The upper bound of minimum moment of inertia of equi-area convex domains

This problem is a generalization of the classical problem of the stability of a spinning rigid body. We obtain the stability chart by using: (i) the computer algebra system MACSYMA in conjunction with a perturbation method, and (ii) numerical integration based on Floquet theory. We show that the form of the stability chart is different for each of the three cases in which the spin axis is the minimum, maximum, or middle principal moment of inertia axis. In particular, a rotation with arbitrarily small angular velocity about the maximum moment of inertia axis can be made unstable by appropriately choosing the model parameters. In contrast, a rotation about the minimum moment of inertia axis is always stable for a sufficiently small angular velocity. The MACSYMA program, which we used to obtain the transition curves, is included in the Appendix.

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The Gravitational Compression of an Elastic Sphere

Australian Journal of Physics ◽

10.1071/ph550167 ◽

1955 ◽

Vol 8 (1) ◽

pp. 167

Author(s):

A Keane

Keyword(s):

Lower Bound ◽

Elastic Material ◽

Upper Bound ◽

Moment Of Inertia ◽

Elastic Sphere ◽

The Moment ◽

Gravitational Compression ◽

Gravitational Equilibrium

On considering a sphere in hydrostatic gravitational equilibrium, composed of a homogeneous elastic material for which the variation of incompressibility x with pressure p is given by dx/dp=n, a constant, we find that there is an upper bound to the radius R of the sphere provided n=2, and that for all values of n there is a lower bound to the value of I/MR2, where I is the moment of inertia about a diameter and M is the mass of the sphere.

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Diabetes is Associated with a Lower Minimum Moment of Inertia Among Older Women: An Analysis of 3D Reconstructions of Clinical CT Scans

Journal of Biomechanics ◽

10.1016/j.jbiomech.2021.110707 ◽

2021 ◽

pp. 110707

Author(s):

Lauren N. Heckelman ◽

Benjamin R. Wesorick ◽

Louis E. DeFrate ◽

Richard H. Lee

Keyword(s):

Older Women ◽

Moment Of Inertia ◽

Ct Scans ◽

3D Reconstructions ◽

Minimum Moment

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A sharp upper bound for the first Dirichlet eigenvalue and the growth of the isoperimetric constant of convex domains

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-08-09399-4 ◽

2008 ◽

Vol 136 (08) ◽

pp. 2997-3006 ◽

Cited By ~ 18

Author(s):

Pedro Freitas ◽

David Krejčiř{í}k

Keyword(s):

Upper Bound ◽

Convex Domains ◽

Dirichlet Eigenvalue ◽

First Dirichlet Eigenvalue ◽

Isoperimetric Constant

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An Elastorelaxometer for Large Reversible Deformation of Elastomeric Systems and Polymer Solutions

Rubber Chemistry and Technology ◽

10.5254/1.3540186 ◽

1961 ◽

Vol 34 (1) ◽

pp. 165-175 ◽

Cited By ~ 1

Author(s):

A. A. Trapeznikov

Keyword(s):

Polymer Solutions ◽

Coaxial Cylinder ◽

Moment Of Inertia ◽

Colloidal Systems ◽

Elastic Recoil ◽

Elastic Deformations ◽

Yield Value ◽

New Instrument ◽

Colloidal Gel ◽

Minimum Moment

Abstract 1. A new instrument, the elastorelaxometer (based on the coaxial-cylinder principle) has been developed, for studies of large high-elastic deformations in relaxing colloidal gel systems and polymer solutions. 2. The effects of the following were investigated : a) width of the gap between the cylinders ; b) moment of inertia of the cylinder (with rapidly relaxing colloidal systems, cylinders of the minimum moment of inertia must be used) ; c) nature of the liquid in the bottom of the cylinder ; d) nature of the motion of the inner cylinder at different ultimate deformations. 3. Values of elastic recoil εc for different predetermined deformations ε have been determined in dilute aluminum naphthenate gels in decalin. It is shown that εc passes through a maximum, associated with transition beyond the yield value of the structure, with increase of ε. It is shown that εc can reeah 6000% in 2% gels.

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An upper bound for nonlinear eigenvalues on convex domains by means of the isoperimetric deficit

Archiv der Mathematik ◽

10.1007/s00013-010-0102-8 ◽

2010 ◽

Vol 94 (4) ◽

pp. 391-400 ◽

Cited By ~ 8

Author(s):

Barbara Brandolini ◽

Carlo Nitsch ◽

Cristina Trombetti

Keyword(s):

Upper Bound ◽

Convex Domains ◽

Isoperimetric Deficit ◽

Nonlinear Eigenvalues

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Curves of Minimum Moment of Inertia with Respect to a Point

Annals of Mathematics ◽

10.2307/1967249 ◽

1906 ◽

Vol 7 (4) ◽

pp. 165 ◽

Cited By ~ 1

Author(s):

Max Mason

Keyword(s):

Moment Of Inertia ◽

Minimum Moment

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A short note on Helmholtz decompositions for bounded domains in $\mathbb{R}^3$

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-114908 ◽

2019 ◽

Vol 125 (2) ◽

pp. 227-238

Author(s):

Immanuel Anjam

Keyword(s):

Upper Bound ◽

Short Note ◽

Type Inequality ◽

Upper Bounds ◽

Bounded Domains ◽

Two Dimensional ◽

Convex Domains ◽

Unknown Constant ◽

Local Equivalent ◽

Poincaré Constant

In this short note we consider several widely used $\mathsf {L}^{2}$-orthogonal Helmholtz decompositions for bounded domains in $\mathbb {R}^3$. It is well known that one part of the decompositions is a subspace of the space of functions with zero mean. We refine this global property into a local equivalent: we show that functions from these spaces have zero mean in every subdomain of specific decompositions of the domain. An application of the zero mean properties is presented for convex domains. We introduce a specialized Poincaré-type inequality, and estimate the related unknown constant from above. The upper bound is derived using the upper bound for the Poincaré constant proven by Payne and Weinberger. This is then used to obtain a small improvement of upper bounds of two Maxwell-type constants originally proven by Pauly. Although the two dimensional case is not considered, all derived results can be repeated in $\mathbb {R}^2$ by similar calculations.

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