<div>This study develops two compartmental models to analyze the co-dynamics between measles and hand, foot, and mouth disease (HFMD): a four-compartment model and a seven-compartment HFMD-Measles co-infection model. For the four-compartment model, we systematically analyzed the co-dynamics of measles and hand, foot, and mouth disease (HFMD), and employed the next-generation matrix method to calculate the basic reproduction number of measles and that of HFMD. Through the analytical study of these two types of basic reproduction numbers, we rigorously determined the existence of the disease equilibrium points, with their quantitative relationship were clearly illustrated through graphical representations. The global asymptotic stability of these equilibria is established by applying LaSalle invariance principle, with stability regions of the four equilibrium points precisely defined. The analysis reveals that within the stability region of the disease-free equilibrium, both diseases will eventually die out, preventing any outbreaks. In the stability region corresponding to the measles equilibrium, HFMD is eliminated while measles remains endemic. Conversely, in the stability region of the HFMD-only equilibrium, measles dies out whereas HFMD persists. Finally, within the stability region of the coexistence equilibrium, both diseases persist and become endemic. Numerical simulations further validate the consistency and reliability of these theoretical results. For the seven-compartment infectious disease model, we calculated the basic reproduction number and verified the threshold theorem. We derived the conditions for both local and global asymptotic stability of the disease-free equilibrium. In particular, the disease-free equilibrium is locally stable when the basic reproduction number is less than one, and we also provided conditions for its global stability. Model validation is performed by fitting empirical data from China on HFMD and measles cases.</div>
Let U : C → D U:\mathcal {C}\rightarrow \mathcal {D} be a strong monoidal functor between abelian monoidal categories admitting a right adjoint R R , such that R R is exact, faithful and the adjunction U ⊣ R U\dashv R is coHopf. Building on the work of Balan [Appl. Categ. Structures 25 (2017), pp. 747–774], we show that R R is separable (resp., special) Frobenius monoidal if and only if R ( 1 D ) R(\mathbb {1}_{\mathcal {D}}) is a separable (resp., special) Frobenius algebra in C \mathcal {C} . If further, C , D \mathcal {C},\mathcal {D} are pivotal (resp., ribbon) categories and U U is a pivotal (resp., braided pivotal) functor, then R R is a pivotal (resp., ribbon) functor if and only if R ( 1 D ) R(\mathbb {1}_{\mathcal {D}}) is a symmetric Frobenius algebra in C \mathcal {C} . As an application, we construct Frobenius monoidal functors going into the Drinfeld center Z ( C ) \mathcal {Z}(\mathcal {C}) , thereby producing Frobenius algebras in it.
We consider GLn (Fq)-analogues of certain factorization problems in the symmetric group Sn: ratherthan counting factorizations of the long cycle(1,2, . . . , n) given the number of cycles of each factor, we countfactorizations of a regular elliptic element given the fixed space dimension of each factor. We show that, as in Sn, the generating function counting these factorizations has attractive coefficients after an appropriate change of basis.Our work generalizes several recent results on factorizations in GLn (Fq) and also uses a character-based approach.We end with an asymptotic application and some questions.
For any Coxeter system (W, S) of rank n, we introduce an abstract boolean complex (simplicial poset) of dimension 2n − 1 which contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples (J,w,K), where J and K are subsets of the set S of simple generators, and w is a minimal length representative for the double parabolic coset WJ wWK . There is exactly one maximal face for each element of the group W . The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the h-polynomial is given by the “two-sided” W -Eulerian polynomial, i.e., the generating function for the joint distribution of left and right descents in W .
We prove a conjecture of J.-C. Novelli, J.-Y. Thibon, and L. K. Williams (2010) about an equivalence of two triples of statistics on permutations. To prove this conjecture, we construct a bijection through different combinatorial objects, starting with a Catalan based object related to the PASEP.
At the end of the 1960s, Knuth characterised in terms of forbidden patterns the permutations that can be sorted using a stack. He also showed that they are in bijection with Dyck paths and thus counted by the Catalan numbers. Subsequently, Pratt and Tarjan asked about permutations that can be sorted using two stacks in parallel. This question is significantly harder, and the associated counting question has remained open for 40 years. We solve it by giving a pair of equations that characterise the generating function of such permutations. The first component of this system describes the generating function $Q(a,u)$ of square lattice loops confined to the positive quadrant, counted by the length and the number of North-West and East-South factors. Our analysis of the asymptotic number of sortable permutations relies at the moment on two intriguing conjectures dealing with this series. Given the recent activity on walks confined to cones, we believe them to be attractive $\textit{per se}$. We prove these conjectures for closed walks confined to the upper half plane, or not confined at all.
We study Newton polytopes of cluster variables in type $A_n$ cluster algebras, whose cluster and coefficient variables are indexed by the diagonals and boundary segments of a polygon. Our main results include an explicit description of the affine hull and facets of the Newton polytope of the Laurent expansion of any cluster variable, with respect to any cluster. In particular, we show that every Laurent monomial in a Laurent expansion of a type $A$ cluster variable corresponds to a vertex of the Newton polytope. We also describe the face lattice of each Newton polytope via an isomorphism with the lattice of elementary subgraphs of the associated snake graph.
Connection matrices for graph parameters with values in a field have been introduced by M. Freedman, L. Lovász and A. Schrijver (2007). Graph parameters with connection matrices of finite rank can be computed in polynomial time on graph classes of bounded tree-width. We introduce join matrices, a generalization of connection matrices, and allow graph parameters to take values in the tropical rings (max-plus algebras) over the real numbers. We show that rank-finiteness of join matrices implies that these graph parameters can be computed in polynomial time on graph classes of bounded clique-width. In the case of graph parameters with values in arbitrary commutative semirings, this remains true for graph classes of bounded linear clique-width. B. Godlin, T. Kotek and J.A. Makowsky (2008) showed that definability of a graph parameter in Monadic Second Order Logic implies rank finiteness. We also show that there are uncountably many integer valued graph parameters with connection matrices or join matricesof fixed finite rank. This shows that rank finiteness is a much weaker assumption than any definability assumption.
AbstractStability conditions on triangulated categories were introduced by Bridgeland as a ‘continuous’ generalisation of t-structures. The set of locally-finite stability conditions on a triangulated category is a manifold that has been studied intensively. However, there are mainstream triangulated categories whose stability manifold is the empty set. One example is Dc(k[X]/(X2)), the compact derived category of the dual numbers over an algebraically closed field k. This is one of the motivations in this paper for introducing co-stability conditions as a ‘continuous’ generalisation of co-t-structures. Our main result is that the set of nice co-stability conditions on a triangulated category is a manifold. In particular, we show that the co-stability manifold of Dc(k[X]/(X2)) is ℂ.
Sarah Miracle, Dana Randall, Amanda Pascoe Streib
et al.
Given a planar triangulation, a 3-orientation is an orientation of the internal edges so all internal vertices have out-degree three. Each 3-orientation gives rise to a unique edge coloring known as a $\textit{Schnyder wood}$ that has proven useful for various computing and combinatorics applications. We consider natural Markov chains for sampling uniformly from the set of 3-orientations. First, we study a "triangle-reversing'' chain on the space of 3-orientations of a fixed triangulation that reverses the orientation of the edges around a triangle in each move. We show that (i) when restricted to planar triangulations of maximum degree six, the Markov chain is rapidly mixing, and (ii) there exists a triangulation with high degree on which this Markov chain mixes slowly. Next, we consider an "edge-flipping'' chain on the larger state space consisting of 3-orientations of all planar triangulations on a fixed number of vertices. It was also shown previously that this chain connects the state space and we prove that the chain is always rapidly mixing.
We exhibit a canonical connection between maximal $(0,1)$-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable and thus a shellable sphere. In particular, this implies a positivity result for Schubert polynomials. For Ferrers shapes, we moreover construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between $k$-triangulations of the $n$-gon and $k$-fans of Dyck paths. Using this, we translate a conjectured cyclic sieving phenomenon for $k$-triangulations with rotation to $k$-flagged tableaux with promotion.
We consider Buch's rule for K-theory of the Grassmannian, in the Schur multiplicity-free cases classified by Stembridge. Using a result of Knutson, one sees that Buch's coefficients are related to Möbius inversion. We give a direct combinatorial proof of this by considering the product expansion for Grassmannian Grothendieck polynomials. We end with an extension to the multiplicity-free cases of Thomas and Yong.
We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph $G_{T,\gamma}$ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph $G_{T,\gamma}$ .
Motivated by the theory of operads, we introduce new combinatorial objects, called shrubs, that generalize forests of rooted trees. We show that the species of shrubs is isomorphic to the species of series-parallel posets.
In type $A$, the $q,t$-Fuß-Catalan numbers $\mathrm{Cat}_n^{(m)}(q,t)$ can be defined as a bigraded Hilbert series of a module associated to the symmetric group $\mathcal{S}_n$. We generalize this construction to (finite) complex reflection groups and exhibit some nice conjectured algebraic and combinatorial properties of these polynomials in $q$ and $t$. Finally, we present an idea how these polynomials could be related to some graded Hilbert series of modules arising in the context of rational Cherednik algebras. This is work in progress.
We study graph weights (i.e., graph invariants) which arise naturally in Mayer's theory and Ree-Hoover's theory of virial expansions in the context of a non-ideal gas. We give special attention to the Second Mayer weight $w_M(c)$ and the Ree-Hoover weight $w_{RH}(c)$ of a $2$-connected graph $c$ which arise from the hard-core continuum gas in one dimension. These weights are computed using signed volumes of convex polytopes naturally associated with the graph $c$. Among our results are the values of Mayer's weight and Ree-Hoover's weight for all $2$-connected graphs $b$ of size at most $8$, and explicit formulas for certain infinite families.
The sandpile group of a graph $G$ is an abelian group whose order is the number of spanning trees of $G$. We find the decomposition of the sandpile group into cyclic subgroups when $G$ is a regular tree with the leaves are collapsed to a single vertex. This result can be used to understand the behavior of the rotor-router model, a deterministic analogue of random walk studied first by physicists and more recently rediscovered by combinatorialists. Several years ago, Jim Propp simulated a simple process called rotor-router aggregation and found that it produces a near perfect disk in the integer lattice $\mathbb{Z}^2$. We prove that this shape is close to circular, although it remains a challenge to explain the near perfect circularity produced by simulations. In the regular tree, we use the sandpile group to prove that rotor-router aggregation started from an acyclic initial condition yields a perfect ball.