Cointegration, Root Functions and Minimal Bases

This paper discusses the notion of cointegrating space for linear processes integrated of any order. It first shows that the notions of (polynomial) cointegrating vectors and of root functions coincide. Second, it discusses how the cointegrating space can be defined (i) as a vector space of polynomial vectors over complex scalars, (ii) as a free module of polynomial vectors over scalar polynomials, or finally (iii) as a vector space of rational vectors over rational scalars. Third, it shows that a canonical set of root functions can be used as a basis of the various notions of cointegrating space. Fourth, it reviews results on how to reduce polynomial bases to minimal order—i.e., minimal bases. The application of these results to Vector AutoRegressive processes integrated of order 2 is found to imply the separation of polynomial cointegrating vectors from non-polynomial ones.

Levi Factors and Admissible Automorphisms

Let $\mathfrak {g}$ g be a complex simple Lie algebra. We consider subalgebras $\mathfrak {m}$ m which are Levi factors of parabolic subalgebras of $\mathfrak {g}$ g , or equivalently $\mathfrak {m}$ m is the centralizer of its center. We introduced the notion of admissible systems on finite order $\mathfrak {g}$ g -automorphisms 𝜃, and show that 𝜃 has admissible systems if and only if its fixed point set is a Levi factor. We then use the extended Dynkin diagrams to characterize such automorphisms, and look for automorphisms of minimal order.

Assessment of point and line contact stiffness formulations leading to the initiation of hammering type brake squeal

This article proposes a refined nonlinear mathematical model to conceptually investigate the brake pad kinematics and dynamics in order to reveal certain important aspects that have been ignored in prior studies. In particular, the proposed model is formulated as a three degree-of-freedom mass positioned on a rigid frictional surface moving at constant velocity. The mass is assumed to make planar motion in vertical plane, two translations and one rotation. The interfacial contact is first examined by a point contact model with linear translational springs at edges and then the line contact is defined over the entire interface. Furthermore, kinematic and clearance nonlinearities are included. The nonlinear governing equations with point contacts at edges are numerically solved at certain angular arrangements of normal force vectors. Then, the line contact interface is solved again for the same normal force vector arrangements. Comparison reveals that the line contact approach provides more meaningful results. Finally, a linearized system model and the existence of quasi-static sliding motion are examined over a range of the normal force vector arrangements. Overall, inclusion of the rotational degree of freedom in the source model is crucial and the importance of pad-disc separation is clearly explained by the proposed formulation. This leads to a better understanding of the hammering type brake squeal source mechanisms while overcoming the limitation of prior minimal order models

A diagrammatic approach to the AJ Conjecture

The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the $$\hat{A}$$ A ^ polynomial), with a classical invariant, namely the defining polynomial A of the $${\mathrm {PSL}_2(\mathbb {C})}$$ PSL 2 ( C ) character variety of a knot. More precisely, the AJ Conjecture asserts that the set of irreducible factors of the $$\hat{A}$$ A ^ -polynomial (after we set $$q=1$$ q = 1 , and excluding those of L-degree zero) coincides with those of the A-polynomial. In this paper, we introduce a version of the $$\hat{A}$$ A ^ -polynomial that depends on a planar diagram of a knot (that conjecturally agrees with the $$\hat{A}$$ A ^ -polynomial) and we prove that it satisfies one direction of the AJ Conjecture. Our proof uses the octahedral decomposition of a knot complement obtained from a planar projection of a knot, the R-matrix state sum formula for the colored Jones polynomial, and its certificate.

Simplification by pruning as a model order reduction approach for RF-MEMS switches

Purpose This paper proposes an algorithm for the extraction of reduced order models of MEMS switches, based on using a physics aware simplification technique. Design/methodology/approach The reduced model is built progressively by increasing the complexity of the physical model. The approach starts with static analyses and continues with dynamic ones. Physical phenomena are introduced sequentially in the reduced model whose order is increased until accuracy, computed by assessing forces that are kept in the reduced model, is acceptable. Findings The technique is exemplified for RF-MEMS switches, but it can be extended for any device where physical phenomena can be included one by one, in a hierarchy of models. The extraction technique is based on analogies that are carried out for both the multiphysics and the full-wave electromagnetic phenomena and their couplings. In the final model, the multiphysics electromechanical phenomena is reduced to a system with lumped components with nonlinear elastic and damping forces, coupled with a system with distributed and lumped components which represents the reduced model of the RF electromagnetic phenomena. Originality/value Contrary to the order reduction by projection methods, this approach has the advantage that the simplified model can be easily understood, the equations and variables have significance for the user and the algorithm starts with a model of minimal order, which is increased until the approximation error is acceptable. The novelty of the proposed method is that, being tailored to a specific application, it is able to keep physical interpretation inside the reduced model. This is the reason why, the obtained model has an extremely low order, much lower than the one achievable with general state-of-the-art procedures.

Some finite p-groups with central automorphism group of minimal order

Let [Formula: see text] be a finite [Formula: see text]-group and let [Formula: see text] be the set of all central automorphisms of [Formula: see text] For any group [Formula: see text], the center of the inner automorphism group, [Formula: see text], is always contained in [Formula: see text] In this paper, we study finite [Formula: see text]-groups [Formula: see text] for which [Formula: see text] is of minimal possible, that is [Formula: see text] We characterize the groups in some special cases, including [Formula: see text]-groups [Formula: see text] with [Formula: see text], [Formula: see text]-groups with an abelian maximal subgroup, metacyclic [Formula: see text]-groups with [Formula: see text], [Formula: see text]-groups of order [Formula: see text] and exponent [Formula: see text] and Camina [Formula: see text]-groups.