### M₂-rank differences for overpartitions

Jeremy Lovejoy, Robert Osburn (2010)

Acta Arithmetica

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Jeremy Lovejoy, Robert Osburn (2010)

Acta Arithmetica

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Bernard Aupetit, H. Mouton (1996)

Studia Mathematica

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We show that the trace and the determinant on a semisimple Banach algebra can be defined in a purely spectral and analytic way and then we obtain many consequences from these new definitions.

Tian, Y. (2003)

Acta Mathematica Universitatis Comenianae. New Series

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A. Lachlan (1980)

Fundamenta Mathematicae

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A. Lachlan (1974)

Fundamenta Mathematicae

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Yong Ge Tian, George P. H. Styan (2002)

Commentationes Mathematicae Universitatis Carolinae

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It is shown that $$\text{rank}\left({P}^{*}AQ\right)=\text{rank}\left({P}^{*}A\right)+\text{rank}\left(AQ\right)-\text{rank}\left(A\right),$$ where $A$ is idempotent, $[P,Q]$ has full row rank and ${P}^{*}Q=0$. Some applications of the rank formula to generalized inverses of matrices are also presented.

Sebastien Ferenczi, Mariusz Lemańczyk (1991)

Studia Mathematica

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G. S. Rogers (1983)

Applicationes Mathematicae

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Irina Gelbukh (2015)

Czechoslovak Mathematical Journal

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For a finitely generated group, we study the relations between its rank, the maximal rank of its free quotient, called co-rank (inner rank, cut number), and the maximal rank of its free abelian quotient, called the Betti number. We show that any combination of the group's rank, co-rank, and Betti number within obvious constraints is realized for some finitely presented group (for Betti number equal to rank, the group can be chosen torsion-free). In addition, we show that the Betti number...