Using the Markov distance and the Ptolemy inequality introduced by Lee et al. (2023), we completely determine the monotonicity of generalized Markov numbers along lines of a given slope. This confirms several conjectures posed by Lee et al. (2023).
A class of Lotka-Volterra competitive systems on a finite graph (network) is investigated, where both diffusive and advective movements are considered. By the theory of graphs, the theory of monotone dynamical systems and the theory of principal eigenvalues, the global dynamics is determined in terms of critical competition coefficients, where the key is to establish the linear stability of all possible positive equilibria. This work can be seen as a further development of Slavík (2020) and Chen et al. (2022), where there is diffusion only (no advection).
In this paper, we give an estimate for the sum of Fourier coefficients λf(n) of Hecke-Maass cusp form f over a fractional sequence. Our main result is S_f(x) = ∑_n ⩽ xλ_f([x/n]) = ∑_n=1^∞λ_f(n)/n(n+1) x + O(x^1/2-3/160+ε). This is a breakthrough to the barrier 1/2 for the error term estimates. Our method is also used to give a new result on a problem initiated by Bordellés et al. (2019).
In this paper, we study the Schur algebra and its Lusztig subalgebra associated with the affine flag variety of type D. We show that the ıquantum group corresponding to a subalgebra of the Lusztig algebra forms a quantum symmetric pair together with U_q( s l_n) . We further construct a monomial basis and the canonical bases of its idempotented version and prove the positivity properties of the canonical basis with respect to multiplication and the bilinear pairing.
We study emergent behaviors of a discrete multi-cluster Cucker-Smale (MC-CS) model. The MC-CS model can be derived from the Cucker-Smale model under the a priori assumptions on the number of clusters and agents in each cluster. For the proposed model, we present a sufficient framework in terms of system parameters and initial data for the asymptotic emergence of a multi-clustering flocking configuration. Under the proposed sufficient framework, we show that the multi-cluster flocking configuration can emerge depending on the decay mode of the communication weight function. We also provide several numerical simulations and compare them with theoretical results.
In this paper, without the assumption of bounded growth, we prove that if an evolution family is sufficiently close to one which has an exponential dichotomy, then it also has an exponential dichotomy. It was proved by Henry (1981) in the case where the evolution family has bounded growth. The difficulty caused by the absence of bounded growth, which is an essential condition in Henry’s proof, is resolved by proving a result on system-dependence of extended Green functions. We also extend this result to exponential dichotomies on a half line ℝ+ or ℝ− as well as on any nonempty closed interval of ℝ.
We study nonnegative solutions to the following higher-order fractional Yamabe and Hardy-Hénon equations: (-Δ)^σu=|x|^-αu^p in ℝ^n\{0}, where 0 < σ < n/2, −∞ < α < 2σ, p > 1, and the origin may be a singularity. When the singularity is removable, we establish optimal Liouville-type theorems for all σ ∈ (0, n/2). Moreover, we prove the existence of positive solutions in the supercritical case. For non-removable singularities, we demonstrate the radial symmetry of solutions, which is useful in studying the higher-order fractional singular Yamabe problem.
In this paper, we study an algebraic structure called an AE-algebra, which involves three linear maps. First, we introduce the notion of an AE-algebra, defined as a BiHom-associative algebra equipped with an algebra endomorphism, along with the concept of a dual representation of a representation of an AE-algebra. Subsequently, we develop the bialgebra theory for AE-algebras. Specifically, by combining an AE-algebra and an AE-coalgebra with compatibility conditions derived from dual representations, we obtain the concept of an AE-associative D-bialgebra. We then show that such a bialgebra can be characterized either by a matched pair of AE-algebras or by a double construction of AE-Frobenius algebras. Based on the notion of AE-associative D-bialgebras, we introduce mixed homomorphisms between BiHom-associative D-bialgebras, which leads to a categorical interpretation of the relationships among three classes of objects: BiHom-associative D-bialgebras, matched pairs of BiHom-associative algebras and double constructions of BiHom-Frobenius algebras. Furthermore, we present several special cases of AE-associative D-bialgebras through associative BiHom-Yang-Baxter equations, O -operators and AE-dendriform algebras. Finally, we demonstrate that a pure AE-associative D-bialgebra naturally gives rise to a pure AE-Lie bialgebra.
In this paper, we solve a class of distributionally robust optimization (DRO) problems with Lipschitz continuous loss functions and weakly compact ambiguity sets via sublinear expectation introduced by Peng (2009). We reformulate the DRO problem as a minimization problem by using sublinear expectation, and introduce a discrete approximation by grouping samples. We prove that optimal values and optimal solutions of the discrete problem converge to those of the DRO problem with probability 1 under capacity. We show that the discrete form is an asymptotic unbiased estimator for the sublinear expectation of the loss function, and provide the quantification of the difference between the discrete problem and the DRO problem with a special moment ambiguity set. Numerical experiments of two real life data sets are conducted. Our preliminary numerical results show that the sublinear expectation method outperforms the existing duality method, especially from the perspective of reliability.
In this paper, we study the nonlinear asymptotic stability of the 2D Boussinesq equations near the Couette flow in 𝕋×ℝ under the condition that the Richardson number satisfies γ^2 > 1 4 . We allow for different viscosity ν and thermal diffusivity μ and establish stability in the regime μ^3≤ν≤μ^1 3 . We show that if the initial perturbation is sufficiently small in HN+1 × HN+2 (with N ⩾ 6) in the sense that, for some sufficiently small α > 0 (0 < α ≪ 1), ∥ v_in-(y,0)∥_H^N+1+∥ρ_in+γ^2y-1∥_H^N+2≤ε_0min{ν,μ}^1 2+4 3α, then the Couette flow is asymptotically stable. The recent result of Zhai and Zhao (2023) represents an important contribution to this problem, and our work constitutes a substantial improvement.
Deep learning approaches have achieved great success in both applications and theoretical studies in recent years. In this paper, we study a partially linear regression model for longitudinal data, with the nonparametric component approximated by a deep neural network. The proposed method circumvents the curse of dimensionality while facilitating the interpretability of linear effects. A maximum likelihood estimation approach is introduced, and a two-step iterative algorithm is developed for optimization. The convergence rate and the asymptotic properties of the resulting estimator are established. The performance of the method is demonstrated through simulation studies and an application to a yeast cell-cycle gene expression dataset.
In this paper, we study tilting and cotilting subcategories of the category of representations of a quiver. Let M be an abelian category, Q be a rooted quiver, and Rep(Q,M) be the category of M -valued representations of Q . By using some recent results about cotorsion torsion triples (resp. torsion cotorssion triples), under certain assumptions, we show that if T is a 1-tilting (resp. 1-cotilting) subcategory of M , then the monomorphism category Φ(T) (resp. the epimorphism category Ψ(T) ) is a 1-tilting (resp. 1-cotilting) subcategory of Rep(Q,M) . Then, we study another types of induced subcategories in Rep(Q,M) and, by using nice descriptions of monomorphism and epimorphism categories, show that if T is a tilting (resp. cotilting) subcategory of M , then the epimorphism category Ψ(T) (resp. the monomorphism category Φ(T) ) is a tilting (resp. cotilting) subcategory of Rep(Q,M) for every finite acyclic quiver Q . This result is a generalization of a lemma due to Zhang (2011) about induced cotilting modules and some recent results due to Bauer et al. (2020). We finally extend Zhang’s reciprocity of the monomorphism operator and the left perpendicular operator for cotilting modules to cotilting subcategories. The results give us a systematic method to create new tilting and cotilting subcategories.
In this paper, we consider the Cauchy problem for the planar magnetohydrodynamics (MHD) system with both constant viscosity and constant resistivity but without heat conductivity. Global well-posedness of strong solutions in the presence of natural far field vacuum, due to the finiteness of the mass, is established for any large initial data of suitable smoothness. Density discontinuity and interior vacuum that are either point-like or piecewise-like are also allowed. Technically, the entropy-type energy inequality, which is commonly used as a basic tool in the existing literature on the planar MHD system, is not workable in this paper, as it is not consistent with the far field vacuum. Instead, besides making full use of advantages of the effective viscous flux, a new coupling structure, between the longitudinal velocity u and the transversal magnetic field h, is exploited to recover the dissipative estimate on h.
In this paper, we completely classify constantly curved holomorphic immersions from the two-sphere into a general complex Grassmann manifold G(k, N) with the norm of the second fundamental form satisfying the equality case of the Simons-type integral inequality for holomorphic curves in G(k, N) established by Wang et al. (2022). It is proven that any such immersion can be decomposed as the “direct sum” of some “foundation stones” up to congruence.
For piecewise-smooth differential systems, in this paper, we focus on crossing limit cycles and sliding loops bifurcating from a grazing loop connecting one high multiplicity tangent point. For the low multiplicity cases considered in previous publications, the method is to define and analyze return maps following the classic idea of Poincaré. However, the high multiplicity causes the domains or properties of return maps to become unclear under perturbations. To overcome these difficulties, we unfold grazing loops by functional parameters and functional functions, and analyze this unfolding along some specific parameter curve. Relationships between the multiplicity and the numbers of crossing limit cycles and sliding loops are given, and our results not only generalize the previous results from multiplicity one to any odd multiplicity, but also are new for any even multiplicity.
A family of subsets F⊆([n] k) is called intersecting if any two of its members share at least one common element. For an intersecting family, a natural extremal problem is to determine its maximum size, whereas the corresponding inverse problem seeks to characterize the extremal structures and establish their stability. The celebrated Erdős-Ko-Rado theorem addresses both the direct and inverse problems and initiated the study of intersection problems in finite set systems. In this paper, we consider a quantitative intersection problem, which can be viewed as an inverse problem in the spirit of Erdős-Ko-Rado-type theorems. For F⊆([n] k) , define its total intersection number as I( F) = ∑_F_1,F_2∈ F|F_1 ∩F_2| . We ask: what is the structure of F when it maximizes the total intersection number among all families of the same size in ([n] k) ? We provide two structural characterizations of optimal families that maximize the total intersection number. As a consequence, when n is sufficiently large and F has an appropriate size, these characterizations imply that the optimal family F is t-intersecting for some t ⩾ 1. To some extent, this reveals a connection between being intersecting and maximizing the total intersection number. Moreover, we establish upper bounds on I( F) for various ranges of | F| , and determine the unique optimal structures for certain family sizes.
In this paper, we are devoted to the study of energy conservation for weak solutions of the inhomogeneous incompressible Euler equations. First, inspired by the strategy outlined in Yu (2009), we develop a unified approach that recovers several classical results on energy conservation in the homogeneous incompressible setting, including those results obtained by Constantin et al. (1994), Duchon and Robert (2000) and Cheskidov et al. (2008). Second, building upon this framework, by analyzing the properties of Besov spaces and employing mollification techniques, we derive new energy conservation results for the inhomogeneous case. Our results not only improve upon the existing studies of Feireisl et al. (2017) and Chen and Yu (2019), but also refine, to some extent, the recent developments of Nguyen et al. (2020).
We first show that every isoparametric hypersurface in 𝕊^n×ℝ^m or ℍn × ℝm possesses a constant angle function with respect to the canonical product structure. Exploiting this rigidity, we achieve a complete classification of isoparametric and homogeneous hypersurfaces in these product spaces. Furthermore, we prove that an isoparametric hypersurface in 𝕊^n×ℝ^m or ℍn × ℝm also has constant principal curvatures.
