In this paper we introduce and study a certain intricate Cantor-like set $\mathscr{C}$ contained in unit interval. Our main result is to show that the set $\mathscr{C}$ itself, as well as the set of dissipative points within $\mathscr{C}$, both have Hausdorff dimension equal to 1. The proof uses the transience of a certain non-symmetric Cauchy-type random walk.

A fake weighted projective space X is a Q-factorial toric variety with Picard number one. As with weighted projective space, X comes equipped with a set of weights (λ₀, ..., λ_n). We see how the singularities of P (λ₀, ..., λ_n) influence the singularities of X, and how the weights bound the number of possible fake weighted projective spaces for a fixed dimension. Finally, we present an upper bound on the ratios λ_j/Σλ_i if we wish X to have only terminal (or canonical) singularities.

In this paper we study n-dimensional complete spacelike submanifolds with constant normalized scalar curvature immersed in semi-Riemannian space forms. By extending Cheng-Yau's technique to these ambients, we obtain results to such submanifolds satisfying certain conditions on both the squared norm of the second fundamental form and the mean curvature. We also characterize compact non-negatively curved submanifolds in De Sitter space of index p.

We show that the random Fourier-Stieltjes (RFS) series associated with a stochastic process of independent and symmetric increments whose laws belong to the domain of symmetric stable distribution converges in the mean to a stochastic integral. We also show that the conjugate RFS series converges in the mean to a stochastic integral. Both the series are also shown to be Abel summable.

Mustaţă, Takagi and Watanabe define F-thresholds, which are invariants of a pair of ideals in a ring of characteristic p > 0. In their paper, it is proved that F-thresholds are equal to jumping numbers of test ideals on regular local rings. In this note, we give formulas of F-thresholds and F-jumping coefficients on toric rings. By these formulas, we prove that there exists an inequality between F-jumping coefficients and F-thresholds. In particular, we observe a difference between F-pure thresholds and F-thresholds on certain rings. As applications, we give a characterization of regularity for toric rings defined by simplicial cones, and we prove the rationality of F-thresholds on certain rings.

In this paper, the authors prove that the commutator [b,L] is a compact operator in L^p(Rⁿ) if and only if b $\in$ VMO(Rⁿ), where L denotes the Littlewood-Paley operators, such as the Littlewood-Paley g-function, Lusin area integral and Littlewood-Paley g_λ^* function.

We consider a real analytic map-germ (f, g):(Rⁿ,0) → (R²,0) such that the fibres of f are simultaneously parallelizable. We call such a map a partially parallelizable map. We establish degree formulas for the following quantities: $$χ (\{f = α\} \cap \{g = δ\} \cap B_ε^n),\\ χ (\{f = α\} \cap \{g ≥ δ\} \cap B_ε^n) − χ (\{f = α\} \cap \{g ≤ δ\} \cap B_ε^n),$$ where (α, δ) is a regular value of (f, g) and 0 < |(α, δ)| « ε « 1.

For a transcendental holomorphic curve and a subset of C^{n + 1} − {0} in subgeneral position, we consider the truncated defect relation by using a generalization of Nochka weight function introduced in [12] and its supplement in Section 3. When it is not extremal, we estimate the sum of defects and when it is extremal, we investigate the number of vectors each defect of which is equal to 1 or the structure of vectors each defect of which is positive.