Linear equations over finite abelian groups
One of the oldest problems in algebra is the solution of a system of linear equations over certain domains. The Gaussian elimination algorithm provides an effective solution over fields. The Smith normal form algorithm yields a method over the integers. Here we consider another variation of this theme. Let A be a finite (additive) abelian group, let αi,j ∈ End(A) for 1 ≤ i ≤ n and 1 ≤ j ≤ m and let b1, . . . , bm ∈ A. Then we want to determine the set S of all (x1, . . . , xn) ∈ A n with
Solving linear equations over finitely generated abelian groups
We discuss various methods and their effectiveness for solving linear equations over finitely generated abelian groups. More precisely, if $\varphi\colon G\to H$ is a homomorphism of finitely generated abelian groups and $b\in H$, we discuss various algorithms for checking whether $b\in \im\varphi$ holds and if so, for computing a pre-image of $b$ in $G$ together with the kernel of $\varphi$.
On a New Subclass of p-Valent Close-to-Convex Mappings Defined by Two-Sided Inequality
<jats:p>A new subclass <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math>-valent close-to-convex mappings defined by two-sided inequality is introduced. Some sufficient conditions for functions to be in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> are given.</jats:p>