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For any positive integers k , m , n with m ≥ 2 and k ≤ n , let A ( m , n , k ) be the sum of all the multiple alternating zeta values of weight mn and depth k with arguments being multiples of m, i.e., A ( m , n , k ) = ∑ s 1 + s 2 + … + s k = n s j ∈ N ζ ( m s 1 ‾ , m s 2 ‾ , … , m s k ‾ ) . In this note, we obtain the evaluation A ( m , n , k ) = ∑ a + b = n a , b ∈ N 0 ( − 1 ) a − k ( a k ) ⋅ ζ ( { m ‾ } a ) ⋅ ζ ⋆ ( { m ‾ } b ) , where ζ ( { m ‾ } a ) and ζ ⋆ ( { m ‾ } b ) denote the multiple alternating zeta values and multiple alternating zeta-star values respectively. Moreover, we give the evaluations of ζ ( { 2 m ‾ } n ) and ζ ⋆ ( { 2 m ‾ } n ) for arbitrary positive integers m , n .

A series of sub-parallel linear glacial scours are identified on the crest of the Baoshi Seamount in the Northwind Abyssal Plain by compiling new multibeam data acquired during the 9th Chinese Arctic Research Expedition (CHINARE-Arc9) in 2018 and previously published data. The new data reveal scours that developed at water depths of 850–1 030 m with an orientation of about 75°/255°. The maximum water depth occurs in the southernmost scour and is deeper than that from previous investigations, which showed a maximum scouring depth of about 900 m on the seamount. The topographic and geomorphological characteristics suggest that these scours resulted from erosion by the ice shelf extending from the Chukchi margin and/or Laurentide Ice Sheet that grounded on the crest of the seamount and moved in a NE-SW direction. Other possibilities of their genesis include armadas of large icebergs/multi-keel icebergs calved from the Chukchi Shelf or the Laurentide Ice Sheet. The new data provide new constraints for assessing the extent and volume of the ice sheet in the Chukchi area during glacial maxima.

φe(n) is a function similar to Euler function φ(n). We discussed and obtained all the odd solutions of the equations φe(xy) = φe(x) + 2φe(y), φe(xy) = 2φe(x) + 3φe(y) and φe(xyz) = φe(x) + φe(y) + φe(z) by using the methods and techniques of elementary number theory.

Marine gravimeter has been proved to be the primary technique to efficiently obtain middle-to-short wavelength signals of the earth’s gravity field in geodesy, geodynamics and marine sciences research. In recent years, some prototypes of inertial platform and strapdown marine gravimeters have been developed, where the inertial platform gravimeter systems include CHZ-II and ZL11, and strapdown gravimeter systems include SAG-2M and SGA-WZ. In order to validate the performance of these marine gravimeter prototypes, a synchronous test with the widely used gravimeters GT-2M and LCR arranged on the same vessel was carried out in the north of South China Sea. All the data are processed according to the survey standard flow, and the performance is estimated by analyzing the errors of the repeat lines and the crossover points under the same environment. The compared results show that all the six gravimeters can meet the precision requirement of marine gravity survey. Meanwhile, the precision results of the improved gravimeters can get close to the precision of gravimeter GT-2M, higher than the precision gravimeter LCR.

Let p be an odd prime, the harmonic congruence such as , and many different variations and generalizations have been studied intensively. In this note, we consider the congruences involving the combination of alternating harmonic sums, where PP denotes the set of positive integers which are prime to p. And we establish the combinational congruences involving alternating harmonic sums for positive integer n=3,4,5.

We conducted a 9-d seismic experiment in October 2015 at Laohugou Glacier No. 12. We identified microseismic signals using the short-term/long-term average trigger algorithm at four stations and classified them as long and short-duration events based on waveform, frequency, duration and magnitude characteristics. Both categories show systematical diurnal trends. The long-duration events are low-frequency tremor-like events that mainly occurred during the daytime with only several events per day. These events lasted tens of seconds to tens of minutes and are likely related to resonance of daytime meltwater. The dominant short-duration events mostly occurred during the night time with a peak occurrence frequency of ~360 h−1. Their short-duration (

In 2014, Wang and Cai established the following harmonic congruence for any odd prime p and positive integer r, ∑ i + j + k = p r i , j , k ∈ P p 1 i ⁢ j ⁢ k ≡ - 2 ⁢ p r - 1 ⁢ B p - 3 ( mod p r ) , $$\sum_{\begin{subarray}{c}i+j+k=p^{r}\\ i,j,k\in\mathcal{P}_{p}\end{subarray}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} \quad\quad(\text{mod} \,\, {p^{r}}),$$ where P n $ \mathcal{P}_{n} $ denote the set of positive integers which are prime to n. In this note, we obtain the congruences for distinct odd primes p, q and positive integers α, β, ∑ i + j + k = p α ⁢ q β i , j , k ∈ P 2 ⁢ p ⁢ q 1 i ⁢ j ⁢ k ≡ 7 8 ⁢ ( 2 - q ) ⁢ ( 1 - 1 q 3 ) ⁢ p α - 1 ⁢ q β - 1 ⁢ B p - 3 ( mod p α ) $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{2pq}\end{subarray}}\frac{1}{ijk}\equiv\frac{7}{8}\left(2-% q\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{% \alpha}} $$ and ∑ i + j + k = p α ⁢ q β i , j , k ∈ P p ⁢ q ( - 1 ) i i ⁢ j ⁢ k ≡ 1 2 ⁢ ( q - 2 ) ⁢ ( 1 - 1 q 3 ) ⁢ p α - 1 ⁢ q β - 1 ⁢ B p - 3 ( mod p α ) . $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{pq}\end{subarray}}\frac{(-1)^{i}}{ijk}\equiv\frac{1}{2}% \left(q-2\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}% \pmod{p^{\alpha}}. $$

The equivalent magnetization (EM) and mantle Bouguer anomaly (MBA) were calculated along the ultraslow-spreading Mohns Ridge axis in the Norwegian-Greenland Sea. The magnetic anomaly and the associated EM were compared with the bathymetry, MBA, seismically determined crustal structure and geochemical data at both the inter-segment scale (>60 km) and the intra-segment scale (20–60 km). At the inter-segment scale, the magnetic highs at the segment centers are independent of the MBA. Of the 13 segments, 9 with magnetic anomalies >700 nT coincide with axial volcanic ridges identified from multibeam bathymetry maps, which suggests that the magnetic highs at the segment centers may be more associated with the extrusive lavas rather than the amount of magma supply. With few exceptions, the magnetic anomaly lows associated with MBA highs at the segment ends increase from south to north. This trend might be explained by thickened extrusive basalts and/or more serpentinized peridotites at the segment ends in the north. At the intra-segment scale, the most prominent features are the decreases in the magnetic anomalies and associated EMs from the segment centers to the ends. The intra-segment magnetic anomalies have positive and negative correlations with the bathymetry and MBA, respectively. The magnetic signal modeled by the seismically determined layer 2A with an assumed constant magnetization is remarkably consistent with the observed magnetic anomaly, which strongly suggests that the thickness of the extrusive basalts dominates the magnetic structure in each segment along the Mohns Ridge. In general, the thickness of the extrusive basalts dominates the magnetic structure along the Mohns Ridge, whereas the contributions from serpentinized peridotites may be significant at the segment ends and may produce long-wavelength magnetic variations. The magnetic data can be used as an indicator of the thickness of the extrusive basalts within segments along the ultraslow-spreading Mohns Ridge.

For any positive integers k , m , n with m ≥ 2 and k ≤ n , let T ( m , n , k ) be the sum of all the multiple Hurwitz zeta values ζ ( s 1 , s 2 , … , s k ; − 1 2 , − 1 2 , … , − 1 2 ) of weight mn and depth k with arguments being multiples of m, i.e., T ( m , n , k ) = ∑ s 1 + s 2 + … + s k = n s j ∈ N ζ ( m s 1 , m s 2 , … , m s k ; − 1 2 , − 1 2 , … , − 1 2 ) . In this note, we obtain the evaluation T ( m , n , k ) = ∑ a + b = n a , b ∈ N 0 ( − 1 ) a − k ( a k ) ⋅ ζ ( { m ; − 1 2 } a ) = ⋅ ζ ⋆ ( { m ; − 1 2 } b ) , where ζ ( { m ; − 1 2 } a ) and ζ ⋆ ( { m ; − 1 2 } b ) denote the multiple Hurwitz zeta values and multiple Hurwitz zeta-star values respectively. Moreover, we give the evaluations of ζ ( { m ; − 1 2 } n ) and ζ ⋆ ( { m ; − 1 2 } n ) when m is even.

The effective elastic thickness ( T e ) is a proxy for lithospheric strength, and it depends primarily on the thermal gradient and composition of the lithosphere. Accordingly, spatial variations in T e reflect changes in lithospheric properties and can be used to better understand the structure and tectonics of particular regions. In this paper, we investigate the Ross Sea and Transantarctic Mountains in terms of T e using gravity and topographic data and the fan wavelet transform technique. The results reveal that relatively high T e values dominate in the extensional basins of the Ross Sea and the hinterland of Transantarctic Mountains, whereas very low T e values occur along the Transantarctic Mountain Front and in the deep ocean basin, with the lowest T e values are found the vicinity of Ross Island and onshore in northern Victoria Land. In addition, the spatial variations in T e correlate well with lithospheric structure at the regional scale. By combining these findings with published seismic and heat flow data, we conclude that the presence of a zone of anomalously low T e values parallel to the coast indicates that the lithosphere beneath the Transantarctic Mountain Front is extremely weak due to Cenozoic volcanism and extension. The T e values increase from the Transantarctic Mountain Front (7 km) toward the center of the continent (~ 80 km), which indicates that the continental lithosphere underlying East Antarctica belongs to the classic Gondwanan craton. The increase in T e indicates that the Transantarctic Mountain Front marks the continent-continent boundary between East Antarctica and West Antarctica. The T e values in the other extensional basins of the Ross Sea exhibit little variation and average approximately 35 km. The relatively high T e values are interpreted to indicate that the lithosphere cooled and became mechanically stronger between late Cretaceous extension and Eocene-Neogene deposition.

We show that the Diophantine system $$\begin{aligned} f(z)=f(x)f(y)=f(u)f(v) \end{aligned}$$ has infinitely many nontrivial positive integer solutions for \(f(X)=X^2-1\), and infinitely many nontrivial rational solutions for \(f(X)=X^2+b\) with nonzero integer b.

In 2014, Wang and Cai established the following harmonic congruence for any odd prime [Formula: see text] and positive integer [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] denote the set of positive integers which are prime to [Formula: see text]. In this paper, we obtain an unexpected congruence for distinct odd primes [Formula: see text], [Formula: see text] and positive integers [Formula: see text], [Formula: see text] and the necessary and sufficient condition for [Formula: see text] Finally, we raise a conjecture that for [Formula: see text] and odd prime power [Formula: see text], [Formula: see text], [Formula: see text] However, we fail to prove it even for [Formula: see text] with three distinct prime factors.

The global ionospheric model based on the reference stations of the Global Navigation Satellite System (GNSS) of the International GNSS Services is presently the most commonly used products of the global ionosphere. It is very important to comprehensively analyze and evaluate the accuracy and reliability of the model for the reasonable use of this kind of ionospheric product. In terms of receiver station deployment, this work is different from the traditional performance evaluation of the global ionosphere model based on observation data of ground-based static reference stations. The preliminary evaluation and analysis of the the global ionospheric model was conducted with the dynamic observation data across different latitudes over the southern oceans. The validation results showed that the accuracy of the global ionospheric model over the southern oceans is about 5 TECu, which deviates from the measured ionospheric TEC by about −0.6 TECu.

Let $p\geq 7$ be a prime, $l\geq 0$ be an integer and $k,~m$ be two positive integers, we obtain the following congruences, \[ \sum\limits_{s=lp}^{(l+1)p-1}\binom{kp-1}{s}^m\equiv\begin{cases}\binom{k-1}{l}^m2^{km(p-1)}\pmod{p^3},&\text{if}~2\nmid m,\\ \binom{k-1}{l}^m\binom{kmp-2}{p-1}\pmod{p^{4}},&\text{if}~2\mid m; \end{cases} \] and \[ \sum\limits_{s=lp}^{(l+1)p-1}(-1)^s\binom{kp-1}{s}^m\equiv\begin{cases}(-1)^l\binom{k-1}{l}^m2^{km(p-1)}\pmod{p^3}, &\text{if}~2\mid m,\\ (-1)^l\binom{k-1}{l}^m\binom{kmp-2}{p-1}\pmod{p^{4}}, &\text{if}~2\nmid m. \end{cases} \] Let $p$ and $q$ are distinct odd primes and $k$ be a positive integer, we have \[ \binom{kpq-1}{(pq-1)/2}\equiv \binom{kp-1}{(p-1)/2}\binom{kq-1}{(q-1)/2}\pmod {pq}. \]

The East Asian monsoon forms a significant element of the global climatic system. We here present detailed sedimentologic investigations of the Oligocene to early Miocene succession in the Sikouzi section, northern China, to determine monsoon evolution during this time. The particle-size characteristics and grain surface structures, in combination with the large-scale cross-bedding arrangements, graded bedding laminations, and small-scale deformation structures, type the Oligocene red beds in association I as eolian, attributable to large dune forms. Rare earth element results imply that the Oligocene sediments are possibly sourced from a local provenance in the West Qinling Mountains, excluding a more regional wind system during this time. Packages of thick-bedded dark-red mudstone/silty mudstone as well as subordinate medium-bedded pale-gray fine- to medium-grained sandstone and thin-bedded gypsum, with the massive mudstone/silty mudstone taking dominance, point to a lacustrine deposition with d...

Spreading rate is a primary factor of mantle melting and tectonic behavior of the global mid-ocean ridges. The spreading rate of the Gakkel ridge decreases gradually from west to east. However, the Gakkel ridge can be divided into four thick-and-thin zones with varying crustal thicknesses along ridge axis. This phenomenon indicates that mantle melting of the Gakkel ridge is not a simple function of spreading rate. Mantle temperature, water content, mantle composition, and other factors are important in crustal accretion processes. Based on gravity-derived crustal thickness and wet melting model, we estimate that the mantle potential temperatures of the four zones are 1 270, 1 220, 1 280, and 1 280°C (assuming that mantle water content equals to global average value), with corresponding mantle water contents of 210, 0, 340, and 280 mg/kg (assuming that mantle potential temperature is 1 260°C), respectivly. The western thinned crust zone is best modeled with low mantle temperature, whereas the other zones are mainly controlled by the enhanced conduction caused by the slower spreading rate. Along the Gakkel ridge, the crustal thickness is consistent with rock samples types. Predominated serpentinized peridotite and basalt are found in the area with crustal thickness 〈1.5 km and 〉2.5 km, respectively. The rock samples are including from basalt to peridotite in the area with crustal thickness between 1.5 and 2.5 km. Based on this consistency, the traditional magmatic accretion zone accounted for only 44% and amagmatic accretion accounted for 29% of the Gakkel ridge. The amagmatic accretion is a significant characteristic of the ultra-slow spreading ridge.

In this paper, first, we prove that the Diophantine system \[f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)\] has infinitely many integer solutions for $f(X)=X(X+a)$ with nonzero integers $a\equiv 0,1,4\pmod{5}$. Second, we show that the above Diophantine system has an integer parametric solution for $f(X)=X(X+a)$ with nonzero integers $a$, if there are integers $m,n,k$ such that \[\begin{cases} \begin{split} (n^2-m^2) (4mnk(k+a+1) + a(m^2+2mn-n^2)) &\equiv0\pmod{(m^2+n^2)^2},\\ (m^2+2mn-n^2) ((m^2-2mn-n^2)k(k+a+1) - 2amn) &\equiv0 \pmod{(m^2+n^2)^2}, \end{split} \end{cases}\] where $k\equiv0\pmod{4}$ when $a$ is even, and $k\equiv2\pmod{4}$ when $a$ is odd. Third, we get that the Diophantine system \[f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)=\frac{f(r)}{f(s)}\] has a five-parameter rational solution for $f(X)=X(X+a)$ with nonzero rational number $a$ and infinitely many nontrivial rational parametric solutions for $f(X)=X(X+a)(X+b)$ with nonzero integers $a,b$ and $a\neq b$. At last, we raise some related questions., 18 pages. Submitted for publication

In this note, we obtain the following identities, ∑ a + b + c = n ζ ( 2 a , 2 b , 2 c ) = 5 8 ζ ( 2 n ) − 1 4 ζ ( 2 ) ζ ( 2 n − 2 ) , for n > 2 , ∑ a + b + c + d = n ζ ( 2 a , 2 b , 2 c , 2 d ) = 35 64 ζ ( 2 n ) − 5 16 ζ ( 2 ) ζ ( 2 n − 2 ) , for n > 3 . Meanwhile, some weighted version of sum formulas are also obtained.

In order to study the pre-Cenozoic tectonics of the northern South China Sea (SCS), we have calculated the reduction to the pole (RTP) anomalies, apparent magnetization and pseudo-gravity anomalies using the latest shipboard magnetic data in the north of the South China Sea. The density and magnetic susceptibility along three OBS profiles were determined by inversion calculation. The result shows that the high magnetic anomaly belt (HMAB) of the Dongsha Uplift (DSU) is the continuation of the volcanic belt in the coastal area of Zhejiang and Fujian Provinces, and is dislocated by the NW trending NW fault F10 which may be the old transformation boundary. The NW trending fault F2 is the south boundary of the HMAB, and limits the north boundary of underplating. The fault F3 is a week zone before the crustal extension and thinning, and deep thermal states are different on both sides, which may reveal the location of the Late Mesozoic subduction. The magnetization of magnetic layer reduces in the magnetic quiet zone between faults F2 and F3. We ascribe the reducing magnetization to the hydrothermal alteration while high temperature mantle materials being underplated. However, on the south side of F3, the magnetization increases and magnetic source blocks of the Magnetic Quiet Zone (MQZ) are fragmented. The cause of weak positive magnetic anomalies on the surface is due to the suppression between positive and negative magnetic anomalies under oblique magnetization. The unique characteristics of the magnetic layer in this area may indicate the existence of “ancient oceanic crust”.

Formation of Mesozoic western China, which was dominated by tectonic amalgamation along its southern margin and associated intracontinental tectonisms, holds a key for interpreting the succedent Cenozoic evolution. This paper presents new data including lithology, sedimentary facies, stratigraphic contact, seismic interpretation and paleo-structures within the Upper Jurassic–Lower Cretaceous strata in the northern Qaidam Basin, NW China. These data all account for a contractional tectonic deformation in the earliest Cretaceous. The South Qilian Shan, according to the sedimentary features and provenance analysis, reactivated and exhumated during the deformation, controlling the deposition of the Lower Cretaceous sequences. A simplified model for the Late Jurassic–Early Cretaceous paleogeography and tectonics of the northern Qaidam Basin is accordingly proposed. The results also support a ∼25° clockwise rotation of the Qaidam Basin since the Early Cretaceous and a more accurate Mesozoic evolution process for the basin. This earliest Cretaceous deformation, associated with the reactivation of the South Qilian Shan at the time, are part of the intracontinental tectonisms in central Asia during the Mesozoic, and probably driven by both the closure of the Mongol-Okhostk Ocean to the north and the collision of the Lhasa and the Qiangtang blocks to the south.

An angular unconformity has been identified between the upper and the lower Youshashan Formation based on the study of the regional geology, drillhole, and geophysical data of the Altyn Tagh Mountain foreland, western Qaidam Basin. The distribution of the unconformity, the contacts between the upper and the lower Youshashan Formation, and its relation to the characteristics of the paleo-structure belts suggest a close relationship between the unconformities in the southern flank of the Altyn Tagh Mountain in the west of the Qaidam Basin and the activity of the Altyn Tagh Fault. It was the same time when the Atlyn Tagh Fault Zone was transformed from regional uplifting to rapid-strike-slipping movement during the Miocene, resulting in the cessation of the regional uplift and subsequent massive regional transgression during the deposition of the upper Youshashan Formation, which unconformably overlies (onlaps) the underlying strata of different ages.

Fault related folds and fractures, which always improve reservoirs and trap capacity, especially in the oil and gas fields in western China, are of extreme importance for oil and gas exploration. Based on four assumptions and fault related fold theory, we propose four conceptual models for fracturing in fault related folds, i.e., a simple-step parallel fault bend anticline, a simple-step parallel fault propagation anticline, a multiple-bend fault-bend fold and a break-forward imbricate. Fracture conditions depend on structural evolution and specific site conditions. A case study of the Kulongshan Anticline in the Jiuquan Basin shows that our conceptual models match reality data very well.

The Late Cretaceous tectonic upheaval was an important event during the evolution of the Qaidam Basin, resulting in the omission of the Upper Cretaceous in the whole basin and unconformities between the Paleogene sequence and pre-K2 strata. Integrating geological and geophysical data, two different groups of Late Cretaceous faults were recognized in the study area, one group consisting of E-W extending strike-slip faults (e.g., the Maxian and Yema-Jinan faults in the Mahai area, which caused an E-W omission zone of Mesozoic), while the other one has NW-SE thrust faults, resulting in NW-SE fold-and-thrust belts. Considering the different strikes and scale of these two groups, a simple-shear model has been employed to explain this structural phenomenon. The NW-SE thrust faults were thought to be subsidiary to the E-W strike-slip faults. Putting this into the framework of the Cretaceous paleogeographic environment of central Asia, it is inferred that this tectonic event of the Qaidam Basin is a response to the continuous northward drifting of the India plate.

In this paper, by using the theory of elliptic curves, we discuss several Diophantine equations related with the so-called figurate primes. Meanwhile, we raise several conjectures related with figurate primes and Hilbert's 8th problem, including Goldbach's conjecture, twin primes conjecture and Catalan's conjecture as well., Comment: 7 pages, 1 figure