scholarly journals Intra- and interpopulation variability of cysts and adults of Artemia (Branchiopoda: Anostraca) in Siberian populations (morphometry)

2021 ◽  
Vol 6 (2) ◽  
pp. 33-51
Author(s):  
L. I. Litvinenko ◽  
K. V. Kutsanov ◽  
L. F. Razova ◽  
A. Sh. Gadiadullina ◽  
A. G. Gerasimov ◽  
...  
Keyword(s):  
Water Body ◽  
Food Resource ◽  
Commercial Fishing ◽  
Important Indicator ◽  
Mean Values ◽  
Interpopulation Variability ◽  
The Mean ◽  
The Absolute ◽  
Artemia Cysts ◽  
Almost All

The size of Artemia cysts is an important indicator of their value as a food resource; to some extent, it allows to identify populations. The data on cysts of Artemia parthenogenetic populations (diameter, chorion thickness, and presence of spots on a shell), sampled in hyperhaline lakes of Western Siberia in different years, were analyzed, as well as the data on the morphometry of Artemia adults, reared from cysts at the same salinity. Significant intra- and interpopulation variability in the indicators was established. The absolute values of the cyst diameter were in the range of 210–330 µm, the mean values for the samples – 243.5–282.9 µm, the mean values for populations – 257.8–279.6 µm; the absolute values of the decapsulated cyst diameter were in the range of 196–294 µm, the mean values for the samples – 236.5–262.6 µm, the mean values for populations – 239.9–253.2 µm; the absolute values of the cyst chorion thickness were in the range of 3.3–16.9 µm, and the mean values for populations – 6.6–12.4 µm. In the main commercial fishing lakes, which account for about 70 % of the total catch of Artemia cysts in Russia, the cysts had similar mean population sizes: 262–268 µm. The absence of intrapopulation correlation between the cyst diameter and chorion thickness is concluded; so, they cannot serve as reliable indicators, identifying Siberian populations. Statistically significant correlation (r = −0.5) was established between salinity of a mother water body and Artemia embryo diameter. Cyst spotting, not exceeding 5 % in almost all the samples, reached 24 % in cysts of Kuchukskoye Lake. Analysis of morphometric indicators of Artemia adults, reared from cysts, showed as follows: the mean body length (9.27–11.63 mm), abdomen width (0.53–0.69 mm), and distance between eyes (1.36–1.52 mm) were closely correlated with salinity of a water body (r values were of −0.76; −0.62; and −0.67, respectively). Cluster analysis of a set of morphometric indicators of Artemia adults showed the unification of populations based on salinity.

scholarly journals Survey of Phenolic Acids, Flavonoids and In Vitro Antioxidant Potency Between Fig Peels and Pulps: Chemical and Chemometric Approach

Molecules ◽  
2021 ◽  
Vol 26 (9) ◽  
pp. 2574
Author(s):  
Lahcen Hssaini ◽  
Francisca Hernandez ◽  
Manuel Viuda-Martos ◽  
Jamal Charafi ◽  
Rachid Razouk ◽  
...  
Keyword(s):  
Phenolic Acids ◽  
Radical Scavenging ◽  
Fruit Peel ◽  
Mean Values ◽  
Local Cultivar ◽  
Ic50 Values ◽  
The Mean ◽  
Lipid Peroxidation Inhibition ◽  
Almost All

In the present study, chromatic coordinates, phenolic acids, flavonoids and antioxidant capacity assessed by 1,1-diphenyl-2-picrylhydrazyl (DPPH), 2,2′-azino-bis (3-ethylbenzothiazoline-6-sulfonate (ABTS) and lipid peroxidation inhibition capacity (LPIC) essays and their relative IC50 were investigated in 25 fig cultivars growing in Morocco. The aims of this study were to determine (i) the variation in these compounds among light and dark-colored cultivars, (ii) their partitioning between fruit peel and pulp and (iii) to display network connections among these variables. Twelve phenolic compounds (PCs) were isolated in peel extract versus eight in pulp samples. Anthocyanins, mainly cyanidin-3,5-diglucoside and cyanidin-3-O-rutinoside, were the predominant compounds in peels, where the mean concentrations were 75.90 ± 18.76 and 77.97 ± 18.95 µg/g dw, respectively. On the other hand, (−)-epicatechin and cyanidin-3-O-rutinoside were the major compounds in the pulp extracts, where the mean values were 5.23 ± 4.03 and 9.01 ± 5.67 µg/g dw, respectively. A two-dimensional hierarchically clustered heatmap was applied to the dataset to explore correlations in the dataset and similarities between cultivars, without dimensionality reduction. Results showed that anthocyanins, particularly pelargonidin-3-O-rutinoside, cyanidin-3,5-diglucoside and cyanidin-3-O-rutinoside, were the main contributors to the peels’ free radical scavenging capacity. This capacity was particularly higher in the peel of dark-colored figs compared to the fruit pulp. The local cultivar “INRA 1301” showed the most promising phenolic profile due to its very high levels of almost all detected PCs, especially (−)-epicatechin, quercetin-3-O-rutinoside, quercetin-3-O-glucoside, cyanidine-3,5-diglucoside, cyanidine-3-O-rutinoside and pelargonidin-3-O-rutinoside (54.66, 141.08, 35.48, 494.08, 478.66, 12.56 µg/g dw, respectively). Having the darkest figs in the collection (L* = 25.72, c* = 22.09 and h° = 20.99), this cultivar has also combined promising IC50 values, which were of 19.85, 40.58 and 124.78 µg/mL for DPPH, ABTS and LPIC essays, respectively.

scholarly journals Variability of morphometric leaf characteristics of black poplar from the area of Vojvodina

2014 ◽  
pp. 63-72 ◽  
Author(s):  
Dijana Cortan ◽  
Mirjana Sijacic-Nikolic ◽  
Radmila Knezevic
Keyword(s):  
Morphological Differentiation ◽  
Morphometric Parameters ◽  
Specific Gene ◽  
Considerable Variability ◽  
Mean Values ◽  
Interpopulation Variability ◽  
Black Poplar ◽  
Significant Difference ◽  
The Mean ◽  
Two Populations

A morphological study of intra and interpopulation variability of black poplar leaves was conducted in two populations in the area of Vojvodina. Nine morphometric parameters of leaves have been analyzed. The results indicate that there is considerable variability within and between the populations. Morphological differentiation is clearly represented with the average values of analyzed parameters and a statistically significant difference between the mean values. On the basis of the obtained data, it has been concluded that the variability within the two populations is much more expressed than the variability between them. The uniformity of environmental conditions of the studied locations indicates that this variability is a consequence of the specific gene pool of the studied populations.

scholarly journals Intercomparison of slant column measurements of NO<sub>2</sub> and O<sub>4</sub> by MAX-DOAS and zenith-sky UV and visible spectrometers

2010 ◽  
Vol 3 (6) ◽  
pp. 1629-1646 ◽  
Author(s):  
H. K. Roscoe ◽  
M. Van Roozendael ◽  
C. Fayt ◽  
A. du Piesanie ◽  
N. Abuhassan ◽  
...  
Keyword(s):  
Cross Sections ◽  
Measuring Instruments ◽  
Mean Values ◽  
Straight Line ◽  
Tropospheric No2 ◽  
The Mean ◽  
Almost All ◽  
Level Of Agreement ◽  
Selection Of ◽  
Differential Spectroscopy

Abstract. In June 2009, 22 spectrometers from 14 institutes measured tropospheric and stratospheric NO2 from the ground for more than 11 days during the Cabauw Intercomparison Campaign of Nitrogen Dioxide measuring Instruments (CINDI), at Cabauw, NL (51.97° N, 4.93° E). All visible instruments used a common wavelength range and set of cross sections for the spectral analysis. Most of the instruments were of the multi-axis design with analysis by differential spectroscopy software (MAX-DOAS), whose non-zenith slant columns were compared by examining slopes of their least-squares straight line fits to mean values of a selection of instruments, after taking 30-min averages. Zenith slant columns near twilight were compared by fits to interpolated values of a reference instrument, then normalised by the mean of the slopes of the best instruments. For visible MAX-DOAS instruments, the means of the fitted slopes for NO2 and O4 of all except one instrument were within 10% of unity at almost all non-zenith elevations, and most were within 5%. Values for UV MAX-DOAS instruments were almost as good, being 12% and 7%, respectively. For visible instruments at zenith near twilight, the means of the fitted slopes of all instruments were within 5% of unity. This level of agreement is as good as that of previous intercomparisons, despite the site not being ideal for zenith twilight measurements. It bodes well for the future of measurements of tropospheric NO2, as previous intercomparisons were only for zenith instruments focussing on stratospheric NO2, with their longer heritage.

Physiological roles of thyrotrophin-releasing hormone and vasoactive intestinal peptide on the pulsatile secretory patterns of prolactin in pituitary-grafted female rats

1994 ◽  
Vol 142 (3) ◽  
pp. 581-586 ◽  
Author(s):  
A Lafuente ◽  
J Marcó ◽  
A I Esquifino
Keyword(s):  
Vasoactive Intestinal Peptide ◽  
Prolactin Secretion ◽  
Female Rats ◽  
Thyrotrophin Releasing Hormone ◽  
Mean Values ◽  
Pulsatile Secretion ◽  
Releasing Hormone ◽  
Absolute Amplitude ◽  
The Mean ◽  
The Absolute

Abstract Much is known about the fact that thyrotrophin-releasing hormone (TRH) and vasoactive intestinal peptide (VIP) stimulate prolactin secretion but areas of uncertainty remain. This work was undertaken to describe the effects of TRH and VIP on the pulsatile secretion pattern of prolactin, in adult sham-operated and pituitary-grafted hyperprolactinaemic female rats. Two pulses of TRH (1 μg/rat) or one pulse of VIP (20 μg/rat) were given 60 or 120 min after the period of blood sampling. Pituitary grafting increased the mean values of prolactin, absolute amplitude and duration of the peaks and decreased their frequency, compared with control animals. In sham-operated rats, TRH elevated prolactin levels by increasing the absolute and relative amplitudes and duration of the pulses, along with a decrease in their frequency. No priming effects of TRH were observed in this study. Hyperprolactinaemia blunted TRH effects on the pulsatile secretion pattern of prolactin. In sham-operated rats, VIP administration increased the absolute and relative amplitudes of the prolactin peaks. None of the other parameters studied were changed. In pituitary-grafted animals, VIP administration increased the absolute and relative amplitudes of the prolactin peaks but to a lesser extent compared with controls. These data suggest that TRH and VIP affect prolactin pulsatility differentially. The effects of TRH and VIP were blunted to some extent by exposure to previously elevated circulating prolactin levels. Journal of Endocrinology (1994) 142, 581–586

scholarly journals Measurements of CFC-11, CFC-12, and HCFC-22 total columns in the atmosphere at the St. Petersburg site in 2009–2019

2020 ◽  
Author(s):  
Alexander Polyakov ◽  
Anatoly Poberovsky ◽  
Maria Makarova ◽  
Yana Virolainen ◽  
Yuri Timofeyev
Keyword(s):  
Seasonal Variability ◽  
Climate Model ◽  
Independent Data ◽  
Mean Values ◽  
Absolute Value ◽  
Qualitative And Quantitative ◽  
Observational Site ◽  
Retrieval Strategies ◽  
The Mean ◽  
The Absolute

Abstract. The retrieval strategies for deriving the atmospheric total columns (TCs) of CFC-11 (CCl3F), CFC-12 (CCl2F2), and HCFC-22 (CHClF2) from ground–based measurements of IR solar radiation have been improved. We demonstrate the advantage of using the Tikhonov-Phillips regularization approach for solving the inverse problem of the retrieval of these gases and give the optimized values of regularization parameters. The estimates of relative systematic and random errors amount to 7.61 % and 3.08 %, 2.24 % and 2.40 %, 5.75 % and 3.70 %, for CFC-11, CFC-12, and HCFC-22, respectively. We analyze the time series of the TCs and mean molar fractions (MMFs) of CFC-11, CFC-12, and HCFC-22 measured at the NDACC site St. Petersburg located near Saint Petersburg, Russia for the period of 2009–2019. Mean values of the MMFs for CFC-11, CFC-12, and HCFC-22 total 225, 493, and 238 pptv, respectively. Estimates of the MMFs trends for CFC-11, CFC-12, and HCFC-22 account for −0.40 ± 0.07 %/yr, -0.49  ±0.05 %/yr, and 2.12±0.13 %/yr, respectively. We have compared the mean values, trends and seasonal variability of CFC-11, CFC-12, and HCFC-22 MMFs measured at the St. Petersburg site in 2009–2019 to that of 1) near–ground volume mixing ratios (VMRs) measured at the observational site Mace Head, Ireland (GVMR); 2) the mean in the 8–12 km layer VMRs measured by ACE–FTS and averaged over 55–65° N latitudes (SVMR); and the MMFs of the Whole Atmosphere Community Climate Model for the St. Petersburg site (WMMF). The means of the MMFs are less than that of the GVMR for CFC-11 by 9 pptv (3.8 %), for CFC-12 by 24 pptv (4.6 %); for HCFC-22, the mean MMFs does not differ significantly from the mean GVMR. The absolute value of the trend estimates of the MMFs is less than that of the GVMR for CFC-11 (−0.40 vs −0.53 %/yr) and CFC-12 (−0.49 vs −0.59 %yr); the trend estimate of the HCFC-22 MMFs does not differ significantly from that of the GVMR. The seasonal variability of the GVMR for all three gases is much lower than the MMFs variability. The means of the MMFs are less than that of the SVMR for CFC-11 by 10 pptv (4.3 %), for CFC-12 by 33 pptv (6.3 %), and for HCFC-22 by 2 pptv (0.8 %). The absolute value of the trend estimates of the MMFs is less than that of the SVMR for CFC-11 (−0.40 vs −0.63 %/yr) and CFC-12 (−0.49 vs −0.58 %/yr); the trend estimate of the HCFC-22 MMFs does not differ significantly from that of the SVMR. The MMF and SVMR values show nearly the same qualitative and quantitative seasonal variability for all three gases. The means of the MMFs are greater than that of the WMMF for CFC-11 by 22 pptv (10 %), for CFC-12 by 15 pptv (3.1 %), and for HCFC-22 by 23 pptv (10 %). The absolute value of the trend estimates of the MMFs is less than that of the WMMF for CFC-11 (−0.40 vs −1.68 %/yr), CFC-12 (−0.49 vs −0.84 %/yr), and HCFC-22 (2.12 %/yr vs 3.40 %/yr). The MMFs and WMMF values show nearly the same qualitative and quantitative seasonal variability for CFC-11 and CFC-12, whereas the seasonal variability of the WMMF for HCFC-22 is essentially less than that of the MMFs. In general, the comparison of the MMFs with the independent data shows a good agreement of their means within the systematic error of considered measurements. The observed trends over the St. Petersburg site demonstrate the smaller decrease rates for CFC-11 and CFC-12 TCs than that of the independent data, and the same decrease rate for HCFC-22. The suggested retrieval strategies can be used for analysis of the IR solar spectra measurements using Bruker FS125HR spectrometers, e.g. at other IRWG sites of the NDACC observational network.

scholarly journals Analysis of Thermodynamic Parameter Variability in a Chamber of a Furnace for Thermo-Chemical Treatment

Energies ◽  
2021 ◽  
Vol 14 (10) ◽  
pp. 2903
Author(s):  
Magda Joachimiak
Keyword(s):  
Boundary Conditions ◽  
Chemical Treatment ◽  
Mean Value ◽  
Mean Values ◽  
Cylinder Length ◽  
Unsteady Heat Conduction ◽  
The Mean ◽  
The Absolute ◽  
The Difference ◽  
Relative Differences

This paper presents results of research on unevenness of cylinder heating in a furnace for thermo-chemical treatment. Experimental research was conducted with respect to nitriding. Various heating speeds and settings of the fan operation in the furnace were considered. Boundary conditions were calculated in the form of temperature and the heat transfer coefficient (HTC) on the cylinder boundary in four planes along the cylinder length. Calculations were performed with the use of the inverse problem for non-linear and unsteady heat conduction equations. Boundary conditions from individual planes were compared with the mean value of them all. The variability of the calculated boundary conditions (temperature and HTC) along the cylinder length was investigated based on values of the absolute and relative differences for temperature and HTC. Estimates: mean value, mean value from the absolute value and the maximum values for the absolute and the relative differences of temperature and HTC were also calculated. Estimates were the measurements of the unevenness of cylinder heating in a furnace for thermo-chemical treatment. Based on the results of our research, it was found that an increase of the fan rotational speed from 50% to 100%, with the same heating speed, resulted in a significant leveling of temperature in the analyzed planes. The difference in temperature along the cylinder length was reduced from 6.8 °C to 3.3 °C. The increase of the heating speed from 5 °C/min to 10 °C/min resulted in an increase of the unevenness of the cylinder heating. Values of the absolute differences of temperature in the analyzed planes with reference to the mean temperature changed from an interval from −2.7 °C to 2.3 °C to a range from −4 °C to 5 °C. In processes with a heating speed greater than 5 °C/min, more intensive heating in the end part of the cylinder (close to the cylinder) was achieved than it was in other planes. It was proven by temperature values, which were higher, even, by 5.4 °C, and by HTC values, higher by 11.4 W/m2K, when compared with mean values. Obtained results can form the basis for nitriding process optimization.

scholarly journals Toothbrush-Dentifrice Abrasion of Dental Sealants: An In Vitro Study

2021 ◽  
Author(s):  
Angkhana Sangpanya ◽  
Pornpoj Fuangtharnthip ◽  
Vanida Nimmanon ◽  
Praewpat Pachimsawat
Keyword(s):  
Abrasive Wear ◽  
In Vitro Study ◽  
Mean Values ◽  
Flowable Composite ◽  
Dental Sealants ◽  
The Mean ◽  
D Values ◽  
Almost All ◽  
One Way Anova

Abstract Objective This study sought to investigate the toothbrush-dentifrice abrasion of dental sealants. Materials and Methods Weight loss (∆W) and depth loss (∆D) were used as abrasion indicators. Sealant samples from nine products were soaked in dentifrice slurry and abraded by using a toothbrushing machine with a brushing force of 300 g. The mean percentages of ∆W and mean values of ∆D after 24,000 and 48,000 strokes of brushing were compared by using paired t-test. A comparison of these mean values among sealant products was performed by using one-way ANOVA and multiple comparison analysis (Scheffe's test). Results Abrasive wear was observed in all sealants. Teethmate F-1 (Kuraray Noritake, Tokyo, Japan)—a fluoride-releasing unfilled sealant—exhibited the maximum abrasive wear, with ∆W and ∆D values of 1.14% ± 0.37% and 12.84 ± 4.28 µm, respectively. Delton (Dentsply Sirona, Charlotte, North Carolina, United States), a light-cured unfilled sealant, showed the minimum abrasive wear, with ∆W and ∆D values of 0.41% ± 0.09% and 2.93 ± 1.23 µm, respectively. No statistical differences were observed among unfilled sealants except when compared with Teethmate F-1. Similarly, no differences were observed when comparing among filled sealants and flowable composite. Conclusion Abrasive wear occurred in all sealants after brushing with dentifrice. Almost all unfilled sealants showed less wear compared with both filled sealants and flowable composite. However, the low abrasive values of all sealants after brushing with dentifrice implied that there is no clinical significance to this finding.

scholarly journals Pengaruh Macam dan Waktu Aplikasi Bahan Organik pada Tanaman Ubi Jalar (Ipomoea batatas L.) Var. Kawi

Jurnal Agro ◽  
10.15575/166 ◽  
2015 ◽  
Vol 2 (1) ◽  
pp. 15-28 ◽  
Author(s):  
Nur Edy Suminarti ◽  
Susanto Susanto
Keyword(s):  
Organic Matter ◽  
Ipomoea Batatas ◽  
Soil Analysis ◽  
Food Resource ◽  
Growth And Yield ◽  
Application Time ◽  
Urban Waste ◽  
Mean Values ◽  
Main Plot ◽  
Almost All

Diversifikasi pangan merupakan langkah tepat untuk mengantisipasi kondisi rawan pangan.  Hal ini menjadi penting karena setiap tahun luas lahan basah telah mengalami penyusutan sekitar 0,1% dari total luas lahan di Indonesia. Sehubungan dengan hal tersebut, dan dalam upaya untuk memenuhi kebutuhan pangan nasional, maka pemanfaatan ubi jalar sebagai sumber bahan pangan alternatif perlu dilakukan. Penelitian yang bertujuan untuk mendapatkan informasi tentang macam dan waktu aplikasi bahan organik yang tepat telah dilakukan di Desa Landungsari, Kabupaten Malang. Rancangan yang digunakan adalah Petak Terpisah, macam bahan organik ditempatkan pada petak utama, terdiri dari 3 macam, yaitu  pupuk kandang sapi, kompos azolla dan kompos sampah kota.  Waktu aplikasi bahan organik sebagai anak petak, terdiri dari 3 macam, yaitu : 30 hari sebelum tanam, 15 hari sebelum tanam dan bersamaan tanam. Pengumpulan data dilakukan secara destruktif, meliputi  komponen pertumbuhan dan panen, analisis pertumbuhan tanaman dan analisis tanah. Uji F taraf 5% ditujukan untuk menguji pengaruh perlakuan, sedang perbedaan diantara rata-rata perlakuan didasarkan pada nilai BNT taraf 5%. Interaksi nyata terjadi pada sebagian besar parameter yang diamati, dan hasil umbi tertinggi didapatkan pada kompos sampah kota yang waktu aplikasinya dilakukan 30 hari sebelum tanam : 28,03 ton umbi ha-1. Food diversification is a precise step to anticipate food shortage condition. It becomes important because wetland area is decreasing of approximately 0,1% each year of total area in Indonesia. Related to that point, also in order to meet food national demand, utilization of sweet potato as alternative food resource needs to be done. Research aimed to get information about proper kind and application time of organic matter had been done at Landungsari, Malang. The design used was Split Plot, kind of organic matter was placed as main plot, consisting of 3 kinds, ie: cow manure, azzola compost, and urban waste compost. Application time was set as subplot, consisting of 3 times, ie: 30 days before planting, 15 days before planting, and simultaneous with planting. Data collection was carried out destructively, including growth and yield components, plant growth analysis and soil analysis. F test on 5% level intended to test treatment effect, meanwhile difference between mean values based on LSD 5%. Real interaction occurred on almost all parameters observed, moreover the highest tuber yield was on urban waste compost which time application was 30 days before planting : 28,03 ton tuber ha-1.

scholarly journals Intercomparison of slant column measurements of NO<sub>2</sub> and O<sub>4</sub> by MAX-DOAS and zenith-sky UV and visible spectrometers

2010 ◽  
Vol 3 (4) ◽  
pp. 3383-3423 ◽  
Author(s):  
H. K. Roscoe ◽  
M. Van Roozendael ◽  
C. Fayt ◽  
A. du Piesanie ◽  
N. Abuhassan ◽  
...  
Keyword(s):  
Cross Sections ◽  
Measuring Instruments ◽  
Mean Values ◽  
Straight Line ◽  
Tropospheric No2 ◽  
The Mean ◽  
Almost All ◽  
Level Of Agreement ◽  
Selection Of ◽  
Differential Spectroscopy

Abstract. In June 2009, 22 spectrometers from 14 institutes measured tropospheric and stratospheric NO2 from the ground for more than 11 days during the Cabauw Intercomparison campaign of Nitrogen Dioxide measuring Instruments (CINDI), at Cabauw, NL (51.97° N, 4.93° E). All visible instruments used a common wavelength range and set of cross sections for the spectral analysis. Most of the instruments were of the multi-axis design with analysis by differential spectroscopy software (MAX-DOAS), whose non-zenith slant columns were compared by examining slopes of their least-squares straight line fits to mean values of a selection of instruments, after taking 30-min averages. Zenith slant columns near twilight were compared by fits to interpolated values of a reference instrument, then normalised by the mean of the slopes of the best instruments. For visible MAX-DOAS instruments, the means of the fitted slopes for NO2 and O4 of all except one instrument were within 10% of unity at almost all non-zenith elevations, and most were within 5%. Values for UV MAX-DOAS instruments were almost as good, being 12% and 7%, respectively. For visible instruments at zenith near twilight, the means of the fitted slopes of all instruments were within 5% of unity. This level of agreement is as good as that of previous intercomparisons, despite the site not being ideal for zenith twilight measurements. It bodes well for the future of measurements of tropospheric NO2, as previous intercomparisons were only for zenith instruments focussing on stratospheric NO2, with their longer heritage.

scholarly journals Restricted mean value theorems and the metric theory of restricted Weyl sums

2020 ◽  
Author(s):  
Changhao Chen ◽  
Igor E Shparlinski
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Square Root ◽  
Mean Values ◽  
Line Segments ◽  
Exceptional Sets ◽  
Natural Measure ◽  
Measure Spaces ◽  
The Mean ◽  
Almost All

Abstract We study the behaviour of Weyl sums on a subset ${\mathcal X}\subseteq [0,1)^d$ with a natural measure µ on ${\mathcal X}$. For certain measure spaces $({\mathcal X}, \mu),$ we obtain non-trivial bounds for the mean values of the Weyl sums, and for µ-almost all points of ${\mathcal X}$ the Weyl sums satisfy the square root cancellation law. Moreover, we characterize the size of the exceptional sets in terms of Hausdorff dimension. Finally, we derive variants of the Vinogradov mean value theorem averaging over measure spaces $({\mathcal X}, \mu)$. We obtain general results, which we refine for some special spaces ${\mathcal X}$ such as spheres, moment curves and line segments.

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