scholarly journals Optimizing quadratic forms of adjacency matrices of trees and related eigenvalue problems

2001 ◽  
Vol 325 (1-3) ◽  
pp. 191-207
Author(s):  
Wai-Shun Cheung ◽  
Chi-Kwong Li ◽  
D.D. Olesky ◽  
P. van den Driessche
Keyword(s):  
Quadratic Forms ◽  
Eigenvalue Problems ◽  
Adjacency Matrices

scholarly journals PONTRJAGIN FORMS AND INVARIANT OBJECTS RELATED TO THE Q-CURVATURE

2007 ◽  
Vol 09 (03) ◽  
pp. 335-358 ◽  
Author(s):  
THOMAS BRANSON ◽  
A. ROD GOVER
Keyword(s):  
Quadratic Forms ◽  
Eigenvalue Problems ◽  
Conformal Invariants ◽  
Action Functional ◽  
Conformally Invariant ◽  
Direct Sum ◽  
De Rham ◽  
Lagrangian Subspaces ◽  
Set Up ◽  
Natural Operators

It was shown by Chern and Simons that the Pontrjagin forms are conformally invariant. We show them to be the Pontrjagin forms of the conformally invariant tractor connection. The Q-curvature is intimately related to the Pfaffian. Working on even-dimensional manifolds, we show how the k-form operators Qk of [12], which generalize the Q-curvature, retain a key aspect of the Q-curvature's relation to the Pfaffian, by obstructing certain representations of natural operators on closed forms. In a closely related direction, we show that the Qk give rise to conformally invariant quadratic forms Θk on cohomology that interpolate, in a suitable sense, between the integrated metric pairing (at k = n/2) and the Pfaffian (at k = 0). Using a different construction, we show that the Qk operators yield a map from conformal structures to Lagrangian subspaces of the direct sum Hk ⊕ Hk (where Hk is the dual of the de Rham cohomology space Hk); in an appropriate sense this generalizes the period map. We couple the Qk operators with the Pontrjagin forms to construct new natural densities that have many properties in common with the original Q-curvature; in particular these integrate to global conformal invariants. We also work out a relevant example, and show that the proof of the invariance of the (nonlinear) action functional whose critical metrics have constant Q-curvature extends to the action functionals for these new Q-like objects. Finally we set up eigenvalue problems that generalize to Qk-operators the Q-curvature prescription problem.

Quadratic forms, weighted eigenfunctions and boundary value problems for non-linear second order ordinary differential equations

1989 ◽  
Vol 112 (1-2) ◽  
pp. 145-153 ◽  
Author(s):  
Alessandro Fonda ◽  
Jean Mawhin
Keyword(s):  
Differential Equations ◽  
Variational Methods ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Quadratic Forms ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
Second Order ◽  
Non Linear

SynopsisSome known results for different kinds of boundary value problems for second order ordinary differential equations are generalised. Different approaches are compared with one another, using topological and variational methods and the theory of weighted eigenvalue problems.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

Totally Wild Quadratic Forms of Type E7

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Quadratic Space ◽  
Trace Map ◽  
Quaternion Division Algebra

This chapter assumes that (K, L, q) is a totally wild quadratic space of type E₇. The goal is to prove the proposition that takes into account Λ‎ of type E₇, D as the quaternion division algebra over K whose image in Br(K) is the Clifford invariant of q, and the trace and trace map. The chapter also considers two other propositions: the first states that if the trace map is not equal to zero, then the Moufang residues R₀ and R₁ are not indifferent; the second states that if the trace map is equal to zero, then the Moufang residues R₀ and R₁ are both indifferent.

Linked Tori, II

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Integral Domain ◽  
Quadratic Space ◽  
Small Observation ◽  
Quaternion Division Algebra

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.

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