Dimensions of ‘self-affine sponges’ invariant under the action of multiplicative integers

Let $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\ldots ,m_1-1\} \times \{0,\ldots ,m_2-1\})^{\mathbb {N}^*}$ that are invariant under the action of the semigroup of multiplicative integers. These sets are defined following Kenyon, Peres, and Solomyak and using a fixed integer $q \geq 2$ . We compute the Hausdorff and Minkowski dimensions of the projection of these sets onto an affine grid of the unit square. The proof of our Hausdorff dimension formula proceeds via a variational principle over some class of Borel probability measures on the studied sets. This extends well-known results on self-affine Sierpiński carpets. However, the combinatoric arguments we use in our proofs are more elaborate than in the self-similar case and involve a new parameter, namely $j = \lfloor \log _q ( {\log (m_1)}/{\log (m_2)} ) \rfloor $ . We then generalize our results to the same subsets defined in dimension $d \geq 2$ . There, the situation is even more delicate and our formulas involve a collection of $2d-3$ parameters.

Explicit formulas for Euler polynomials and Bernoulli numbers

In this paper, we give several explicit formulas involving the n-th Euler polynomial E_{n}\left(x\right). For any fixed integer m\geq n, the obtained formulas follow by proving that E_{n}\left(x\right) can be written as a linear combination of the polynomials x^{n}, \left(x+r\right)^{n},\ldots, \left(x+rm\right)^{n}, with r\in \left \{1,-1,\frac{1}{2}\right\}. As consequence, some explicit formulas for Bernoulli numbers may be deduced.

Ideal-Based k-Zero-Divisor Hypergraph of Commutative Rings

Let [Formula: see text] be a commutative ring, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] a fixed integer. The ideal-based [Formula: see text]-zero-divisor hypergraph [Formula: see text] of [Formula: see text] has vertex set [Formula: see text], the set of all ideal-based [Formula: see text]-zero-divisors of [Formula: see text], and for distinct elements [Formula: see text] in [Formula: see text], the set [Formula: see text] is an edge in [Formula: see text] if and only if [Formula: see text] and the product of the elements of any [Formula: see text]-subset of [Formula: see text] is not in [Formula: see text]. In this paper, we show that [Formula: see text] is connected with diameter at most 4 provided that [Formula: see text] for all ideal-based 3-zero-divisor hypergraphs. Moreover, we find the chromatic number of [Formula: see text] when [Formula: see text] is a product of finite fields. Finally, we find some necessary conditions for a finite ring [Formula: see text] and a nonzero ideal [Formula: see text] of [Formula: see text] to have [Formula: see text] planar.

Two Results on Ramsey-Turán Theory

Let $f(n)$ be a positive function and $H$ a graph. Denote by $\textbf{RT}(n,H,f(n))$ the maximum number of edges of an $H$-free graph on $n$ vertices with independence number less than $f(n)$. It is shown that $\textbf{RT}(n,K_4+mK_1,o(\sqrt{n\log n}))=o(n^2)$ for any fixed integer $m\geqslant 1$ and $\textbf{RT}(n,C_{2m+1},f(n))=O(f^2(n))$ for any fixed integer $m\geqslant 2$ as $n\to\infty$.

A result related to derivations on unital semiprime rings

The purpose of this paper is to prove the following result. Let n≥3 be some fixed integer and let R be a (n+1)!2n-2-torsion free semiprime unital ring. Suppose there exists an additive mapping D: R→ R satisfying the relation for all x ∈ R. In this case D is a derivation. The history of this result goes back to a classical result of Herstein, which states that any Jordan derivation on a 2-torsion free prime ring is a derivation.

Tight Upper Bound of the 3-Total-Rainbow Index for 2-(Edge-)Connected Graphs

Let [Formula: see text] be an edge-colored graph with order [Formula: see text] and [Formula: see text] be a fixed integer satisfying [Formula: see text]. For a vertex set [Formula: see text] of at least two vertices, a tree containing the vertices of [Formula: see text] in [Formula: see text] is called an [Formula: see text]-tree. The [Formula: see text]-tree [Formula: see text] is a total-rainbow [Formula: see text]-tree if the elements of [Formula: see text], except for the vertex set [Formula: see text], have distinct colors. A total-colored graph [Formula: see text] is said to be total-rainbow [Formula: see text]-tree connected if for every set [Formula: see text] of [Formula: see text] vertices in [Formula: see text], there exists a total-rainbow [Formula: see text]-tree in [Formula: see text], while the total-coloring of [Formula: see text] is called a [Formula: see text]-total-rainbow coloring. The [Formula: see text]-total-rainbow index of a nontrivial connected graph [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed in a [Formula: see text]-total-rainbow coloring of [Formula: see text]. In this paper, we show a sharp upper bound for [Formula: see text], where [Formula: see text] is a 2-connected or 2-edge-connected graph.

Orders of automorphisms of smooth plane curves for the automorphism groups to be cyclic

For a fixed integer $$d\ge 4$$ d ≥ 4 , the list of groups that appear as automorphism groups of smooth plane curves whose degree is d is unknown, except for $$d=4$$ d = 4 or 5. Harui showed a certain characteristic about structures of automorphism groups of smooth plane curves. Badr and Bars began to study for certain orders of automorphisms and try to obtain exact structures of automorphism groups of smooth plane curves. In this paper, based on the result of T. Harui, we extend Badr–Bars study for different and new cases, mainly for the cases of cyclic groups that appear as automorphism groups.

(ϕ, φ)-derivations on semiprime rings and Banach algebras

Let ℛ be a semiprime ring with unity e and ϕ, φ be automorphisms of ℛ. In this paper it is shown that if ℛ satisfies 2 𝒟 ( x n ) = 𝒟 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) 𝒟 ( x ) + 𝒟 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒟 ( x n - 1 ) 2\mathcal{D}\left( {{x^n}} \right) = \mathcal{D}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right) for all x ∈ ℛ and some fixed integer n ≥ 2, then 𝒟 is an (ϕ, φ)-derivation. Moreover, this result makes it possible to prove that if ℛ admits an additive mappings 𝒟, Gscr; : ℛ → ℛ satisfying the relations 2 𝒟 ( x n ) = 𝒟 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) 𝒢 ( x ) + 𝒢 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒢 ( x n - 1 ) , 2\mathcal{D}\left( {{x^n}} \right) = \mathcal{D}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{G}\left( x \right) + \mathcal{G}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{G}\left( {{x^{n - 1}}} \right), 2 𝒢 ( x n ) = 𝒢 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) D ( x ) + 𝒟 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒟 ( x n - 1 ) , 2\mathcal{G}\left( {{x^n}} \right) = \mathcal{G}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right), for all x ∈ ℛ and some fixed integer n ≥ 2, then 𝒟 and 𝒢 are (ϕ, φ)- derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras.