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Keywords: magnesium alloys, resistance spot welding, welded joints, mechanical properties, microstructure
Published in DiRROS: 24.04.2025; Views: 564; Downloads: 230
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Abstract: Let $T$ be an operator on Banach space $X$ that is similar to $- T$ via an involution $U$. Then $U$ decomposes the Banach space $X$ as $X = X_1 \oplus X_2$ with respect to which decomposition we have $U = \left(\begin{matrix} I_1 & 0 \\ 0 & -I_2 \end{matrix} \right)$, where $I_i$ is the identity operator on the closed subspace $X_i$ ($i=1, 2$). Furthermore, $T$ has necessarily the form $T = \left(\begin{matrix} 0 & * \\ * & 0 \end{matrix} \right)$ with respect to the same decomposition. In this note we consider the question when $T$ is a commutator of the idempotent $P = \left(\begin{matrix} I_1 & 0 \\ 0 & 0 \end{matrix} \right)$ and some idempotent $Q$ on $X$. We also determine which scalar multiples of unilateral shifts on $l^p$ spaces ($1 \le p \le \infty$) are commutators of idempotent operators.
Keywords: Banach spaces, operators, idempotents, commutators
Published in DiRROS: 24.04.2025; Views: 400; Downloads: 224
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Keywords: cold metal transfer welding, 301LN, tensile strength, hardness, contact-tip-to-workpiece distance
Published in DiRROS: 24.04.2025; Views: 475; Downloads: 241
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Keywords: debonding, low-velocity impact, sandwich, polyester pin, MATLAB, digital image analysis
Published in DiRROS: 24.04.2025; Views: 530; Downloads: 231
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Abstract: Let $R$ be a ring and let $n \ge 2$. We discuss the question of whether every element in the matrix ring $M_n(R)$ is a product of (additive) commutators $[x, y] = xy−yx$, for $x,y \in M_n(R)$. An example showing that this does not always hold, even when $R$ is commutative, is provided. If, however, $R$ has Bass stable rank one, then under various additional conditions every element in $M_n(R)$ is a product of three commutators. Further, if $R$ is a division ring with infinite center, then every element in $M_n(R)$ is a product of two commutators. If $R$ is a field and $a \in M_n(R)$, then every element in $M_n(R)$ is a sum of elements of the form $[a, x][a, y]$ with $x, y \in M_n(R)$ if and only if the degree of the minimal polynomial of $a$ is greater than $2$.
Keywords: commutators, matrix ring, division ring, Bass stable rank, L'vov–Kaplansky conjecture
Published in DiRROS: 24.04.2025; Views: 395; Downloads: 232
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