Atomic Decompositions of Weak Hardy Spaces of B-Valued Martingales

The atomic decompositions of weak Hardy spaces of Banach-space-valued martingales are given. With the help of the atomic decompositions, some inequalities for B-valued martingales are established in the case 0〈r≤1. Here the results are connected closely with the p-uniform smoothness and q-uniform convexity of Banach spaces which the martingales take values in.

BOUNDEDNESS OF DYADIC DERIVATIVE AND CESARO MEAN OPERATOR ON SOME B-VALUED MARTINGALE SPACES

In this article, it is proved that the maximal operator of one-dimensional dyadic derivative of dyadic integral I＊ and Cesàro mean operator σ＊ are bounded from the B-valued martingale Hardy spaces pΣα, Dα, pLα, p H#α, pKr to Lα （0 α ∞）, respectively. The facts show that it depends on the geometrical properties of the Banach space.

ATOMIC DECOMPOSITION FOR B-VALUED R.V. SEQUENCE SPACES

In this article,atomic decompositions and the duals of some B-valued r.v.se- quence spaces are investigated.The results show that it closely depends on the geometrical properties of the sequence that take values in.

A Moment Characterization of B-Valued Generalized Functionals of White Noise

Kernel theorems are established for Bananch space-valued multilinear mappings, A moment characterization theorem for Banach space-valued generalized functionals of white noise is proved by using the above kernel theorems. A necessary and sufficient condition in terms of moments is given for sequences of Banach space-valued generalized functionals of white noise to converge strongly. The integration is also discussed of functions valued in the space of Banach space-valued generalized functionals.

Duals of Hardy Spaces of B-valued Martingales for 0

The dual of B-valued martingale Hardy space Hr^s（p） （B） with small index 0 〈 r ≤ 1, which is associated with the conditional p-variation of B-valued martingale, is characterized. In order to obtain the results, a new type of Campanato spaces for B-valued martingales is introduced and the classical technique of atomic decompositions is improved. Some results obtained here are connected closely with the p-uniform smoothness and q-uniform convexity of the underlying Banach space.

A Characterization Theorem of the Differential of Functions Valued in B-Valued Generalized Functionals

Banach space-valued generalized functionals of white noise form an important part of vector-valued generalized functionals of white noise. In this paper, we discuss the differential of abstract function valued in B-valued generalized functional space. A characterized theorem is obtained by using their S-transform.

EQUIVALENCE OF COMPLETE CONVERGENCEAND LAW OF LARGE NUMBERS FOR B-VALUED RANDOM ELEMENTS

Under some conditions on probability, this note discusses the equivalence between the complete convergence and the law of large number for B-valued independent random elements. The results of [10] become a simple corollary of the results here. At the same time, the author uses them to investigate the equivalence of strong and weak law of large numbers, and there exists an example to show that the conditions on probability are weaker.

Complete Conyergence and Strong Laws of Large Numbers-for Weighted Sums of Sequences of Independent B-Valued Random Elements

In this paper, we obtain theorems of complete convergence and strong laws of large numbers for weighted sums of sequences of independent random elements in a Banach space of type p （1 ≤ p ≤ 2）. The results improve and extend the corresponding results on real random variables obtained by [1] and [2].

SOME DISCUSSION ON THE CONDITIONS OF COMPLETE CONVERGENCE FOR SUMS OF B-VALUED RANDOM ELEMENTS

In this paper, we discuss the conditions of complete convergence for partial sums and randomly indexed partial sums of independent but not necessarily identically distributed random elements in type p space, and obtain the characterization of Banach space of type p. Yu. H.’s some results(1989) on the case of real value are extended.

Study on the Physical Process and Seismogenic Mechanism of the Yangbi MS 6.4 Earthquake in Dali,Yunnan Province

The Ms 6.4 earthquake occurred on May 21,2021 in Yangbi County,Dali Prefecture,Yunnan Province,which was the largest earthquake after the 2014 Jinggu Ms 6.6 earthquake,in western Yunnan.After the earthquake,the rapid field investigation and earthquake relocation reveal that there was no obvious surface rupture and the earthquake did not occur on pre-existing active fault,but on a buried fault on the west side of Weixi–Qiaohou–Weishan fault zone in the eastern boundary of Baoshan sub-block.Significant foreshocks appeared three days before the earthquake.These phenomena aroused scholars'intensive attention.What the physical process and seismogenic mechanism of the Yangbi Ms 6.4 earthquake are revealed by the foreshocks and aftershocks?These scientific questions need to be solved urgently.

Some Statistical Aspects for Algerian Earthquake Catalogue

The scope of this study is to analyze some aspects of the Algerian earthquake catalogue between 1980 and 2009.Seismicity analysis is based on reliable compilation of earthquake catalogs obtained from different agencies.All intensities and magnitudes were converted to Ms magnitude using appropriate relationships.Dependent events were removed using adapted time and space windows.In addition,the completeness of the catalogue as a function of magnitude was determined from the standard deviation of occurrence rate plots,using the Stepp’s methodology.The remaining 2,016 independent earthquakes with Ms 2.2 were used to obtain various parameters(b-value,z-value)to characterize the temporal and spatial seismic activity for the entire northern part of Algeria.Finally,the obtained results are discussed to explain parameters variability.

Feasibility study of computed vs measured high b-value(1400 s/mm2) diffusion-weighted MR images of the prostate

AIM: To evaluate the impact of computed b = 1400 s/mm2(C-b1400) vs measured b = 1400 s/mm2(M-b1400) diffusion-weighted images(DWI) on lesion detection rate, image quality and quality of lesion demarcation using a modern 3T-MR system based on a small-field-of-view sequence(sFOV). METHODS: Thirty patients(PSA: 9.5 ± 8.7 ng/mL; 68 ± 12 years) referred for magnetic resonance imaging(MRI) of the prostate were enrolled in this study. All measurements were performed on a 3T MR system.For DWI, a single-shot EPI diffusion sequence(b = 0, 100, 400, 800 s/mm2) was utilized. C-b1400 was cal-culated voxelwise from the ADC and diffusion images. Additionally, M-b1400 was acquired for evaluation and comparison. Lesion detection rate and maximum lesion diameters were obtained and compared. Image quality and quality of lesion demarcation were rated accord-ing to a 5-point Likert-type scale. Ratios of lesion-to-bladder as well as prostate-to-bladder signal intensity(SI) were calculated to estimate the signal-to-noise-ratio(SNR). RESULTS: Twenty-four lesions were detected on M-b1400 images and compared to C-b1400 images. C-b1400 detected three additional cancer suspicious lesions. Overall image quality was rated significantly better and SI ratios were significantly higher on C-b1400(2.3 ± 0.8 vs 3.1 ± 1.0, P < 0.001; 5.6 ± 1.8 vs 2.8 ± 0.9, P < 0.001). Comparison of lesion size showed no significant differences between C- and M-b1400(P = 0.22). CONCLUSION: Combination of a high b-value extrap-olation and sFOV may contribute to increase diagnostic accuracy of DWI without an increase of acquisition time, which may be useful to guide targeted prostate biopsies and to improve quality of multiparametric MRI(mMRI) especially under economical aspects in a pri-vate practice setting.

Laboratory investigation on relationship between degree of saturation, B-value and P-wave velocity

Laboratory tests were performed on Toyoura sand specimens to investigate the relationship between degree of saturation Sr, B-value and P-wave velocity Vp. Different types of pore water （de-aired water or tap water） and pore gas （air or CO2） as well as different magnitudes of back pressure were used to achieve different Sr （or B-value）. The measured relationship between B-value and Vp was not consistent with the theoretical prediction. The measurement shows that the Vp value in the specimen flushed with de-aired water is independent of B-value （or St） and is always around the one in fully saturated condition. However, the Vp value in the specimen flushed with tap water increases with B-value, but the shape of the relationship between Vp and B-value is quite different from the theoretical prediction. The possible explanation for the discrepancy between laboratory measurement and theoretical prediction lies in that the air exists in the water as air bubbles and therefore the pore fluid （air-water mixture） is heterogeneous instead of homogenous assumed in the theoretical prediction.

Spatial Variations of b-Values in the Coastal Area of Guangdong

We used earthquake catalogs recorded by Guangdong Seismological Network from 2008 to 2014 to resolve the spatial variations of b-values in the coastal area of Guangdong, particularly in three key research areas(Yangjiang, Heyuan, and offshore Nanao Island) with strong seismicity. Our results revealed that b-values exhibited significant spatial variations, and zones with low b-values could indicate the most likely seismogenic area of large earthquakes. We observed three clear low b-value patches in the offshore Nanao Island. We found a distinct high b-value peak at the depth of 11 km and two minimum peaks at about 14 and 7–8 km in the Yangjiang area. The overall b-values generally decrease with depth in the Heyuan area. The spatial variations of b-values reflect tectonic anomalies; that is, the ‘low-high-low' distribution of b-values in the offshore Nanao Island and the Yangjiang area may indicate the anomaly of the crustal structure with a weak layer. The b-values of reservoir-induced seismicity are obviously lower than that induced by tectonism. This finding indicates that the reservoir area is generally at high stress state under the condition of high pore pressure. We inferred that large earthquakes might be prone to occur at 10–12 km depth in the offshore Nanao Island, at 12–15 km depth in the Yangjiang area, and at the lower part of the seismic activity zone in the Heyuan area. Moreover, the upstream area of the Xinfengjiang reservoir is the most likely area of future large earthquakes in the Heyuan area.

Continuous-cyclic variations in the b-value of the earthquake frequency-magnitude distribution

Seismicity of the Earth （M ≥ 4.5） was compiled from NEIC, IRIS and ISC catalogues and used to compute b-value based on various time windows. It is found that continuous cyclic b-variations occur on both long and short time scales, the latter being of much higher value and sometimes in excess of 0.7 of the absolute b-value. These variations occur not only yearly or monthly, but also daily. Before the occurrence of large earthquakes, b-values start increasing with variable gradients that are affected by foreshocks. In some cases, the gradient is reduced to zero or to a negative value a few days before the earthquake occurrence. In general, calculated b-values attain maxima 1 day before large earthquakes and minima soon after their occurrence. Both linear regression and maximum likelihood methods give correlatable, but variable results. It is found that an expanding time window technique from a fixed starting point is more effective in the study of b-variations. The calculated b-variations for the whole Earth, its hemispheres, quadrants and the epicentral regions of some large earthquakes are of both local and regional character, which may indicate that in such cases, the geodynamic processes acting within a certain region have a much regional effect within the Earth. The b-variations have long been known to vary with a number of local and regional factors including tectonic stresses. The results reported here indicate that geotectonic stress remains the most significant factor that controls b-variations. It is found that for earthquakes with Mw ≥ 7, an increase of about 0.20 in the b-value implies a stress increase that will result in an earthquake with a magnitude one unit higher.

The spatiotemporal variation of the b-value and its tectonic implications in North China

In this study, we adopt an improved Bayesian approach based on free-knot B-spline bases to study the spatial and temporal distribution of the b-value. Synthetic tests show that the improved Bayesian approach has a superior performance compared to the Bayesian approach as well as the widely used maximum likelihood estimation （MLE） method in fitting the real variation of b-values. We then apply the improved Bayesian approach to North China and find that the b-value has a clear relevance to seismicity. Temporal changes of b-values are also investigated in two specific areas of North China. We interpret sharp decreases in the b-values as useful messages in earthquake hazard analysis.

The b-value and fractal dimension of local seismicity around Koyna Dam(India)

Earthquakes began to occur in Koyna region （India） soon after the filling of Koyna Dam in 1962. In the present study, three datasets 1964-1993, 1993-1995, and 1996-1997 are analyzed to study the b-value and fractal dimension. The b-value is calculated using the Gutenberg- Richter relationship and fractal dimension Dcorr. using correlation integral method. The estimated b-value and Dcorr. of this region before 1993 are found to be in good agreement with previously reported studies. In the sub- sequent years after 1995, the b-value shows an increase. The estimated b-values of this region are found within the limits of global average. Also, the pattern of spatial clustering of earthquakes show increase in clustering and migration along the three zones called North-East Zone, South-East Zone （SEZ）, and Warna Seismic Zone. The earthquake events having depth ≤5 km are largely confined to SEZ. After 1993, the Dcorr. shows decrease, implying that earth- quake activity gets clustered. This seismic clustering could be helpful for earthquake forecasting.

Spatio-temporal distribution of earthquake occurrence in Eastern Himalaya and vicinity(26°N-31°N and 87°E-98°E)based on b-value and fractal dimension

This study investigates the spatial and temporal variation of fractal dimension and b-value for the eastern part of the Himalaya and adjoining area(26°N–31°N and 87°E–98°E).The analysis is carried out on the earthquake dataset of 1373 events(Mc=4.0)by sliding window technique for the period 1964 to 2020.The region is divided into three sub regions A(87°E–92°E),B(92°E–94°E)and C(94°E–98°E).The b-value computed for the region A comprising eastern Nepal is smaller compared to other two regions which infers the possible high stress and asperities in the region.High spatial fractal dimension(Dc>1.5)and low temporal fractal dimension(Dt<0.31)are computed for the regions.High spatial fractal dimension may indicate that fractures generating earthquakes are approaching a 2D structure and low temporal fractal dimension implies high clustering of earthquake’s epicenters.The b value shows a weak negative correlation with Dc for regions A and C while a weak positive correlation is observed for the region B.Based on b-value and fractal dimension,this study explains the frequency of earthquakes and heterogeneity of the seismogenic structure in this part of the Himalaya.

b-value and fractal dimension variations in Iran

Iran sits on a region with a high intrinsic level of seismic activity due to its tectonic setting.Through statistical examination of the earthquakes listed in the catalogue from International Institute of Earthquake Engineering and Seismology(IIEES),this research attempted to calculate some seismicity factors and find correlation between them.A preliminary analysis indicated changes in theb-value of the Gutenberg-Richter relationship over the study region.Thus,the study area was divided into five zones(Alborz,Zagros,Azerbaijan,Central and East)andb-value was computed for each zone.Considering faulting mechanism styles and theb-values in the region,it was found that the lowestb-values belong to the thrust events and strike-slip faulting earthquakes have intermediate values.These findings support previous studies.Furthermore,results ofb-value calculation were used for the estimation of accumulated differential stresses(σ1–σ2)over each zone.Overall,theb-value for Iran is averagely low which signifies the high stress tectonic regime in this region.Also,by having calculated fractal dimension(D)in each zone,a correlation obtained showing that in Iran region,theb-value correlates to fractal dimension byD=4.2b–2 relation which does not support Aki's(1981)speculation ofD=3b/c.

Correcting the Gutenberg–Richter b-value for effects of rounding and noise

The effects of magnitude rounding and of the presence of noise in the rounded magnitudes on the estimation of the Gutenberg-Richter b-value are explored, and the ways to correct for these effects are proposed. For typical values, b = 1 and rounding interval △M = 0.1, the rounding error is approximately -10^-3 and it can be corrected to a negligible approximately -10^-5. For the same typical values, the effect of noise can be larger, depending on the characteristics of the noise distribution; for normally distributed noise with standard deviation σ = 0.1, the correct b-value may be underestimated by a factor - 0.97.