator $F$ to map into negative-order fractional spaces to incorporate the convection term, we prove sharp strong convergence rates of order $O(h^\beta + k^{\beta/2})$ for $\beta \in (0, 2)$ and $O\bigl(h^2 + k(1+\ln(t_n/k))^{1/2}\bigr)$ at the critical endpoint $\beta = 2$ for any fixed $t_n > 0$, without requiring the regularity of the initial data to match that of the noise. Furthermore, these convergence rates are achieved under a relaxed differentiability condition requiring only a Lipschitz continuous first Gâteaux derivative of $F$. Numerical experiments validate the theoretical results across both trace-class and space-time white noise regimes.

Study of the structure of groups by variation in the arithmetic conditions on conjugacy classes and character degrees has produced several interesting results and open problems. In continuation of such work, Dolfi and Lucido in \cite{MR1826493} introduced a property for groups. For primes $p,q$, a group $G$ is said to have property $P(p,q)$ if every $p'$-element in $G$ has $q'$-class size. They obtained several results on the structure of $G$ and of some subgroups when $G$ satisfies the property $P(p,q)$. Motivated by this work, we introduce a vanishing analogue of the above property: for primes $p \neq q$, a finite group $G$ is said to have the property $P_v(p,q)$ if every vanishing $p'$-element of prime power order in $G$ has conjugacy class size not divisible by $q$. We show that no finite simple group satisfies the property $P_v(p,q)$ for primes $p\neq q$ dividing $|G|$. We use this result to show that if a finite group $G$ satisfies the property $P_v(p,q)$ with $p \neq q$ and $p > 2$, then $O^{q'}(G)$ (subgroup generated by all Sylow $q$-subgroups of $G$) is solvable. This generalises a result of Dolfi and Lucido under weaker conditions.

We prove that the inclusion of left-invariant forms into the Dolbeault complex of a compact nilmanifold $M$ endowed with a left-invariant complex structure $J$ induces an isomorphism in cohomology in every bidegree, settling a long-standing conjecture. As consequences, we show that small deformations of $J$ are still invariant, as conjectured by Hasegawa. We also prove that Bott–Chern, Aeppli, and Frölicher invariants are computed by invariant forms and are independent of the lattice, settling a conjecture of Angella on Bott–Chern cohomology.

We prove that every tower of connected quasi-torsor covers of a connected normal globally $F$-regular projective variety over an algebraically closed field of positive characteristic stabilizes, and that the Nori fundamental group scheme of its regular locus is finite and linearly reductive. The linear reductivity asserts that any quasi-torsor cover is tame, while the finiteness is a positive-characteristic analogue of theorems of Xu and Braun on the finiteness of the fundamental group of the smooth locus of a log Fano complex projective variety.

The compressible Euler-Riesz equations arise in the modeling of a wide range of physical phenomena, including stellar dynamics, plasma physics, and mathematical biology. In this paper, we investigate the nonlinear stability and instability of steady states for the multidimensional compressible Euler-Riesz equations under general pressure laws. In the polytropic case, we establish the nonlinear instability of steady states in the mass-supercritical regime for attractive potentials; this is achieved by analyzing the concavity of the free energy along mass-preserving dilations. At the mass-critical exponent, we show that, for any steady state, there exist solutions that start arbitrarily close to it, but develop growing support. For general pressure laws, we employ a concentration-compactness approach to prove the existence of energy minimizers and establish the nonlinear stability of steady states. Moreover, we quantify the finite-time stability by deriving a relative entropy bound for finite-energy solutions, without requiring uniform pointwise upper and lower bounds on the density. We further exploit the convexity of the second moment to obtain quantitative growth estimates for solutions with positive energy, thereby proving the local nature of the stability result. Finally, we prove the global existence of finite-energy weak solutions to the compressible Euler-Riesz equations with spherical symmetry for general pressure laws via the compensated compactness method, thereby yielding unconditional stability around steady states within the class of weak solutions. The approach developed in this paper should be useful for solving other nonlinear partial differential equations involving similar difficulties.

Let $(X,\chi,k)$ be a triple consisting of a smooth, compact hyperbolic Riemann surface $X$ of genus $g$, and an $m$ dimensional unitary multiplier system $\chi$ of admissible weight $k$. Our first result establishes an analogue of the prime geodesic theorem for the weighted prime geodesic counting function associated to $(X,\chi,k)$. The error term we obtain is explicit with effectively computable constants which depend solely on the genus of $X$, the dimension of $\chi$, the length of shortest geodesic on $X$ and the smallest non-zero eigenvalues of the weighted Laplacian $\Delta_{2k}$ as well that of the scalar Laplacian $\Delta_{0}$. Our second result studies the asymptotic behavior of the spectral determinant $\det\Delta_{2k_n}$ for a sequence $(X_{n}, \chi_{n}, k_{n})$ for which the genus of $X_n$ tends to infinity. Under reasonably general circumstances, namely the existence of a weak spectral gap, a uniform discreteness of the underlying Fuchsian group, and a type of non-accumulation of bounded geodesics, we prove that $\log\det\Delta_{2k_n}/\mathrm{vol}(X_{n})$ converges to a constant $C_{\alpha}$ which depends only on $\alpha=\lim_{n\to\infty} k_n$. Our result is deterministic and is compatible with the three well-studied probabilistic models, namely Weil-Petersson, Brooks-Makover, and random covers model.

In this paper we prove that Dubois–Prade's eight possibilistic operators in Formal Concept Analysis arise canonically from Kan extensions of the underlying boolean profunctor. This provides a conceptual explanation for the result that $N\Pi$-pairs are the formal concepts of the complement context. We further prove that the FCA closure operator and the $N\Pi$-pairs are the only symmetric or asymmetric operator compositions that give formal concepts. Finally we use these eight possibilistic operators to construct new closure operators on a formal context via standard categorical arguments.

For a natural number $a$, let $E_{2a}$ be the Mordell elliptic curve $X^3 +Y^3=2a$. We give an explicit construction of (mock) Heegner point on the Mordell curve $E_{2p}$ for a prime $p\equiv 4 \mod 9$ and $E_{2p^2}$ for a prime $p\equiv 7 \mod 9$, under the assumption that $2$ is not a cube modulo $p$. We also verify the explicit Gross-Zagier formula for these curves and go on to show that the BSD formula holds for these curves up to a $2$-adic unit. Using a result of Burungale-Flach, we show that the full BSD formula holds for the rank zero curve $E_{2p}$ for $p\equiv 7 \mod 9$ and $E_{2p^2}$ for $p\equiv 4 \mod 9$, whenever $2$ is not a cube modulo $p$.

This paper investigates the classification of spatio-temporal patterns generated by linear cellular automata with uniform weights (LCA-UW) over the ring ${\mathbb Z} / n {\mathbb Z}$. While these systems are governed by the state size $n$ and a transition coefficient $c$, their combined influence produces a vast array of patterns that are difficult to organize through exhaustive observation. We introduce a binary projection operator $\mathcal{B}$ to focus on the fundamental structural evolution (infinite binary orbits) of these automata. Our main result demonstrates a fundamental reduction principle. For any coefficient $c$ that shares prime factors with $n$, the generated infinite binary orbit eventually coincides with the orbit of an LCA-UW with some reduced state size and a unit coefficient $c=1$. We prove that for a fixed $n$, there exist exactly $2^m - 1$ distinct types of binary orbits, where $m$ is the number of distinct prime factors of $n$. This theorem effectively collapses the two-dimensional parameter space $(n, c)$ into a one-dimensional search over $n$, providing a streamlined framework for the topological and fractal classification of LCA-UW dynamics.

Compactly Supported Radial Basis Functions (CS-RBFs) are a fundamental tool in multivariate approximation theory. However, their use in statistics and probability modeling remains underexplored, having been used mainly to express covariance functions in Gaussian processes or as kernel functions. This work explores CS-RBFs as a novel parametric family of probability density functions, focusing in particular on Wendland $\mathscr{C}^2$ kernels. The primary contribution of this work is the derivation of analytical expressions for various statistical properties, such as moments and the cumulative distribution function, of CS-RBFs as univariate and conditional densities. The approach comprises two alternative scenarios: when the CS-RBF support lies entirely within the variable's domain (untruncated support) and when part of it is outside (truncated support). Mixture models employing CS-RBFs are also analyzed, and their main properties are detailed. Furthermore, we introduce an incremental learning algorithm for density estimation with CS-RBF mixture models, in which centers are determined using k-means and weights and shape parameters are optimized by stochastic gradient descent. Experiments on synthetic and real-world datasets show that CS-RBF densities provide competitive results in terms of likelihood and model complexity in comparison with Gaussian mixture models. In addition, these CS-RBF densities allow the exact computation of key distributional properties in univariate and conditional settings.

In this paper, we revisit the Poincaré polynomials, Betti numbers, and Euler characteristics of the Deligne-Mumford moduli spaces $\overline{\mathcal M}_{0,n}$ of stable $n$-pointed rational curves. We give elementary derivations of two recent closed formulas for their Poincaré polynomials, due respectively to Aluffi-Marcolli-Nascimento (arXiv:2406.13095) and to Eur-Ferroni-Matherne-Pagaria-Vecchi (arXiv:2504.16776). Our approach shows that both formulas are already implicit in the generating-series results of Getzler and Manin, and can be extracted from them by elementary manipulations of generating functions, the binomial series, and standard identities for Stirling numbers. Beyond these new derivations, the same method also yields new linear recurrence relations for refined invariants associated with these Poincaré polynomials, namely distinguished summands and a bivariate refinement. As a further consequence, we obtain two additional formulas for the Betti numbers, not previously recorded in this form. We also study the Euler characteristics $\chi(\overline{\mathcal M}_{0,n})$. Using the Taylor expansion of a suitable branch of the Lambert $W$-function, we show that their sequence is obtained by evaluating complete Bell polynomials at an explicit auxiliary integer sequence. This Bell-polynomial representation yields Hessenberg determinantal formulas and a new linear recursion, distinct from the well-known quadratic Keel-Manin recursion. It also provides an explicit extraction of the Euler characteristics from the Lambert $W$-function expression considered by Aluffi-Marcolli-Nascimento. Finally, we refine the Manin-Zagier asymptotic estimate for these Euler characteristics by computing the full asymptotic expansion.

We introduce a new approach to the reconstruction of hidden structures from incomplete data, unifying techniques from geometric integration and topological analysis within the frameworks of Vaisman and Neifeld. Our method employs a refined geometric decomposition of configuration spaces into invariant foliations and moment maps, thereby addressing the intrinsic ambiguities of underdetermined inverse problems. By combining Vaisman's insights into symmetry with Neifeld's analytical methodologies, we establish a robust, noise-resistant framework that ensures computational tractability while providing a unified perspective on reconstruction in imaging and structural analysis. This approach enables applications across diverse scientific domains and highlights the interplay between geometry and topology in the solution of inverse problems.

We study a Cucker–Smale type system with bonding forces on complete Riemannian manifolds with uniformly bounded curvature. On general manifolds, the time variation of parallel transport between moving agents produces curvature-dependent terms, so the standard energy-dissipation argument does not directly yield asymptotic velocity alignment. The bonding energy confines all pairwise distances below the injectivity radius, providing global well-posedness and time integrability of the transported velocity discrepancies. To overcome the remaining geometric obstruction, we combine the variation formula for parallel transport with a uniform endpoint estimate for Jacobi fields along moving minimizing geodesics. This yields the uniform regularity needed to convert energy dissipation into asymptotic alignment. Under an energy-dependent injectivity condition and a positive communication bound on the dynamically relevant distance range, we establish asymptotic flocking. Numerical simulations illustrate the resulting dynamics in a nonconstant-sectional-curvature setting.

We give a self contained and concise proof of the following result, which was first proved by Glauberman as his $ZJ$-Theorem. Let $p$ be an odd prime and $S \not= 1$ a $p$-group. Then there exists a characteristic subgroup $W(S) \not= 1$ of $S$ with the property: whenever $G$ is a finite $Qd(p)$-free group of characteristic $p$ and $S$ is a Sylow subgroup of $G$ then $W(S) \unlhd G$.

An operator system is similar to a module over a ring, with the role of scalar multiplication played by the action of completely positive maps. Using enriched category theory, we make this analogy into a precise categorical equivalence, namely between a certain category of nonunital operator systems and a certain category of left modules over the category of matrix algebras enriched over regularly ordered Banach spaces. Using right modules instead yields an equivalence with a certain category of nonunital dual operator systems. We also develop general separation, representation and extension theorems for modules in enriched category theory. Specializing these to our nonunital operator systems recovers results which partly recover the corresponding classical theorems for operator systems.

The birational invariants introduced by Katzarkov-Kontsevich-Pantev-Yu allows one to obtain irrationality results for varieties whose quantum cohomology is well-behaved. We observe that under certain cohomological conditions, we can deduce irrationality of a very general member from the theory of atoms without actually computing them, using only monodromy equivariance of quantum multiplication and irreducibility of the monodromy representation. Our criterion applies to the very general cubic and Gushel-Mukai fourfolds, whose irrationalities were already known, but also to the very general Küchle fourfold of type (c5), which is a Fano manifold of index one.

We propose a discrete Markov-chain approximation of diffusion processes on networks with both Kirchhoff and sticky vertex conditions. Stickiness is modeled by a probabilistic residence mechanism at the vertex, while the motion along the edges follows an Euler-Maruyama-type update at the diffusive scale. We prove that the associated time-interpolated chain converges in distribution to the limiting diffusion in the Skorokhod space using the Ethier-Kurtz framework. Based on this construction, we derive a fully discrete semi-Lagrangian scheme for Hamilton-Jacobi-Bellman equations on networks and establish its convergence using viscosity solution techniques.

We study dynamical phase retrieval for Schr\”odinger evolutions on finite connected graphs. Let \[ H\_Q=\Delta\_G+Q \] be a graph Schr\”odinger operator with a real diagonal potential. We investigate when phaseless data obtained from the associated Schr\”odinger evolution \[ |e^{-itH\_Q}u\_0(j)|, \qquad 0\leq t\leq T,\ j\in V, \] determines the initial state $u\_0\in\C^V$ up to a global phase. We give a uniqueness criterion in terms of the eigenvalues and eigenvectors of $H\_Q$. The assumptions are a $B\_2$ condition on the spectrum, meaning that the sums $\lambda\_j+\lambda\_k$ determine the unordered pair $\{j,k\}$, invertibility of the squared-eigenvector matrix $\bigl(\phi\_k(j)^2\bigr)\_{j,k}$ and an overlap condition on the supports of pairs of eigenvectors. Under these hypotheses, the phaseless Schr\”odinger data determine every initial state uniquely, modulo global phase. We then show that the criterion is both realized and generic. Every finite connected graph admits an explicit real diagonal potential for which the criterion holds. Moreover, for every finite connected graph, dynamical phase retrieval holds for Lebesgue-almost every real potential $Q\in\R^V$ and every $T>0$. We also give several obstructions to uniqueness.

We consider the dissipative generalized surface quasi-geostrophic equations with a dispersive forcing term and study the behavior of solutions in the limit of vanishing viscosity. Using the approach of relative energy inequality, we prove convergence results in different dispersive regimes that extend the results obtained by Kosloff et al. (Appl. Math. Lett. (2019) 88 243-249) to more singular active scalar equations.

We prove that any $\mathbb C$-convex domains with $\C^{1,\alpha}$ boundary admits an increasing exhaustion by bounded $C^\infty$ strongly linearly convex domains.

We prove a local weak Harnack inequality for nonnegative weak super-solutions to the anisotropic Trudinger equation. As an application, we show a proof of the Harnack inequality that bypasses any sort of Krylov-Safonov covering argument. We further develop an analysis of global sub-potential lower bounds which ultimately leads to the identification of a Barenblatt-type profile.

In this article, we use the expression of the Schwarzian derivative to set up differential equations to find answers to three fundamental questions in the context of QFT in curved spacetime, specifically in two dimensions. One of the ways in which one can derive the Unruh effect in two dimensions is to use the anomalous transformation law of the energy-momentum tensor for a CFT that involves a Schwarzian derivative (Virasoro Anomaly). We answer the following three questions. The first question is as follows: If we have a spacetime with a massless scalar field in vacuum, what are all the subsets of spacetime such that the subset has a thermal distribution of particles for the left-moving and/or right-moving sectors? We obtain a general solution to this question by setting up and solving a third-order nonlinear differential equation based on the expression of Schwarzian. Based on the general solution, we can generate various subsets of the given spacetime that have a thermal flux/density of particles, of which the Rindler spacetime is one. The second question is an inverse question in which we suppose we are given a spacetime with a thermal distribution of particles; what are the possible purifying spacetimes (the “parent” spacetimes with the field in vacuum state whose reduced state in the given spacetime yields the observed particle content)? We similarly obtain a general class of solutions by setting up and solving a second differential equation. In this context, we also define “partial purification” where we obtain a spacetime that purifies only the left-moving or right-moving sector. The third question concerns locating spacetimes with the same particle content starting from the same “parent” spacetime. These sibling spacetimes are generated again by obtaining the general solution of a third differential equation based on the expression of Schwarzian.

We study a graph-limit problem for Banach-decorated graphs. Given a sequence of $X$-decorated graphs whose homomorphism densities converge against all $X^*$-decorated test graphs, we ask whether the limiting densities are represented by an $X$-valued graphon. The results connect this graph-limit problem with Banach-space structure. If $X^*$ is separable, then the graphon representation property for graph sequences uniformly bounded in $L^p$ for every finite $p$ holds if and only if $X$ is reflexive. For Banach lattices, it is equivalent to the Radon–Nikodým property. For dual Banach spaces, it is equivalent to the conjunction of the Radon–Nikodým property and weak sequential completeness. In the bounded setting, the same characterization extends to arbitrary Banach spaces: for every Banach space $X$, the representation property for uniformly $L^\infty$-bounded graph sequences holds if and only if $X$ has the Radon–Nikodým property and is weakly sequentially complete.

In this paper, based on the structure of higher algebraic $K$-groups of the rings of integers of number fields, we introduce new equivalence relations between number fields called $K$-equivalence and almost $K$-equivalence, and investigate their relationships with arithmetical equivalence and local integral equivalence. Building upon the classical properties of arithmetically equivalent number fields studied by R. Perlis and on the pioneering work by Komatsu, we apply the Rost-Voevodsky theorem (the Quillen-Lichtenbaum conjecture) for odd primes $p$, thereby analyzing algebraic $K$-groups within the framework of continuous étale cohomology from a more modern perspective. As our main results, by utilizing permutation representations of global Galois groups and the data of local decomposition groups, we refine Komatsu's previous results in the range $p \neq 2$ and describe the conditions for number fields to be (almost) $K$-equivalent. Through these developments, we clarify how the arithmetic information reflected in higher $K$-groups resonates with the special values of zeta functions and Galois representations.