This paper investigates the variation bounds and mixed norm bounds for the dilated averages operator. The established bounds are shown to be sharp, except for certain endpoint cases.

Suppose that

is a 2-(v, k, λ) design with (v − 1, k − 1) = 2 and that G is a flag-transitive group of automorphisms of

\mathcal{D}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cal{D}$$\end{document}

. It is shown in this paper that either G is 2-transitive on points or G is a subgroup of the group AΓL(1, v) of 1-dimensional semilinear affine transformations, so that

\mathcal{D}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cal{D}$$\end{document}

has v = p^d points (p is a prime). Further, we classify such type of designs admitting an almost simple automorphism group G with socle a Suzuki group Sz(q), by considering the classification under the (even weaker) assumption that k is odd. As its application, we construct two new families of flag-transitive 2-designs with socle Sz(q).

Let G be a simple graph. We say that a hypergraph

is a Berge-G if there is a bijection

\psi :E(G)\to E(\mathcal{H}))

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\psi : E(G)\rightarrow E({\cal{H}}))$$\end{document}

such that e ⊆ ψ(e) for all e ∈ E(G). For any r-uniform hypergraph

\mathcal{H}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cal{H}$$\end{document}

and a real number p ≥ 1, the p-spectral radius of

\mathcal{H}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cal{H}$$\end{document}

is defined as A keyring C_n(k) is a graph of order n obtained from a cycle of length n − k by appending k leaves to one of vertices of the cycle. In this paper, we obtain the 3-uniform hypergraphs with maximum p-spectral radius for p ≥ 1 among 3-uniform Berge-G hypergraphs when G is a keyring.

In this paper we evaluate several determinants involving quadratic residues modulo primes. For example, for any prime p > 3 with p ≡ 3 (mod 4) and a, b ∈ ℤ with p ∤ ab, we prove that

denotes the Legendre symbol. We also pose some conjectures for further research.

The notion of differential algebra of weight λ unifies those of usual differential algebras of zero weight and difference algebras. In this paper, we study modules over differential algebra of weight λ, with emphasis on the role played by the differential operators and the difference between differential modules and modules over an algebra in the usual sense. We introduce the concepts of free, projective, injective and flat differential modules. Furthermore, we present a construction of free differential modules and show that there are enough projective, injective and flat differential modules to provide the corresponding resolutions for derived functors.

In this paper, we describe transposed Poisson algebra structures on the Heisenberg–Virasoro algebra L of rank two and its generalization L(p, q), where p, q ∈ ℂ. We show that transposed Poisson algebra structures are trivial on L as well as on L(p, q) when (p, q) ∉ ℤ × ℤ. We then describe all nontrivial transposed Poisson algebra structures on L(p, q) for (p, q) ∈ ℤ × ℤ. Specifically, we classify the nontrivial transposed Poisson algebra structures on L(0, 0).

In this paper, we study the following quasilinear elliptic equation

where N ≥ 3, V(x) is a potential function and f satisfies some suitable conditions. We prove the existence of a positive ground state solution via Pohožaev manifold.

In this paper, we consider the two-phase flow model with slip boundary condition in a three-dimensional simply connected bounded domain with C^∞ boundary ∂Ω. The pressure depends on two different variables from the continuity equation. After discovering some new estimates on the boundary related to the slip boundary condition, we are able to obtain that the classical solutions to the initial-boundary-value problem of two-phase flow model exist globally in time provided that the initial energy is suitably small. As we know, this is the first result concerning the global existence of classical solutions to the compressible two-phase flow model with slip boundary condition and the density containing vacuum initially for general 3D bounded smooth domains.

Motivated by the work of Chow and Luo [J. Differential Geom., 2003, 63(1): 97–129], Ge and his collaborators ([Trans. Amer. Math. Soc., 2018, 370(2): 1377–1391], [Differential Geom. Appl., 2016, 47: 86–98], [Adv. Math., 2018, 333: 523–538]) introduced the combinatorial Calabi flow to study circle patterns in Euclidean and hyperbolic background geometries. Recently, Popelensky [Filomat, 2023, 37(25): 8675–8681] further developed a weighted combinatorial Ricci flow. Inspired by these contributions, we define a weighted combinatorial Calabi flow to investigate circle patterns. This paper addresses two cases: Euclidean and hyperbolic background geometries. In Euclidean background geometry, we prove that the flow exists for all time, and the flow converges if and only if a constant curvature circle pattern metric exists. Moreover, we establish that the prescribed flow converges if and only if the prescribed curvature is attainable. In hyperbolic background geometry, we prove that the flow exists for all time, and the flow converges if and only if a zero curvature circle pattern metric exists. Additionally, in hyperbolic background geometry, for a prescribed curvature, we show that the prescribed flow converges if and only if the curvature is attainable.

We follow the idea of gluing theory in instanton moduli spaces and discuss the case when there is a finite group Γ acting on the 4-manifolds X₁, X₂ with x₁, x₂ as isolated fixed points, how to glue two Γ-invariant ASD connections over X₁, X₂ together to get a Γ-invariant ASD connection on the connected sum X₁ # X₂.