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We prove a homogenization result for a family of time-dependent Hamilton-Jacobi equations, rescaled by a parameter $\varepsilon$ tending to zero, posed on a periodic network, with a suitable notion of periodicity that will be defined. As $\varepsilon$ becomes infinitesimal, we derive a limiting Hamilton-Jacobi equation in a Euclidean space, whose dimension is determined by the topological complexity of the network and is independent of the ambient space in which the network is embedded. Among the key contributions of our analysis, we extend to the setting of networks and graphs Mather's result on the asymptotic behavior of the average minimal action functional, as time tends to infinity. Additionally, we establish the well-posedness of the approximating problems, representing a nontrivial generalization of existing results for finite networks to a non-compact setting.

We describe the cohomology ring of toric wonderful models for arbitrary building set, including the case of non well-connected ones. Our techniques are based on blowups of posets, on Gr\"obner basis over rings and admissible functions., Comment: 30 pages, 5 figures. Comments are welcome

This paper presents a bridge between the theories of wonderful models associated with toric arrangements and wonderful models associated with hyperplane arrangements. In a previous work, the same authors noticed that the model of the toric arrangement of type $A_{n-1}$ is isomorphic to the one of the hyperplane arrangement of type $A_{n}$; it is natural to ask if there exist similar isomorphisms between other families of arrangements. The aim of this paper is to study one such family, namely the family of arrangements defined by graphs. The main result states that there is indeed an isomorphism between the model of a toric arrangement defined by a graph $\Gamma$ and the model of a hyperplane arrangement defined by the cone of $\Gamma$, provided that a suitable building set is chosen., Comment: 19 pages, 2 figures

Mutually intersecting varieties of a finite Desarguesian projective space are varieties with the same size, such that the intersection of two of them contains the same number of points. Here, we show the existence of a new family of mutually intersecting varieties by using certain quasi-Hermitian varieties of $\mathrm{PG}(n,q^2)$, where $q$ is any prime power. With the help of these quasi-Hermitian varieties we provide a new construction of $5$-dimensional MDS codes over ${\mathbb F}_q$ as well as an infinite family of simple orthogonal arrays $OA(q^{2n-1},q^{2n-2},q,2)$ of index $\mu=q^{2n-3}$., Comment: 13 pages

The chromatic quasisymmetric functions (csf) of Shareshian and Wachs associated to unit interval orders have attracted a lot of interest since their introduction in 2016, both in combinatorics and geometry, because of their relation to the famous Stanley-Stembridge conjecture (1993) and to the topology of Hessenberg varieties, respectively. In the present work we study the csf associated to the larger class of interval orders with no restriction on the length of the intervals. Inspired by an article of Abreu and Nigro, we show that these csf are weighted sums of certain quasisymmetric functions associated to the increasing spanning forests of the associated incomparability graphs. Furthermore, we define quasisymmetric functions that include the unicellular LLT symmetric functions and generalize an identity due to Carlsson and Mellit. Finally we conjecture a formula giving their expansion in the type 1 power sum quasisymmetric functions which should extend a theorem of Athanasiadis., Comment: 28 pages, 5 figures

A new algorithm for time dependent Hamilton Jacobi equations on networks, based on semi Lagrangian scheme, is proposed. It is based on the definition of viscosity solution for this kind of problems recently given in. A thorough convergence analysis, not requiring weak semilimits, is provided. In particular, the check of the supersolution property at the vertices is performed through a dynamical technique which seems new. The scheme is efficient, explicit, allows long time steps, and is suitable to be implemented in a parallel algorithm. We present some numerical tests, showing the advantage in terms of computational cost over the one proposed in [7]

In this paper we study the BM quasi-Hermitian varieties introduced in [A. Aguglia, A. Cossidente, G. Korchm\`aros, On quasi-Hermitian Varieties, J. Combin. Des. 20 (2012) 433-447.] in characteristc $2$ and dimension $3$. After a brief investigation of their combinatorial properties, we first show that all of these varieties are projectively equivalent in non-zero even characteristic, exhibiting a behavior which is strikingly different from what happens in odd characteristic, see [A. Aguglia, L. Giuzzi, On the equivalence of certain quasi-Hermitian varieties, J. Combin. Des. 1-15 (2022)]. This completes the classification project started in that paper. Next, by using previous results, we explicitly determine and investigate the structure of the full collineation group stabilizing these varieties. Finally, as a byproduct of our investigation, we also construct and a family of simple orthogonal arrays $OA(q^5,q^4,q,2)$ with entries in ${\mathbb F}_q$ where $q$ is a power of $2$., Comment: 22 pages/submitted version. Work supported by INDAM. A. Aguglia and V. Siconolfi have been partially supported by the EU under the Italian NRRP of NextGenerationEU, partnership on "Telecommunications of the Future" (PE00000001 - program RESTART, CUP: D93C22000910001)

We describe an effective procedure to compute the local subgroup zeta functions of the free class-$2$-nilpotent groups on $d$ generators, for all $d$. For $d=4$, this yields a new, explicit formula. For $d\in\{4,5\}$, we compute the topological subgroup zeta function. We also obtain general results about the reduced and topological subalgebra zeta functions. For the former, we determine the behaviour at one and for the latter, the degree and behaviours at zero and infinity. Some of these results confirm, in the relevant special cases, general conjectures by T. Rossmann., Comment: 42 pages, minor revisions, including referee's suggestions. To appear in Journal of Combinatorial Algebra

In this paper we find monomial bases for the integer cohomology rings of compact wonderful models of toric arrangements. In the description of the monomials various combinatorial objects come into play: building sets, nested sets, and the fan of a suitable toric variety. We provide some examples computed via a SageMath program and then we focus on the case of the toric arrangements associated with root systems of type A. Here the combinatorial description of our basis offers a geometrical point of view on the relation between some Eulerian statistics on the symmetric group., Comment: 39 pages, 7 figures

We provide a Lax-Oleinik-type representation formula for solutions of time-dependent Hamilton-Jacobi equations, posed on a network with a rather general geometry, under standard assumptions on the Hamiltonians. It depends on a given initial datum at $t=0$ and a flux limiter at the vertices, which both have to be assigned in order the problem to be uniquely solved. Previous results in the same direction are solely in the frame of junction, namely network with a single vertex. An important step to get the result is to define a suitable action functional and prove existence as well as Lipschitz-continuity of minimizers between two fixed points of the network in a given time, despite the fact that the integrand lacks convexity at the vertices., Comment: 28 pages

The aim of the paper is to recover some results of Cardaliaguet-Nolen-Souganidis in \cite{CNS} and Xin-Yu in \cite{XY} about the homogenization of the G--equation, using different and simpler techniques. The main mathematical issue is the lack of coercivity of the Hamiltonians. In our approach we consider a multivalued dynamics without periodic invariants sets, a family of intrinsic distances and perform an approximation by a sequence of coercive Hamiltonians.

We study well posedness of time--dependent Hamilton--Jacobi equations on a network, coupled with a continuous initial datum and a flux limiter. We show existence and uniqueness of solutions as well as stability properties. The novelty of our approach is that comparison results are proved linking the equation to a suitable semidiscrete problem, bypassing doubling variable method. Further, we do not need special test functions, and perform tests relative to the equations on different arcs separately.

We study the Ricci curvature of the Hasse diagrams of the Bruhat order of finite irreducible Coxeter groups. For this purpose we compute the maximum degree of these graphs for types $B_n$ and $D_n$. The proof uses a new graph $\Gamma(\pi)$ defined for any element $\pi$ in the corresponding group., Comment: 32 pages

We formulate Aubry-Mather theory for Hamiltonians/Lagrangians defined on graphs and discuss its relationship with weak KAM theory developed in [24]., Comment: 51 pages

We consider the notion of discrete Ricci curvature for graphs defined by Schmuckenschl{\"a}ger \cite{shmuck} and compute its value for Bruhat graphs associated to finite Coxeter groups. To do so we work with the geometric realization of a finite Coxeter group and a classical result obtained by Dyer in \cite{Dyer}. As an application we obtain a bound for the spectral gap of the Bruhat graph of any finite Coxeter group and an isoperimetric inequality for them. Our proofs are case-free., Comment: 12 pages. arXiv admin note: text overlap with arXiv:2102.10134

We express the discrete Ricci curvature of a graph as the minimal eigenvalue of a family of matrices, one for each vertex of a graph whose entries depend on the local adjaciency structure of the graph. Using this method we compute or bound the Ricci curvature of Cayley graphs of finite Coxeter groups and affine Weyl groups. As an application we obtain an isoperimetric inequality that holds for all Cayley graphs of finite Coxeter groups., Comment: 25 pages

In this paper, we introduce and investigate a class P of continuous and periodic functions on R. The class P is defined so that second-order central differences of a function satisfy some concavity-type estimate. Although this definition seems to be independent of nowhere differentiable character, it turns out that each function in P is nowhere differentiable. The class P naturally appear from both a geometrical viewpoint and an analytic viewpoint. In fact, we prove that a function belongs to P if and only if some geometrical inequality holds for a family of parabolas with vertexes on this function. As its application, we study the behavior of the Hamilton Jacobi flow starting from a function in P. A connection between P and some functional series is also investigated. In terms of second-order central differences, we give a necessary and sufficient condition so that a function given by the series belongs to P. This enables us to construct a large number of examples of functions in P through an explicit formula.

We study the asymptotic behavior of the solutions to a family of discounted Hamilton Jacobi equations, posed in the Euclidean N dimensional space, when the discount factor goes to zero. The ambient space being noncompact, we introduce an assumption implying that the Aubry set is compact and there is no degeneracy at infinity. Our approach is to deal not with a single Hamiltonian and Lagrangian but with the whole space of generalized Lagrangians, and then to define via duality minimizing measures associated to both the corresponding ergodic and discounted equations. The asymptotic result follows from convergence properties of these measures with respect to the narrow topology. We use as duality tool a separation theorem in locally convex Hausdorff spaces, we use the strict topology in the space of the bounded generalized Lagrangians as well.

For any triple given by a positive integer n, a finite group G, and a faithful representation V of G, one can describe a subspace arrangement whose intersection lattice is a generalized Dowling lattice in the sense of Hanlon. In this paper we construct the minimal De Concini-Procesi wonderful model associated to this subspace arrangement and give a description of its boundary. Our aim is to point out the nice poset provided by the intersections of the irreducible components in the boundary, which provides a geometric realization of the nested set poset of this generalized Dowling lattice. It can be represented by a family of forests with leaves and labelings that depend on the triple (n,G,V). We will study it from the enumerative point of view in the case when G is abelian., Comment: In this revised version the bijection in Section 5 has been improved and holds also in the case when G is not abelian

We study discounted Hamilton Jacobi equations on networks, without putting any restriction on their geometry. Assuming the Hamiltonians continuous and coercive, we establish a comparison principle and provide representation formulae for solutions. We follow the approach introduced in 11, namely we associate to the differential problem on the network, a discrete functional equation on an abstract underlying graph. We perform some qualitative analysis and single out a distinguished subset of vertices, called lambda Aubry set, which shares some properties of the Aubry set for Eikonal equations on compact manifolds. We finally study the asymptotic behavior of solutions and lambda Aubry sets as the discount factor lambda becomes infinitesimal., Comment: Corrected typos in the title

Although the stability properties of the wake past impervious bluff bodies have been widely examined in the literature, similar analyses regarding the flow around and through porous ones are still lacking. In this work, the effect of the porosity and permeability on the wake patterns of porous rectangular cylinders is numerically investigated at low to moderate Reynolds numbers in the framework of direct numerical simulation combined with local and global stability analyses. A modified Darcy-Brinkman formulation is employed here so as to describe the flow behavior inside the porous media, where also the convective terms are retained to correctly account for the inertial effects at high values of permeability. Different aspect ratios of the cylinder are considered, varying the thickness-to-height ratios, t/d, from 0.01 (flat plate) to 1.0 (square cylinder). The results show that the permeability of the bodies has a strong effect in modifying the characteristics of the wakes and of the associated flow instabilities, while the porosity weakly affects the resulting flow patterns. In particular, the fluid flows through the porous bodies and, thus, as the permeability is progressively increased, the recirculation regions, initially attached to the rear part of the bodies, at first detach from the body and, eventually, disappear even in the near wakes. Global stability analyses lead to the identification of critical values of the permeability above which any linear instability is prevented. Moreover, a different scaling of the non-dimensional permeability allows to identify a general threshold for all the configurations here studied that ensures the suppression of vortex shedding, at least in the considered parameter space., Comment: 31 pages and 17 figures

We study a class of weakly coupled systems of Hamilton{Jacobi equations at the critical level. We associate to it a family of scalar discounted equation. Using control{theoretic tech- niques we construct an algorithm which allows obtaining a critical solution to the system as limit of a monotonic sequence of subsolutions. We moreover get a characterization of isolated points of the Aubry set and establish semiconcavity properties for critical subsolutions.

We study a one-parameter family of Eikonal Hamilton-Jacobi equations on an embedded network, and prove that there exists a unique critical value for which the corresponding equation admits global solutions, in a suitable viscosity sense. Such a solution is identified, via an Hopf-Lax type formula, once an admissible trace is assigned on an intrinsic boundary. The salient point of our method is to associate to the network an abstract graph, encoding all of the information on the complexity of the network, and to relate the differential equation to a discrete functional equation on the graph. Comparison principles and representation formulae are proven in the supercritical case as well., Comment: 46 pages

Following the random approach of Mitake, Siconolfi,Tran and Yamada, we define a Lax--Oleinik formula adapted to evolutive weakly coupled systems of Hamilton--Jacobi equations. It is reminiscent of the corresponding scalar formula, with the relevant difference that it has a stochastic character since it involves, loosely speaking, random switchings between the various associated Lagrangians. We prove that the related value functions are viscosity solutions to the system, and establish existence of minimal random curves under fairly general hypotheses. Adding Tonelli like assumptions on the Hamiltonians, we show differentiability properties of such minimizers, and existence of adjoint random curves. Minimizers and adjoint curves are trajectories of a twisted generalized Hamiltonian dynamics., Comment: 38 pages

We study a class of weakly coupled systems of Hamilton-acobi equations using the random frame introduced in a previous paper of Mitake-Siconolfi-Tran-Yamada. We provide a cycle condition characterizing the points of Aubry set. This generalizes a property already known in the scalar case.