Papers

We endow spaces of decorated planar rooted trees with new dendriform and tridendrifrom algebra structures and provide their combinatorial description. We then show that the planar counterparts of Arborified Zeta Values are algebra morphisms for these shuffle and quasi-shuffle products of planar rooted trees. We also prove an arborified version of Hoffman's regularisation relation for Arborified Zeta Values. We conjecture that those give every rational relation between Arborified Zeta Values and show that this conjecture implies the regularised double shuffle conjecture for Multiple Zeta Values.

We study the $\mathcal F$-transitivity and topological $\mathcal F$-recurrence of left translation semigroups on weighted $L^p$-spaces over directed metric trees. Motivated by the recent work of Mangino and Vargas-Moreno on hypercyclicity and weak mixing for these semigroups, we investigate the different problem of $\mathcal F$-transitivity, where the entire return-time set is required to belong to a prescribed finitely invariant Furstenberg family. Assuming that the weight is $p$-admissible, we obtain necessary and sufficient integral conditions for both rooted and rootless trees, and establish the equivalence between $\mathcal F$-transitivity and topological $\mathcal F$-recurrence. In the rooted case, the criteria depend only on the weights along descendant edges. In the rootless case, an additional ancestor term and a measurable cancellation construction reflect the backward geometry of the tree. Our approach is based on measurable weighted minimization and quantitative estimates over subsets of the edge parameter interval. Examples on homogeneous rooted trees and rootless trees with a free left end show that branching may determine the dynamics, but forward branching alone does not guarantee $\mathcal F$-transitivity in the rootless setting.

Benjamini and Tzalik obtained an upper bound on the number of shortest paths between two vertices at distance $t$ in a multigraph of maximum degree at most $\Delta$, and proposed a conjecture on the sharp bound. In this paper, we develop a probabilistic counting argument based on probability distributions induced by random walks from the two endpoints. This approach yields a sharp bound for multigraphs and confirms their conjecture. We further determine the exact maximum for simple graphs and thus answer another question of Benjamini and Tzalik. We also investigate the equality cases, describing the structure of the subgraph formed by shortest paths between $x$ and $y$ and giving tight examples.

In this lecture note, we present a concise and self-contained derivation of the discrete-time Kalman filter equations that requires only a basic understanding of least squares estimation. The treatment is designed to minimize mathematical overhead while preserving both rigor and generality.

Metastability is typically formulated for systems with finitely many metastable states, most often in positively recurrent settings. In this article, we extend the resolvent framework for metastability to Markov processes with countably many metastable states, thereby encompassing null-recurrent and transient dynamics. For a family of processes on locally compact Polish spaces, we prove, under a mild boundary regularity assumption, that the asymptotic flatness of microscopic resolvent solutions, supplemented by two compactness conditions, is equivalent to convergence in law of the projected trace processes to a limiting Markov chain and to the negligibility of the time spent outside the metastable sets. We apply this framework to two non-compact stochastic systems. First, we study a condensing inclusion process on a countably infinite, uniformly locally finite graph, in a setting where the process may be null recurrent or transient. We prove that the condensate location converges to a weighted random walk on the underlying graph, while the time spent away from the fully condensed configurations is negligible. Second, we study a small-noise one-dimensional Langevin diffusion with countably many stable equilibria, without assuming ergodicity. In the Eyring-Kramers time scale associated with the minimal energy barrier, we establish local equilibration inside each well, convergence of the well-index process to an explicit nearest-neighbor Markov chain on $\mathbb Z$, and negligibility of inter-well excursions. Together, these results broaden the scope of resolvent-based metastability theory beyond finite metastable state spaces and positive recurrence.

We study the distribution of square-full numbers under the assumptions of Riemann hypothesis and that the zeros of $\zeta(s)$ are simple. Thus the error term is improved to a considerable extent with an extra main term.

In this work we prove a Payne-Weinberger type inequality for quantum dot Dirac operators defined on bounded and simply connected planar domains. This is a sharp upper bound for their first positive eigenvalue depending only on the isoperimetric deficit of the domain. To this end, we use a recently studied connection with the so-called $\overline\partial$-Robin Laplacian and we establish an analogous inequality for its first eigenvalue, relying on the corresponding inequality for the Robin Laplacian.

In many communication networks, the transmission of a packet may automatically trigger the transmission of a subsequent packet from the same source after a (possibly random) delay, without requiring acknowledgment or feedback. Such behavior arises in multi-stage status updating, proactive protocols, and other applications where users generate causally dependent packet streams. In this paper, in order to analyze these systems, we introduce the $\mathrm{M^T/G/1}$ queue. In this model, primary customers arrive according to a Poisson process, and each primary customer triggers a secondary customer to join the queue after an independent delay. This arrival mechanism falls outside the scope of classical queueing models with renewal arrival processes. When the triggering delays follow an exponential distribution, we exploit the memoryless property to set up a tractable Markov description. By truncating the number of pending secondary customers, we derive a finite system of linear algebraic equations in the Laplace–Stieltjes transform domain and solve them using matrix-analytic methods. Based on the resulting workload distribution, we compute class-specific performance metrics using PASTA for primary customers and Palm conditioning for secondary customers. Finally, we validate the accuracy of this truncation through numerical experiments.

The cycle space of a directed graph is generated by a cycle basis, where, in general, cycles are allowed to have both forward and backward arcs. In a forward cycle, all arcs must follow the given direction. Several open questions remain regarding the complexity of the minimum cycle basis problem, in particular the minimum-weight integral cycle basis problem, and the minimum-weight weakly and strictly fundamental forward cycle basis problems. In this paper, we address these open questions. First, we study the existence, structure, and computational complexity of minimum-weight forward cycle bases. We give a complete structural characterization of digraphs that admit weakly fundamental (and hence integral) forward cycle bases. We further provide a characterization when a strongly connected digraph admits a forward fundamental cycle basis, proving that such a basis exists if and only if the set of directed cycles has cardinality equal to the cycle rank; in this case, the basis is unique. Lastly, we show that while minimum-weight forward fundamental cycle bases can be found in polynomial time whenever they exist, the minimum-weight forward weakly fundamental cycle basis problem is APX-hard via an L-reduction from the minimum-weight weakly fundamental cycle basis problem on digraphs with metric weights. Second, we introduce opt-in graphs, i.e., the family of graphs for which minimum cycle bases are integral for any weight function. We show that this family is minor-closed and hence, by the Robertson-Seymour theorem, is characterized by a finite set of forbidden minors, so that the opt-in recognition problem is solvable in polynomial time. Lastly, we present an algorithm to check whether a graph is opt-in, and if not, to identify which of its minors belong to the set of forbidden minors. Applying this algorithm, we show that the complete graph $K_n$ is opt-in if and only if $n \leq 7$.

We establish a sharp rigidity–chaos dichotomy for planar two-centre billiards, motivated by a natural analogue of the Birkhoff–Poritsky conjecture: the only tables integrable at every energy should be ellipses confocal with the two centres. Let $\Omega$ be a bounded domain with $\mathcal C^1$ boundary containing the segment joining the centres. At every fixed energy $h\geq 0$, if $\partial\Omega$ is not a confocal ellipse, we construct billiard trajectories that shadow the stable and unstable manifolds of the collision–reflection orbit and realise arbitrarily prescribed sequences of sufficiently large winding numbers around the segment. This yields an invariant set semiconjugate to the full shift on a countable alphabet, periodic trajectories with prescribed finite itineraries, and compact invariant subsystems with arbitrarily large topological entropy. If, in addition, $\partial\Omega$ is real-analytic, every real-analytic function on the fixed-energy phase space $M_h$ that is invariant under the billiard map is constant. This establishes the real-analytic form of the two-centre Birkhoff–Poritsky conjecture throughout the non-negative-energy regime.

Let $P=\{0,a,b\}$, where $0<a<b$ and $\gcd(a,b)=1$. For a finite set $A\subset\mathbb Z$, let $M_P^+(A)$ count the copies $x,x+ad,x+bd\in A$ with $d>0$, and let $M_P(A)$ count the copies with any $d\ne0$. We prove that every such three-point pattern other than the arithmetic progression $\{0,1,2\}$ satisfies \[ M_P^+(A)\le \frac{99}{400}|A|^2+O(|A|), \qquad M_P(A)\le \frac{13}{28}|A|^2+O_P(|A|). \] For the particular pattern $P=\{0,1,3\}$ – the subject of a question raised by Ganguly and recorded as Problem 24 in Green's list of open problems – we sharpen the bound allowing both signs of the dilation to \[ M_{\{0,1,3\}}(A)\le \frac{47}{122}|A|^2+O(|A|). \]

Recently, Kupers–Rudenko–Sierra introduced the Goncharov Lie coalgebra. They gave a presentation using correlator generators and four types of relations: homogeneity, cyclic symmetry, shuffle relations, and decomposition relations. We show that the first three families of relations follow from the decomposition relations.

We consider eigenvalue counting functions of Robin Laplace operators on cuboids where the Robin parameter and the spectral cut-off $\lambda$ are coupled. Our main focus is the critical regime, in which the Robin parameter is proportional to $\sqrt{\lambda}$. We obtain a two-term asymptotic expansion for the counting functions in this coupled setting. In the critical regime, the second term in the asymptotic expansion depends non-trivially on the proportionality constant and interpolates continuously between the corresponding second terms for the Dirichlet and Neumann Laplacians. Outside the critical regime, one recovers the corresponding Dirichlet or Neumann asymptotics. We also establish a Pólya-type inequality for the counting function whenever the ratio of the Robin parameter and $\sqrt{\lambda}$ exceeds a dimension-dependent threshold.

We extend the global existence results for critical parameters of [Bianchini and Leccese, J. Math. Anal. Appl. (2024)] to the multidimensional problem \begin{equation*} \partial_t u + \text{div } (b(t,x) u^{1+k}) = \Delta u \end{equation*} for $b:\mathbb R\times{\mathbb R}^d\to{\mathbb R}^d$ non-autonomous fields, in the case \begin{equation*} (\text{div } b)_-\in L^\infty_{\mathrm{loc}} \big([0,\infty);L^{p,\infty}({\mathbb R}^d)\big), \qquad p>d. \end{equation*} The proof of global existence follows the parabolic symmetrization approach of [Bandle, J. Analyse Math. (1976)] and is based on the study of the function \begin{equation*} m(t,s)=\int_0^s u^*(t,r)\,dr. \end{equation*}

End-user-centric collaborative MIMO (UE-CoMIMO) lets nearby devices form a virtual multi-antenna terminal to overcome the antenna limitations of individual user equipment. Extending such cooperation to communication, sensing, computing, and task-relevant information exchange requires a control layer that can interpret user intent, select cooperation mechanisms, and replan as conditions change. This article introduces Agentic UE-CoMIMO, in which device micro-agents, a smartphone or CPE hub agent, and edge/network agents coordinate device participation, relay modes, traffic splitting and duplication, compute placement, semantic-token exchange, and topology reconfiguration. Two system-level scenario studies on creator-centric live streaming and wearable-collaborative blind-spot sensing compare the proposed controller with capability-matched adaptive baselines. The results show that, by anticipating changes and preparing cooperation and fallback actions in advance, agentic control sustains high-quality streaming for longer and maintains blind-spot warnings through device outages. We also discuss the associated standardization, interoperability, trust, and validation challenges.

Among four points of a CAT(0) space, the planar symmetric trapezoids are configurations on which Ptolemy's and Reshetnyak's inequalities are both equalities. We show that the symmetric trapezoids remain extremal for the entire family of inequalities interpolating between the two, indexed by the nondecreasing convex functions with concave derivative, and that this function class is characterized by this property. The result is qualitatively stronger than the known quadruple inequalities for this class of functions, and recovers them with their optimal constants, which were previously known only for power functions. Moreover, we extend the analysis to CAT($\kappa$) spaces with $\kappa>0$. We derive variants of Reshetnyak's quadrilateral comparison and of Ptolemy's inequality under positive upper curvature bounds, each with the optimal constant. These two inequalities yield the trapezoid comparison inequality in CAT($\kappa$) spaces, where the product of the bases carries an additional constant factor compared to the $\kappa=0$ case. Again the constant is optimal.

The graph coloring game is a two-player game in which the players alternately color an uncolored vertex of a graph $G$. The game chromatic number is the minimum number of colors needed for the first player to guarantee a win. We investigate this parameter for generalized Mycielski graphs $M_k(G)$, where $G$ is a path $P_n$ or a cycle $C_n$ with $n$ vertices. For every $k\geq2$ and $n\geq5$, we establish $4\leq\chi_g\bigl(M_k(P_n)\bigr)\leq5$ and $4\leq\chi_g\bigl(M_k(C_n)\bigr)\leq5$. We also determine the exact values $\chi_g\bigl(M_2(P_5)\bigr)=\chi_g\bigl(M_2(P_6)\bigr)=4$. The proofs of the lower bounds use a configuration in which Bob can create two threats simultaneously, while the four-color upper bounds in the two exact cases are proved using the double-doctor lemma. Thus the number of layers and the order of the base graph may grow, but the game chromatic number remains bounded by five.

This paper is about disjoint F-transitive operators on Hilbert C*-modules, where F is a finitely invariant Furstenberg family of subsets of N. We characterize disjoint F-transitivity for generalized bilateral weighted shift operators on the standard Hilbert module over C*-algebra of compact operators on the separable Hilbert space. Then we give a sufficient condition for these operators to be disjoint chaotic and give concrete examples. We also characterize disjoint F-transitivity for generalized translation operators on the non-unitary C*-algebra and provide some applications.

We introduce the notion of Frobenius Alexander quandles associated with abelian varieties over finite fields, and study them from both knot-theoretic and arithmetic viewpoints. We determine exactly for which rational primes a knot admits a nonconstant coloring by these quandles, in terms of the resultant of its Alexander polynomial and the Frobenius characteristic polynomial of the abelian variety. This leads to the notion of persistent colorability and its characterization, as well as applications to fibered knots and knots of small genus. We also prove a rigidity result showing that the isomorphism class of a Frobenius Alexander quandle determines the similarity class of the Frobenius endomorphism on the corresponding torsion subgroup. As a consequence, one sufficiently large Frobenius Alexander quandle determines the characteristic polynomial of the Frobenius endomorphism, and hence, determines the isogeny class of the abelian variety over the given finite base field.

Heat transfer in a dense gas between two parallel plates is studied for assessing the quantitative difference of two variants of the Enskog equation, i.e., the original Enskog equation and the Enskog equation with a modified Enskog factor recently proposed in Phys. Rev. E 111, 065108 (2025). The main advantage of the latter is that the H theorem, which the former lacks, has been established. The influence of the modification is assessed by comparing the macroscopic quantities in detail. It is found that the impact of this modification is quite limited and the two variants reasonably agree as a whole.

Let $X$ be a closed hyperbolic surface. We prove that the number of topological types of primitive closed geodesics of length at most $L$ is asymptotic to \[ \frac{1}{|\Isom(X)|}\frac{e^L}{2L}. \] as $L$ grows. Thus Huber's asymptotic remains unchanged after quotienting by topological type, up to the finite symmetry factor coming from the isometry group of $X$.

Following B. Yu's work on Lyapunov graphs of non-singular Smale flows on $S^1\times S^2$, we characterize the Lyapunov graphs of non-singular Morse-Smale flows on $S^1 \times S^2$ by using filtrating neighborhoods as the local data attached to the vertices. More precisely, we determine which oriented graphs with vertices labeled by filtrating neighborhoods can be realized as such Lyapunov graphs.

We study locally conformally flat (LCF) Riemannian manifolds with nonnegative or positive scalar curvature (PSC), using the conformal boundary of the developing image. For closed oriented LCF $n$-manifolds with PSC and infinite fundamental group, $n\ge5$, we bound the macroscopic dimension of their Riemannian universal covers by $\lfloor(n-1)/2\rfloor$, establish the existence of a nontrivial homotopy group above the middle dimension, and, under mild additional hypotheses, bound the Hausdorff dimension of the limit sets of their Kleinian groups in the interval $(1,(n-2)/2)$. In particular, no closed aspherical manifold admits an LCF metric with PSC. If the scalar curvature is at least $n(n-1)$ and the manifold is not isometric to the round sphere, then every smooth nonzero-degree map to the sphere expands somewhere when its Kleinian group is elementary, while the developing map expands somewhere whenever the fundamental group is infinite. We prove that the space of LCF metrics with PSC and its moduli space are contractible in dimension three for finite fundamental group and for $S^2\times S^1$, and that the moduli space is empty or contractible for smooth manifolds homeomorphic to spherical space forms but not diffeomorphic to $S^n$ in dimensions $n\ge4$. For complete open simply connected LCF manifolds of nonnegative scalar curvature, we obtain Euclidean rigidity under additional topological hypotheses at infinity (and, in dimension three, from vanishing second homology alone). We also show that, in dimensions $n\ge4$, the Euclidean conclusion can fail when neither of the two additional topological hypotheses is assumed: we construct complete contractible examples of PSC that are not homeomorphic to $\mathbb{R}^n$.

This article is concerned with generalizations of pseudo-differential operators in $L^2(\mathbb{R}^n)$, $n\geq 1$. The definition of the new calculi depends only on bounded measures on the phase space $\mathbb{R}^{2n}$ and each measure gives rise to a specific calculus. Quantizations of anti-Wick, Weyl, classical and Born-Jordan are particular cases of the general calculi. Classes of symbols in this framework are then studied. The Gevrey class of parameter 1/2 is a class of symbol that is common to all the general calculi, that is, a class of symbols independent on the bounded measures parametrizing the quantizations. Precise additional hypotheses on the measures are necessary in the aim to consider the larger class of symbols $L^{\infty}(\mathbb{R}^{2n})$. This result can be applied for anti-Wick but not for Weyl quantization. Concerning Weyl pseudo-differential calculus, we recover the standard class of Sjöstrand and Gröchening. Then, we prove that probability measures of Lévy processes on the phase space $\mathbb{R}^{2n}$ with diffusion larger than 1/4 are natural examples of measures satisfying the latter additional hypotheses in order to consider $L^{\infty}(\mathbb{R}^{2n})$ symbols. This relation is derived using the Lévy-Khintchine formula. Composition laws in that general context are next investigated. In that purpose, we give a formula for the composition of two symbols in some precise class of symbols valid for all general quantizations. General calculi are relying on Wick quantization which is therefore primarily examined for some precise classes of symbols. Additional results in that context are provided, such as Mizrahi series expansions and Banach algebra isomorphisms between operators and symbol classes.

We introduce a compactly supported multinode Shepard operator for the interpolation of scattered data on the torus embedded in $\mathbb{R}^3$. The method combines local polynomial interpolation of total degree $d\in \mathbb{N}$ with compactly supported Shepard-type weights, so that the approximation at each evaluation point depends only on neighbouring stencils of nodes. The torus is treated as an algebraic surface defined by a quartic polynomial, and Gröbner bases are used to construct reduced polynomial spaces on the surface by removing the redundancy induced by the defining equation. This yields local Vandermonde systems adapted to the toroidal geometry. We discuss the metric structure of the torus, show the local equivalence between the periodic parameter distance and the Euclidean distance inherited from $\mathbb{R}^3$, and use this equivalence to motivate the compact support construction. We establish a uniform error estimate in terms of the maximal support radius and the local Lebesgue constants. Under uniform locality and stability assumptions, the method converges with order $d+1$ with respect to the fill distance. Numerical experiments on analytical test functions confirm polynomial reproduction and exhibit an error decay consistent with the theoretical analysis. The approach is also tested on Computational Fluid Dynamics data mapped onto the torus, including the interpolation of the velocity components, the reconstruction of the velocity magnitude from the interpolated components, and the reconstruction of a tangent velocity field through an orthonormal lifting of the interpolated components.

We prove uniform endpoint estimates, killed local limits, and conditioned-bridge invariance principles for triangular arrays of centered, aperiodic, integer-valued random walks with uniform exponential tails. The endpoint estimates cover all nonnegative heights, while the local limits treat positive diffusive and zero endpoints. The increment law may vary with the scaling parameter.

We study the graph $\mathcal{AO}(G)$ of acyclic orientations of a graph $G$. Two acyclic orientations are adjacent in this graph if they disagree on the orientation of a single arc. In particular, we focus on the Hamiltonicity of the graphs $\mathcal{AO}(G)$. Using two methods of pattern lacing which generalize the zig-zag method of Brenner, Cardinal, McConville, Merino and Mütze, we characterize which multipaths are $\mathcal{AO}$-Hamiltonian. Moreover, we give a criterion for the gluing of a multipath on a given graph to preserve $\mathcal{AO}$-Hamiltonicity. Building towards an inductive certification of $\mathcal{AO}$-Hamiltonicity via the (open) ear decomposition of 2-connected graphs, we propose three ways of gluing several multipaths to a given graph. In addition, we define the acyclic polynomials to encapsulate both the number of acyclic orientations of a graph and the "parity problem" proposed by Savage, Squire and West: if $-1$ is not a root of the acyclic polynomial of $G$, then $G$ is not $\mathcal{AO}$-Hamiltonian. We explore numerous properties of the acyclic polynomials, proving that they are not instances of the famous Tutte-Whitney polynomials, but that they too exhibit a partial deletion-contraction phenomenon.

Cycles $C_1,\ldots,C_k$ in a graph are called nested without geometric crossings if they are pairwise edge-disjoint, $V(C_k)\subseteq\cdots\subseteq V(C_1)$, and each pair of consecutive cycles induces the same cyclic order on the vertices of the inner cycle, up to reversal. Let $f_k(n)$ be the least number of edges that forces such a family in every $n$-vertex graph. Gil Fernández, Kim, Kim and Liu proved that $f_2(n)=O(n)$, answering a question of Erdős, and asked whether $f_k(n)=O_k(n)$ for every fixed $k$. We prove this for all $k$. The proof selects the inner cycles together with a disjoint subgraph that supplies their external neighbours. A reselection argument gives disjoint paths from every inner-cycle vertex to any sufficiently large target set. Sublinear expansion and a rooted clique minor then allow the vertices to be joined in the required cyclic order.

In this paper, we develop the method of moving planes entirely in the weak formulation for the fractional $p$-Laplacian in the singular range $1<p\leq2$. We first establish a small region principle for antisymmetric functions and apply it to prove radial symmetry and monotonicity of nonnegative weak solutions of fractional $p$-Laplacian equations in a bounded domain. We also consider the nonlocal quasilinear Lane–Emden equation $(-\Delta)^s_pu=u^q$ in $\mathbb R^n$. In the Sobolev critical case, we establish radial symmetry, monotonicity, and precise asymptotic behavior at infinity for finite-energy weak solutions. Under a suitable decay condition, we also obtain radial symmetry for the full range $q>p-1$. Our results complete those of Chen-Li (Adv. Math., 2018: 735-758), where analogous results were obtained for $C^{1,1}$ solutions in the pointwise sense.

We prove two simple bivariate recurrences for the number of permutations with a long increasing subsequence. The two recurrences imply D-finiteness of the sequence in a certain range. As a consequence, we also obtain a proof of a conjecture posed by Kauers and Koutschan in 2023.