Embedding properties of network realizations of dissipative reduced order models Jörn Zimmerling, Mikhail Zaslavsky,Rob Remis, Shasri Moskow, Alexander Mamonov, Murthy Guddati, Vladimir Druskin, and Liliana Borcea Mathematical Sciences Department, Worcester Polytechnic Institute https://www.wpi.edu/people/vdruskin Abstract Realizations of reduced order models of passive SISO or MIMO LTI problems can be transformed to tridiagonal and blocktridiagonal forms, respectively, via dierent modications of the Lanczos algorithm. Generally, such realizations can be interpreted as ladder resistorcapacitorinductor (RCL) networks. They gave rise to network syntheses in the rst half of the 20th century that was at the base of modern electronics design and consecutively to MOR that tremendously impacted many areas of engineering (electrical, mechanical, aerospace, etc.) by enabling ecient compression of the underlining dynamical systems. In his seminal 1950s works Krein realized that in addition to their compressing properties, network realizations can be used to embed the data back into the state space of the underlying continuum problems. In more recent works of the authors Krein's ideas gave rise to socalled nitedierence Gaussian quadrature rules (FDGQR), allowing to approximately map the ROM statespace representation to its full order continuum counterpart on a judicially chosen grid. Thus, the state variables can be accessed directly from themore »
Stabilization of the cohomology of thickenings
For a local complete intersection subvariety $X = V (I)$ in $P^n$ over a field of characteristic zero, we show that, in cohomological degrees smaller than the codimension of the singular locus of $X$, the cohomology of vector bundles on the formal completion of $P^n$ along $X$ can be effectively computed as the cohomology on any sufficiently high thickening $X_t = V (I^t)$; the main ingredient here is a positivity result for the normal bundle of $X$. Furthermore, we show that the Kodaira vanishing theorem holds for all thickenings $X_t$ in the same range of cohomological degrees; this extends the known version of Kodaira vanishing on $X$, and the main new ingredient is a version of the Kodaira AkizukiNakano vanishing theorem for $X$, formulated in terms of the cotangent complex.
 Award ID(s):
 1800355
 Publication Date:
 NSFPAR ID:
 10159157
 Journal Name:
 American journal of mathematics
 ISSN:
 10806377
 Sponsoring Org:
 National Science Foundation
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