Let G be a simple, connected graph. The leaf number, L(G), is the maximum number of leaf vertices contained in a spanning tree of G. The cardinality of the smallest connected dominating set in G is the connected domination number, γ_c(G). The girth g(G) is the length of a shortest cycle in a graph. If G has girth g(G) ≥ 5, minimum degree 𝛿(G) ≥ 3, order n(G), leaf number L(G) and connected domination number γ_c(G), we prove that $L(G){\geq}{\frac{2}{5}}n(G)+{\frac{4}{5}}$. That is, ${\gamma}_c(G){\leq}{\frac{3}{5}}n(G)-{\frac{4}{5}}$ and the bound is asymptotically sharp. This settles completely, a conjecture by Griggs, Kleitman and Shastri.

We show that every single ample divisor on a complex smooth projective variety can recapture the essential information about the sheaves on the small Zariski site of the variety. We then use it to characterize ample divisors among nef divisors topos-theoretically.

We determine the sharp bounds of the second Hankel determinants of logarithmic coefficients of Ma-Minda type functions.

In this paper, we study some non-Riemannian curvature properties of general spherically symmetric Finsler metrics. First, we find the necessary and sufficient condition under which a general spherically symmetric Finsler metric be weakly Berwaldian. Then we find the necessary and sufficient condition under which a general spherically symmetric Finsler metric be of isotropic Berwald curvature. Finally, we find the necessary and sufficient condition under which a general spherically symmetric Finsler metric be Landsbergian.

We improve the existing lower bounds on the order of counterexamples to a conjecture by P. Schmid, determine some properties of the possible counterexamples of minimum order for each prime, and the isomorphism type of the center of the Frattini subgroup for the counterexamples of order 256. We also show that nonabelian metacyclic p-groups, nonabelian groups of maximal nilpotency class and 2-groups of coclass two satisfy the conjecture.

Circles in the Minkowski space 𝕃ⁿ are a natural generalization of circles in Euclidean space. In this paper, d𝕊³ or ℍ³ are supposed to be embedded in 𝕃⁴. A surface M in d𝕊³ or in ℍ³ is said to be circlefoliated if M is the union of a smooth one-parameter family of circles in 𝕃⁴. Similarly, we say that M is ruled if M is the union of a smooth one-parameter family of geodesics. We completely classify minimal surfaces in d𝕊³ or in ℍ³ that are circle-foliated or ruled. We show that a minimal surface in ℍ³ foliated by circles is either ruled, rotationally symmetric or part of a pseudo-great sphere. We also show that a minimal surface in d𝕊³ foliated by circles is either ruled, rotationally symmetric or part of a pseudo-great sphere. We give explicit parametrizations of all ruled minimal surfaces in d𝕊³ or ℍ³. Finally, we show that there is no ruled surface of non-zero constant mean curvature in d𝕊³ or in ℍ³.

In this paper, we introduce and investigate rings in which every S-finite ideal is S-finitely presented, called strongly S-coherent rings. Any strongly S-coherent ring is S-coherent. It is clear that a ring is Noetherian if and only if it is S-Noetherian and strongly S-coherent. We study the transfer of this notion to various contexts of commutative ring extensions, such as direct products, trivial ring extensions, and pullbacks. Our results generate new families of examples of S-coherent rings that are not strongly S-coherent, and we show that the S-Noetherian and strongly S-coherent properties are not comparable.

In this paper, biconservative submanifolds Mⁿ_r with parallel normalized mean curvature vector field (pnmc) in pseudo-Riemannian space forms are studied under the assumption that Mⁿ_r has diagonalizable shape operator. We first study the scalar curvature rigidity of such non-cmc submanifolds due to the seminar papers by S. S. Chern et al. and prove that the scalar curvature is a fractional power polynomial concerning the mean curvature with nonzero constant coefficients. Second, we discuss the nonexistence problem of non-cmc biconservative submanifolds.

In theory of univalent functions, the estimates on higher-order Schwarzian derivatives of univalent analytic functions f in the unit disk 𝗗 have received wide attention and lots of sharp results on these have been given, such as the starlike and convex function classes. In this paper, we investigate the bounds for two kinds of higher-order Schwarzian derivatives of f at zero with f(0) = f'(0) - 1 = 0 when f(z) satisfies ${\Re}({\frac{zf^{\prime}(z)}{f(z)-f(-z)})>0$ or ℜf'(z) > 0, where z ∈ 𝗗.

In this paper, we introduce and study the q-Krull dimension of a commutative ring via its q-operation. A new characterization of τ_q-von Neumann regular rings is obtained, and some properties of rings with q-Krull dimension 0 are studied. Moreover, we characterize the q-Krull dimension of τ_q-Noetherian rings in terms of the Krull dimension of certain total quotient rings. Additionally, we show that every ring with w-Krull dimension 0 is a DW ring.

In this paper, we study the submodularity of the covolume function in global function fields. The submodular property is often needed in the study of homogeneous dynamics, especially to define a Margulis function. We prove that the covolume function is submodular when the class group of the global function field is trivial.

A knot is a closed loop in space without self-intersection. Two knots are equivalent if there is a self homeomorphism of space bringing one onto the other. An arc presentation is an embedding of a knot in the union of finitely many half planes with a common boundary line such that each half plane contains a simple arc of the knot. The minimal number of such half planes among all arc presentations of a given knot is called the arc index of the knot. A knot is usually presented as a planar diagram with finitely many crossings of two strands where one of the strands goes over the other. A grid diagram is a planar diagram which is a non-simple rectilinear polygon such that vertical edges always cross over horizontal edges at all crossings. It is easily seen that an arc presentation gives rise to a grid diagram and vice versa. It is known that the arc index of an alternating knot is two plus its minimal crossing number. There are 4,878 prime alternating knots with minimal crossing number 13. We obtained minimal arc presentations of them in the form of grid diagrams having 15 vertical segments. This is a continuation of the works on prime alternating knots of 11 crossings and 12 crossings [5, 6].

A family of complex-valued close-to-convex harmonic mappings, Ψ_c,𝛽 : 𝔻 → ℂ (c > 0, 𝛽 ≥ 0), defined on the open unit disc 𝔻, is constructed and the convolution Ψ_c,𝛽 * f is studied for a harmonic mapping f = h + ḡ. It is proved that this convolution is locally univalent and sense-preserving whenever h ± 𝜖g are starlike, for any fixed 𝜖 with |𝜖| = 1. Some conditions on univalent harmonic mapping f = h + ḡ are also determined so that the convolution, Ψ_c,𝛽 * f, is close-to-convex or is convex in certain direction for every 𝛽 ≥ 1. Apart from proving an Alexander-type result for the mapping, Ψ_c,𝛽 * f, it is also established that this convolution decomposes into a convex combination of two harmonic mappings. Finally, weak subordination properties of the mapping Ψ_c,𝛽 are examined in the case when 𝛽 is a positive integer.

In this paper, we consider two classes K_S and S^∗_S of convex and starlike functions with respect to symmetric points respectively. We determine bounds on Hankel determinants and Hermitian-Toeplitz determinants involving initial coefficients, initial inverse and initial logarithmic coefficients for such functions with missing second coefficient. Analogous results are also established for two subclasses of S^∗_S which are associated with the lemniscate of Bernoulli and the exponential function, respectively.

Within this article, our focus lies on positive integers represented as pq and 2pq, where p and q are prime numbers adhering to the conditions p ≡ q ≡ 1 (mod 8). Additionally, we will operate under the assumption that q is a quadratic non-residue modulo p. Our aim is to present a condition that ensures that n is a non-congruent number, and we will substantiate the sufficiency of this condition through proof.

Suppose n ≥ 2 and 𝔹_n is the unit ball in 𝔹ⁿ. Let Φ : 𝔹_n → 𝔹_n be a quasiconformal mapping. In this paper, we give a necessary and sufficient condition such that the composition operator C_Φ is bounded between the harmonic Bergman space b^p(𝔹_n) and b^q(𝔹_n) for 1 < p, q < ∞. Our results generalize Fang-Guo-Wang's recent work to harmonic settings.

This paper considers the existence of multiple normalized solutions to the following fractional choquard equation with Hardy-Littlewood-Sobolev upper critical exponent $$\begin{cases}(-{\Delta})^su+V({\varepsilon}x)u={\lambda}u+(I_{\alpha}{\ast}{\mid}u{\mid}^p){\mid}u{\mid}^{p-2}u+{\mu}(I_{\alpha}{\ast}{\mid}u{\mid}^{2^*_{{\alpha},s}}){\mid}u{\mid}^{2^*_{{\alpha},s}-2}u,\;in\;{\mathbb{R}}^N,\\{\int}_{\mathbb{R}^N}\;{\mid}u{\mid}^2dx=a^2,\end{cases}$$ where 0 < s < 1, 0 < α < N, 𝜀, µ, a > 0, N ≥ 2s, λ ∈ ℝ, ${\frac{N+{\alpha}}{N}}\;<\;p\;<\;{\frac{N+2s+{\alpha}}{N}}\;<\;2^*_{{\alpha},s$, $2^*_{{\alpha},s}\;:=\;{\frac{N+{\alpha}}{N-2s}}$ is the upper Hardy-Littlewood-Sobolev critical exponent, I_α is the Riesz potential, V : ℝ^N → [0, ∞) is a continuous function. Since it has local minimizer, we will use the truncated skill to deal with critical term. When 𝜀 is small enough, we prove that the numbers of normalized solutions are related to the topology of the set where the potential V attains its minimum value by applying Lusternik-Schnirelmann category.

We consider a vortex shedding model to describe the unsteady separated flow from a body. Both the separated flow and body are regarded as vortex sheets. The solution and unsteady Kutta condition of the model were presented in several different forms. We clarify ambiguities in the solutions and Kutta conditions of the model. We show that the edge circulations between the different approaches are essentially the same, thereby yielding the same vorticity field. We also identify the unsteady Kutta conditions for the solutions of the model. In the analytic solutions, the Kutta condition is the continuity of the sheet strength at the body edges, whereas in the blob-regularized model, the Kutta condition is set to zero of the sheet strength.

In this paper, we prove that the class of Gorenstein coflat comodules is preserved under Frobenius functors. As a generalization of the Frobenius functors and Frobenius morphism of coalgebras, we obtain that if φ : C → D is a right quasi-Frobenius morphism of coalgebras and _DC is a cogenerator, then X is a Gorenstein injective right C-comodule if and only if X is a Gorenstein injective right D-comodule.