We assume the generalized Riemann hypothesis and prove an asymptotic formula for the number of primes for which Fp can be generated by r given multiplicatively independent numbers. In the case when the r given numbers are primes, we express the density as an Euler product and apply this to a conjecture of Brown–Zassenhaus (J. Number Theory 3 (1971), 306–309). Finally, in some examples, we compa...

 The category of the title is called $mathcal{W}$. This has all free objects $F(I)$ ($I$ a set). For an object class $mathcal{A}$, $Hmathcal{A}$ consists of all homomorphic images of $mathcal{A}$-objects. This note continues the study of the $H$-closed monoreflections $(mathcal{R}, r)$ (meaning $Hmathcal{R} = mathcal{R}$), about which we show ({em inter alia}): $A in mathcal{A}$ if and  only if...

The aim of this paper is to present a proof of the hyperbolicity of the family $f_c(x)=c(x-frac{x^3}{3}), |c|>3$, on an its invariant subset of $mathbb{R}$.

It is then easily verified that RG satisfies the ring axioms; in fact, RG is a linear algebra over R. (We write all groups multiplicatively, and denote group identities by 1; we also use 1 for the unit element of R if there is one.) If R, in addition to being a ring, is a Banach algebra (i.e., an algebra over the complex field K, with a submultiplicative norm which makes R a Banach space), then...

Abstract We show that if a closed Lipschitz surface in $${\mathbb {R}}^n$$ R n has bounded Kolasinski–Menger energy, then it can be triangulated with triangles whose number is by the energy and area. Each of an image subset plane under diffeomorph...

In this paper, we consider a generalized Boussinesq-type equation posed in $ (0,\infty)\times \Omega $, where \Omega\subset \mathbb{R}^N $. The considered arises many physical models including the description of nonstationary processes crystalline semiconductors. We will handle two cases: = \overline{\mathbb{R}^N\backslash{B_1}} and B_1\backslash\{0\} B_1 is closed unit ball Using unified appro...

in this paper, we introduce and study the concept of $r$-fuzzy regular semi open (closed) sets in smooth topological spaces. by using $r$-fuzzy regular semi open (closed) sets, we define a new fuzzy closure operator namely $r$-fuzzy regular semi interior (closure) operator. also, we introduce fuzzy regular semi continuous and fuzzy regular semi irresolute mappings. moreover, we investigate the ...

Definition 2.1. Let R be a ring. A left R-module is an abelian group (M,+) together with an R-action · : R ×M → M which satisfies M1 1 ·m = m for all m ∈ M; M2 r(s ·m) = (r s) ·m for all r, s ∈ R and m ∈ M; M3 r · (m + n) = (r ·m) + (s · n) A submodule N of M is a subset which is closed under + and satisfies r · n ∈ N for all r ∈ R and n ∈ N. A right R-module is defined similarly, but modules w...

Let A be a closed-interval subset of R, let f be a function from A into R, let S be a non empty Division of A, and let D be an element of S. A finite sequence of elements of R is said to be a middle volume of f and D if it satisfies the conditions (Def. 1). (Def. 1)(i) len it = lenD, and (ii) for every natural number i such that i ∈ domD there exists an element r of R such that r ∈ rng(f divset...

Abstract Let $X \subset \mathbb{C}^{n}$ be an algebraic variety, and let $\Lambda a discrete subgroup whose real complex spans agree. We describe the topological closure of image $X$ in $\mathbb{C}^{n} / \Lambda $, thereby extending result Peterzil–Starchenko case when $ is cocompact. also obtain similar extension $X\subset \mathbb{R}^{n}$ definable o-minimal structure with no restrictions on a...