In this note, we present several generalized Poincaré inequalities on Finsler metric measure manifolds. As applications, we prove the general Sobolev inequality and the general Nash inequality on Finsler manifolds with certain conditions.

Complex manifolds include different subclasses, and one of the most significant ones is the class of Kähler manifolds. These types of manifolds have valuable curvature properties that are related to almost complex structures. For this study, we focused on Kähler manifolds having semi-symmetry conditions associated with H-curvature tensors. Another important subclass is of IK-normal complex contact metric manifolds, which are complex manifolds with a contact structure. Since the complex structure of this type of manifold is Kähler, it has important applications. In the last section, we used the results obtained for a Kähler manifold to an IK-normal complex contact metric manifold.

Let (ℤ, T_k) be a topological space on the set of integers, where the topology T_k is generated by the set S_k as a subbase, where k ∈ ℤ and S_k := {S_k,t | S_k,t := {2t, 2t + 1, 2t + 2k + 1}, t ∈ ℤ}. Then we call (ℤ, T_k) a T_k-topological space. For the set of positive integers k₁, k₂ ∈ ℕ, the paper initially proves that (ℤ, T_k1) is topologically embedded into (ℤ, T_k2) if and only if k₁ ≤ k₂, i.e., ∃ an embedding h_(k1,k2) : (ℤ, T_k1) → (ℤ, T_k2) ⇔ k₁ ≤ k₂. Furthermore, in the case of k₁ ≤ k₂, the subspace (ℤ \ Im(h_(k1,k2)), (T_k2)_{ℤ\Im(h(k1,k2))}) induced from (ℤ, T_k2) has (k₂ - k₁) components. Besides, each of the components is homeomorphic with the Khalimsky (K-, for brevity) topological line. Finally, we obtain some embeddings of (ℤⁿ, κⁿ) into (ℤⁿ, (T_k)ⁿ), the n-fold product topological space of a T_k-topological space, k ∈ ℕ, where (ℤⁿ, κⁿ) is the n-dimensional K-topological space. Since a T_k-topological space is an Alexandroff space, an embedding problem of a T_k-topological space plays an important role in pure and applied topology.

In this paper, we define an extended arctangent function (called B-arctangent function) by iterated integrals. Then applying the B-arctangent function, we define and study an alternating variant of Kaneko-Tsumura 𝜓-values, called alternating Kaneko-Tsumura 𝜓-values. We evaluate a few families of integrals involving the B-arctangent function in terms of alternating multiple mixed values (AMMVs), alternating multiple T-harmonic sums (AMTHSs) and alternating multiple S-harmonic sums (AMSHSs). Further, we investigate some special cases of alternating Kaneko-Tsumura 𝜓-values via alternating multiple mixed values, alternating multiple T-values (AMTVs) and alternating multiple S-values (AMSVs). Finally, applying Au's Mathematica package, we can find many explicit evaluations of alternating Kaneko-Tsumura 𝜓-values.

Let x₁, …, x_s be a filter regular sequence in a local ring (R, 𝖒). Denote by R_x1,…,xs the Koszul complex of x₁, …, x_s over R. In this paper, we give an explicit number N such that the sum of lengths $\sum{_{i=1}^{s}}\,(-1)^i{\ell}(H_i(R_{x_1,{\ldots},x_s}))$ is preserved when we perturb the sequence x₁, …, x_s by 𝜀₁, …, 𝜀_s ∈ 𝖒^N. Applying this result and the main theorem of Eisenbud [4], we show that there exists N > 0 such that for all i ≥ 1 the length of H_i(R_x1,…,xs) is preserved under small perturbation.

Given a smooth compact strictly convex initial hypersurface, we consider smooth compact convex hypersurfaces expanding by the curvature flow with its speed equal to a linear combination of mean curvature and inverse mean curvature in Euclidean space. Then we show that there exists a unique solution to the flow and that the hypersurfaces expand to infinity using the curvature bounds with decay estimates.

In this paper, we investigate some chaotic dynamics of a dynamical system that possesses an irregular recurrence point, i.e., a point that is Banach recurrent but not almost-periodic. We establish that a dynamical system with an irregular recurrence point is Li-Yorke chaotic for any two positive integers r, s, where r < s. Moreover, we demonstrate that a transitive system is thickly sensitive, and is also multi-sensitive if it admits a transitive irregular recurrence point and has at least two distinct minimal sets. Additionally, we point out that the existence of at least two distinct minimal sets is necessary.

The main object in this paper is to deliberate the characterizations of an almost *-Schouten soliton and gradient almost *-Schouten soliton on Kenmotsu manifold. In particular, we have proved that if a Kenmotsu manifold admits an almost *-Schouten soliton with non-zero potential vector field V collinear to the Reeb vector field ζ, then the manifold becomes an η-Einstein manifold. Furthermore, it is shown that if a (κ, µ)' -almost Kenmotsu manifold admitting an almost *-Schouten soliton with κ < 1, then it is locally isometric to ℍⁿ⁺¹(-4) × ℝⁿ. Lastly, we show that if a metric g of Kenmotsu manifold endows with a gradient almost *-Schouten soliton and ζ leaves the scalar curvature r invariant, then the manifold is an Einstein manifold with constant scalar curvature r = -2n(2n + 1). Lastly, we provide an example of 5-dimensional Kenmotsu manifolds admitting an almost *-Schouten soliton.

Inspired by the foundational work of Shao, Tian, and Zhao [19] about hypergraphs, this paper aims to explore the biharmonic equation and expand application range of biharmonic equation. We focus on a connected finite hypergraph H = (V, E), which generalizes traditional graphs by allowing edges to connect more than two vertices. Our investigation centers on the biharmonic equation $${\Delta}^2{\phi}-div(x(v){\nabla}{\phi})+y(v){\phi}=\frac{f(v,{\phi})}{{\delta}_{e}^{\alpha}}+{\beta}g(v)$$, where x(v) and y(v) are two functions defined on V , α ≥ 0, β > 0. Here, δ_e denotes the degree of hyperedge e ∈ E, defined as δ_e = |e|, which represents the cardinality of the vertices contained in e. Under certain conditions on the functions f and g, we establish the existence of two distinct positive solutions to the biharmonic equation.

In this paper, we study almost Ricci-Yamabe soliton of (α, β)-type which admitting concurrent-recurrent vector field. We show that if the potential vector field is (i) conformal vector field, or (ii) pointwise collinear to the concurrent-recurrent vector field, or (iii) gradient of a smooth function, then the soliton is Einstein or almost β-Yamabe soliton. At the end of this paper, we give explicitly some manifolds which admit (or do not admit) concurrent-recurrent vector field.

Let R be a commutative ring with identity. We call a proper ideal I of R integral primary if whenever a, b ∈ R such that ab ∈ I, then either a ∈ I or b ∈ Ī where Ī denotes the integral closure of I. The main purpose of this study is to give new characterizations for some special rings such as présimplifiable rings, UN-rings, von Neumann regular rings and fields by using the concept of integral primary ideals. Furthermore, we apply some properties of integral primary ideals to conclude the integral primary avoidance theorem.

It is proved that vertical graphs and radial graphs are strongly stable for a certain type of densities in Euclidean space ℝⁿ⁺¹. Particular cases of these densities include translators, expanders and singular minimal hypersurfaces. Using techniques of calibrations, it is also proved that for densities depending on a spatial coordinate, stationary vertical graphs are weighted minimizers in a certain class of hypersurfaces.

This paper focuses on different classes of sets in the dual of a Banach lattice such as V-sets, L-sets, Right sets, L-limited sets and almost L-sets and then, based on the connections between them, some operator characterizations are obtained. As a result, we show that each almost L-set and Right set is a V-set and a V-set is itself an L-limited set. In particular, it is proved that almost L-sets are relatively weakly compact if and only if E^* has the order continuous norm and also they are relatively compact if and only if E^* is discrete with the order continuous norm.

We consider a class of algebraic interpolation problems involving infinite sequences of points in space. In each instance, we prove the existence of a universal function such that a sequence of points is interpolable if and only if the interpolation degree growth of its initial segments is eventually bounded by the function.

In this paper, we classify the singular fibers F according to their topological index Sign(F) . As an application, we provide a sharp upper bound on the topological signature Sign(S) of a complex algebraic surface S with a genus g ≥ 2 fibration f : S → ℙ¹, where f has two singular fibers. We show that if Sign(S) ≠ 0, then Sign(S) ≤ -4.

We introduce a combinatorial method for computing the spectrum of the singularity at the origin of a homogeneous polynomial defining a curve arrangement in the projective plane. For arrangements with ordinary singularities, our approach counts lattice points in specific regions of ℤ³, naturally separating global contributions determined by the total degree from local contributions at singular points. This geometric interpretation reveals that the spectrum depends only on the total degree and multiplicities at singular points, leading to efficient computations of the Milnor fiber's Euler characteristic and monodromy eigenspace dimensions. Our method suggests potential generalizations to more complex singularities through appropriate modifications of the local regions.

In this paper, we deal with the following fractional Kirchhoff type equation $${\mathcal{M}}\({\int__{\mathbb{R}^N}}\,{\mid}(-{\Delta})^{\frac{s}{2}}u{\mid}^2dx\)(-{\Delta})^su+u\\=g(x){\mid}u{\mid}^{2^*_s-2_u}+{\lambda}h(x){\mid}u{\mid}^{q-2}u,\;in\;{\mathbb{R}}^N,$$, in ℝ^N where N > 2s, 𝒨(t) = a + bt with a ≥ 0 and b > 0, $s{\in}({\frac{1}{2}},\,1)$, $h(x)={\frac{{\mid}x{\mid}^{\beta}}{1+{\mid}x{\mid}^{\beta}}}$, 0 < β < N and $2^*_s\,<\,q\,<\,{\frac{2(N+{\beta})}{N-2s}}$, $2^*_s\,:=\,{\frac{2N}{N-2s}}$ is the fractional critical Sobolev exponent, λ > 0 is a real parameter. By using the Nehari manifold and variational methods, we prove that for λ small enough, the number of nontrivial solutions depends on the profile of g(x) for the above equation. Also, for λ sufficiently large, we establish the existence of least energy sign-changing solutions. Some novel results improve and generalize the existing ones in the literature.

In this paper, we establish a curvature estimate of convex solutions for k-th shifted mean curvature equation in hyperbolic space. Additionally, we provide a new proof of Tu's work [22] on global curvature estimates for shifted Gauss curvature equations.

In this paper, we study the Cauchy problem for a chemotaxis-Euler equations modeling coral fertilization in ℝ². We first establish the local well-posedness of this model in Sobolev space. In the latter, motivated by the paper [4, 20], by using the 𝑊^1,p-energy estimate (2 < p < ∞) and the Brézis-Wainger inequality, we can extend these local solutions to global ones under the small initial data assumption.