Morse Theory for S-balanced Configurations in the Newtonian n-body Problem

For the Newtonian (gravitational) n-body problem in the Euclidean d-dimensional space, the simplest possible solutions are provided by those rigid motions (homographic solutions) in which each body moves along a Keplerian orbit and the configuration of the n-body is a (constant up to rotations and scalings) central configuration. For $$d\le 3$$ d ≤ 3 , the only possible homographic motions are those given by central configurations. For $$d \ge 4$$ d ≥ 4 instead, new possibilities arise due to the higher complexity of the orthogonal group $$\mathrm {O}(d)$$ O ( d ) , as observed by Albouy and Chenciner (Invent Math 131(1):151–184, 1998). For instance, in $$\mathbb {R}^4$$ R 4 it is possible to rotate in two mutually orthogonal planes with different angular velocities. This produces a new balance between gravitational forces and centrifugal forces providing new periodic and quasi-periodic motions. So, for $$d\ge 4$$ d ≥ 4 there is a wider class of S-balanced configurations (containing the central ones) providing simple solutions of the n-body problem, which can be characterized as well through critical point theory. In this paper, we first provide a lower bound on the number of balanced (non-central) configurations in $$\mathbb {R}^d$$ R d , for arbitrary $$d\ge 4$$ d ≥ 4 , and establish a version of the $$45^\circ $$ 45 ∘ -theorem for balanced configurations, thus answering some of the questions raised in Moeckel (Central configurations, 2014). Also, a careful study of the asymptotics of the coefficients of the Poincaré polynomial of the collision free configuration sphere will enable us to derive some rather unexpected qualitative consequences on the count of S-balanced configurations. In the last part of the paper, we focus on the case $$d=4$$ d = 4 and provide a lower bound on the number of periodic and quasi-periodic motions of the gravitational n-body problem which improves a previous celebrated result of McCord (Ergodic Theory Dyn Syst 16:1059–1070, 1996).

A Comprehensive Video Codec Comparison

In this paper, we compare the video codecs AV1 (version 1.0.0-2242 from August 2019), HEVC (HM and x265), AVC (x264), the exploration software JEM which is based on HEVC, and the VVC (successor of HEVC) test model VTM (version 4.0 from February 2019) under two fair and balanced configurations: All Intra for the assessment of intra coding and Maximum Coding Efficiency with all codecs being tuned for their best coding efficiency settings. VTM achieves the highest coding efficiency in both configurations, followed by JEM and AV1. The worst coding efficiency is achieved by x264 and x265, even in the placebo preset for highest coding efficiency. AV1 gained a lot in terms of coding efficiency compared to previous versions and now outperforms HM by 24% BD-Rate gains. VTM gains 5% over AV1 in terms of BD-Rates. By reporting separate numbers for JVET and AOM test sequences, it is ensured that no bias in the test sequences exists. When comparing only intra coding tools, it is observed that the complexity increases exponentially for linearly increasing coding efficiency.

Computing the bounds on the loss rates

We consider an example network where we compute the bounds on cell loss rates. The stochastic bounds for these loss rates using simple arguments lead to models easier to solve. We proved, using stochastic orders, that the loss rates of these easier models are really the bounds of our original model. For ill-balanced configurations these models give good estimates of loss rates.

Sensitivity analysis of balanced robotic manipulators

SUMMARYThe payload variation has been one of the principal reasons in reducing path tracking accuracy and complicating controller design. In this paper, the trajectory and input sensitivities for the payload variation are investigated for three different robot configurations which include unbalanced, inaccurately balanced and balanced configurations. Based upon the sensitivity theory, simulation studies were made to evaluate the sensitivities of these configurations with respect to payload variations. The simulation results are discussed in detail and compared for the three configurations.

The Role of Intermolecular Forces in the Mechanism of High-Elastic Deformation. I. The Molecular Mechanism and an Equation for the Kinetics of High Elastic Deformation

1. The molecular mechanism of the relaxation of deformation of high-elastic polymers has been studied. 2. It is shown that the slow relaxation, which is typical of high-elastic polymers, may be best explained as a restoration process, which either partial or complete (depending on the degree of development of side chains in the molecular structure formed by the main valence chains) of the balanced configurations of the molecular chains. 3. It is shown that the rate of the relaxation process in this case is determined by the molecular activity of the particular polymer. 4. An approximate equation for the kinetics of high-elastic deformation which expresses qualitatively the mechanical properties of high-elastic polymers is proposed. 5. Hypotheses concerning the relation between the time of relaxation and the unbalanced stress are advanced. Equation (2) is derived as characteristic of this relation. 6. It is shown that the joint application of Equations (1) and (2) makes it possible to describe qualitatively the relaxation of stress at constant deformation.