Search results for "Quadratic form"
showing 8 items of 18 documents
Lower bounds for eigenvalues of a quadratic form relative to a positive quadratic form
1968
Abstract : A method is presented for the calculation of lower bounds to eigenvalues of operators that arise from variational problems for one quadratic form relative to a positive definite quadratic form. Eigenvalue problems of this kind occur, for example, in the theory of buckling of continuous linear elastic systems. The technique used is a modification of one introduced earlier, (1) sections II and IVB, for the determination of lower bounds to eigenvalues of semi-bounded self-adjoint operators. Other methods for the latter problem can be carried over without essential changes. The particular difficulty in the case we consider is that some operators which enter the calculation for the lo…
Mass of astrophysically relevantCl31and the breakdown of the isobaric multiplet mass equation
2016
The mass of $^{31}\mathrm{Cl}$ has been measured with the JYFLTRAP double-Penning-trap mass spectrometer at the Ion Guide Isotope Separator On-Line (IGISOL) facility. The determined mass-excess value, $\ensuremath{-}7034.7(34)$ keV, is 15 times more precise than in the Atomic Mass Evaluation 2012. The quadratic form of the isobaric multiplet mass equation for the $T=3/2$ quartet at $A=31$ fails $({\ensuremath{\chi}}_{n}^{2}=11.6)$ and a nonzero cubic term, $d=\ensuremath{-}3.5(11)$ keV, is obtained when the new mass value is adopted. $^{31}\mathrm{Cl}$ has been found to be less proton-bound, with a proton separation energy of ${S}_{p}=264.6(34)$ keV. Energies for the excited states in $^{31…
Representation Theorems for Indefinite Quadratic Forms Revisited
2010
The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensuring the second representation theorem to hold is proved. A new simple and explicit example of a self-adjoint operator for which the second representation theorem does not hold is also provided.
Are compliance constants ill-defined descriptors for weak interactions?
2013
Just as the potential energy can be written as a quadratic form in internal coordinates, so it can also be expanded in terms of generalized forces. The resulting coefficients are termed compliance constants. In this article, the suitability of compliance constants as non-covalent bond strength descriptors is studied (a) for a series of weakly bound hydrogen halide–rare gas complexes applying a configuration interaction theory, (b) for a double stranded DNA 4-mer using approximate density functional methods and finally (c) for a double stranded DNA 20-mer using empirical force fields. Our results challenge earlier studies, which concluded the inappropriateness of compliance constants as soft…
The Tan 2Θ Theorem in fluid dynamics
2017
We show that the generalized Reynolds number (in fluid dynamics) introduced by Ladyzhenskaya is closely related to the rotation of the positive spectral subspace of the Stokes block-operator in the underlying Hilbert space. We also explicitly evaluate the bottom of the negative spectrum of the Stokes operator and prove a sharp inequality relating the distance from the bottom of its spectrum to the origin and the length of the first positive gap.
Subdivisions of Ring Dupin Cyclides Using Bézier Curves with Mass Points
2021
Dupin cyclides are algebraic surfaces introduced for the first time in 1822 by the French mathematician Pierre-Charles Dupin. A Dupin cyclide can be defined as the envelope of a one-parameter family of oriented spheres, in two different ways. R. Martin is the first author who thought to use these surfaces in CAD/CAM and geometric modeling. The Minkowski-Lorentz space is a generalization of the space-time used in Einstein’s theory, equipped of the non-degenerate indefinite quadratic form $$Q_{M} ( \vec{u} ) = x^{2} + y^{2} + z^{2} - c^{2} t^{2}$$ where (x, y, z) are the spacial components of the vector $$ \vec{u}$$ and t is the time component of $$ \vec{u}$$ and c is the constant of the spee…
A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature
2017
This paper is a survey of some arithmetic applications of techniques in the geometry and ergodic theory of negatively curved Riemannian manifolds, focusing on the joint works of the authors. We describe Diophantine approximation results of real numbers by quadratic irrational ones, and we discuss various results on the equidistribution in \(\mathbb{R}\), \(\mathbb{C}\) and in the Heisenberg groups of arithmetically defined points. We explain how these results are consequences of equidistribution and counting properties of common perpendiculars between locally convex subsets in negatively curved orbifolds, proven using dynamical and ergodic properties of their geodesic flows. This exposition…
High-precision mass measurements for the isobaric multiplet mass equation atA= 52
2017
Masses of $^{52}$Co, $^{52}$Co$^m$, $^{52}$Fe, $^{52}$Fe$^m$, and $^{52}$Mn have been measured with the JYFLTRAP double Penning trap mass spectrometer. Of these, $^{52}$Co and $^{52}$Co$^m$ have been experimentally determined for the first time and found to be more bound than predicted by extrapolations. The isobaric multiplet mass equation for the $T=2$ quintet at $A=52$ has been studied employing the new mass values. No significant breakdown (beyond the $3\sigma$ level) of the quadratic form of the IMME was observed ($\chi^2/n=2.4$). The cubic coefficient was 6.0(32) keV ($\chi^2/n=1.1$). The excitation energies for the isomer and the $T=2$ isobaric analogue state in $^{52}$Co have been d…