Row reduced matrices and annihilator semigroups1

The set of row reduced matrices (and of echelon form matrices) is closed under multiplication. We show that any system of representatives for the $\text{Gl}_{n}(\mathbb{K})$ action on the $n\times n$ matrices, which is closed under multiplication, is necessarily conjugate to one that is in simultaneous echelon form. We call such closed representative systems Grassmannian semigroups. We study internal properties of such Grassmannian semigroups and show that they are algebraic semigroups and admit gradings by the finite semigroup of partial order preserving permutations, with components that are naturally in one–one correspondence with the Schubert cells of the total Grassmannian. We show that there are infinitely many isomorphism types of such semigroups in general, and two such semigroups are isomorphic exactly when they are semiconjugate in $M_{n}(\mathbb{K})$. We also investigate their representation theory over an arbitrary field, and other connections with multiplicative structures on Grassmannians and Young diagrams.

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Modeling Support Effects: Finite Element and Experimental Modal Methods

Volume 5: Manufacturing Materials and Metallurgy; Ceramics; Structures and Dynamics; Controls, Diagnostics and Instrumentation; Education; General ◽

10.1115/96-gt-403 ◽

1996 ◽

Cited By ~ 2

Author(s):

Jonathan R. Buckles ◽

Keith E. Rouch ◽

John R. Baker

Keyword(s):

Finite Element ◽

High Speed ◽

Transfer Functions ◽

Coupling Effects ◽

Support Effects ◽

Critical Speeds ◽

Modal Data ◽

The Stability ◽

Direct Use ◽

Reduced Matrices

The effects of support/foundation dynamics are often significant in high speed turbomachinery, and can affect the stability and response to unbalance. In some cases additional critical speeds are introduced, related to resonances in the foundation or interaction with rotor resonances of foundation resonances. This paper reviews several methods for representing these effects, including (1) reduced matrices from finite element substructures (ANSYS, for example), (2) matrices generated from modal data, and (3) direct use of experimental transfer functions. These methods are implemented in a finite element rotor program in a PC-DOS environment. The application of the methods to two laboratory rotor configurations described and results presented. Situations with a foundation resonance above and near the rotor critical are included. The importance of including coupling effects between supports is shown.

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