Distortion Coefficients of the $$\alpha $$-Grushin Plane

We compute the distortion coefficients of the $$\alpha $$ α -Grushin plane. They are expressed in terms of generalised trigonometric functions. Estimates for the distortion coefficients are then obtained and a conjecture of a measure contraction property condition for the generalised Grushin planes is suggested.

New Results on F-Contractions in Complete Metric Spaces

The main purpose of this paper is to improve, generalize, unify, extend and enrich the recent results established by Dung and Hang (2015), Piri and Kumam (2014, 2016), and Singh et al. (2018). In our proofs, we only use the property (F1) of Wardowski’s F-contraction, while the many authors in their papers still use all tree properties of F-contraction as well as two new properties introduced by Piri and Kumam. Our approach in this paper indicates that for most results with F-contraction, property (F1) is sufficient. It is interesting to investigate whether (F1) is sufficient in the case of multi-valued mappings.

Simulating quantum computations with Tutte polynomials

We establish a classical heuristic algorithm for exactly computing quantum probability amplitudes. Our algorithm is based on mapping output probability amplitudes of quantum circuits to evaluations of the Tutte polynomial of graphic matroids. The algorithm evaluates the Tutte polynomial recursively using the deletion–contraction property while attempting to exploit structural properties of the matroid. We consider several variations of our algorithm and present experimental results comparing their performance on two classes of random quantum circuits. Further, we obtain an explicit form for Clifford circuit amplitudes in terms of matroid invariants and an alternative efficient classical algorithm for computing the output probability amplitudes of Clifford circuits.

Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability statement concerning such bounds and - in the Riemannian curvature-dimension (RCD) setting - characterize the cases of equality.

Effects of 3D Moving Platform Exercise on Physiological Parameters and Pain in Patients with Chronic Low Back Pain

Background and objectives: Patient-handling activities predispose women to chronic low back pain (CLBP), but sufficient evidence is not available on whether a 3D moving platform, made for core stability exercise, affects pain, trunk flexibility, and static/dynamic muscle contractions in CLBP patients. Materials and Methods: The participants were twenty-nine women who were randomly divided into a control group (CON) and a 3D exercise group (3DEG), which took part in 3D moving exercise three times a week for 8 weeks. Both groups measured a visual analog scale (VAS) about their CLBP. Body composition, forward and backward trunk flexibilities, static muscle contraction property in rectus abdominis, and erector spinae were measured by tensiomyography, which found contraction time (Tc) and maximal displacement (Dm). Dynamic muscle contraction property in the abdomen and back were measured with an isokinetic device, which could measure peak torque (Pt) and work per repetition (Wr), before and after the trial. Results: The 3DEG had a significantly decreased fat mass and waist/hip ratio, as well as improved static muscle contractions of the erector spinae. The Wr of trunk extensor of 3D exercise group were also significantly increased. In the VAS, although the scores showed a significant change in some variables, while others did not. The Δ% in feeling pain at rest or at night, during exercise, walking, sitting in a hard chair, sitting in a soft chair, and lying down in 3DEG were significantly changed after 8 weeks. This indicates that the platform exercise provided a greater reduction of pain for activities that are done on a daily basis. Conclusions: This study confirms that the 3D moving platform exercise can provide the similar effect of the core stability exercise used in previous studies. Moreover, this study suggests that 3D moving platform exercise is a suitable means to reduce fatness, to increase trunk extensor, and to increase trunk backward flexibility, which led to reduced back pain in the women with CLBP.

Contraction for large perturbations of traveling waves in a hyperbolic–parabolic system arising from a chemotaxis model

We consider a hyperbolic–parabolic system arising from a chemotaxis model in tumor angiogenesis, which is described by a Keller–Segel equation with singular sensitivity. It is known to allow viscous shocks (so-called traveling waves). We introduce a relative entropy of the system, which can capture how close a solution at a given time is to a given shock wave in almost [Formula: see text]-sense. When the shock strength is small enough, we show the functional is non-increasing in time for any large initial perturbation. The contraction property holds independently of the strength of the diffusion.