Seyed Morteza Tayebi

Ayoub Saeidi

Mohamad Fashi

Seyed Pouya

Jafari Hokmabadi

Abedin Khosravi

Hossein Shirvani

Mehdi Ahmadian

Abderraouf Ben Abderrahman

Anthony C Hackney

Hassane Zouhal
email: hassane.zouhal@univ-rennes2.fr.

Plasma retinol-binding protein-4 and tumor necrosis factor-α are reduced in postmenopausal women after combination of different intensities of circuit resistance training and Zataria supplementation

Keywords: Strength training, Herbal supplementation, Retinol-binding protein-4 (RBP-4), Tumor necrosis factor-α (TNF-α), Women

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Introduction

Menopause in women is a reproductive phenomenon associated with aging. Studies have shown that it significantly increases the likelihood of diseases, such as obesity, metabolic syndrome, type 2 diabetes mellitus, and cardiovascular disease in women [START_REF] Zuo | Prevalence of metabolic syndrome and factors associated with its components in Chinese adults[END_REF].

The development of abdominal obesity and as a result increased incidence of metabolic syndrome and type 2 diabetes mellitus in women after menopause are mainly associated with the loss of the protective role of estrogen and a relative increase in androgen circulation [START_REF] Stefanska | Association of folliclestimulating hormone and sex hormone binding globulin with the metabolic syndrome in postmenopausal women[END_REF]. On the other hand, intra-abdominal fat produces substances that affect inflammatory responses, cause insulin resistance, and increase cardiovascular diseases' risk. Some of the substances that are directly produced by fat cells are known as adipokine [START_REF] Stachowiak | Metabolic disorders in menopause[END_REF]. Specific to this study, retinol-binding protein-4 (RBP-4), an adipokine, is derived from adipocyte. Studies have confirmed the relationship between the circulation level of RBP-4 and various aspects of obesity [START_REF] Haider | Serum retinolbinding protein 4 is reduced after weight loss in morbidly obese subjects[END_REF][START_REF] Jia | Association of Serum retinol binding protein 4 and visceral adiposity in chinese diabetes and non-diabetes subjects[END_REF] and metabolic syndrome [START_REF] Jia | Association of Serum retinol binding protein 4 and visceral adiposity in chinese diabetes and non-diabetes subjects[END_REF]. Furthermore, increased RBP-4 in the serum increases hepatic expression of the gluconeogenic enzyme (phosphoenol-pyruvate carboxykinase) and causes impairing insulin signaling in muscles. Therefore, it would be beneficial to reduce insulin sensitivity (resistance) and risk factors related to cardiovascular diseases in high-risk patients if RBP-4 can be reduced [START_REF] Bukovská | Effects of a combination of thyme and oregano essential oils on TNBS-induced colitis in mice[END_REF].

Tumor necrosis factor-α (TNF-α) can regulate many cellular and biological processes, such as immune function, cell differentiation, proliferation, apoptosis, and cell metabolism. TNFα released from lipid tissue can lead to dyslipidemia and other metabolic disorders as well as through stimulation of insulin resistance of adipocytes can impair glucose uptake in adipocytes [START_REF] Cawthorn | TNF-α and adipocyte biology[END_REF]. Studies have reported increased levels of TNF-α in obesity, while weight loss through diet and/or exercise has been associated with a reduction in levels of TNF-α [START_REF] Olszanecka-Glinianowicz | Body fat changes and activity of tumor necrosis factor α system-a 5-year follow-up study[END_REF].

Healthcare providers recognize the importance of exercise in reducing inflammation and inflammatory factors [START_REF] Lujan | Physical activity, by enhancing parasympathetic tone and activating the cholinergic anti-inflammatory pathway, is a therapeutic strategy to restrain chronic inflammation and prevent many chronic diseases[END_REF]. Recently, resistance training or weight training has become a prevalent exercise method to improve health and body com-position (i.e., increase muscle mass) [START_REF] Saremi | The Effect of resistance training on serum myostatin level and insulin resistance in obese-overweight men 11[END_REF]. Furthermore, it has been suggested that high-intensity resistance training may compensate for the reduction in bone mass and muscle strength in age-related changes found in postmenopausal women [11,[START_REF] Karaarslan | Effects of different intensity resistance exercise programs on bone turnover markers, osteoprotegerin and receptor activator of nuclear factor kappa β ligand in post-menopausal women[END_REF].

On the other hand, the combination of diet and exercise adjustment can have a positive effect on health-related quality of life more than anyone intervention alone [START_REF] Imayama | Dietary weight loss and exercise interventions effects on quality of life in overweight/obese postmenopausal women: a randomized controlled trial[END_REF]. In particular, nutrient supplementations are extremely popular and are used as adjunctive therapy in the aging population. One such popular supplement in some cultures is Zataria. Zataria is a plant belonging to the Labiatae family with the scientific name Zataria multiflora [START_REF] Mohammadi | Zataria multiflora increases insulin sensitivity and PPARγ gene expression in high fructose fed insulin resistant rats[END_REF]. Zataria is used as a tea, spice, and herbal medicine for treatment of asthma and is used as an antiseptic of the stomach and urinary tract as well as diuretic and anti-inflammatory in traditional medicine [START_REF] Anvari | The effect of thyme (Zataria multiflora Boiss.) decoction on pregnancy in rats[END_REF]. Research showed that compounds containing Zataria are able to inhibit inflammatory reactions' media-tors and serve as an antioxidant, antispasmodic, and antipain [START_REF] Sodouri | Effects of Zataria multi-flora, Shirazi thyme, on the severity of premenstrual syndrome[END_REF]. Recently, in animal models, thymol and carvacrol, found in Zataria, have been demonstrated as bioactive compounds, which reduce pro-inflammatory and increasing antiinflammatory cytokines [START_REF] Bukovská | Effects of a combination of thyme and oregano essential oils on TNBS-induced colitis in mice[END_REF][START_REF] Ocaña | Effects of thyme extract oils (from Thymus vulgaris, Thymus zygis, and Thymus hyemalis) on cytokine production and gene expression of oxLDL-stimulated THP-1macrophages[END_REF]. It seems that the com-bination of high-intensity resistance exercises and Zataria supplementation may have beneficial effects on systemic inflammation, antioxidant defense and ultimately induces well-being [START_REF] Tayebi | Intense circuit resistance training along with Zataria multiflora supple-mentation reduced plasma retinol binding protein-4 and tumor necrosis factor-α in postmenopausal females[END_REF]19]. However, to the best of our knowledge, the effects of low to moderate intensities of resistance training accompany with Zataria supplementation have not been studied yet. Therefore, the purpose of the present study was to investigate the potential additive effects of 8 weeks of low (35% 1-RM) and moderate (55% 1-RM) intensities of circuit resistance training along with Zataria supplementation on plasma RBP-4 and TNF-α levels in postmenopausal women.

Measurements

Eighty-eight postmenopausal women volunteered from throughout public centres and offices in Tehran. A gynecologist examined all volunteers prior to entry into the study. Before participating in the study, all procedures were explained to volunteers and written informed consent was obtained. Among the 88 potential participants, 72 within the age range of 48-65 years were selected after being reviewed by a cardiologist. These selected research participants were divided into six equal groups (for each, n = 12): Control [START_REF] Mcginley | Influence of training intensity on adaptations in acid/base transport proteins, muscle buffer capacity, and repeated-sprint ability in active men[END_REF], Training 35% (T35%), Training 55% (T55%), Zataria [START_REF] Özgünen | Effect of hot environmental conditions on physical activity patterns and temperature response of football players[END_REF], Zataria/Training 35% (ZT35%), and Zataria/Training 55% (ZT55%). This study was approved by the Education Ethical Research Committee of Babol University of Medical Sciences, Iran and was conducted in accordance with the Declaration of Helsinki.

Anthropometric measurements

The following anthropometric measurements were recorded: height (accuracy of 0.1 cm; Hotain, UK) and body mass (0.1 kg; Tanita BF683W, Munich, Germany). Body mass index was calculated as weight (kg) divided by height 2 (m). An estimation of total body fat and muscle mass was obtained by bio-impedance body composition analyzer (Medigate Company Inc., Dan-dong Gunpo, Korea).

GC-MS

Zataria leaves were collected in March around the gardens of Eghlid, Shiraz, Iran. Then, they were dried in the shade for 10 days. Zataria leaves were dried in an oven for 48 h at a temperature of 32 °C and then made into powder in a Chinese pounder. 50 g of the sample powder was extracted for 3 h using water distillation method in Clevenger and at a boiling point. The essential was filtered, then dried on anhydrous sodium sulfate, and finally transferred into a glass container with a lid closed and stored at 4 °C. The essential yield was calculated as dry essential volume divided by the initial dry powder weight multiplied by 100. The yield cal-culated was 3% [START_REF] Saei-Dehkordi | Chemical composition of essential oils in Zataria multiflora Boiss. From different parts of Iran and their radical scavenging and antimicrobial activity[END_REF]. To isolate and identify components of Zataria essential oil, GC-MS was used. GC-MS analysis was done using the Agilent 5975 mass spectrometer detector (MSD) coupled with gas chromatography model of Agilent USA GC 7890A MS 5975C. The column used was of welded silica HP-5 (5% phenyl, 1.95% polydimethyl siloxane) with a profile of 30 × 0.25 mm 2 id and the film thickness of 0.25 μm. Helium was used for carrying gas and flow rate of the gradient was 1 ml/min. The thermal program used was as follows: first column temperature was set at 50 °C for 5 min, then increased at 3 °C per minute to 240 °C, then increased at 5 °C per minute to 300 °C, and finally set for 3 min at this temperature. Essential oil samples stored were diluted by nhexane at a ratio of 1:10 and 1 μl was injected in gas chromatography. The temperature at the injector and detector was fixed at 290 °C. Compounds in Zataria essential oil were identified using fragmentation pattern in the database of wiley7n land NIST08 and also using retention time in chromatography column [START_REF] Tayebi | Intense circuit resistance training along with Zataria multiflora supple-mentation reduced plasma retinol binding protein-4 and tumor necrosis factor-α in postmenopausal females[END_REF]19]. For each combination, the ratio of the level below the peak was determined to the total levels below the peak of all compounds and the results are summarized in Table 1.

Supplementation 500 mg of dry Zataria [START_REF] Tayebi | Intense circuit resistance training along with Zataria multiflora supple-mentation reduced plasma retinol binding protein-4 and tumor necrosis factor-α in postmenopausal females[END_REF]19] leaves and powder was placed in capsules via a Chinese oven and readied for use. In a double-blinded fashion, the ZG, ZT35%, and ZT55% groups received 500 mg of Zataria every day after breakfast, one capsule (500 mg) with 100 ml of water. CG, T35%, and T55% also consumed 100 ml of water with placebo capsules (500 mg capsule) after breakfast.

Resistance training

Exercise instructions were given before the circuit resistance training began to insure that participants became familiar with the environment and exercises (1 week). Next, one repetition maximum (1RM) for each of the exercises was determined. Exercises included:

(1) squat, (2) chest press, (3) leg press, (4) standing military press, (5) knee extension, (6) seated cable rowing, (7) knee Curl, (8) biceps curl, (9) standing calf raise, [START_REF] Saremi | The Effect of resistance training on serum myostatin level and insulin resistance in obese-overweight men 11[END_REF] triceps press, (11) back extension, and ( 12) crunch [START_REF] Tayebi | Intense circuit resistance training along with Zataria multiflora supple-mentation reduced plasma retinol binding protein-4 and tumor necrosis factor-α in postmenopausal females[END_REF]19]. The 1RM for each exercise was calculated using the Brzycki equation method. Participants in T35% and ZT35% performed the training movements with 35% of 1RM, whereas participants in T55% and ZT55% per-formed the same training movements with 55% of 1RM, for 8 weeks (three sessions per week). Each resistance training session included 5 min warm-up followed by 12 stations with duration of 30 s for each station. In each session, two sets (turn) of 12 movements were carried out such that between each turn, there was a 3 min active rest [START_REF] Ghanbari-Niaki | Effects of circuit resistance training with crocus sativus (saffron) supplementation on plasma viscosity and fibrinogen[END_REF].

Blood samples were taken from a forearm vein in a sitting position; specifically, 48 h before the beginning of training procedure and 48 h after the last training session and specimens were placed in test tubes containing EDTA and sterile tubes without anticoagulant. Then, specimens were spun at 3000 rpm for 10 min by centrifugation. Obtained plasma was used for the separate analysis of RBP-4 and TNF-α. RBP-4 and TNF-α of plasma were measured using commercial ELISA kits of Abcam Co., (Cat. No. ab108897 and ab46087) with a sensitivity of 6 ng/mL and 10 pg/mL by ELISA method, respectively.

Statistics

Kolmogorov-Smirnov and Leven tests were used to deter-mine the normality of data and homogeneity of variances, respectively. To compare data of the six groups, analysis of covariance (ANCOVA) was used. A paired t test was used to assess the differences in measured variables pre-and post study in each group separately. Alpha was considered at the level of 0.05. All data were analyzed using software SPSS.

Results

Demographic parameters

Body mass and BMI decreased significantly at the post-test in ZT35%, T55%, ZT55% compared to the pre-test (Table 2). Similarly, the percentage of body fat decreased significantly after the training period in ZG, T35%, ZT35%, T55%, ZT55% (Table 2). In contrast, muscle mass increased significantly in T55% and ZT55% compared to pre-test (Table 2). After training, the percentage of body fat was significantly lower (P < 0.05) in all groups in comparison with CG, except for ZG and T35% (Table 2). Similarly, muscle mass was significantly higher in ZT55% in comparison to ZG (Table 2).

RBP-4

RBP-4 significantly decreased (P < 0.05) at post compared to pre in all groups except for ZG (P = 0.201) and CG (P = 0.596) (Fig. 1). There was no significant difference in RBP-4 between any of the groups at pre (P = 0.860). A significant difference was noted in RBP-4 between groups at post (P < 0.001), with RBP-4 which was significantly lower in all groups as compared to CG post, except for ZG (P = 0.819) (Fig. 1). Moreover, significantly lower values were noted in T55%, ZT35%, and ZT55% as compared to ZG at post (Fig. 1). However, we found no evidence that performing the circuit resistance training at 55% of 1RM was more effective than performing the movements at 35% of 1RM in decreasing post-RBP-4 levels (T35% and T55%, P = 0.931). Similarly, no significant effect was found between the combinations of two training intensities with Zataria supplement in decreasing RBP-4 post (ZT35% and ZT55%, P = 0.996).

TNF-α

TNF-α values significantly (P < 0.05) decreased post as compared to pre in the ZT35% and ZT55% groups (Fig. 2). There was no significant difference in TNF-α between any of the groups at pre (P = 0.300). A significant difference was noted in TNF-α between groups at post (P = 0.002), with TNF-α which was significantly lower in ZT55% as compared to CG and T35% at post (Fig. 2). However, again we found no evidence that performing the circuit resistance training movements at 55% of 1RM was more effective than performing these movements at 35% of 1RM for TNF-α post (T35% and T55%, P = 0.901). Similarly, no significant effect was found between combinations of the two training intensities with Zataria supplement in decreasing TNF-α post (ZT35% and ZT55%, P = 0.187).

Discussion

In this study, we examined the effects of 8 weeks of circuit resistance training at different intensities along with Zataria supplementation on plasma levels of RBP-4 and TNF-α in postmenopausal women. The main findings show that per-forming circuit resistance training, both at low and moderate intensities, and consuming Zataria induce positive effects on RBP-4 and TNF-α levels in postmenopausal women. In addition, a significant decrease was observed in BMI and the percentage of body fat. The reduction in BMI and body fat percent associated with potential anti-inflammatory effects of exercise [START_REF] Petersen | The anti-inflammatory effect of exercise[END_REF] that makes circuit resistance training and Zataria supplementation as an appropriate method for the improvement of health.

In the present study, circuit resistance training, both at low and moderate intensities, induced a significant decrease of BMI and percentage of body fat, and an increase in muscle mass. These results are in accordance with other previous studies [START_REF] Bryner | Effects of resistance vs. aerobic training combined with an 800 calorie liquid diet on lean body mass and resting metabolic rate[END_REF][START_REF] Prabhakaran | Effect of 14 weeks of resistance training on lipid profile and body fat percentage in premenopausal women[END_REF][START_REF] Ghahramanloo | The effect of concurrent training on blood lipid profile and anthropometrical characteristics of previously untrained men[END_REF] investigating the effect of high-intensity circuit resistance training.

Our findings showed that the RBP-4 values significantly decreased at the post as compared to pre in all groups except for ZG. Additionally, RBP-4 was significantly lower in all groups as compared to CG in the post except for ZG. Increased plasma RBP-4 is one of the factors associated with the development of glucose intolerance disorder and diabetes in human and animal samples. Research has shown that exercise training in sedentary people and those with impaired glucose tolerance and diabetes caused a reduction in RBP-4 concentration and overall body weight [START_REF] Saremi | The Effect of resistance training on serum myostatin level and insulin resistance in obese-overweight men 11[END_REF]. Although, some conflicting results have been reported on changes in plasma levels of RBP-4 after doing different exercise training pro-grams. For instance, Souri, Hasani Ranjbar and Vahabi [START_REF] Souri | Effects of interval aerobic training on serum RBP4 and insulin resistance in type 2 diabetic patients[END_REF], in patients with type-2-diabetes showed that 8 weeks of alternative aerobic exercise programs with the intensity of 70-85% of maximum heart rate significantly decreased the RBP-4, cholesterol, triglycerides, and low-density lipoprotein (LDL), as well as a significant relationship, was observed between baseline levels and changes in RBP-4. In addition, low levels of RBP-4 and triglycerides were associated with improved insulin resistance and increased insulin sensitivity [START_REF] Souri | Effects of interval aerobic training on serum RBP4 and insulin resistance in type 2 diabetic patients[END_REF]. But another study on obese women showed that 12 weeks of combined aerobic and resistance exercise (aerobic exercise with an intensity of 60-75% of maximum heart rate and resistance exercise with an intensity of 50-70% of one repetition maximum) did not lead to significant changes in plasma levels of RBP-4 [START_REF] Choi | Effect of exercise training on A-FABP, lipocalin-2 and RBP4 levels in obese women[END_REF]. It seems changes in plasma levels of RBP-4 (reduction or no change) in adaptation to training is more closely related to changes in body weight and fat percentage [START_REF] Souri | Effects of interval aerobic training on serum RBP4 and insulin resistance in type 2 diabetic patients[END_REF][START_REF] Choi | Effect of exercise training on A-FABP, lipocalin-2 and RBP4 levels in obese women[END_REF][START_REF] Graham | Retinolbinding protein 4 and insulin resistance in lean, obese, and diabetic subjects[END_REF], BMI, waist to hip ratio [START_REF] Choi | Effect of exercise training on A-FABP, lipocalin-2 and RBP4 levels in obese women[END_REF], as well as lipid profile [START_REF] Choi | Effect of exercise training on A-FABP, lipocalin-2 and RBP4 levels in obese women[END_REF][START_REF] Graham | Retinolbinding protein 4 and insulin resistance in lean, obese, and diabetic subjects[END_REF]. Along this line, a significant decrease in BMI and percentage of body fat was observed in the training groups that can, at least in part, explain the results of the current study.

Our finding also showed that the RBP-4 was significantly lower in T55%, ZT35%, and ZT55% as compared to ZG post (Fig. 1). Zataria has several active compounds; the main ones are thymol and carvacrol. The cumulative beneficial effects of the supplement are attributed to these compounds that can lead to the prevention of fat deposit in vessels [START_REF] Okazaki | Human platelet aggregation inhibitors from thyme (Thymus vulgaris L.)[END_REF]. Furthermore, Zataria has terpenoids and flavonoids com-pounds like other plants of Lamiaceae family, which in addition to reducing hyperglycemia effect can cause a reduction in overall levels of blood triglycerides too. In addition, Zataria has a very high antioxidant feature, which aids in a reduction in blood lipid as well as plays a role in inhibiting LDL production [START_REF] Ocaña | Effects of thyme extract oils (from Thymus vulgaris, Thymus zygis, and Thymus hyemalis) on cytokine production and gene expression of oxLDL-stimulated THP-1macrophages[END_REF]. Thus, considering the significant relationship between RBP-4 changes and potential changes in levels of blood triglycerides, it seems that Zataria may also reduce dyslipidemia, which can cause a reduction in RBP-4 levels.

Our finding showed that the TNF-α values significantly decreased post-intervention as compared to pre in the ZT35% and ZT55% groups (Fig. 2). Additionally, TNF-α was significantly lower in ZT55% as compared to CG and T35% at post (Fig. 2). A main attribute of the cytokine TNF-α is a fast acting agent. Increased serum level of this TNF-α induces increase in resting energy metabolism and as a result whole body weight loss occurs. This cytokine acts according to its target tissue receptor, which can vary, so the physiological action depends entirely on the type of target cells. TNF-α is considered as the main factor in disorders related to glucose and insulin metabolism because it is expressed more in lipid tissue and muscles of obese people and may inhibit insulin action in a multitude of tissues (i.e., resistance) [START_REF] Saghizadeh | The expression of TNF alpha by human muscle. Relationship to insulin resistance[END_REF]. However, it remains unclear whether the chronic presence of low concentration of TNF-α may inhibit insulin action as changes observed in insulin sensitivity have been detected as the main factor associated with RBP-4 changes. In general, it is well established that the role of exercise in increasing insulin functionality occurs by reducing the accumulation of intracellular triglyceride, increased fatty acid oxidation, and improving mitochondrial biogenesis [START_REF] Saghizadeh | The expression of TNF alpha by human muscle. Relationship to insulin resistance[END_REF]. It is thought that TNF-α produced through fatty tissue may locally affect insulin act by the autocrine or paracrine method and then be released into the serum where it may have endocrine actions [START_REF] Hotamisligil | Tumor necrosis factor α: a key component of the obesitydiabetes link[END_REF]. Exercise can directly reduce cytokines' production in lipid tissue, muscle, and mononuclear cells and thus indirectly increase insulin sensitivity and improve endothelial function. Hence exercise training causes protection against insulin resistance caused by TNF-α. It also leads to reduced levels of TNF-α and IL-6 and increased levels of anti-inflammatory cytokines, such as IL-4 and IL-10 [START_REF] Bouskela | Effects of resistance training on cytokines[END_REF]. Reduced TNF-α as a result of exercise training may also be a result of IL-6-dependent and non-dependent paths [START_REF] Fiers | Tumor necrosis factor characterization at the molecular, cellular and in vivo level[END_REF][START_REF] Keller | Exercise normalises overexpression of TNF-α in knockout mice[END_REF]. Weight loss caused by doing training leads to reduced volume and the number of adipocytes and reduced number of macrophages and endothelial cells. Increased production of anti-inflammatory mediators by adipocytes and production of hepatic fibrinogen and other pre-inflammatory mediators are other results of weight loss caused by training. However, the mechanism of the effect of circuit resistance training on reducing inflammation is not fully known. It seems that intensive resistance training results in a significant reduction in plasma levels of TNF-α, even if no tangible and significant change is observed in body composition and body fat mass. Results showed that doing a combination of aerobic and resistance training with high intensity for 12 months as compared to aerobic training alone reduces the levels of TNF-α more [START_REF] Balducci | Antiinflammatory effect of exercise training in subjects with type 2 diabetes and the metabolic syndrome is dependent on exercise modalities and independent of weight loss[END_REF]. An important feature of the pre-sent study is that the body weight of subjects is the same as before the start of the study. This makes the effect of exercise separate from the effect of changes in body weight-mass alone. Also, there is a relationship between physical activity and physical fitness and lower inflammation independent of adiposity (i.e., obesity) [START_REF] Pollock | Resistance exercise in individuals with and without cardiovascular disease: benefits, rationale, safety, and prescription an advisory from the committee on exercise, rehabilitation, and prevention, council on clinical cardiology[END_REF]. Thus, according to the present study, which used matched subjects, it seems there are other mechanisms by which physical activities cause a reduction in inflammation.

In addition to the positive role of resistance training on the reduction of inflammatory factors, such as reducing the values of TNF-α and RBP-4 in-group exercise, when Zataria intervention was added to the study; the effect of exercise was essentially doubled and caused further reduction in these two indices. Zataria is a herb that has a very high antioxidant feature, and its anti-inflammatory and oxidative anti-stress effects have been well reported [39]. In regard to the reduced levels of RBP-4 and TNF-α, in the present study, it seems that consuming Zataria essence along with doing resistance exercise can be effective in reducing inflammatory and injury-causing factors and the prevention and treatment of a variety of chronic inflammatory diseases, such as cardiovascular diseases and diabetes. However, we did find no evidence that the performing movements of circuit resistance training at 55% of 1RM were more effective than that of circuit resistance training at 35% of 1RM in decreasing RBP-4 in post-test. Similarly, no significant effect was found between combinations of two training intensities with Zataria supplement in decreasing both the RBP-4 and TNF-α in post-test. Taken together, these results suggest that improvements in RBP-4 and TNF-α can apparently be obtained by adding a herbal supplement such as Zataria to an exercise program, with no clear discernable effect of exercise intensity.

Conclusion

Our findings showed that there is a positive effect of per

  Plasma retinol-binding protein-4 and tumor necrosis factor-α are reduced in postmenopausal women after combination of different intensities of circuit resistance training and Zataria supplementation S.M. Tayebi, A. Saeidi, M. Fashi, S. Pouya Jafari Hokmabadi, A. Khosravi, H. Shirvani, M. Ahmadian, A.B. Abderrahman, A.C. Hackney, Hassane Zouhal To cite this version: S.M. Tayebi, A. Saeidi, M. Fashi, S. Pouya Jafari Hokmabadi, A. Khosravi, et al.. Plasma retinolbinding protein-4 and tumor necrosis factor-α are reduced in postmenopausal women after combination of different intensities of circuit resistance training and Zataria supplementation. Sport Sciences for Health, 2019, 15 (3), pp.551-558. 10.1007/s11332-019-00544-2. hal-02094918

Table 1

 1 Compounds of Zataria multiflora

	-forming circuit resistance training,

Plasma RBP-4 in pre-and post-test in all groups. CG control group, ZG Zataria group, T35% training 35% group, T55% training 55% group, ZT35% Zataria/training 35% group, ZT55% Zataria/training 55% group. Data are expressed as means ± SD. *P < 0.05, **P < 0.01, ***P < 0.001 versus pre-test in same group. ## P < 0.01, ### P < 0.001 versus CG in post-test. ¥ P < 0.05, ¥¥ P < 0.01 versus ZG in post-test Fig. 2. Plasma TNF-α in pre-and post-test in all groups. CG control group, ZG Zataria group, T35% training 35% group, T55% training 55% group, ZT35% Zataria/training 35% group, ZT55% Zataria/train-ing 55% group. Data are expressed as means ± SD. *P < 0.05 versus pre-test in same group. ## P < 0.01 versus CG in post-test. ¥ P < 0.05 versus T30% in post-test Sport Sci Health. Author manuscript; available in PMC 2019 December 27.

Table 2

 2 Demographics parameters of participants volunteering in the study at pre-and post-test evaluations

	Groups Age (

years) Height (cm) Body mass (kg) BMI (kg/m 2 ) Body fat (%) Muscle mass (kg)

		27.6 ± 2.5	27.6 ± 2.3		26.5 ± 2.5	26.8 ± 2.1		27.7 ± 2.7	28.7 ± 1.6		26.7 ± 2.6	27.8 ± 2		26.8 ± 2.8	28.4 ± 2.2 #		28.0 ± 1.9	29.5 ± 1.9 ###£
		20.6 ± 1.7	20.5 ± 1.6		19.9 ± 1.9	18.6 ± 1.4 # ¥		20.7 ± 2	18.7 ± 1.2 ### ¥		20 ± 1.9	17.5 ± 1.4 ***###		20.1 ± 2.1	17.1 ± 1.9 ***###		21.0 ± 1.4	18 ± 1.1 **###
		27.9 ± 2.2	27.9 ± 1.8		25.6 ± 2.2	25.4 ± 1.6		27.8 ± 2.9	27.0 ± 1.7 *		27.6 ± 2.7	26.6 ± 1.9 #		26.7 ± 3.1	25.6 ± 2.3 #		27.6 ± 2.7	26.4 ± 2.1 ##
		68.7 ± 5.9	68.7 ± 4.7		66.4 ± 6.4	65.9 ± 5.1		69.3 ± 6.7	67.5 ± 3.9		66.8 ± 6.5	64.4 ± 5 #		67.1 ± 7.2	64.3 ± 5.6 #		69.9 ± 5.7	67.0 ± 4 ##
	Control	Pre 56.5 ± 4.2 156.7 ± 2	Post --	ZG	Pre 54.4 ± 3.9 160.9 ± 2.9	Post --	T35%	Pre 57.7 ± 3.6 158.1 ± 2.2	Post --	ZT35%	Pre 55.8 ± 4.5 155.5 ± 2.5	Post --	T55%	Pre 58.0 ± 4.7 158.7 ± 2.4	Post --	ZT55%	Pre 53.8 ± 6 159.2 ± 3.2	Post --	T training, ZG Zataria multiflora	* P < 0.05,	** P < 0.01,	*** P ≤ 0.001 versus control at post-test	# P < 0.05,	## P < 0.01,	### P < 0.001 versus pre-test in same group

Sport Sci Health. Author manuscript; available in PMC 2019 December 27.

Sport Sci Health. Author manuscript; available in PMC 2019 December 27.

£ P < 0.05 versus ZG at post-test Sport Sci Health. Author manuscript; available in PMC 2019 December 27.


J Friedrich

S Gallon

A Pumir

R Grauer

Multi-point fractional Brownian bridges and their applications

We propose and test a method to interpolate sparsely sampled signals by a stochastic process with a broad range of spatial and/or temporal scales. To this end, we extend the notion of a fractional Brownian bridge, defined as fractional Brownian motion with a given scaling (Hurst) exponent H and with prescribed start and end points, to a bridge process with an arbitrary number of prescribed intermediate and non-equidistant points. We demonstrate the validity of our method on a signal from fluid turbulence in a high Reynolds number flow. Furthermore, we discuss possible extensions of the present work to include the non-self-similar character of the signal. The derived method could be instrumental within a variety of fields such as astrophysics, particle tracking, specific tailoring of surrogate data, and spatial planning.

Many non-equilibrium phenomena in physics involve random fluctuations with a wide range of spatial and/or temporal scales [1,2]. The theory of stochastic processes provides a conceptual framework to describe such phenomena [3,4]. The most emblematic example is provided by Brownian motion, which results from random uncorrelated collisions acting on a particle, and can be described by a simple Wiener process [5]. By contrast, several complex systems in nature also involve long-or short-range correlations, which require a description in terms of fractional Brownian motion [START_REF] Lévy | Random functions: general theory with special reference to Laplacian random functions[END_REF][START_REF] Mandelbrot | [END_REF]. Examples include velocity fluctuations in turbulence [8][9][10], magnetic field fluctuations in the solar wind [11], random amoebid motion [12], or heart interbeat fluctuations [13]. In the following, we will restrict ourselves to temporal processes, or more generally to processes depending on only one variable. Fractional Brownian motion (fBm) X(t) is a nonstationary centered Gaussian process, and thus entirely characterized by its covariance

X(t)X(t ) = 1 2 t 2H + t 2H -|t -t | 2H . (1)
which implies that (X(t) -X(t )) 2 = |tt | 2H . The Hurst exponent H determines negative (antipersistent) correlations for 0 < H < 1/2, whereas positive (persistent) correlations prevail for 1/2 < H < 1.

In general, the initial value X(t = 0) of a fBm X(t) is fixed and no further conditions can be imposed during the entire evolution, for 0 < t ≤ t 1 , where t 1 is the final time. Nevertheless, some applications such as the tracking of animal movement [14], information-based financial models [15], the number of neutrons involved in reactor diffusion [16,17], or cosmic ray propagation in astrophysical turbulence [18] require that the stochastic process follows certain constraints.

To be more specific, we focus on the important example of cosmic ray propagation in turbulent magnetic fields [19,20]. In order to overcome the overwhelming problem of resolving the wide range of scales involved in these extremely high Reynolds number astrophysical flows, several methods for generating synthetic turbulent fields were developed in the last decades [21][22][23][24]. Such methods are frequently implemented and used in major cosmic ray propagation codes (see e.g. [18]). To capture large anisotropies due to the geometry of galaxies (spiral arms, outflow regions, bow shocks), synthetic turbulent fields must be embedded in large-scale magnetohydrodynamic (MHD) simulations of the turbulent interstellar or intergalactic plasma. Therefore, in this problem we face two challenges: i.) fBm, which is used to generate synthetic fields, must be constrained to match the values on the numerical grid of the MHD simulation and ii.) scaling properties, represented by the Hurst exponent of fluctuations of the coarse-grained MHD simulation must be determined from sparse grid data, in order to allow for an "optimal stochastic interpolation" of sparsely-sampled data. In this letter, we address precisely these two issues.

We first present a method to construct a fBm X(t), which takes specific values X i at times t i , and discuss its application as an optimal stochastic interpolation of a sparsely-sampled real time signal. Accordingly, we start with the well-known notion of a fractional Brownian bridge [25,26], which is defined as fBm starting from 0 at t = 0, ending at X 1 at t = t 1 , and possessing the same statistical (including scaling (1)) properties as X(t). Such a fractional Brownian bridge (fBb) can be constructed from X(t) according to

X B (t) = X(t) -(X(t 1 ) -X 1 ) X(t)X(t 1 ) X(t 1 ) 2 . (2)
It is possible to generalize this ordinary fBb to an arbitrary number of prescribed intermediate grid points in the following manner: First, we consider the n-times con-ditional moments

X(t)|{X i , t i } = X(t) n i=1 δ(X(t i ) -X i ) n i=1 δ(X(t i ) -X i ) , (3)
X(t)X(t )|{X i , t i } = X(t)X(t ) n i=1 δ(X(t i ) -X i ) n i=1 δ(X(t i ) -X i ) . (4)
We then demand that our bridge process X B (t) is conditional on X i at t i for i = 1, . . . , n, which is equivalent to the process possessing the conditional moments (3)(4).

For a Gaussian process with zero mean and covariance X(t)X(t ) , the conditional moments read

X(t)|{X i , t i } = X(t)X(t i ) σ -1 ij X j , (5)
and

X(t)X(t )|{X i , t i } = X(t)X(t ) (6) -X(t)X(t i ) σ -1 ij -σ -1 ik X k X l σ -1 jl X(t )X(t j ) ,
where we implied summation over equal indices and where σ ij = X(t i )X(t j ) denotes the covariance matrix. As shown in [27], the multi-point fractional Brownian bridge

X B (t) = X(t) -(X(t i ) -X i )σ -1 ij X(t)X(t j ) , (7)
possesses one-and two-point moments which are identical to (5-6) and we thus conclude that X B (t) is the stochastic process X(t) conditioned on points X i at times t i . We indeed obtain 1(a) depicts the multi-point fBb [START_REF] Mandelbrot | [END_REF] for three different Hurst exponents with 16 equidistant fixed points (black).

X B (t k ) = X(t k ) -(X(t i ) -X i )σ -1 ij σ jk = X(t k ) -(X(t i ) -X i )δ ik = X k . Fig.
In order to check that the simulated bridge processes possess the desired properties (1) and (5-6), we have carried out numerical calculations of the second order structure functions S 2 (τ ) = (X B (t + τ ) -X B (t)) 2 for three different Hurst exponents (H = 0.33, 0.5, 0.66) with a total number of N = 32768 total grid points. From these we prescribed Ñ = 32 equidistant points generated from fBm (1) with a Hurst exponent H = 0.5.

The results are shown in Fig. 1(b) and are in agreement with the prediction S 2 (τ ) = |τ | 2H (dashed black lines) for small τ . For such a case, where the prescribed points also follow fBm with Hurst exponent H, we can obtain an explicit formula for S 2 (τ ) from Eq. ( 7), namely

S 2 (τ ) = |τ | 2H (8) -δ τ X(t)X(t i ) σ -1 ij -σ -1 ik σkl σ -1 lj δ τ X(t)X(t j ) ,
where δ τ X(t) = X(t + τ ) -X(t) and where σij = X i X j denotes the covariance matrix of the prescribed points with H.

(τ ) = (X B (t + τ ) -X B (t))
2 calculated from 100 realizations of the multi-point fractional Brownian bridge [START_REF] Mandelbrot | [END_REF] for N = 32768 total grid points and Ñ = 32 prescribed grid points. Dashed lines correspond to the explicit formula (8), whereas thin grey lines correspond to the ordinary scaling of fBm |τ | 2H and extend up to the grid length of the coarse grid d t = 1/32. The prescribed grid points were drawn as fBm with H = 0.5. Accordingly, the fBb with H = 0.5 (orange curve) is the optimal bridge that belongs to the prescribed points. (c) Cost curve Hemp(H) -H for three under-sampled time series Ñ = 128 (fBm with H = 0.33, 0.5, 0.66 for samples 1-3). Each time series is embedded into bridges [START_REF] Mandelbrot | [END_REF] with varying Hurst exponents H and resolution N = 4096. Therefore, the fit for Hemp relies on the left part of (b). FBm was generated by the Davies-Harte method [28].

series {X i , t i } that possess a self-similar part governed by H, the bridge with H = H can be considered as the optimal stochastic interpolation of this time series. Therefore, as already highlighted in the introduction, we are now in the position to describe an optimization procedure that allows us to estimate the Hurst exponents from sparsely sampled time series. The basic idea is to embed a given time series {X i , t i } into fBbs [START_REF] Mandelbrot | [END_REF] with varying Hurst exponents H. For each of these bridges we determine the empirical Hurst exponent H emp as a function of H by fitting the second order structure function up to the smallest time scale of the time series d t, (i.e., fitting only the left part up to dt coarse in Fig. 1(b)). This procedure ensures that we only measure deviations from the scaling |τ | 2H in the interpolated region (grey lines in Fig. 1(b)) and are not directly contaminated by correlations contained in {X i , t i }. We have tested our optimiza- The Hurst exponent for the fBb H = 0.376 was determined from randomized samples of the original turbulent signal such as the one depicted in (b). Due to the self-similarity of these samples, an optimization procedure similar to the one depicted in Fig. 1(c) could be applied. The bridge (orange) is in much better agreement with the self-similar signal (blue) in (b) than with the original signal (blue) in (a).

tion for three different samples of fBm with Hurst exponents H = 0.33, 0.5, 0.66 and Ñ = 128 grid points. Each of the samples was embedded in fBbs with varying Hurst exponents H and N = 4096 grid points. Fig. This effect can be attributed to finite sample sizes, and the corresponding deviations remain rather small, which is quite appealing given the fact that common methods (rescaled range analysis [29] or wavelets [30]) yield erroneous results for such sparsely sampled time series. The proposed method can thus roughly be considered as the extrapolation of self-similar properties of a given time series to finer scales.

In the examples discussed so far, we have systematically chosen the synthetic signal X i , as well as the process X(t) used in Eqs. (1,[START_REF] Mandelbrot | [END_REF], to be normalized in the same manner. In order to apply our optimization to real signals, we examine a turbulent velocity time series obtained from hot wire anemometry in the superfluid high Reynolds von Kármán experiment (SHREK) at CEA-Grenoble [31]. The particular experimental setup is a von Kármán cell with two-counter rotating disks (-0.12 Hz on top, +0.18 Hz on bottom) in normal Helium (see [START_REF] Kharche | iTi Conference on Turbulence[END_REF] for further specifications). The temporal resolution is 50kHz and the attained Taylor-Reynolds number was Re λ = 2737. We applied Taylor's hypothesis of frozen turbulence [START_REF] Monin | Statistical Fluid Me-chanics: Mechanics of Turbulence[END_REF] to relate single-point velocity measurement at time t to scales x = u t, where u is the mean velocity. Furthermore, a key prerequisite for the above mentioned optimization procedure is the standardization of the signal by lim r→L (u(x + r)u(x)) 2 = 2 u(x) 2 = √ 2σ. This standardization ensures the correct large-scale limit of the second-order structure function in Fig. 1(b), which was necessarily fulfilled by the synthetic samples of fBm (1) in the previous study. The blue curve in Fig. 2(a) depicts an extract of the velocity field u(x) standardized by σ of the total signal for around one integral length scale L.

By contrast to the above optimization procedure for data from synthetic samples, the present analysis is complicated by: i.) the existence of different scaling regimes in the flow, namely a dissipative and integral range of scales, and ii.) non-self similar (intermittent) features in the signal. Turning to point i.), we chose sample sizes of length L and determined the subset of points in order to guarantee that their grid length lies within the inertial range of scales (here we choose ≈ 100η). As far as point ii.) is concerned, intermittent features manifest themselves in form of strongly varying H opt for different samples [27]. An example of such an intermittent fluctuation is well visible in the signal shown in Fig. 2(a), at x ≈ 0.8L. In this region our interpolation procedure does not work very well and the bridge process [START_REF] Mandelbrot | [END_REF] has to be generalized to a non-Gaussian process. In this letter, however, we are solely interested in the self-similar part of the signal and thus perform a randomization of Fourier phases of the turbulent signal. A snapshot of the resulting randomized signal is depicted in Fig. 2(b). Strikingly, and contrary to Fig. 2(a), our interpolation procedure leads to much better results than for the original signal. The randomization procedure has effectively suppressed intermittency, and made the signal essentially Gaussian. We fed several samples of this randomized signal into our optimization routine which reduced fluctuations of the Hurst exponent to an extend comparable to the ones observed in synthetic signals [27]. Moreover, we obtain a Hurst exponent of the randomized signal H opt = 0.3786 ± 0.0251 (evaluated from the optimization of 100 snapshots [27]). To put this result into context, we consider the log-normal model of turbulence [START_REF] Kolmogorov | [END_REF]35] which suggests scaling exponents ζ n = n/3µn(n -3)/18 for the scaling of velocity structure functions (u(x + r)u(x)) n ∼ |r| ζn , where µ denotes the intermittency coefficient. Hence, the selfsimilar part in the model is given by H K62 = (2 + µ)/6 and our analysis suggests that µ = 0.2716 ± 0.1506 which is comparable to µ = 0.2913 ± 0.0853 acquired from an analysis of the entire turbulent signal [27].

To conclude, we have presented a generalization of a fractional Brownian bridge to a stochastic process with an arbitrary number of prescribed points. Furthermore, we devised an optimization method which allowed us to estimate the Hurst exponent of a sparsely sampled time series. Our method has proven reliable even in the presence of strong anomalous fluctuations, i.e., non-self-similar features, at small scales. In order to address such features, which are visibly not captured by the multi-point fBb in Fig. 2(a), it will be a task for the future to construct multi-point bridge processes with multiscaling properties (i.e., which are non-self-similar and potentially possess a dissipative scale [36,37]). A generalization of the bridge process [START_REF] Mandelbrot | [END_REF] to an arbitrary number of dimensions is straight-forward and might be of potential interest for the construction of various synthetic fields in several physical contexts. In turbulence, the full spatio-temporal (though non-intermittent) Eulerian velocity field u(x, t) can possibly be reconstructed from a set of Lagrangian trajectories X(y, t) where Ẋ(y, t) = u(X(y, t), t). Latter application could be of considerable interest for particle tracking measurements, which sometimes require a certain knowledge of the flow field in the vicinity of tracer particles [START_REF] Machicoane | Flowing Matter[END_REF]. Furthermore, the optimization procedure may help to shed light into the ongoing discussion about the inertial range power spectrum in the solar wind [11,[START_REF] Boldyrev | [END_REF]. Last, we mention possible applications to the widely different domain of urban decision making, where our method could be 1 Univ. Lyon, ENS de Lyon, Univ. Claude Bernard, CNRS, Laboratoire de Physique, F-69342, Lyon, France 2 Institute for Theoretical Physics I, Ruhr-University Bochum, Universitätsstr. 150, D-44801 Bochum, Germany (Dated: May 18, 2020)

I. CALCULATION OF CONDITIONAL MOMENTS FOR CENTERED GAUSSIAN PROCESS

Although the conditional moments of a multivariate Gaussian distribution are well known, in what follows, we want to give a brief derivation. To this end, we consider the characteristic functional ϕ[α] = e i dtα(t)X(t) .

(

) 1
For a centered Gaussian process X(t), the characteristic functional reduces to (see, for instance [1])

ϕ[α] = e -1 2 dt dt α(t) X(t)X(t ) α(t ) . (2)
The n-point probability density function (PDF)

f n (X 1 , t 1 ; . . . ; X n , t n ) = n i=1 δ(X(t i ) -X i ) , (3)
can thus be re-expressed according to

f n (X 1 , t 1 ; . . . ; X n , t n ) = dk 1 2π . . . dk n 2π e -i n i=1 kiXi e i n i=1 kiX(ti) = dk 1 2π . . . dk n 2π e -i n i=1 kiXi ϕ α(t) = n i=1 k i δ(t -t i ) = dk 1 2π . . . dk n 2π e -i n i=1 kiXi exp - 1 2 n i=1 n i =1 k i k i X(t i )X(t i ) .
Therefore, we obtain a multivariate Gaussian distribution

f n (X 1 , t 1 ; . . . ; X n , t n ) = 1 (2π) n det(σ) e -1 2 X T σ -1 X , (4)
where

X = (X 1 , . . . , X n ) T , (5)
and where

σ ij = X(t i )X(t j ) , (6)
denotes the covariance matrix. Next, we can calculate the n + 1-point quantity,

n i=1 X(t)δ(X(t i ) -X i ) = dk 1 2π . . . dk n 2π e -i n i=1 kiXi X(t)e i n i=1 kiX(ti) = dk 1 2π . . . dk n 2π e -i n i=1 kiXi δϕ[α] δiα(t) α(t)= n i=1 kiδ(t-ti) = i dk 1 2π . . . dk n 2π e -i n i=1 kiXi n i=1 k i X(t)X(t i ) e -1 2 n i=1 n i =1 kik i X(ti)X(t i ) = - n i=1 X(t)X(t i ) d dX i f n (X 1 , t 1 ; . . . ; X n , t n ) , (7)
arXiv:2005.07591v1 [physics.data-an] 15 May 2020 as well as the n + 2-point quantity

n i=1 X(t)X(t )δ(X(t i ) -X i ) = dk 1 2π . . . dk n 2π e -i n i=1 kiXi X(t)X(t )e i n i=1 kiX(ti) = dk 1 2π . . . dk n 2π e -i n i=1 kiXi δ 2 ϕ[α] δiα(t)δiα(t ) α(t)= n i=1 kiδ(t-ti) = dk 1 2π . . . dk n 2π e -i n i=1 kiXi X(t)X(t ) - n i=1 n i =1 k i k i X(t)X(t i ) X(t )X(t i ) e -1 2 n i=1 n i =1 kik i X(ti)X(t i ) = X(t)X(t ) + n i=1 n i =1 X(t)X(t i ) X(t )X(t i ) d 2 dX i dX i f n (X 1 , t 1 ; . . . ; X n , t n ) . , (8)
The n-times conditional moments

X(t)|{X i , t i } = X(t) n i=1 δ(X(t i ) -X i ) n i=1 δ(X(t i ) -X i ) , (9)
X(t)X(t )|{X i , t i } = X(t)X(t ) n i=1 δ(X(t i ) -X i ) n i=1 δ(X(t i ) -X i ) , (10)
thus read

X(t)|{X i , t i } = σ -1 ij X(t)X(t i ) X j , (11)
and

X(t)X(t )|{X i , t i } = X(t)X(t ) -σ -1 ij -σ -1 ik X k X l σ -1 jl X(t)X(t i ) X(t )X(t j ) . (12)
where we made use of the symmetry of the covariance matrix and where we imply summation over equal indices.

II. PROOF FOR MOMENTS OF MULTI-POINT FRACTIONAL BROWNIAN BRIDGE

In this section, we want to proof that the multi-point fBb

X B (t) = X(t) -(X(t i ) -X i )σ -1 ij X(t)X(t j ) , (13)
possesses identical one and two-points moments to the moments fBm process X(t) conditioned on {X i , t i }, i.e., eqs. (11)(12). Therefore, we first calculate the mean of the fBb in eq. ( 13)

X B (t) = X(t) =0 -( X(t) =0 -X i )σ -1 jk X(t)X(t j ) = X i σ -1 jk X(t)X(t j ) , (14)
where we assumed that the fBm possesses zero mean. Next, the correlation function of the generalized fBb reads

X B (t)X B (t ) = X(t)X(t ) -2 X(t)X(t i ) σ -1 ij X(t)X(t j ) + X(t k )X(t l ) σ kl σ -1 ik X(t)X(t i ) X(t )X(t j ) σ -1 jl + σ -1 ik X k σ -1 jl X l X(t)X(t i ) X(t )X(t j ) = X(t)X(t ) -σ -1 ij X(t)X(t i ) X(t )X(t j ) + σ -1 ik X k σ -1 jl X l X(t)X(t i ) X(t )X(t j ) , (15)
where we made use of the identity σ kl σ -1 ki = δ il .

III. VARIATIONS OF THE OPTIMAL HURST EXPONENT

An optimization procedure similar to Fig. 1(c) of the main paper for different synthetic samples drawn as fBm with Hurst index H yields slightly different results for the optimal Hurst exponent H opt . In order to compare to the optimization procedure for the turbulent signal later, we choose 100 synthetic samples of fBm with Hurst exponent H = 0.3786 with a spatial resolution of Ñ = 128 points. Fig. 1(a) depicts the histograms for the optimal Hurst exponent obtained via optimization procedured as discussed in the main paper. The histogram is clearly peaked at the prescribed values H and we obtain H opt = 0.3783 ± 0.013. These slight variations are likely due to finite-size effects. Turning next to the turbulent signal in Fig. 2(a) of the main paper, we first perform a Fourier transform of the entire signal and then randomize Fourier phases in order to get rid of strong (intermittent) correlations. The resulting signal possesses Gaussian statistics which can be verified by computing the PDF of velocity increments δ r u = u(x + r)u(x) at different scales r, as it has been done in Fig. 2(a). For comparison, Fig. 2(b) depicts the velocity increment PDF of the original turbulent signal for the same scale separations r as in (a). As to be expected the PDFs develop pronounced tails at small scales, a key signature of small-scale intermittency. Due to the self-similar property of the randomized signal in Fig. 2(a), the corresponding fluctuations can be treated by the same optimization routine as the synthetic fBm. However, as explained in the main text, one has to take care of the existence of a dissipation scale in both the turbulent and randomized signal. To this end, we consider plateaus in logderivatives of structure functions and determine the small-scale cut-off to be approximately 96η, where η denotes the Kolmogorov microscale. We thus perform the optimization procedure for 100 different sub-samples of the randomized signal. The result is shown in Fig. 1(b) and shows many similarities to the one obtained from synthetic samples in Fig. 1(a), although the standard deviation is slightly higher. In fact, we obtain H opt = 0.3786 ± 0.0251. In order to assess this result, we consider the K62 model of turbulence [2,3], which predicts structure functions The estimated value H opt implies that µ = 0.2716 ± 0.1506. This result, which has been obtained from the optimization procedure of the randomized signal can now be compared to the original turbulent case as follows: First, the velocity increment PDF of the K62 model is accessible via a Mellin transform of the scaling exponents ( 16), which yields [4] f

(δ r u) = 1 2πδ r vu √ ln r b ∞ -∞ dx e -x 2 exp   - ln δru r a √ 2x 2 4b ln r    , (17)
This PDF is now used as a fit for the velocity increment PDFs in Fig. 2(b) after a symmetrization (the PDFs in Fig. 2(b) are strongly skewed, a direct consequence of the turbulent energy transfer). The symmetrization is depicted in Fig. 2(c) and fitting the whole range of scales that were used for the optimization procedure from above, we obtain µ = 0.2913 ± 0.0833. Examples of the corresponding PDF (17) for different scales r are indicated as the dashed curves in Fig. 2(c). Hence, the intermittency coefficient determined from the optimization procedure of the randomized signal and the intermittency coefficient from the turbulent signal agree fairly well. Finally, we applied the optimization procedure to the original turbulent signal in keeping the same range of scales and number of sub-samples as for the randomized signal. The result is shown in Fig. 1(c). Compared to (a) and (b) the optimal Hurst exponent exhibits stronger sample-to-sample fluctuations. The mean value, however, agrees very well with the Hurst exponent of the randomized signal and we obtain H opt = 0.3797 ± 0.0368. In order to further categorize these fluctuations, which can be considered as a direct signature of intermittency (i.e., non-self-similarity), we determined the joint PDF of the optimal Hurst parameter and the energy dissipation rate ε = 2ν ∂u(x) ∂x 2 averaged over each sample. The result is depicted in Fig. 3 (a). Even with limited statistics, we observe a clear negative correlation between energy dissipation and the determined value of the optimal Hurst exponent.

FIG. 2 .

 2 FIG. 2. (a)Turbulent velocity field measurements (blue) in a von Kármán experiment using normal Helium. The number of points N = 16384 corresponds roughly to one integral length scale L. The corresponding fBb (orange) was constructed from Ñ = 64 points of the signal (black) and possesses a resolution of 1024 points. The smallest scale of the bridge process therefore corresponds approximately to the Taylor scale of the flow which ensures that the fBb and the velocity field u(x) possess comparable inertial ranges. Small-scale turbulent fluctuations in the velocity field (blue) cannot be reproduced by the fBb due to its restricting Gaussian properties. The Hurst exponent for the fBb H = 0.376 was determined from randomized samples of the original turbulent signal such as the one depicted in (b). Due to the self-similarity of these samples, an optimization procedure similar to the one depicted in Fig.1(c) could be applied. The bridge (orange) is in much better agreement with the self-similar signal (blue) in (b) than with the original signal (blue) in (a).

FIG. 1 .

 1 FIG. 1. (a) Histogram of the optimal Hurst exponent evaluated from 100 synthetic samples of fBm with H = 0.3786. (b) Same plot as in (a), but now for 100 sub-samples of the randomized turbulent signal (Fig. 2(b) in main paper). (c) Same plot, but now for 100 sub-samples of the turbulent signal (Fig. 2(a) in main paper).

FIG. 2 .

 2 FIG. 2. (a)PDFs of velocity increments δru (log(r/L) = -6.41, -5.60, -4.66, -3.64, -2.75, -1.86, -0.87, 0.12 from top to bottom) of the randomized turbulent signal which has been obtained by random rotation of Fourier phases from the original turbulent signal in (b). The PDFs in (a) exhibit self-similarity across different scales and the randomized signal can thus be used in the described optimization procedure via fBbs. (c) symmetrization of the PDFs in (b). The dashed curves correspond to the prediction of the K62 model of turbulence with µ = and are only shown for PDFs with scale separation r in the inertial range.

(FIG. 3 .

 3 FIG. 3. (a) Joint histogram of energy dissipation rate and Hurst exponent evaluated from 320 different sub-samples of the turbulent signal. The axis of the energy dissipation has been divided by the standard deviation determined from all sub-samples. (b) Histogram of the energy dissipation rate.

  1(c) depicts the minimization of H emp (H) for the three samples. It can be clearly seen that optimal Hurst exponents H opt are recovered with high accuracy, although slight variations (∆H opt ≈ 0.015 from 20 different samples of the same H) between different samples can be observed [27].

applied to model land price fields [40].

We are grateful to the SHREK collaboration [31,32] for providing us with their hot wire anemometry measurements. J.F. acknowledges funding from the Humboldt Foundation within a Feodor-Lynen fellowship and also benefited from financial support of the Project IDEXLYON of the University of Lyon in the framework of the French program "Programme Investissements d'Avenir" (ANR-16-IDEX-0005). S.G.'s visit to Lyon was financially supported by the German Academic Exchange Service through a PROMOS scholarship.

Supplemental Material for "Multi-point fractional Brownian bridges and their applications"