Unique geodesics for Thompson's metric

In this paper a geometric characterization of the unique geodesics in Thompson's metric spaces is presented. This characterization is used to prove a variety of other geometric results. Firstly, it will be shown that there exists a unique Thompson's metric geodesic connecting $x$ and $y$ in the cone of positive self-adjoint elements in a unital $C^*$-algebra if, and only if, the spectrum of $x^{-1/2}yx^{-1/2}$ is contained in $\{1/\beta,\beta\}$ for some $\beta\geq 1$. A similar result will be established for symmetric cones. Secondly, it will be shown that if $C^\circ$ is the interior of a finite-dimensional closed cone $C$, then the Thompson's metric space $(C^\circ,d_C)$ can be quasi-isometrically embedded into a finite-dimensional normed space if, and only if, $C$ is a polyhedral cone. Moreover, $(C^\circ,d_C)$ is isometric to a finite-dimensional normed space if, and only if, $C$ is a simplicial cone. It will also be shown that if $C^\circ$ is the interior of a strictly convex cone $C$ with $3\leq \dim C<\infty$, then every Thompson's metric isometry is projectively linear.Comment: 30 page

Order isomorphisms between cones of JB-algebras

In this paper we completely describe the order isomorphisms between cones of atomic JBW-algebras. Moreover, we can write an atomic JBW-algebra as an algebraic direct summand of the so-called engaged and disengaged part. On the cone of the engaged part every order isomorphism is linear and the disengaged part consists only of copies of $\mathbb{R}$. Furthermore, in the setting of general JB-algebras we prove the following. If either algebra does not contain an ideal of codimension one, then every order isomorphism between their cones is linear if and only if it extends to a homeomorphism, between the cones of the atomic part of their biduals, for a suitable weak topology

Midpoints for Thompson's metric on symmetric cones

We characterise the affine span of the midpoints sets, $M(x,y)$, for Thompson's metric on symmetric cones in terms of a translation of the zero-component of the Peirce decomposition of an idempotent. As a consequence we derive an explicit formula for the dimension of the affine span of $M(x,y)$ in case the associated Euclidean Jordan algebra is simple. In particular, we find for $A$ and $B$ in the cone positive definite Hermitian matrices that $dim(aff M(A,B)) = q^2$, where $q$ is the number of eigenvalues $\mu$ of $A^{-1}B$, counting multiplicities, such that $\mu ≠ max\{\lambda_+(A^{-1}B),\lambda_-(A^{-1}B)^{-1}\},$ where $\lambda_+(A^{-1}B) := max\{\lambda:\lambda \in \sigma(A^{-1}B)\}$ and $\lambda_-(A^{-1}B) := min\{\lambda:\lambda \in \sigma(A^{-1}B)\}$. These results extend work by Y. Lim [18]

Passion and Pacing in Endurance Performance

Endurance sports are booming, with sports passionates of varying skills and expertise battering city streets and back roads on their weekly or daily exercise rounds. The investments required for performing in endurance exercise are nevertheless considerable, and passion for their sport might explain the efforts endurance athletes are willing to make. Passion may be defined as a strong motivational force and as such might be related to the neurophysiological basis underlying the drive to exercise. A complex relationship between the brain and other systems is responsible for athletes' exercise behavior and thus performance in sports. We anticipate important consequences of athletes' short term choices, for example concerning risk taking actions, on long term outcomes, such as injuries, overtraining and burnout. We propose to consider athletes' type of passion, in combination with neurophysiological parameters, as an explanatory factor inunderstanding the apparent disparity in the regulation of exercise intensity during endurance sports. Previous research has demonstrated that athletes can be passionate toward their sport in either a harmonious or an obsessive way. Although both lead to considerable investments and therefore often to successful performances, obsessive passion may affect athlete well-being and performance on the long run, due to the corresponding inflexible exercise behavior. In this perspective we will thus examine the influence of passion in sport on athletes' short term and long term decision-making and exercise behavior, in particular related to the regulation of exercise intensity, and discuss the expected long term effects of both types of passion for sport

High-order discontinuous Galerkin method for elastohydrodynamic lubrication line contact problems

In this paper a high-order discontinuous Galerkin method is used to solve steady-state isothermal line contact elastohydrodynamic lubrication problems. This method is found to be stable across a wide range of loads and is shown to permit accurate solutions using just a small number of degrees of freedom provided suitable grids are used. A comparison is made between results obtained using this proposed method and those from a very large finite difference calculation in order to demonstrate excellent accuracy for a typical highly loaded test problem

Diagnosing overtraining in athletes using the two-bout exercise protocol

Objective: In this work, whether a two-bout exercise protocol can be used to make an objective, immediately available distinction between non-functional over reaching (NFO) and overtraining syndrome (OTS) was studied. Design: Underperforming athletes who were diagnosed with the suspicion of NFO or OTS were included in the study. Recovery of the athletes was monitored by a sports physician to retrospectively distinguish NFO from OTS. Setting: Sports medicine laboratory Participants: The protocol was started and completed by 10 underperforming athletes. NFO was retrospectively diagnosed in five athletes, and OTS was diagnosed in five athletes. Interventions: A two-bout maximal exercise protocol was used to measure physical performance and stressinduced hormonal reactions. Main outcome measurements: Exercise duration, heart rate and blood lactate concentration were measured at the end of both exercise tests. Venous concentrations cortisol, adrenocorticotrophic hormone (ACTH), prolactin and growth hormone were measured both before and after both exercise tests. Results: Maximal blood lactate concentration was lower in OTS compared with NFO, while resting concentrations of cortisol, ACTH and prolactin concentrations were higher. However, sensitivity of these measures was low. The ACTH and prolactin reactions to the second exercise bout were much higher in NFO athletes compared with OTS and showed the highest sensitivity for making the distinction. Conclusions: NFO might be distinguished from OTS based on ACTH and prolactin reactions to a two-bout exercise protocol. This protocol could be a useful tool for diagnosing NFO and OTS; however, more data should be collected before this test can be used as the gold standard

Drive in sports: How mental fatigue affects endurance performance

Performance in endurance sports relies on athletes' drive, which is the sum of all factors pushing athletes to exert effort during exercise. Mental fatigue can influence endurance performance by decreasing athletes' drive to exercise. From a psychological point of view, mental fatigue has two separate components: it can affect drive by increasing the perceived effort necessary for a given task ("I cannot do this, I am too exhausted"), or by decreasing the perceived value of the reward that can be obtained ("I do not want to do this, it is not worth it"). Neurophysiological theories confirm this dual nature of mental fatigue. It is suggested that mental fatigue can activate the inhibition centers of the brain, increasing perceived effort for a given task, hence decreasing drive and willingness to act. On the other hand, it may also deactivate facilitative brain centers (normally responsible for motivated behavior and increased drive toward a reward), also resulting in decreased drive. In this Perspective we will adopt a multidimensional approach, describing how mental fatigue interacts with drive and performance in endurance exercise. We aim to show how mental fatigue affects endurance performance via two main mechanisms: perceived effort and reward. We will study the interaction between mental fatigue and other factors impacting on drive, such as perceived exertion and motivation, and examine how these factors combined result in athletes' exercise behavior (such as pacing) and performance. This will provide researchers, coaches, and athletes with useful tools in order to understand, influence and enhance athletes' drive in exercise, which is of high relevance in elite endurance sports, where mental fatigue, motivation, and stakes all are of the highest level

Isometries of infinite dimensional Hilbert geometries

In this paper we extend two classical results concerning the isometries of strictly convex Hilbert geometries, and the characterisation of the isometry groups of Hilbert geometries on finite dimensional simplices, to infinite dimensions. The proofs rely on a mix of geometric and functional analytic methods

Hot and hypoxic environments inhibit simulated soccer performance and exacerbate performance decrements when combined.

The effects of heat and/or hypoxia have been well-documented in match-play data. However, large match-to-match variation for key physical performance measures makes environmental inferences difficult to ascertain from soccer match-play. Therefore, the present study aims to investigate the hot (HOT), hypoxic (HYP) and hot-hypoxic (HH) mediated-decrements during a non-motorised treadmill based soccer-specific simulation. Twelve male University soccer players completed three familiarisation sessions and four randomised crossover experimental trials of the intermittent Soccer Performance Test (iSPT) in normoxic-temperate (CON: 18oC 50% rH), HOT (30oC; 50% rH), HYP (1,000m; 18oC 50% rH) and HH (1,000m; 30oC; 50% rH). Physical performance and its performance decrements, body temperatures (rectal, skin and estimated muscle temperature), heart rate (HR), arterial blood oxygen saturation (SaO2), perceived exertion, thermal sensation (TS), body mass changes, blood lactate and plasma volume were all measured. Performance decrements were similar in HOT and HYP [Total Distance (-4%), High-speed distance (~-8%) and variable run distance (~-12%) covered] and exacerbated in HH [total distance (-9%), high-speed distance (-15%) and variable run distance (-15%)] compared to CON. Peak sprint speed, was 4% greater in HOT compared with CON and HYP and 7% greater in HH. Sprint distance covered was unchanged (p > 0.05) in HOT and HYP and only decreased in HH (-8%) compared with CON. Body mass (-2%), temperatures (+2-5%) and TS (+18%) were altered in HOT. Furthermore, SaO2 (-8%) and HR (+3%) were changed in HYP. Similar changes in body mass and temperatures, HR, TS and SaO2 were evident in HH to HOT and HYP, however, blood lactate (p < 0.001) and plasma volume (p < 0.001) were only significantly altered in HH. Perceived exertion was elevated (p < 0.05) by 7% in all conditions compared with CON. Regression analysis identified that absolute TS and absolute rise in skin and estimated muscle temperature (r = 0.82, r = 0.84 r = 0.82, respectively; p <0.05) predicted the hot-mediated-decrements in HOT. The hot, hypoxic and hot-hypoxic environments impaired physical performance during iSPT. Future interventions should address the increases in TS and body temperatures, to attenuate these decrements on soccer performance