In this work, we propose a new generalization of the notion of convexity of functions. Namely, for a function g :ℝ→ℝ , we introduce the class of functions f :J →ℝ (J is an interval of ℝ ) satisfying f(x)≤g(u_2-x)/g(u_2-u_1)f(u_1)+g(x-u_1)/g(u_2-u_1)f(u_2), u_1≤ x≤ u_2, for every subinterval [u_1,u_2]⊂ J with u_1
The study of cardinal characteristics 𝔟 and 𝔡 is well known in set-theoretic and general topology. In this paper, we present characterizations of 𝔟 and 𝔡 using selective properties, sub-bounded and sub-dominating families. In addition, we define infinitely long two-player games to present other characterizations of the critical cardinalities 𝔟 and 𝔡 .
Building on the work of Dou, Fan and Qiu, who introduced Bowen entropy for flows via reparametrization ball techniques and established a variational principle for fixed-point free flows, this paper further develops the variational principle of Bowen entropy in such systems. We introduce two new notions of measure-theoretic entropy tailored to fixed-point free flows and establish both a variational principle and an inverse variational principle relating these entropies to Bowen entropy. Finally, we give some calculations about Bowen entropy.
In this paper, we obtain the following improved upper bound on the size of k-wise ℒ -intersecting families and ℒ -differencing Sperner families modulo prime powers by employing linear algebra methods:
Let G be a finite group and H a subgroup of G. We say that H is an ℋ -subgroup of G if N_G(H)∩ H^g≤ H for all g∈ G . The subgroup H is called an ℋN -subgroup of G if there exists a normal subgroup T with (|H|,|G:HT|)=1 such that N_T(H)∩ H^g≤ H for all g∈ G . In this paper, some new criteria for a group G to be p-nilpotent and supersolvable are given when certain subgroups of prime power orders are ℋN -subgroups of G. Our results improve and generalize some recent results in the literature.
Given a unit vector v⃗∈ℝ^3 and λ ,α∈ℝ , a surface Σ⊂ℝ^3 is called λ -singular minimal surface if its mean curvature H satisfies H(p)=α⟨ N(p),v⃗⟩/⟨ p,v⃗⟩+λ for all p∈Σ , where N is the unit normal to Σ . In this paper, we study the axisymmetric solutions of this equation, identifying two types of surfaces depending on whether the rotation axis is parallel or orthogonal to v⃗ . For the first type, we prove the existence of surfaces that orthogonally intersect the rotation axis. For the second type, we provide a complete geometric description.
The theory of uniform Diophantine approximation concerns the study of Dirichlet improvable (or non-improvable) numbers. The metrical aspect of this theory leads to the study of the products of partial quotients in continued fractions. It is known that the dimension of the set of Dirichlet non-improvable numbers depends upon the number of partial quotients in the product string. However, the Hausdorff dimension is the same when partial quotients in the product are separated with a gap of fixed length. In this paper, we provide a detailed Hausdorff dimension analysis of the set of real numbers with a large product of partial quotients with indices in arithmetic progressions. This poses significant challenges as opposed to the considerations of consecutive partial quotients. As an application of our general theorem, in particular, we obtain the Hausdorff dimension of the set E(ψ ):={ x∈ [0, 1): n a_n(x)a_2n(x)⋯ a_n^2(x)≥ψ (n) for infinitely many n∈ℕ} .
In this paper, we consider the logarithmic Sobolev capacity Cap^ℍ^n_log ,γ ,p in the Heisenberg group ℍ^n , a capacity generated by the logarithmic Sobolev space W^log ,γ_p(ℍ^n) , where (γ , p)∈ (0,∞ )× [1,∞ ) . In particular, we investigate several properties of the space W^log ,γ_p(ℍ^n) , including completeness, min-max estimation and density. In addition, we investigate the properties of the logarithmic Sobolev capacity and the logarithmic perimeter in the Heisenberg group. Furthermore, we deal with the relationship between Cap^ℍ^n_log ,γ ,p and the Hausdorff capacity in the Heisenberg group. As an application, we prove the corresponding capacity estimate and the tracing principle.
For any x,y∈ [0,1] and α∈ [0,1) , we consider the set E_α(x,y) of β >1 such that the orbits of x under T_α ,β are bounded away from y, i.e., E_α(x,y)={β>1: inf _i≥ 1|T_α ,β^i(x)-y|>0}, where T_α ,β is the (α ,β ) -transformation defined by T_α ,β(x)=β x+α (mod 1) . We prove that E_α(x,y) is always 1/2-winning in the sense of Schmidt’s game if (x,y) (0,α ) , which extends the result of Langeveld and Samuel [14] to the parameter space and improves upon some known results about E_α(x,y) . As an application, a question mentioned in [20] on the Hausdorff dimension of the set of β for which T_α ,β has the specification property for a given α is solved.
In this note, we prove that there do not exist three consecutive powerful numbers, one of which is of the form x^n± 1 and whose cubic square-free part (i.e., the unique square-free integer b such that the number can be written as a^2b^3 ) is either 1 or the product of distinct primes.
We establish that, for any Tychonoff space X, at least one of the spaces C-p(X) and CpCp(X) has a dense subspace of countable pseudocharacter. Under MA, we give an example of a space X such that C-p(X) does not have a dense subspace of countable functional tightness. We also show that there exists a compact zero-dimensional space K such that C-p(K, {0, 1}) is exponentially separable while K is not a Sokolov space. For compact scattered spaces K of countable dispersion index, we show that C-p(K) has a dense exponentially separable subspace if and only if K is omega-monolithic. Our results provide answers to several published open questions.
In this paper, we investigate the spectrality of a class of Moran measures μ _{R_n},{B_n} generated by a sequence of integers {R_n}_n=1^∞ and a sequence of product-form digit sets {B_n}_n=1^∞ , where R_n=N^q_n , B_n={0,1,… ,N-1}⊕ N^p_n{0,1,… ,N-1} with N≥ 2 and q_n, p_n are positive integers for n≥ 1 . We give some sufficient conditions for μ _{R_n},{B_n} to be a spectral measure, i.e., there exists a countable set Λ such that {e^2π iλ· x:λ∈Λ} is an orthonormal basis in L^2(μ _{R_n},{B_n}) . Our results partially extend the work of Liu et al. [19], where p_n=p and q_n=q for all n≥ 1 . Furthermore, we show that our results can be extended to higher-dimensional settings.
Denote by H a regular hexagon with sides of length 1. Let S be a square with a side parallel to a side of H and let {S_n} be a collection of the homothetic copies of S. In this note a tight lower bound of the sum of the areas of squares from {S_n} that can parallel cover H is determined.
A well-known theorem of Noble states that each Tychonoff space X is homeomorphic to a closed subspace of a pseudocompact k_ℝ -space. We strengthen this result by showing that any Tychonoff space X is homeomorphic to a closed subspace of an abelian pseudocompact k_ℝ -group G such that w(G)≤ℵ _1· w(X) , and if, in addition, X is a precompact group, then X is topologically isomorphic to a closed subgroup of G. It is constructed the first examples of pseudocompact groups G_1 and G_2 (in fact, they are even countably compact and of weight ℵ _2 ) such that G_1 is Ascoli but not a k_ℝ -space, and G_2 is a k_ℝ -space but not a k-space. Under MA+¬ CH , we show that any pseudocompact group of weight ℵ _1 is Ascoli. These results are proved using topological properties of pseudocompact spaces X of weight ℵ _1 and of Σ -products in products of compact spaces. Being motivated by these results and the countably compact part of Noble’s theorem, it is shown by a well-known technique that each countably compact infinite group has a separable countably compact subgroup of cardinality continuum.
The cohomological classification of orbit spaces of free G=ℤ_2 or G=𝕊^1 actions on the mod 2 or rational cohomology product of three spheres 𝕊^n×𝕊^m ×𝕊^l,1≤ n≤ m≤ l , has been discussed in [5, 14]. In this paper, we discuss orbit spaces of free G=𝕊^3 actions on a finitistic space X whose rational cohomology is isomorphic to the product of three spheres. As an application, we determine Borsuk-Ulam type theorems.
This note is intended to strengthen the result in the paper mentioned in the title, which states that for join-compact T_0 quasi-uniform spaces, the quasi-uniform entropy of a uniformly continuous self-map coincides with the quasi-uniform entropy of its extension to the bicompletion.
