Franck Brocherie
email: franck.brocherie@unil.ch

Olivier Girard

Grégoire P Millet

Millet Gp

Girard

Psycho-Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia

Keywords: Acclimation, altitude, high-intensity exercise, team sports, perceived peripheral discomfort, ratings of perceived exertion. "Psycho-Physiological Responses to Repeated-Sprint Training

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Introduction

A growing number of team-sport athletes are using altitude training with the belief that it can promote greater physiological adaptations compared to similar training conducted near sea level, in turn further improving in-game physical performance [START_REF] Girard | Position statement--altitude training for improving team-sport players' performance: current knowledge and unresolved issues[END_REF] . Time constraints during the yearly competitive calendar render the implementation of 'traditional' altitude training (e.g., 3 weeks of 'live high-train high' at natural altitude or 'live high-train low' in altitude dormitories) challenging. Innovative hypoxic methods such as repeated-sprint training in hypoxia (RSH), defined as the repetition of several short (≤ 30 s) 'all-out' exercise bouts in deprived-O2 environments interspersed with incomplete recoveries (exercise-to-rest ratio ≤ 1:5) [START_REF] Faiss | Advancing hypoxic training in team sports: from intermittent hypoxic training to repeated sprint training in hypoxia[END_REF] , encounter an expanding popularity in the team-sport community [START_REF] Millet | Hypoxic training and team sports: a challenge to traditional methods?[END_REF] . By adding hypoxic stimuli to repeated-sprint training (2-6 weeks), larger benefits have been reported on in-game physical performance-related variables [e.g., Yo-Yo intermittent recovery and/or repeated-sprint ability (RSA) tests] when compared to similar training in normoxia (RSN) [START_REF] Brocherie | High-intensity intermittent training in hypoxia: a double-blinded, placebo-controlled field study in youth football players[END_REF][START_REF] Galvin | Repeated sprint training in normobaric hypoxia[END_REF] .

In the available literature, most of RSH studies are laboratory-based through the use of standard hypoxic facilities (e.g., O2-filtration chambers/tents or breathing hypoxic mixtures with a mask) while sprinting on a cycle ergometer [START_REF] Faiss | Significant molecular and systemic adaptations after repeated sprint training in hypoxia[END_REF] , a treadmill [START_REF] Galvin | Repeated sprint training in normobaric hypoxia[END_REF] , over-ground using short shuttles [START_REF] Gatterer | Shuttle-run sprint training in hypoxia for youth elite soccer players: a pilot study[END_REF] or when treadmill and over-ground runs were combined [START_REF] Brocherie | High-intensity intermittent training in hypoxia: a double-blinded, placebo-controlled field study in youth football players[END_REF] . At top level, coaches and athletes are naturally looking for sport-specific exercises with a direct relevance (high ecological validity) to their programmes. New technologies such as the mobile inflatable hypoxic marquees [START_REF] Girard | On the use of mobile inflatable hypoxic marquees for sport-specific altitude training in team sports[END_REF] now offer advancements of hypoxic training practical applications, notably with the opportunity to train (e.g., repeated sprinting over 30 m, smallsided games, resistance training in hypoxia) under field-based hypoxic conditions (e.g., on natural/artificial grass in footballers wearing their football cleats). We have recently demonstrated superior effects of training inside these marquees using a RSH paradigm as compared to RSN, when combined with residing at moderate simulated altitude (≥ 14 h.d - a simulated altitude of 2800-3000 m), for improving RSA in elite field-hockey players during an in-season 2-wk training camp [START_REF] Brocherie | Live High-Train Low and High" Hypoxic Training Improves Team-Sport Performance[END_REF] .

Determining the day-to-day athletes' tolerance to the prescribed training load is fundamental for optimal periodization and adaptations (i.e., maximize performance gains and minimize overreaching or injury risks) [START_REF] Bishop | Sports-science roundtable: does sports-science research influence practice?[END_REF] . In team-sports, heart rate (HR) and rate of perceived exertion (RPE) are common indicators of players' degree of physical strain [START_REF] Impellizzeri | Use of RPE-based training load in soccer[END_REF] .

Although the RPE is a recognized marker of subjective assessment of exercise intensity, particularly if this is known by the participants [START_REF] Eston | Use of ratings of perceived exertion in sports[END_REF] , we have recently reported that the perception of peripheral discomfort is independent from the sense of effort during submaximal and maximal cycling bouts, despite a hypoxic-induced exaggeration of the physiological responses and higher ratings of perceived difficulty breathing [START_REF] Christian | The role of sense of effort on selfselected cycling power output[END_REF] . Single sprint performance requires, by nature, a maximal 'sense of effort' or 'perception of effort' (i.e., depending on the definition used by authors [START_REF] Marcora | Perception of effort during exercise is independent of afferent feedback from skeletal muscles, heart, and lungs[END_REF] ), but produces quite moderate perceived discomfort values due to mild sensory inputs from heart, lungs and active muscles [START_REF] Billaut | Prolonged repeated-sprint ability is related to arterial O2 desaturation in men[END_REF] .

However, when such maximal efforts are repeated (i.e., at a remaining maximal 'sense of effort') peripheral discomfort increases substantially [START_REF] Billaut | Prolonged repeated-sprint ability is related to arterial O2 desaturation in men[END_REF] . While this highlights the necessity to distinguish and individually report the accompanying perceptions of perceived discomfort (i.e., overall peripheral discomfort, difficulty breathing and lower-limb discomfort), our understanding of responses to RSH training so far is limited to mucosal immune function [START_REF] Born | Circadian variation of salivary immunoglobin A, alpha-amylase activity and mood in response to repeated double-poling sprints in hypoxia[END_REF] .

Our aim was therefore to compare changes in sprint performance along with physiological [HR and arterial oxyhemoglobin saturation (SpO2)] and perceptual responses (RPE, overall peripheral discomfort, difficulty breathing and lower-limb discomfort) to six repeated-sprint sessions in normobaric hypoxia (RSH) and normoxia (RSN) in elite teamsport players, conducted during a 2-wk 'altitude' training camp.

Methods

Participants

After being informed of the potential risks and benefits involved, twenty-three lowland elite field hockey players (Table 1) provided their written consent to participate in this study.

Experimental design/Protocol

This study is part of a large experiment conducted during an in-season competition phase (i.e., December-February, including the winter break) and additional details about the research design can be find elsewhere [START_REF] Brocherie | Live High-Train Low and High" Hypoxic Training Improves Team-Sport Performance[END_REF] . Participants were matched for initial fitness level and playing position and randomly assigned to a normobaric hypoxia (RSH, n = 11) or normoxia (RSN, n = 12) group. In addition to their usual field hockey practice, all participants undertook six repeated-sprint training sessions (at least 36 h apart), while residing under normobaric hypoxic conditions (≥ 14 h.d -1 at 2800-3000 m; FiO2 14.5-14.2%), during a 2-wk in-season training camp.

Sprint performance was evaluated during RSH/RSN training sessions (Fig. 1). Before and after each of the four sets of each training session (see below), several variables were collected to assess psycho-physiological strain associated with the completion of the proposed intervention: HR and SpO2 together with RPE, overall perceived peripheral discomfort, lower-limb discomfort and difficulty breathing.

Specific repeated-sprint training

Each session lasted 50 min, including a 15-min warm-up [i.e., low-intensity running with athletic (e.g., skipping, high knee runs) and acceleration drills (e.g., shorts bursts of progressive accelerations and maximal 10-m straight-line sprints)], the repeated-sprint training routine and a 10-min recovery phase (i.e., cool down and passive stretching).

Specifically, the repeated-sprint training routine included four sets of 5 × 5-s maximal sprints in alternating directions interspersed with 25 s of passive recovery with 5 min of standing rest between sets. With an exercise-to-rest ratio of 1:5, up to six players trained together with an invariant execution order. Practically, each participant departed for sprinting every 5 s, as controlled by audio beeps. Because of space restriction inside the marquee, they had to decelerate on a short distance (i.e., ~10 m). During all training sessions, participants were constantly reminded to assume a standing ready position for 2 s before starting, and were vigorously encouraged to perform 'all-out' efforts in trying to reach peak acceleration and to maintain the highest possible running speed for every sprint.

Training sessions were completed on an indoor synthetic grass inside a mobile inflatable simulated hypoxic equipment (Altitude Technology Solutions Pty Ltd, Brisbane, Queensland, Australia), as previously described [START_REF] Girard | On the use of mobile inflatable hypoxic marquees for sport-specific altitude training in team sports[END_REF] . Briefly, it comprised a polyvinyl chloride inflatable running lane tunnel (length = 45 m, width = 1.8m and height = 2.5m) and a stateof-the-art hypoxic trailer (55 kW screw compressor), generating over 3000 Lpm of hypoxic air with FiO2 between 21% and 10% (a simulated altitude up to 5100 m). Temperature and humidity were maintained constant at ~25°C and ~55% relative humidity, owing to the exercise-generated humidity compensated by the dry air supplied inside the marquee. For RSH, ambient air was mixed with nitrogen (from pressurized tanks) to reduce FiO2 to ~14.5% in order to simulate an altitude of ~3000 m. While up to six participants trained simultaneously in the marquee (Fig. 1), they were all assigned to different teammates during the six sessions, in order to reinforce motivation. For blinding purposes, participants were told (head coach request) that they were all training under hypoxic conditions but had no accurate information about actual simulated altitude levels. The total hypoxic exposure corresponded exactly to the training duration, i.e., with RSH / RSN participants breathing hypoxic / normoxic air from the time of entrance to exit of the marquee.

Sprinting timing

During training, the straight-line sprints were performed back and forth. Sprint performance (timing) during the first and the last (sprint 1 and 5, respectively) sprint of each of the four sets was measured to the nearest 0.01 s using photocells connected to an electronic timer (Polifemo Radio light, Microgate, Bolzano, Italy). Timing gates were positioned at 0and 20-m distance intervals, with additional turned-off gates placed at a distance of ~35 m (corresponding to ~5-s duration sprint) in order to make sure participants produced maximal effort (i.e., no deceleration) throughout each sprint bout. Performance fatigability (i.e., percent increase in time between the fastest and slowest sprints) was calculated using the following formula: Fatigue index (FI, %) = [(sprint 5sprint 1) / sprint 1] × 100 [START_REF] Girard | Repeated-sprint ability -part I: factors contributing to fatigue[END_REF] .

Physiological responses to exercise

HR and SpO2

HR and SpO2 were monitored (i.e., participants were blinded to values), respectively, via a wireless Polar monitoring system (Polar Electro Oy, Kempele, Finland) and noninvasive pulse oximeter using a finger probe (GO2TM Achieve 9570-A, Nonin, Plymouth, MN, USA). These values were first recorded prior to and immediately upon completion of the standardized warm-up (i.e., 15 min following entry to the mobile hypoxic marquee).

Afterwards, physiological responses were collected exactly 1 min and 4 min into the 5-min resting period following each of the four repeated sprints sets.

Perceived Exertion Measures

Rating of perceived exertion and training load

As an index of overall feeling of subjective perceived exertion, session RPE was assessed within 10 min of completing each training session with the 6-20 Borg scale.

Training loads (arbitrary units, a.u.) were calculated for all players as total training session duration (min) × session RPE [START_REF] Foster | Monitoring training in athletes with reference to overtraining syndrome[END_REF] .

Perceived discomfort scales

Based on modified Borg CR10 scales, ratings of overall perceived peripheral discomfort, perceived lower-limb discomfort and perceived difficulty breathing [START_REF] Christian | The role of sense of effort on selfselected cycling power output[END_REF] were recorded (invariant order) at the same time intervals as physiological responses (i.e., 1 min and 4 min after each of the four exercise sets). Beforehand, players were thoroughly instructed that the 'perceived discomfort' scales, visible to participants at all times, were used to evaluate their subjective perception of (i) overall peripheral discomfort ('How uncomfortable do you feel overall?'), (ii) specific lower-limb discomfort ('How uncomfortable do your legs feel?') and (iii) difficulty breathing ('How uncomfortable does it feel to breathe?') (Fig. 1).

Statistical analysis

Normal distribution was examined using Kolmogorov-Smirnov normality test.

Separate two-way analyses of variance (ANOVA), with repeated measures for condition (RSH vs. RSN), time [(session 1, 2, 3, 4, 5 vs. 6) or (set 1, 2, 3 vs. 4)] and possible interaction between these two factors was conducted. When ANOVA revealed a significant effect, post hoc comparisons were made using the Bonferroni method. Averaged values (mean of the 6 sessions) and percentage changes (relative to session 1) were compared between the two groups using a one-way ANOVA. For each ANOVA, effect size (ES) was calculated (Cohen's d) with the following criteria: an ES of < 0.2 is classified as a 'trivial', 0.2-0.4 as a 'small', 0.5-0.7 as a 'moderate' and > 0.8 as a 'large' effect. Data are presented as means ± SD. Statistical analyses were performed using Sigmaplot 11.0 software (Systat Software, San Jose, CA, USA). Statistical significance was set at P < 0.05.

Results

Prior to entering the mobile hypoxic marquee, HR and SpO2 did not differ among conditions (P = 0.06 and 0.12, respectively) and between sessions (P = 0.15 and 0.23) (Fig. 2). Following the 15-min warm-up period, there was a hypoxia-induced increase in HR and decrease in SpO2 values (both P < 0.001). Sprint performance outcomes along with psychophysiological responses to RSH and RSN are presented in Figs. 2 and3 with the corresponding statistical analyses (i.e., averaged data of all sets) displayed in Table 2.

During the first session, SpO2 remained depressed in RSH (P < 0.001) while HR increased across sets (P < 0.001), independently of the conditions (Fig. 2). Thereafter, both SpO2 and HR remained similar across sessions for each condition. While first sprint time was similar between conditions, sets and sessions (Fig. 2 and Table 2), last sprint time and sprint 1-5 fatigue index significantly decreased across sets and sessions (time effect, P < 0.01 and < 0.05, respectively) leading to significant lower averaged sprint 5 time values (P < 0.001) in RSH compared with RSN (Fig. 2 and Table 2). At the first session, ratings of overall perceived discomfort, difficulty breathing and lower-limb discomfort were higher (P < 0.05) in RSH compared with RSN (Fig. 3 and Table 2), with an effect of time (P < 0.001) on ratings of overall perceived discomfort and difficulty breathing. During subsequent sessions, overall perceived discomfort (time and condition effects; P < 0.001 and 0.05, respectively), difficulty breathing (time effect; P < 0.001) and lower-limb discomfort (condition and interaction effects; P < 0.001 and 0.05, respectively) values decreased to a larger extent in RSH vs. RSN (Fig. 3 and Table 2).

While session RPE did not differ between conditions at the first session (P = 0.10), perceived exertion was significantly lower (P < 0.05) in RSH compared with RSN, with a significant different magnitude of change (P < 0.05) (Table 2). Statistical analysis revealed no main effect of condition (P = 0.93), time (P = 0.80) and interaction (P = 0.18) for training load (pooled values for the six sessions: 59268 and 59369 in RSH and RSN, respectively) monitored during the 2-wk period.

Discussion

The main findings of the present investigation were as follows: (i) despite a higher hypoxia-induced physiological stress (i.e., SpO2) and higher-perceived ratings, sprint performance responses were comparable for RSH and RSN at the first session; and (ii) as training sessions progressed, perceptual responses improved for RSH, indicative of an effective altitude 'acclimation'. Effective tolerance to this innovative RSH training was accompanied by an improved repeated sprints performance (i.e., lower averaged sprint 5 time).

Sprint performance

Single sprint performance of short duration (< 10 s) is generally well-preserved under hypoxic (2000-3800 m) vs. normoxic conditions [START_REF] Bowtell | Acute physiological and performance responses to repeated sprints in varying degrees of hypoxia[END_REF] . In line with these observations, in our study, the first sprint performance was also similar between conditions, sets and sessions. This is presumably due to an increase in the rate of anaerobic energy release, which compensates for the limited oxidative contribution when sprinting in O2-deprived environments [START_REF] Weyand | Highspeed running performance is largely unaffected by hypoxic reductions in aerobic power[END_REF] . As expected, as sprints were repeated, a significant time effect was found for last sprint time and performance fatigability, which may primarily to a decline in adenosine triphosphate resynthesis from phosphocreatine hydrolysis and non-oxidative glycolysis [START_REF] Girard | Repeated-sprint ability -part I: factors contributing to fatigue[END_REF] . However, whereas hypoxia has been associated with earlier and larger RSA performance decrements [START_REF] Bowtell | Acute physiological and performance responses to repeated sprints in varying degrees of hypoxia[END_REF][START_REF] Goods | Effect of different simulated altitudes on repeat-sprint performance in team-sport athletes[END_REF] , in our study, no difference was observed between conditions. These discrepant findings may result from the smaller number of sprints repetition within a set and/or the shorter distances monitored (i.e., first and last effort of 5 repetitions over 20 m) vs. those actually completed (5-s sprints or 30 m sprints distance). This methodological choice was guided by the ease of comparisons with our companion paper where a 8 × 20 m RSA test was used during pre-and post-tests [START_REF] Brocherie | Live High-Train Low and High" Hypoxic Training Improves Team-Sport Performance[END_REF] . Regardless, the significant improvement in sprint 5 performance over successive sessions confirms that RSH (and to a lesser extent RSN) is an effective time-efficient training strategy to improve RSA performance [START_REF] Taylor | The effects of repeated-sprint training on field-based fitness measures: a meta-analysis of controlled and non-controlled trials[END_REF] . This corroborates previous findings from our research group using isolated-6 , mixed-4 or combined-RSH training [START_REF] Brocherie | Live High-Train Low and High" Hypoxic Training Improves Team-Sport Performance[END_REF] .

Physiological and perceptual responses

In the present study, exposure to hypoxia during training (i.e., RSH) induced lower SpO2 values compared to similar training near sea level in normoxia (i.e., RSN), also in line with available RSH literature [START_REF] Brocherie | High-intensity intermittent training in hypoxia: a double-blinded, placebo-controlled field study in youth football players[END_REF][START_REF] Galvin | Repeated sprint training in normobaric hypoxia[END_REF][START_REF] Bowtell | Acute physiological and performance responses to repeated sprints in varying degrees of hypoxia[END_REF] . Lower SpO2 values are generally accompanied by increased values for HR, minute ventilation and O2 debt [START_REF] Bowtell | Acute physiological and performance responses to repeated sprints in varying degrees of hypoxia[END_REF] . Although HR is not always a valid marker for prolonged anaerobic exercise, in the present study, HR increased with exercise and across sets. When ten trained male team-sport athletes completed 3 sets of repeated sprints (9 × 4 s) on a non-motorized treadmill at sea level and at simulated altitudes of 2000, 3000, and 4000 m, total work across all sets was highest at sea level with also correspondingly lower mechanical performance at each successive altitude [START_REF] Goods | Effect of different simulated altitudes on repeat-sprint performance in team-sport athletes[END_REF] . In line with our study, however, HR values did not differ between conditions. Potentially different time points for measuring HR values during recovery periods post-exercise may have led to different HR readings between conditions.

In addition to an exacerbated rate of peripheral fatigue development [START_REF] Girard | Neuro-mechanical determinants of repeated treadmill sprints -Usefulness of an "hypoxic to normoxic recovery" approach[END_REF][START_REF] Billaut | Interaction of central and peripheral factors during repeated sprints at different levels of arterial O2 saturation[END_REF] , an acute repeated-sprint exercise in hypoxia is known to elevate subjective fatigue sensations compared to normoxia [START_REF] Christian | The role of sense of effort on selfselected cycling power output[END_REF][START_REF] Billaut | Interaction of central and peripheral factors during repeated sprints at different levels of arterial O2 saturation[END_REF] . Accordingly, the higher overall session RPE and difficulty breathing perceived in RSH compared with RSN at the first session might relate to the higher ventilation response that is generally observed for any given absolute exercise intensity in hypoxia [START_REF] Aliverti | Respiratory and leg muscles perceived exertion during exercise at altitude[END_REF] . This was further accompanied by higher leg discomfort observed in RSH compared with RSN at the first session, which possibly reflects the greater accumulation of metabolic by-products (e.g., blood lactate concentration) [START_REF] Pollak | Exogenously applied muscle metabolites synergistically evoke sensations of muscle fatigue and pain in human subjects[END_REF] when repeated sprints were performed under acute normobaric hypoxia [START_REF] Girard | Repeated-sprint ability -part I: factors contributing to fatigue[END_REF][START_REF] Bowtell | Acute physiological and performance responses to repeated sprints in varying degrees of hypoxia[END_REF][START_REF] Goods | Effect of different simulated altitudes on repeat-sprint performance in team-sport athletes[END_REF][START_REF] Girard | Neuro-mechanical determinants of repeated treadmill sprints -Usefulness of an "hypoxic to normoxic recovery" approach[END_REF][START_REF] Billaut | Interaction of central and peripheral factors during repeated sprints at different levels of arterial O2 saturation[END_REF] . Although blood lactate concentration was not measured here, larger values under RSH may have stimulated the sensory feedback from group III/IV muscle afferents to a larger extent [START_REF] Amann | Group III and IV muscle afferents contribute to ventilatory and cardiovascular response to rhythmic exercise in humans[END_REF] , as previously observed after arterial occlusion [START_REF] Shibata | The relationships between movement-related cortical potentials and motor unit activity during muscle contraction[END_REF] . Bearing in mind that these neurons play a twofold rolei.e., limiting motoneuronal output, thereby modulating central fatigue; and/or enhancing circulatory and ventilatory response, thereby preventing peripheral fatiguequantifying their influence in the regulation of the cardiorespiratory response to RSH deserves further attention. That said, the higher perceptual responses (RPE, overall peripheral discomfort, difficulty breathing and lower-limb discomfort) observed in RSH vs. RSN at the first session (acute effect) were attenuated from the second session onwards with essentially similar values in reference to RSN over the remaining five sessions ('acclimation' effect). This is contrasting with previous findings demonstrating that normobaric hypoxia and exercise when combined exacerbate perceived levels of exertion [START_REF] Aliverti | Respiratory and leg muscles perceived exertion during exercise at altitude[END_REF] . This discrepancy may be due to difference in exercise mode between studies (i.e., repeated sprints vs. Wingate-based highintensity training vs. 4-6 × 30 s of 'all-out' cycling efforts). Regardless, the attenuated perceptual responses reported in RSH after the initial session are likely due to an improved tolerance to the hypoxic stimulus, possibly via an increased hypoxic ventilatory response mediated by improved chemosensitivity [START_REF] Townsend | Living high-training low increases hypoxic ventilatory response of well-trained endurance athletes[END_REF][START_REF] Dempsey | Role of chemoreception in cardiorespiratory acclimatization to, and deacclimatization from, hypoxia[END_REF] . One cannot rule out that the prolonged exposure to hypoxia during the nights has improved the chemosensitivity and therefore accelerated the 'acclimation' during the RSH sessions.

Despite lower SpO2 values in RSH vs. RSN, ratings of RPE, overall peripheral discomfort and difficulty breathing remained rather low even under hypoxic exposure (i.e., values never exceeding 7 on CR10 scales), which may explain why we failed to report significant difference between the two conditions after the first 'acclimation' session. On the one hand, it could be argued that longer sprints, shorter recoveries or combination of both (i.e., more intense exercise-to-rest ratios) and/or the use of more severe hypoxic conditions [also causing larger performance fatigability; 21 ] might lead to significant difference between RSH and RSN. On the other hand, this also suggests that a manipulation of such trainingrelated factors (i.e., exercise-to-rest ratios and/or hypoxic levels) is necessary to optimize individual responses to RSH [START_REF] Millet | Hypoxic conditions and exercise-to-rest ratio are likely paramount[END_REF] . The increased lower-limb discomfort observed in RSN may be attributed to possible inter-session fatigue appearance. However, this would have also been observed in RSH, unless hypoxic 'tolerance' had lessened this phenomenon. Despite being constantly reminded about what perceived lower-limb discomfort actually measured, it remains possible that several players rated the discomfort of their legs with a variety of sensationse.g., muscle pain, joint and foot pressure, and muscle tension [START_REF] Gatterer | Shuttle-run sprint training in hypoxia for youth elite soccer players: a pilot study[END_REF] experienced during repeated sprinting, not just how uncomfortable their legs felt.

Previous studies have documented that short-term intermittent normobaric/hypobaric hypoxia (e.g., 1-2 h.day -1 over 7-10 consecutive days) leads to ventilatory 'acclimation/acclimatization' to hypoxia, characterized by a progressive recovery in physiological functions (i.e., increase in SpO2 and decrease in HR values) during hypoxic exercise [START_REF] Katayama | Effect of a repeated series of intermittent hypoxic exposures on ventilatory response in humans[END_REF] . In this study, however, SpO2 remained stable and the decrease in HR was not significant across sessions notably for RSH, mainly due to high individual variability in response to hypoxic exercise. Pending confirmatory research, it is possible that an increased hypoxic ventilatory response may have been counteracted by other likely RSH-induced physiological mechanisms as hemodynamic [START_REF] Boning | Altitude and hypoxia training--a short review[END_REF] and/or muscular adaptations [START_REF] Hoppeler | Muscle tissue adaptations to hypoxia[END_REF] .

Practical applications

We observed that ratings of overall perceived discomfort, difficulty breathing and lower-limb discomfort decreased after the first RSH session to reach comparable values with RSN, while sprint performance achieved with or without hypoxic exposure followed similar trends across sessions. Together with larger performance improvement (i.e., lower averaged sprint 5 time), these positive perceived adaptations for RSH may have played a role in the post-intervention performance gains [START_REF] Brocherie | Live High-Train Low and High" Hypoxic Training Improves Team-Sport Performance[END_REF] . This indicates that the proposed RSH modality (i.e., four sets of 5 × 5-s maximal sprints interspersed with 25 s of passive recovery with 5 min of standing rest between sets at FiO2 ~14.5%/~3000 m), combined with normobaric hypoxic residence (≥ 14 h.d -1 at FiO2 14.5-14.2%; 2800-3000 m), is sufficient in severity, duration and/or frequency to elicit a significant hypoxic 'acclimation'. This study therefore adds to the recent results of Born et al. [START_REF] Born | Circadian variation of salivary immunoglobin A, alpha-amylase activity and mood in response to repeated double-poling sprints in hypoxia[END_REF] , where mucosal immune function was not further impaired with RSH, by showing that players tolerance to RSH is effective after only one session. Since RSH does not negatively alter psycho-physiological responses when compared to RSN, coaches and their athletes are encouraged to implement such time-efficient hypoxic interventions in order to gain a competitive edge [START_REF] Brocherie | Live High-Train Low and High" Hypoxic Training Improves Team-Sport Performance[END_REF] . By closely monitoring training loads and psycho-physiological responses of players in response to each session, this investigation Notes: Values are mean ± SD. C, T, Icondition, time and interaction effects, respectively.

* significantly different from session number 1, P < 0.05, ** < 0.01, *** < 0.001. # significantly different from RSN, P < 0.05, ### < 0.001.

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  Psycho-Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia " by Brocherie F, Millet GP, Girard O International Journal of Sports Physiology and Performance © 2016 Human Kinetics, Inc.

  The experiment was approved by the Anti-Doping Lab Qatar institutional review board (Agreement SCH-ADL-070) and conformed to the current Declaration of Helsinki guidelines. Participants practiced ~7.0-9.0 h.wk -1 of training (5-7 weekly sessions) and competitive matches (1-2 matches per week) during the season. Training sessions included aerobic, anaerobic and strength training in isolation or combined with sport-specific training (i.e., technical and tactical skills). All participants were born and raised at altitudes < 1000 m and had not travelled to elevations > 2000 m for more than 48 h in the 6 months prior to the study.

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  Psycho-Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia " by Brocherie F, Millet GP, Girard O International Journal of Sports Physiology and Performance © 2016 Human Kinetics, Inc.

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  Psycho-Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia " by Brocherie F, Millet GP, Girard O International Journal of Sports Physiology and Performance © 2016 Human Kinetics, Inc.

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  Psycho-Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia " by Brocherie F, Millet GP, Girard O International Journal of Sports Physiology and Performance © 2016 Human Kinetics, Inc.

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  Psycho-Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia " by Brocherie F, Millet GP, Girard O International Journal of Sports Physiology and Performance © 2016 Human Kinetics, Inc.

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  Psycho-Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia " by Brocherie F, Millet GP, Girard O International Journal of Sports Physiology and Performance © 2016 Human Kinetics, Inc.

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  Psycho-Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia " by Brocherie F, Millet GP, Girard O International Journal of Sports Physiology and Performance © 2016 Human Kinetics, Inc.

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  Psycho-Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia " by Brocherie F, Millet GP, Girard O International Journal of Sports Physiology and Performance © 2016 Human Kinetics, Inc.

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  Psycho-Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia " by Brocherie F, Millet GP, Girard O International Journal of Sports Physiology and Performance © 2016 Human Kinetics, Inc.

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  Psycho-Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia " by Brocherie F, Millet GP, Girard O International Journal of Sports Physiology and Performance © 2016 Human Kinetics, Inc. opens new frontiers to optimize future hypoxic training applications (e.g., appropriate periodization to implement isolated RSH and/or in combination with prolonged residence at simulated altitude) in team sports. Conclusion This is the first study examining psycho-physiological responses to repeated-sprint training (6 sessions) in hypoxia and normoxia in elite team-sport players over a 2-wk inseason 'live high-train low' training camp. During the first session, hypoxia exposure during training elevated physiological and perceptual responses, whereas sprint performance (timing) was similar. Despite higher hypoxia-induced physiological stress (e.g., lower SpO2 values), improved tolerance to the hypoxic training (effective hypoxic 'acclimation') led to a decrease in perceptual responses (that were not maximal in both groups) in RSH, so as to become essentially similar to those measured in RSN. These positive adaptations could have played a role in the larger RSH-related improvement in repeated sprint performance.

Figure 1 .

 1 Figure 1. Illustration of the mobile inflatable simulated hypoxic equipment used for repeated-sprint training sessions. External (A, overall view of the 45-m long marquee) and inside tunnel view with players completing sprints (B) and monitoring their psychophysiological responses (C).

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  Psycho-Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia " by Brocherie F, Millet GP, Girard O International Journal of Sports Physiology and Performance © 2016 Human Kinetics, Inc.

Figure 2 .

 2 Figure 2. Sprint times (A), arterial oxyhemoglobin saturation (SpO2; B) and heart rate (HR; C) for the four sets of repeated-sprint exercise performed during each training session. Values are mean ± SD, N = 23. Data is presented for normobaric hypoxia (RSH; FiO2 ~14.5%) and normoxia (RSN; FiO2 21.0%).

Figure 3 .

 3 Figure 3. Ratings of overall peripheral discomfort (A), difficulty breathing (B) and lowerlimb discomfort (C) for the four sets of repeated-sprints performed during each training session. Values are mean ± SD, N = 23. Data is presented for normobaric hypoxia (RSH; FiO2 ~14.5%) and normoxia (RSN; FiO2 21.0%).

  1 at "Psycho-Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia " by Brocherie F, Millet GP, Girard O International Journal of Sports Physiology and Performance © 2016 Human Kinetics, Inc.

Table 1 .

 1 Subjects' characteristics. Psycho-Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia " by Brocherie F, Millet GP, Girard O International Journal of Sports Physiology and Performance © 2016 Human Kinetics, Inc.

		RSH (n = 11)	RSN (n = 12)
	Age (yr)	27.4 ± 4.6	24.9 ± 4.5
	Height (cm)	178.7 ± 8.2	178.9 ± 4.7
	Weight (kg)	77.0 ± 7.7	76.9 ± 8.5
	BMI (kg.m -2 )	24.1 ± 1.6	24.0 ± 1.8
	VO2max (ml.min -1 .kg -1 )	52.7 ± 2.4	52.5 ± 1.8
	HRmax (beats.min -1 )	189 ± 4	192 ± 3
	VO2max = maximal oxygen uptake; HRmax = maximal heart rate.
	Values are mean ± SD.

"

Table 2 .

 2 Sprint performance parameters and psycho-physiological responses (average of all sets) during the 6 sessions of repeated-sprint training in hypoxia (RSH) and in normoxia (RSN). Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia " by Brocherie F, Millet GP, Girard O International Journal of Sports Physiology and Performance © 2016 Human Kinetics, Inc.

		Training			Session number			Average	Changes	ANOVA main
			1	2	3	4	5	6		(%)	effects (effect sizes)
	Sprint performance
		RSH	3.27±0.14	3.25±0.17	3.24±0.13	3.24±0.13	3.24±0.13	3.21±0.15	3.24±0.14	-1.8±3.8	C = 0.99 (0.00);
	Sprint time 1 (s)										T = 0.15 (0.22);
		RSN	3.26±0.08	3.24±0.08	3.24±0.07	3.24±0.06	3.24±0.06	3.23±0.09	3.24±0.07	-0.5±2.9	I =0.95 (0.08)
		RSH	3.38±0.16	3.35±0.19	3.34±0.16	3.32±0.16	3.31±0.17	3.30±0.16*	3.33±0.16###	-2.4±3.8	C = 0.55 (0.23);
	Sprint time 5 (s)										T <0.01 (0.31);
		RSN	3.38±0.08	3.38±0.08	3.36±0.06	3.36±0.06	3.36±0.07	3.33±0.09	3.36±0.07	-1.3±2.9	I = 0.80 (0.11)
		RSH	4.6±1.9	4.3±2.7	3.9±3.2	3.4±2.4	3.0±2.4	2.2±3.6*	3.6±2.8	-34.1±133.9	C = 0.16 (0.32);
	FI (%)										T < 0.05 (0.44);
		RSN	5.6±4.1	5.4±1.2	4.6±2.3	4.5±1.9	4.4±2.5	3.6±3.3*	4.6±2.7	-62.7±127.7	I = 0.99 (0.07)
	Physiological responses
		RSH	92.0±1.6### 92.0±1.4###	92.0±1.1### 91.9±1.0### 91.9±0.9### 91.7±1.2*,### 91.9±1.2###	-0.9±1.4	C < 0.001 (3.48);
	SpO2 (%)										T < 0.001 (0.42);
		RSN	97.0±0.5	96.8±0.6	96.5±0.6	96.7±0.4	96.8±0.7	97.3±0.6	96.9±0.6	0.2±0.6	I < 0.001 (0.39)
		RSH	142±13	144±13	143±10	141±13	139±11	138±14	140±12	0.7±12.9	C = 0.053 (0.53);
	HR (bpm)										T = 0.11 (0.35);
		RSN	148±13	145±12	153±14	145±11	143±13	150±13	147±12	0.9±-9.8	I = 0.55 (0.23)
	Perceptual responses
		RSH	14.6±0.8	13.3±0.9	14.1±1.0	14.1±1.3	13.8±1.7	13.1±1.3	13.8±1	-10.5±15.4#	C < 0.05 (0.61);
	RPE (a.u.)										T = 0.31 (0.28);
		RSN	14.4±2.6	14.1±1.9	14.3±1.8	14.1±0.9	14.3±1.9	14.8±0.9	14.3±1.7	4.0±17.7	I = 0.09 (0.37)
	Overall	RSH	4.5±0.8#	3.4±0.7**	3.8±0.7**	3.6±1.0**	3.5±0.8**	3.2±0.8***	3.7±0.8#	-28.5±22.4#	C < 0.05 (0.55);

Acknowledgements

This research was funded by a Grant awarded by Aspetar (Qatar Orthopedic and Sports Medicine Hospital) at the Aspire Zone Foundation, Qatar (AF/C/ASP1905/11). The authors are grateful to Mr. Julien Rysman for his help during the study. "Psycho-Physiological Responses to Repeated-Sprint Training in Normobaric Hypoxia and Normoxia " by Brocherie F, Millet GP, Girard O International Journal of Sports Physiology and Performance © 2016 Human Kinetics, Inc.


Philippe Jacquet

Dimitrios Milioris

Keywords: Asymptotic analysis, Eulerian paths and circuits, Analytic combinatorics

In this paper, we evaluate the number of Eulerian circuits that can be obtained by an arbitrary rotation in a Markovian string, i.e., corresponding to a given Markovian type. Since all rotations do not result in an Eulerian circuit, but several of them, called Eulerian components; we also investigate the number of Eulerian components that result from a random rotation in a Markovian string. We consider the asymptotic behaviour of those quantities when the size of the string n tends to infinity. In particular we show that the average number of components tends to be in log V , where V is the size of a large alphabet, in the uniform case.

Introduction

In this paper we consider X n , a string of length n generated by a Markov process over an alphabet A of size V . We denote K(A) the set of integer V × V matrices defined on A × A. Let k ∈ K(A) when ∀(a, b) ∈ A 2 the coefficient k ab of k is equal to the number of time symbol b follows symbol a in X n , we say that k is the type of string X n as defined in [START_REF] Jacquet | Markov Types and Minimax Redundancy for Markov Sources[END_REF]. The matrix k can be seen as the adjacency matrix of the multigraph G(X n ) whose vertex set is A and edge set collects all the transitions between consecutive symbols in X n .

The string X n is an Eulerian path in G(X n ), i.e., a path that takes every edge of G(X n ) once and only once. Let a ∈ A and let k a = b∈A k ab . An Eulerian path on X n will visit symbol a, k a times, and every time takes one outgoing edge among the non already visited edge, until all edges have been visited. The order at which the edge are visited is a permutation σ a of the outgoing edges of symbol a. There are ka (k ab ) b∈A = ka! b∈A k ab ! , since we don't distinguish over the edges that connect to the same next symbol. Therefore an Eulerian path is defined by an initial symbol and a V tuple (σ a ) a∈A of such permutations † Partially supported by LINCS ‡ Partially supported by INRIA and École Polytechnique that we call rotation. There are B k rotations with

B k = a∈A k a (k ab ) b∈A . (1)
Of course not every rotation in G(X n ) will generate an Eulerian path, since the path may end on a symbol which has exhausted all its outgoing edges before some other symbols exhaust their. We denote N a k the number of Eulerian path that shares the same type k starting from symbol a, and we naturally have N a k ≤ B k . Notice that all these Eulerian paths corresponds to strings that have the same probability in the Markov model.

Let R n be the average number of rotation for a string a length n generated by the Markov process. We show that R n ∼ Aλ 2n for some A > 0 and λ > 1 that depends on the transition matrix. This quantity is almost impossible to simulate when n is large since the main contributions come from types whose probability is exponentially small. Let L n denote the average of the logarithm of the number of rotations, we show that in general L n = log R n and that in fact we have L n ∼ H(X n ) where H(X n ) is the entropy of the Markov string X n .

When closing the string X n we make it cyclical. In this case it is equivalent to add an edge in G(X n ) between the last and the first symbol of X n , and we get the new graph G(X n ). We show such a graph in Figure 1 for the sentence "to be or not to be that is the question". We call anchored Eulerian circuit when we specify the initial symbol. Several anchored Eulerian circuits may correspond to the same Eulerian circuit, just translated by circular permutations. An anchored Eulerian path corresponds to a rotation in G(X n ) but contrary to Eulerian path it may not have its symbol before last equal to the last symbol of X n , even if the first symbols are the same, thus an anchored Eulerian circuit in G(X n ) is not equivalent to an Eulerian path in G(X n ). To make it equivalent one may consider an anchored Eulerian circuit starting on the last symbol of X n , say symbol b, and force the first edge permutation to start with the edge ba; a being the first symbol of X n .

When a rotation in G(X n ) is not an Eulerian circuit, it can be decomposed into a sequence of paths in G(X n ) that are edge-disjoint, that we call Eulerian components. We will provide an asymptotic estimate of the average number of such components. In particular when the Markov process is memoryless and uniform the probability that a rotation is an Eulerian circuit is asymptotically equal to 1 V and the average number of component is asymptotically equal to H(V ), the harmonic function of V .

The paper is organized as follows: Section 2 investigates the enumeration of Eulerian paths and other parameters via generating functions, while Section 3 applies these results on a Markov generating process and gives an analysis of rotations. Finally, Section 4 presents some experimental results and gives conclusions.

Balanced types, generating functions and Eulerian path

Pioneering work on Eulerian paths and circuits enumeration could be found in [START_REF] Tutte | On unicursal paths in a network of degree 4[END_REF][START_REF] Van Aardenne-Ehrenfest | Circuits and trees in oriented linear graphs[END_REF] but we rely and extend the work and methodology of [START_REF] Jacquet | Markov Types and Minimax Redundancy for Markov Sources[END_REF]. Notice the related works in [START_REF] Mckay | Asymptotic enumeration of Eulerian circuits in the complete graph[END_REF] about Eulerian circuits in complete graphs and [START_REF] Isaev | Asymptotic number of Eulerian circuits in complete bipartite graphs[END_REF] about Eulerian path in bipartite graphs.

Balanced types

Let c ∈ A and a type k we denote k c = d∈A k cd and k c = d∈A k dc respectively the out-degree and in-degree of symbol c in the syntax graph. A type is balanced if ∀a ∈ A: k a = k a . Let F n be the set balanced types k such that

• k α > 0; • a∈A k a = n.
We also denote F * = n F n as in [START_REF] Jacquet | Markov Types and Minimax Redundancy for Markov Sources[END_REF]. If C ⊂ A we denote F * (C) the types such that

k ab = 0 ⇒ (a, b) ∈ C 2 .

Generating functions on types

Let z be a matrix of complex coefficients z ab and let k a type, we denote

z k = (a,b)∈A 2 z k ab ab (2)
As in [START_REF] Jacquet | Markov Types and Minimax Redundancy for Markov Sources[END_REF] we introduce the generating functions of matrix variable, of the form f (z) = k∈K(A) a k z k , where a k is a sequence of indexed by types. In particular

B(z) = k∈K(A) B k z k = a∈A 1 1 -b∈A z ab . (3)
Let F be the operator defined by Ff (z) = k∈F * a k z k , i.e. the generating function restricted to balanced types. In particular we show in [START_REF] Jacquet | Markov Types and Minimax Redundancy for Markov Sources[END_REF] that

FB(z) = 1 det(I -z) . (4)
If C ⊂ A we define by abuse of notation K(C) as the subset of K(A) such that

k ab = 0 ⇒ (a, b) ∈ C 2 . Therefore F * (C) = F * ∩ K(C). We denote G C the operator G C f (z) = k∈K(C) a k z k . ( 5
)
The operators F and G C commute and we have

G C FB(z) = 1 det C (I -z) (6)
where det C (M) for a matrix M defined on A × A is the determinant of the matrix M with only the row and columns indexes in C.

Enumerating Eulerian paths

If X n starts with a and ends with symbol b, and has type k then k + δ ba ∈ F n with δ ab the type with all coefficients at zero except the (a, b) coefficient set at one. Let E k the number of Eulerian circuit that can be made from a string of type k of the syntax graph. We use the following definitions: k * is the matrix whose (a, b) coefficients is k ab ka .

Theorem 1 Let a type k such that k + δ ba ∈ F n (thus X n ends with symbol b), we have

E k = B k+δ ba det A-{a} (I -k * )(1 + O( 1 √ n )) (7)
and

N a k = B k det A-{b} (I -k * )(1 + O( 1 √ n )) (8)
Most of the analysis is in [START_REF] Jacquet | Markov Types and Minimax Redundancy for Markov Sources[END_REF] but for the sake of completeness we again give the proof of this theorem, because it leads to an important insight in the enumeration of types and Eulerian paths. We first concentrate on E k estimate. We denote F a k the number of Eulerian circuit that can be made from a string starting from symbol a with type k after being made cyclic. In other words if k = 0:

F a k = b∈A E k-δ ba . (9)
Let F a (z) its generating function. We also denote F ba k the number of Eulerian paths that can be made starting from symbol a and ending on symbol b, or equivalently the number of Eulerian path starting with edge ba. We have:

F ba k = N a k-δ ba . ( 10
)
Lemma 1 Let f (z) and g(z) be two generating functions. If k is a balanced type:

[z k ]Ff (z)Fg(z) = [z k ]f (z)Fg(z) . (11)
Proof: Since for all balanced type k,

[z k ]f (z) = [z k ]Ff (z), we have: F(f (z)Fg(z)) = Ff (z)Fg(z). 2
Lemma 2 We have the identity

   F a (z) = det A-{a} (I-z) det(I-z) -1 , F ba (z) = F (B(z)z ba ) det A-{b} (I-z) , (12)
and

E k = [z k+δ ba ]B(z)det A-{a} (I -z) , N a k = [z k+δ ba ]B(z)z ba det A-{b} (I -z) . (13)
Proof: Let k be a balanced type. If we take an arbitrary rotation among the B k rotations and we proceed to create the path from symbol a: If k a = 0 there is no path, and therefore k ∈ F * (A -{a}). Otherwise the path creates an anchored sub-Eulerian circuit (i.e., an Eulerian circuit on a subgraph) that will end on symbol a after exhausting all the edges going to and departing from a. If k is the type of this sub-Eulerian circuit then a rotation remains on kk that does not contain symbol a,

thus k -k ∈ F * (A -{a}). Therefore if k / ∈ F * (A -{a}): B k = k-k ∈F * (A-{a}) F a k B k-k . ( 14
)
Thus we have the identity

FB(z) -G A-{a} FB(z) = F a (z)G A-{a} FB(z) . ( 15
)
Concerning the F ba k , we start the rotation from symbol b with the edge ba therefore there are B k-δ ba such rotations, thus we have the identities

B k-δ ba = k-k ∈F * (A-{b} F ba k B k-k , (16)
and

F(B(z)z ba ) = F ba (z)G A-{b} FB(z) . ( 17
)
Thus F a k = [z k ]FB(z)det A-{a} (I -z) F ba k = [z k ]F(B(z)z ba )det A-{a} (I -z) (18)
Since k a balanced type, using lemma 1 achieves the proof. 2

The end of the proof proceeds by the following technical lemma:

Lemma 3 Let a generating function of the form B(z)f (z) = k∈K(A) a k z k . Let k ∈ F n assume that f (z) has no singularities, then the k's coefficient of [z k ]B(z)f (z) satisfies: [z k ]B(z)f (z) = B k f (k * ) + O( 1 n )B k . (19)
Proof: This is a variation of Theorem 1 (iii) in [START_REF] Jacquet | Markov Types and Minimax Redundancy for Markov Sources[END_REF] that we try to generalize with the methods in [START_REF] Pemantle | Asymptotics of multivariate sequences: I. smooth points of the singular variety Journal of Combinatorial Theory[END_REF]. See Appendix. In fact a more careful proof would give an O( 1 n ) error term. 2

Remark When f (z) has singularities one must check via a similar analysis that the contribution of z -k on these singularities does not exceed the order (k * ) -k .

Corollary 1

The probability that a random rotation starting from symbol a gives an Eulerian circuit in a string of length n and type k is equal to det A-{a} (I -k * ) + O( 1 n ). In passing, when k is proportional to 1 this probability tends to 1 V .

Enumeration Eulerian components

To evaluate the number of components we will assume that the symbols A are ordered: A = {a 1 , . . . , a V }.

We denote A i = A -{a 1 , . . . , a i } with the convention A 0 = A. We also have A V = ∅. A rotation on a type k is decomposed the following ways:

• if k a1 = 0 then start the Eulerian circuit on a 2 , etc, on the first a i such that k ai = 0.

• the sub Eulerian circuit anchored at a i is one component and has type k , the remaining type k-k ∈ F * (A i ), if it is non null, we continue the decomposition on a i+1 , etc.

Let k a balanced type and let r be an integer (smaller than or equal to V ). We denote C r k the number of rotations among the B k rotations on k that generate exactly r components. We also denote the enumerating generating function C k (u) = r u r C r k and finally C(z, u) = k∈F * C k (u). Theorem 2 We have the identity

C(z, u) = V i=1 1 + ( G Ai-1 FB(z) G Ai FB(z) -1)u ( 20
)
with the convention G(∅)B(z) = 1.

Proof: We basically show by recursion: if k / ∈ F * (A 1 ) then we have

C k (u) = k-k ∈F * (A1) uF a1 k E k-k (u) . (21)
Going to generating functions we got the equation

C(z, u) = uF a1 (z)G A1 C(z, u) + G A1 C(z, u) , (22)
and the result come by developing the induction (using the fact that G(A 1 )(C(z, u) concerns types in F * (A 1 ) and rotations starting from symbol a 2 . 2

Theorem 3 We have the identity

C k (u) = B k V i=1 (1 + (u -1)d i k ) + O( 1 n )B k (23)
with

d i k = 1 - det Ai-1 (I -k * ) det Ai (I -k * ) . ( 24
)
Remark We have d 1 k → 1 since always det(I -k * ) = 0 (the row sums to 0).

Remark When k is proportional to 1 we have k * = 1 V 1 and det Ai (I -k * ) = i V , thus d i k = 1 i . When V is large C k (u) = B k u exp((u -1)H(V ))G(u)(1 + O( 1 V )) + O( 1 n )B k
where G(u) is the p.g.f of a random variable independent of V . In other words the number of components tends to be the sum of a Poisson variable of mean

H(V ) = log V -γ + O( 1 V ) plus a fixed random variable. Proof: We have C(z, u) = FB(z) V i=1 G Ai FB(z) G Ai-1 FB(z) (1 -u) + u (25)
We identify

G Ai FB(z) = (det Ai (I -z)) -1 thus C k (u) = [z k ]B(z) V i=1 det Ai-1 (I -z) det Ai (I -z) (1 -u) + u (26)
and we end the proof with the application of lemma 3 (but with some care about the singularities of det A i-1 (I-z) det A i (I-z) that we don't detail here). 2

Corollary 2

The average value C k (1) satisfies:

C k (1) = B k V i=1 d i k + O( 1 n )B k . ( 27
)
Remark When k is proportional to 1, C k (1) = B k H(V ) + O( 1 n )B k .
When V is large the average number of components is equivalent to log V .

Analysis of rotations and Eulerian paths in Markov models

In this section we assume that the string X n is generated by a Markov model with transition matrix P.

To simplify we assume that all strings start with the same fixed symbol a. For the sequel of the paper the symbol a is fixed. If k is the type of string X n then P (X n ) = P k .

The number of strings that share the same type k is exactly N a k , thus we have the identity b∈A k+δ ba ∈Fn

N a k P k = 1 , (28)
the sums being split according to the ending symbol of the string, (assuming all strings start with symbol a).

Average logarithm of the number of rotations

We denote L n the average logarithm of the number of rotations that can be done from a random Markov string X n made cyclic. This number is important because it gives the quantity of information that can be carried by a rotation in a Markov string.

Theorem 4

The average quantity of information carried by a rotation in a random Markov string X n is asymptotically equivalent to the entropy H(X n ) of the string.

For c ∈ A we denote k c = d∈A k cd and k c = d∈A k dc respectively the outdegree and indegree of symbol c in the syntaxic graph. Let F n the set of balanced types, i.e. such that ∀c ∈ A : k c = k c , and such (c,d)∈A 2 k cd = n. It is clear that the definition were extended to non positive types then F n would be a coset of the lattice made of types k such that ∀c ∈ A: k c = k c . Therefore F n is the intersection of this coset with the non negative type k such that (c,d)∈A 2 k cd = n. We denote ω V its elementary covolume (the volume between neighboring elements). In order to get into the proof we need several technical lemmas.

Lemma 4 The set F n is a lattice of dimension V 2 -V [5] . Its size a(n) = O(n V 2 -V +1
) and we denote ω the volume of its elementary element.

Proof: The set of integer matrices is embedded in the vector space of real matrices which is a dimension V 2 . The V balance equations are in fact V -1 since any of them can be deduced from the sum of the other. There is a last equation to specify that all coefficients sum to n.

2

We denote F(1) the set of balanced real matrices with positive coefficients that sum to 1. For y a real non negative matrix and s ≥ 0, we denote L(y, s) = (c,d)∈A 2 y c,d log( yc y cd p s cd ). We denote P(s) the matrix made of coefficients (p cd ) s (if p cd = 0 we assume (p cd ) s = 0). We denote λ(s) the main eigenvalues of matrix P(s). Lemma 5 Let s > 0, the maximal value of L(y, s) for y ∈ F(1) -δ ba n is the matrix ỹn (s) which converges to a matrix ỹ(s) ∈ F(1) whose (c, d) coefficient is v c (s)u d (s) p s cd λ(s) with (u c (s)) c∈A and (v c (s)) c∈A being respectively the right and left main eigenvectors of P(s) (with c∈A u c (s)v c (s) = 1). For instance L(ỹ(s), s) = log λ(s) and L(ỹ n (s), s) = L(ỹ(s), s) -1 n (log λ(s) u b (s) ua(s) ).

Proof: See appendix. 2

Proof of Theorem 4: For convenience we will handle only natural logarithms. We have

L n = b∈A L b n with L b n = k+δ ba ∈Fn N a k P k log B k+δ ba = k+δ ba ∈Fn P k B k det A-{a} (I -k * ) log B k+δ ba (1 + O( 1 n )) . ( 29
)
We assume that the order O( 1 n ) is uniform but this is not a necessary assumption since we will only need to consider a shrinking neighborhood around the matrix ỹ(1). Using the Stirling approximation:

k! = √ 2πkk k e -k (1 + O( 1 k ))
and defining (y) = cd y cd log yc y cd we have

L b n = (n + O(1)) 1 n (V -1)V /2 k+δ ba ∈Fn r b (y) exp(nL(y, 1)) (y) +O(log n) (30)
where r b () is a rational function and y is the matrix of coefficients y cd = k cd n . According to Lemma 5 the maximum value of L(y, 1) is log λ(1) = 0 and thus we already have

L b n = O(a(n)) = O(n V 2 -V ). Since ỹ (1)
is the maximum of L(y, 1) over F(1) there exists A > 0 such that ∀y ∈ F(1): L(y, 1) ≤ L(ỹ(1), 1) -A yỹ(1) 2 , and when y + 1 n δ ba ∈ F(1)

L(y, 1) ≤ L(ỹ n (1)) -A y -ỹn (1) 2 (31)
Let B > 0: we define

L b n (B) = n n (V -1)V /2 k+δ ba ∈Fn |y-ỹn |<B log n/ √ n r b (y) exp(nL(y)) (y) (32)
From (31) we have

L b n = L b n (B)(1 + O(n -1
)). Since function L(y, 1) is infinitely derivable and attains its maximum in F(1), which is zero, on ỹn (1) we have for all y ∈ F(1) -

1 n δ ba L(y, 1) = L(ỹ n (1)) -D n (y -ỹn (1)) + O( y -ỹn (1) 3 ) , (33)
where D n () is a positive quadratic form obtained from the second derivative of L(y, 1) on ỹn [START_REF] Tutte | On unicursal paths in a network of degree 4[END_REF]. The value of L n will be attained in the vicinity of the maximum since exp(nL(y, 1)) behaves like a Dirac when n → ∞. Indeed we are in a kind of Saddle point application. Thus, since L(ỹ n (1)) = 1 n log λ(1) u b (1) ua(1) = 0 (λ(1) = 1 and ∀c ∈ A: u a (1) = 1)

L b n (B) = 1 n (V -1)V /2 (1 + O( log 3 n √ n ))
k+δ ba ∈Fn |y-ỹn |<B log n/ √ n r(y)e nDn(y-ỹn(1)) .

(34)

Since 1 n F n is a lattice of degree (V -1)V with elementary volume ωn -V 2 +V and D n () converge to some non negative quadratic form D() on F(1):

L b n (B) = n (ỹ(1)) r(ỹ(1)) n (V -1)V /2 n V 2 -V ω (1 + O( 1 n )) F (1)
e -nD(y-ỹ(1)) dy V (V -1) (35)

We use the property that

F (1) exp(-nD(y -ỹ))dy V (V -1) = 1 n V 2 -V det(πD) (36)
where det(πD) must be understood as the determinant of the quadratic operator πD() on the vector space F(1). Therefore

L n = n (y) 1 det(πD) b∈A r b (ỹ)(1 + O( log 3 n √ n )) . (37)
The same analysis can be done by removing log B k+δ ba and since via (28) we shall get b∈A r b (ỹ(1))

√ det(πD) = 1, we get

L n = n (ỹ(1))(1 + O( log 3 n √ n )) . (38)
We terminate the proof of theorem 4 by the fact that n (ỹ(1)) = H(X n ) (since ∀c ∈ A: u c (1) = 1 and (v c (1)) c∈A is the Markov stationary distribution). 2

Remark The error term in O( log 3 n √ n ) is in fact very rough and could be arranged to be in O( 1 n ). We conjecture that H(X n ) -L n = O(log n) as experimented in the last section.

Average number of rotations

Theorem 5 The average number of rotations in a Markov string X n of length n is equivalent to αλ 2n ( 12 ) for some λ, and α > 0 that can be explicitly computed.

Proof: We proceed equivalently as in the proof of Theorem 4 except that R

n = b∈A R b n with R b n = k+δ ba ∈Fn N a k P k B k+δ ba = k+δ ba ∈Fn P k B 2 k k b + 1 k ba + 1 det A-{a} (I -k * )(1 + O( 1 n )) . (39)
In this paper, we evaluated the number of Eulerian circuits that could be obtained by an arbitrary rotation in a Markovian string given to a Markovian type. We also studied the number of Eulerian components that result from random rotations in Markovian strings. The asymptotic behaviour of those quantities was considered, when the size n of a string X tends to infinity. To simplify we assume that f (z) has no singularities. We apply theorem 3.9 of [START_REF] Pemantle | Asymptotics of multivariate sequences: I. smooth points of the singular variety Journal of Combinatorial Theory[END_REF], the Kernel V is the set of matrices z such that ∀a ∈ A: b∈A z ab = 1, it is of dimension V 2 -V . The function B(z) satisfies all the conditions of theorem 3.9 [START_REF] Pemantle | Asymptotics of multivariate sequences: I. smooth points of the singular variety Journal of Combinatorial Theory[END_REF]. Thus

a k = (z * ) -k f (z * )A(z * )n V -V 2 2 + O((z * ) -k n V -V 2 2 -1 )) , (45)
where z * is the element of V that attains the maximal value of z k and A(.) is a specific function detailed in [START_REF] Pemantle | Asymptotics of multivariate sequences: I. smooth points of the singular variety Journal of Combinatorial Theory[END_REF]. The maximal value of z k on V satisfies for all a ∈ A 2 the quantities ∂ ∂z ab log z k must be identical.

Since ∂ ∂z ab log z k = k ab z ab , (46)
we have z * = k * . Since the estimate of B k is given by replacing f (z) by 1:

B k = (z * ) -k A(z * )n V -V 2 2 + O((z * ) -k n V -V 2 2 -1 )) , (47)
We get the claimed result. 2

Remark When f (z) has singularities one must check via a similar analysis that the contribution of z -k does not exceed the order (k * ) -k .

Proof of lemma 5:

We have for all (c, d) ∈ A 2 the gradient matrix is

∂ ∂y cd L(y, s) = log y c y cd p s cd . (48)
The maximum on F(1) or F(1) -1 n δ ba must be member of the vector space generated by the matrix 1 (made of all one), and the matrices A j , j ∈ A, the coefficients of A j are all zeros excepted 1 on the jth column and -1 on the jth row, and zero on the diagonal. These matrices are the orthogonal matrices that define F(1) (or a translation of it). Membership to this vector space is equivalent to the fact that

Thus L(ỹ(s), s) = log λ(s) and L(ỹ n (s), s) = (1 -1 n ) log λ(s) + 1 n log ua(s) u b (s) . 2

Fig. 1 :

 1 Fig.1: Left: type of "to be or not to be that is the question", Right: Difference H(Xn) -Ln (in bits) versus n.

∂ 2 y

 2 ∂y cd L(y, s) must be of the form α + z c -z d for some α and (z c ) c∈A , which is equivalent to the fact that some λ and (x c ) c∈A . From the fact that d∈A y cd yc = 1 we get λ = λ(s) and (x c ) c∈A = (u c (s)) c∈A , i.e. λx c = d∈A p s cd x d . Consequently L(y, s) = (c,d)∈A cd -y c ) log(x c )

Information in Strings: Enumeration of Eulerian Paths and Eulerian Components in Markov Sequences

The main factor P k B 2 k leads to a factor exp(2nL(y, 1 2 )). Consideration on the order of the n factors leads to the estimate in αλ 2n ( 1 2 ). 2

Remark This average is purely theoretical, in practice it is hardly possible to calculate this result when n is large, since the most important and decisive contributions come from strings with extremely low probabilities.

Eulerian components

Theorem 6 The probability p n that a random rotation on a Markov string of length n builds a full Eulerian path (i.e. results in a single Eulerian component) tends to det A-{a} (I -P) when n → ∞. The p.g.f of the number of Eulerian components built from a random rotation C n (u) tends to

Remark Notice that the first factor is u since det(I -P) = 0.

Proof: We have

Then the proof proceeds as with the proof of theorem 4 but replacing (y) with det A-{a} (I -y * ), thus we get

and we get the result thanks to the identity ỹ(1) = P. Similarly Theorem 4, to get our result. 2

Experimental results and Conclusion

Figure 1 shows I(X n , Y n ), the discrepancy between L n and H(X n ) versus the string length n, when X n is generated by a Markov process of memory 1 based on the statistics of the phrase "to be or not to be that is the question". As predicted the conjectured is quite sub-linear and seems to be in log n as assumed.

Each point has been simulated 100 times.

Appendix

Proof of lemma 3: We give only a sketch. We proceed via multivariable cauchy: