A prime $p$ is said to be a Wolstenholme prime if it satisfies the congruence ${2p-1\choose p-1} \equiv 1 \pmod {p^4}$. For such a prime $p$, we establish an expression for ${2p-1\choose p-1}\pmod {p^8}$ given in terms of the sums $R_i:=\sum _{k=1}^{p-1}1/k^i$ ($i=1,2,3,4,5,6)$. Further, the expression in this congruence is reduced in terms of the sums $R_i$ ($i=1,3,4,5$). Using this congruence, we prove that for any Wolstenholme prime $p$ we have $$ \left ({2p-1\atop p-1}\right ) \equiv 1 -2p \sum _{k=1}^{p-1}\frac {1}{k} -2p^2\sum _{k=1}^{p-1}\frac {1}{k^2}\pmod {p^7}. $$ Moreover, using a recent result of the author, we prove that a prime $p$ satisfying the above congruence must necessarily be a Wolstenholme prime. Furthermore, applying a technique of Helou and Terjanian, the above congruence is given as an expression involving the Bernoulli numbers.
A control system architecture design for an underwater ROV, primarily Class I - Pure Observation underwater ROV is presented in this paper. A non-linear plant model was designed using SolidWorks 3D modeling tool and is imported to MATLAB as a 3D model. The non-linear modeled plant is linearized using the MATLAB linear analysis toolbox to have a linear approximate model of the system. The authors designed controllers for the linear plant model of underwater ROV. PID controllers are utilized as a controller of the modeled plant. The PID tuning tools by MATLAB are utilized to tune the controller of the plant model of underwater ROV. The researchers test the control design of underwater ROV using MATLAB Simulink by analyzing the response of the system and troubleshoot the control design to achieve the objective parameters for the control design of underwater ROV.
Consider the forced higher-order nonlinear neutral functional differential equation \[ \frac{{\mathrm d}^n}{{\mathrm d}t^n}[x(t)+C(t) x(t-\tau )]+\sum ^m_{i=1} Q_i(t)f_i(x(t-\sigma _i))=g(t), \quad t\ge t_0, \] where $n, m \ge 1$ are integers, $\tau , \sigma _i\in {\mathbb{R}}^+ =[0, \infty )$, $C, Q_i, g\in C([t_0, \infty ), {\mathbb{R}})$, $f_i\in C(\mathbb{R}, \mathbb{R})$, $(i=1,2,\dots ,m)$. Some sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general $Q_i(t)$ $(i=1,2,\dots ,m)$ and $g(t)$ which means that we allow oscillatory $Q_i(t)$ $(i=1,2,\dots ,m)$ and $g(t)$. Our results improve essentially some known results in the references.
In this paper, a scheme of signal extraction and modeling for GNSS position time series based on Monte Carlo Multi-channel Singular Spectrum Analysis (MC-MSSA) is introduced, which can effectively consider the spatial correlation of different directions by processing the different components of position time series at the same time. Meanwhile, the Monte Carlo significance test is utilized to distinguish the signal from the colored noise. By comparing with Singular Spectrum Analysis (SSA), it can be confirmed that MSSA has better signal extraction and modeling performance by taking into account the correlation of different channels. Then, taking the LHAZ station as an example, MC-MSSA is utilized to simultaneously model the three components of GNSS position time series, and the trend and periodic signals are respectively identified by Kendall nonparametric test and W-correlation correlation analysis. The result denotes that MC-MSSA can effectively model the tectonic and non-tectonic signals of GNSS position time series, and the modeled signals can more intuitively reflect the dynamic movement of the station. Finally, based on the MC-MSSA-modeled tectonic signal, we characterize the crustal deformations around the eastern Tibetan Plateau, mainly including the crustal movement and strain rate change. The results suggest that the pushing movement of the Tibetan Plateau from the Indian plate is blocked by the South China block, and the crustal movement rate is obviously decreased and appears a right-handed movement trend. Meanwhile, the junction of the Tibetan Plateau and South China block has accumulated a certain amount of stress, and the tectonic activity at the junction is relatively strong and still belongs to the dangerous zone of seismic activity.
The article contains important results of analysis the rainfalls and geographical conditions influence of drainage basins (area, relief, mean gradient and shape, forest cover, hydrogeological conditions) and other morphological characteristic (length of river valley) and their previous changes on the origin of flood in July 1997. With application the method of linear multiple correlation is analytical solution the geographical conditions of drainage basin on the main hydrological characteristics (peak discharge, depth of maximal runnoff and maximal runoff per unit of surface) of the flood in July 1997 in the Czech Republic. and Článek obsahuje hlavní výsledky analýzy vlivů srážek a geografických podmínek povodí (plocha, reliéf, sklon, tvar a lesnatost, hydrogeologické poměry) a dalších morfologických charakteristik (délek údolí toků) a částečně i jejich změn na vznik povodně v červenci 1997. K řešení závislosti povodňových průtokových charakteristik (odtokové výšky, kulminačního průtoku a maximálního specifického odtoku) ve vybraných povodích na nezávisle proměnných (geografických podmínkách) byl použit lineární regresní model.