We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as cross-referencing their analytical behaviour against known general considerations. Spectrally negative Lévy processes and scale functionsLet X = {X t : t ≥ 0} be a Lévy process defined on a filtered probability space (Ω, F, F, P), where {F t : t ≥ 0} is the filtration generated by X satisfying the usual conditions. For x ∈ R denote by P x the law of X when it is started at x and write simply P 0 = P. Accordingly we shall write E x and E for the associated expectation operators. In this paper we shall assume throughout that X is spectrally negative meaning here that it has no positive jumps and that it is not the negative of a subordinator. It is well known that the latter allows us to talk about the Laplace exponent ψ(θ) := log E[e θX1 ] for (θ) ≥ 0 where in particular we have the Lévy-Khintchine representationwhere a ∈ R, σ ≥ 0 is the Gaussian coefficient and Π is a measure concentrated on (−∞, 0) satisfying (−∞,0) (1∧x 2 )Π(dx) < ∞. The, so-called, Lévy triple (a, σ, Π) completely characterises the process X. For later reference we also introduce the function Φ : [0, ∞) → [0, ∞) as the right inverse of ψ on (0, ∞) so that for all q ≥ 0 Φ(q) = sup{θ ≥ 0 : ψ(θ) = q}.(2)

To measure peak velocity in soccer, let the players sprint. J Strength Cond Res 36(1): 273-276, 2022-Expressing externals loads relative to a player's individual capacities has potential to enhance understanding of dose-response. Peak velocity is an important metric for the individualization process and is usually measured during a sprint test. Recently, however, peak velocity was reported to be faster during soccer matches when compared with a 40-m sprint test. With the aim of developing the practice of individualized training prescription and match evaluation, we examined whether the aforementioned finding replicates in a group of elite youth soccer players across a broader range of soccer activities. To do this, we compared the peak velocities of 12 full-time male youth soccer players (age 16.3 6 0.8 years) recorded during a 40-m sprint test with peak velocity recorded during their routine activities (matches, sprints, and skillbased conditioning drills: small-sided games [SSG], medium-sided games [MSG], large-sided games [LSG]). All activities were monitored with 10-Hz global positioning systems (Catapult Optimeye S5, version 7.32) with the highest speed attained during each activity retained as the instantaneous peak velocity. Interpretation of clear between-activity differences in peak velocity was based on nonoverlap of the 95% confidence intervals for the mean difference between activities with sprint testing. Peak velocity was clearly faster for the sprint test (8.76 6 0.39 m•s 21 ) when compared with matches (7.94 6 0.49 m•s 21 ), LSG (6.94 6 0.65 m•s 21 ), MSG (6.40 6 0.75 m•s 21 ), and SSG (5.25 6 0.92 m•s 21 ), but not sprints (8.50 6 0.36 m•s 21 ). Our data show the necessity for 40-m sprint testing to determine peak velocity.

The purpose of the present study was to evaluate the effect of three pacing strategies upon performance of the 400-m sprint. Eight healthy male physical education students participated in this study. Each participant performed a 200-m maximal test (200(MAX)) and three 400-m running tests in a random counterbalanced design. The 400-m tests were run with the first 200-m pace set at 98% (400(98%)), 95% (400(95%)), and 93% (400(93%)), respectively, of the effort for 200(MAX). The stimulation of the lactate system was assessed by post-test blood lactate concentration (BLa). Running speed (RS) was controlled with time-keeping devices. Stride frequency (SF), stride length (SL) and lower extremity kinematics were acquired with video cameras operating at 100 fps at the 125 and 380-m marks of the tests. A two-way analysis of variance (ANOVA) with repeated measures was used to identify modifications caused by the pacing strategies used. Non-significant differences were revealed for BLa. The fastest 400-m race was run in 400(93%), but performance was not significantly different (p > 0.05) among the examined pacing strategies. RS, SF and SL had significantly (p < 0.05) lower values in the 380-m mark when compared with the 125-m mark. In 400(98%), both SF and SL decreased by approximately 13%, while SF and SL dropped 2.4 and 9.2%, respectively, in 400(93%). In conclusion, lower peak BLa and less unfavorable modifications of running mechanics were recorded in 400(93%), where time differential between the halves of the 400-m race was smaller, which eventually resulted in better performance.

The version presented here may differ from the published version or from the version of the record. Please see the repository URL above for details on accessing the published version and note that access may require a subscription.