Vector Sum Research Articles

High spatial resolution φ-OFDR based on frequency-shift averaging and rotating vector summation

The coherent fading noise hinders the spatial resolution enhancements of phase optical frequency domain reflectometry (φ-OFDR). To mitigate the coherent fading noise, this study introduces frequency-shift averaging and rotating vector summation for strain measurements. The original Rayleigh backscattering signal was first divided into numerous sub-segments in frequency domain by using the frequency-shift averaging, and each sub-signal segment was transferred into vector signal with both intensity and phase information by using the rotating vector summation. Furthermore, a spatial position correction algorithm is incorporated to counteract the degradation of phase coherent, facilitating large-scale strain measurements. After optimizing the sub-frequency range and overlap ratio of the sub-signal in the frequency domain, we experimentally demonstrated that this method achieves strain measurement with a broad strain range of 2000 με, a high spatial resolution of 0.54 mm, and a high precision of 0.87%.

Colorimetric Oxygen Sensing by Analyzing Chromaticity Points for Polymer Cobalt Schiff Base in CIELAB

AbstractPolymer membranes of a copolymer ligand (L) coordinated to tetra‐coordinated N,N'‐ethylene bis(salicylideneiminato) cobalt(II) at the fifth coordinating site of the center cobalt(II), Co(II)S‐L, are prepared to measure colorimetrically the color changes in response to rapid reversible oxygen binding simultaneously with slow oxidation under moist conditions to yield Co(III)S‐L. The chromaticity point observed in CIELAB is the endpoint of the vector sum of the unit vectors for Co(II)S‐L, O2‐Co(II)S‐L, and Co(III)S‐L, scaled by their respective abundance ratios: (1 − m)(1 − n), m(1 − n), and n, where m is the ratio of O2‐Co(II)S‐L to the sum of Co(II)S‐L and O2‐Co(II)S‐L and n is the ratio of Co(III)S‐L to the total of the three cobalt complexes. The observed limited area in CIELAB as a stoichiometric Euclidean space is supported by the following two different reaction processes of the polymer membranes: an oxygen‐binding reaction under several different partial pressures of oxygen supplied to the membranes in a short period, during which oxidation of the cobalt(II) complexes is negligible, and an oxidation reaction over a long period at a constant partial pressure of oxygen. The chromaticity points in both processes progress linearly in three different two‐dimensional coordinates: a*b*, a*L*, and b*L*.

The Spatial Correlation between CN-line- and Dust-continuum-emitting Regions in High-mass Star-forming Clouds

Measuring the strength of a three-dimensional (3D) magnetic field vector is challenging as it is not easy to recognize whether its line-of-sight (LOS) and plane-of-sky (POS) components are obtained from the same region. CN (N = 1–0) emission has been used to get the LOS component of a magnetic field (B LOS) from its Zeeman splitting lines, while dust continuum emission has been used to get the POS component of a magnetic field (B POS). We use the CN (N = 1–0) data observed with the Taeduk Radio Astronomy Observatory 14 m telescope and the dust continuum data from the Herschel archive toward six high-mass star-forming regions in order to test whether CN line and dust continuum emission can trace a similar region and thus can be used for inferring 3D magnetic field strength. Our comparison between CN and H2 column densities for all targets indicates that CN line emission tends to be strong toward bright continuum regions. The positions of peak CN column densities are particularly well correlated with those of peak H2 column densities, at least over the H2 column density of 8.0 × 1022 cm−2 within one or two telescope beam sizes in all targets, implying that CN-line- and dust-continuum-emitting regions are likely spatially coincident. This enabled us to make the reliable measurement of the 3D magnetic field strengths of five targets by taking a vector sum of their B LOS and B POS, helping to decide the magnetical criticality of the targets as supercritical or transcritical.

Stability analysis of slopes with stepped failure using a vector sum particle flow method

Stability analysis of slopes with stepped failure using a vector sum particle flow method

Hue and orientation pinwheels in macaque area V4.

Area V4 is an intermediate-level area of the macaque visual cortical hierarchy that serves key functions in visual processing by integrating inputs from lower areas such as V1 and V2 and providing feedforward inputs to many higher cortical areas. Previous V4 imaging studies have focused on differential responses to color, orientation, disparity, and motion stimuli, but many details of the spatial organization of significant hue and orientation tuning have not been fully described. We used support vector machine (SVM) decoding of intrinsic cortical single-condition responses to generate high-resolution maps of hue and orientation tuning and to describe the organization of hue and orientation pinwheels in V4. Like V1 and V2, V4 contains maps of orientation that are organized as pinwheels. V4 also contains maps of hue that are organized as pinwheels, whose circular organization more closely represents the perception of hue than is observed in antecedent cortical areas. Unlike V1, where orientation is continuously mapped across the surface, V4 hue and orientation pinwheels are organized in limited numbers of pinwheel sequences. The organization of these sequences and the distance between pinwheels may provide insight into the functional organization of V4. Regions significantly tuned for hue occupy roughly four times that of the orientation, are largely separated from each other, and overlap by roughly 5%. This spatial organization is largely consistent with segregated inputs arising from V2 thin and interstripes. This modular organization of V4 suggests that further integration of color and shape might occur in higher areas in inferotemporal cortical.NEW & NOTEWORTHY The representation of hue and orientation in macaque monkey area V4 was determined by intrinsic cortical imaging of responses to isoluminant hues and achromatic grating stimuli. Vector summation of support vector machine (SVM) decoded single-condition responses was used to generate hue and orientation maps that, like V1 orientation maps, were both characterized by distinct pinwheel patterns. These data suggest that pinwheels are an important structure to represent different stimulus features across multiple visual cortical areas.

Influence of scene aspect ratio and depth cues on verticality perception bias.

Perceiving verticality is crucial for accurate spatial orientation. Previous research has revealed that tilted scenes can bias verticality perception. Verticality perception bias can be represented as the sum of multiple periodic functions that play a role in the perception of visual orientation, where the specific factors affecting each periodicity remain uncertain. This study investigated the influence of the width and depth of an indoor scene on each periodic component of the bias. The participants were presented with an indoor scene showing a rectangular checkerboard room (Experiment 1), a rectangular aperture on the wall (Experiment 2), or a rectangular dotted room (Experiment 3), with various aspect ratios. The stimuli were presented with roll orientations ranging from 90° clockwise to 90° counterclockwise. The participants were asked to report their subjective visual vertical (SVV) perceptions. The contributions of 45°, 90°, and 180° periodicities to the SVV error were assessed by the weighted vector sum model. In Experiment 1, the periodic components of the SVV error increased with the aspect ratio. In Experiments 2 and 3, only the 90° component increased with the aspect ratio. These findings suggest that extended transverse surfaces may modulate the periodic components of verticality perception.

Blasting Vibration Analysis with Micromate Tools: Experimental Study and Characterization at the PT Semen Padang Mine

This research investigates the impact of vibrations produced by blasting activities at the PT Semen Padang Mine, with a focus on vibration analysis and experimental characterization using the Micromate tool. The main objective of this research is to understand the vibrations resulting from blasting, analyze the impact of variations in PVS (Peak Vector Sum) values, and evaluate the effect on the surrounding environment, especially employee housing around the mining area. The research method used involves several important stages, first of which is the use of a Micromate tool to measure the vibrations produced during the blasting process. Micromate is a tool that can record and analyze vibrations with high accuracy, thus providing reliable data for further analysis. Apart from that, this research also uses power regression analysis to understand the relationship between PVS and Scaled Distance variables. This analysis is important to determine how variations in blasting distance and intensity affect the resulting PVS values. Then, a distance analysis was carried out to obtain a PVS value below 3 mm/s, which is considered a safe threshold to prevent structural damage to the building. The research results show several important findings. Even though the measured vibration values ​​are below the Threshold Values ​​set by environmental regulations, it is important to comply with all existing regulations to avoid long-term negative impacts. The power regression analysis carried out shows that there is a significant relationship between PVS and Scaled Distance, which means that the greater the distance between the blasting point and the measurement point, the smaller the PVS value detected. The validation results of this analysis are also in accordance with the empirical data collected during the research. In addition to these main findings, this research also provides several practical recommendations. One of them is setting the explosive charge to reduce the impact of the resulting ground vibrations. These arrangements include reducing the amount of explosive per blast or changing the blasting technique to reduce vibration intensity. In conclusion, this research provides an in-depth understanding of the factors that influence vibrations from blasting in a mining context. The results of this research have important implications for the development of more accurate and effective vibration measurement methods as well as for mitigating environmental impacts caused by blasting activities. This research not only provides new insights for the mining industry but also helps in formulating better policies to protect the environment and communities around mining areas. Thus, this research contributes to ongoing efforts to achieve a balance between industrial activities and environmental sustainability.

A vector sum analysis method for stability evolution of expansive soil slope considering shear zone damage softening

Slope stability analysis is a classical mechanical problem in geotechnical engineering and engineering geology. It is of great significance to study the stability evolution of expansive soil slopes for engineering construction in expansive soil areas. Most of the existing studies evaluate the slope stability by analyzing the limit equilibrium state of the slope, and the analysis method for the stability evolution considering the damage softening of the shear zone is lacking. In this study, the large deformation shear mechanical behavior of expansive soil was investigated by ring shear test. The damage softening characteristic of expansive soil in the shear zone was analyzed, and a shear damage model reflecting the damage softening behavior of expansive soil was derived based on the damage theory. Finally, by skillfully combining the vector sum method and the shear damage model, an analysis method for the stability evolution of the expansive soil slope considering the shear zone damage softening was proposed. The results show that the shear zone subjected to large displacement shear deformation exhibits an obvious damage softening phenomenon. The damage variable equation based on the logistic function can be well used to describe the shear damage characteristics of expansive soil, and the proposed shear damage model is in good agreement with the ring shear test results. The vector sum method considering the damage softening behavior of the shear zone can be well applied to analyze the stability evolution characteristics of the expansive soil slope. The stability factor of the expansive soil slope decreases with the increase of shear displacement, showing an obvious progressive failure behavior.

Interference fading suppression with a multi-subcarrier pulse in a distributed acoustic sensor

A low-complexity multi-subcarrier pulse generation scheme is proposed to suppress the interference fading in a phase-sensitive optical time-domain reflectometer (Φ-OTDR) based distributed acoustic sensor (DAS) with heterodyne coherent detection. The multi-subcarrier pulse is generated in the digital domain based on the proper clipping operation of a sine signal. The localization and recovery of the disturbance signal are realized by the spectrum extraction and rotated vector sum (SERVS) method. The experimental results show that the occurrences of interference fading can be significantly reduced. The intensity fluctuation is reduced from ∼75 dB to ∼25 dB. Multiple disturbance signals are successfully demodulated to verify the effectiveness of the proposed method.

Neutrophils-derived Parameters as Cost-effective Inflammatory, Disease Activity Marker in Systemic Lupus Erythematosus (SLE) and Role in Prediction of Lupus Nephritis

Introduction: Systemic lupus erythematosus (SLE) is a chronic inflammatory autoimmune disease with various clinical manifestations and approximately 50% of the SLE patients develop lupus nephritis (LN), which increases the risk of renal failure, cardiovascular diseases and overall survival. Objectives: Evaluation of neutrophil-derived parameters (Neut-X, Neut-Y and Neut-Z) as an inflammatory, disease activity marker and as a predictor of nephritis in SLE patients. Material and Methods: In this cross-sectional study, 3 ml K3EDTA blood was taken from 110 SLE patients presented in Department of Clinical Immunology to evaluate neutrophil-derived parameters in Sysmex XT2000i haematology analyser and their correlation with other inflammatory biomarkers like, erythrocyte sedimentation rate (ESR) and C-reactive protein (CRP) as well as SLE disease activity index-2000 (SLEDAI-2K). 24-hours urine protein levels were also estimated as a marker of renal involvement. Records of renal biopsy were available in 73 SLE patients, who belonged to different morphologic classification of LN. Results: After performing bivariate Spearman correlation analysis (SPSS software, version 26.0), Neut-X (measures cytoplasmic granularity by side scattering) and Neut-Z (vector sum of Neut-X and Neut-Y) showed a significant positive correlation ( r > 0.200, P < .05) with ESR, CRP and SLEDAI-2K while Neut-Y (measures nucleic acid content by sideward fluorescence) showed a significant negative correlation ( r > 0.200, P < .05) with ESR and CRP. Receiver-operating characteristic curve analysis was used to evaluate diagnostic value of 24-hour urine protein and predictive values of neutrophil-derived parameters for renal involvement in SLE patients. Among neutrophil-derived parameters, Neut-X was found to be the best predictor of renal dysfunction with highest AUC of 0.710 (95% CI: 0.604-0.816; cut-off: 1344.50; sensitivity: 84.7%; specificity: 50%) followed by Neut-Z with AUC of 0.707 (95% CI: 0.600-0.813; cut-off: 1403.67; sensitivity: 87.5%; specificity: 50%). Conclusion: Our study shows that neutrophil-derived parameters may be used as novel cost-effective, non-invasive biomarker of disease activity as well as for predicting renal involvement in SLE patients.

Constructing predictive models for seismic oscillation parameters using covariance functions and Doppler effect phenomena: A case study of InSight mission V2 data

An ability to construct predictive models for identifying seismic oscillation parameters by using the mathematics of covariance functions and Doppler effect phenomena is examined in this work. In the calculations, the Mars seismic oscillations measurement data from InSight Mission V2, observed in the months May, June and July of 2019, was used. To analyze the observation data arrays the Doppler phenomena and the expressions of covariance functions were employed. The seismic oscillations trend's intensity vectors were assessed by least squares method, and the random errors of measurements at the stations were eliminated partially as well. The estimates of the vector's auto-covariance and cross-covariance functions were derived by altering the quantization interval on the general time scale while varying the magnitude of the seismic oscillation vector on the same time scale. To detect the mean values of z —the main parameter of Doppler expression— we developed a formula by involving the derivatives of cross-covariance functions of a single vector and algebraic sum of the relevant vectors.

Application of microtremor H/V spectral ratio method in determining characteristic parameters of typical laterite sites

Microtremor signals contain rich geophysical information, and the application of microtremor measurements in determining site characteristic parameters is increasingly valued because of their advantages of simple operation, environmental friendliness, and low cost. In this study, systematic microtremor measurements were conducted at a typical laterite site. The microtremor measurement data were processed with different combinations of horizontal components using the H/V spectral ratio method(H: Horizontal frequency domain signal of the microtremor; V: Vertical frequency domain signal of the microtremor). to take the HVSR curve. The period corresponding to the peak of the HVSR curve was considered to be the predominant period of the site, the site classifications were determined, and geotechnical types were divided. This study indicates that the microtremor H/V spectral ratio method is highly consistent with the results of borehole exploration for determining site classifications and dividing geotechnical types. Moreover, using the combination of vectors sum of East-West direction and North-South direction as horizontal components in the microtremor H/V spectral ratio method can more easily and accurately determine the site-predominant period.

Impact of Normalization on Entropy-Based Weights in Hellwig's Method: A Case Study on Evaluating Sustainable Development in the Education Area.

Determining criteria weights plays a crucial role in multi-criteria decision analyses. Entropy is a significant measure in information science, and several multi-criteria decision-making methods utilize the entropy weight method (EWM). In the literature, two approaches for determining the entropy weight method can be found. One involves normalization before calculating the entropy values, while the second does not. This paper investigates the normalization effect for entropy-based weights and Hellwig's method. To compare the influence of various normalization methods in both the EWM and Hellwig's method, a study evaluating the sustainable development of EU countries in the education area in the year 2021 was analyzed. The study used data from Eurostat related to European countries' realization of the SDG 4 goal. It is observed that vector normalization and sum normalization did not change the entropy-based weights. In the case study, the max-min normalization influenced EWM weights. At the same time, these weights had only a very weak impact on the final rankings of countries with respect to achieving the SDG 4 goal, as determined by Hellwig's method. The results are compared with the outcome obtained by Hellwig's method with equal weights. The simulation study was conducted by modifying Eurostat data to investigate how the different normalization relationships discovered among the criteria affect entropy-based weights and Hellwig's method results.

Chloroplast-actin filaments decide the direction of chloroplast avoidance movement under strong light in Arabidopsis thaliana.

Chloroplast-actin (cp-actin) filaments are crucial for light-induced chloroplast movement, and appear in the front region of moving chloroplasts when visualized using GFP-mouse Talin. They are short and thick, exist between a chloroplast and the plasma membrane, and move actively and rapidly compared to cytoplasmic long actin filaments that run through a cell. The average period during which a cp-actin filament was observed at the same position was less than 0.5s. The average lengths of the cp-actin filaments calculated from those at the front region of the moving chloroplast and those around the chloroplast periphery after stopping the movement were almost the same, approximately 0.8µm. Each cp-actin filament is shown as a dotted line consisting of 4-5 dots. The vector sum of cp-actin filaments in a moving chloroplast is parallel to the moving direction of the chloroplast, suggesting that the direction of chloroplast movement is regulated by the vector sum of cp-actin filaments. However, once the chloroplasts stopped moving, the vector sum of the cp-actin filaments around the chloroplast periphery was close to zero, indicating that the direction of movement was undecided. To determine the precise structure of cp-actin filaments under electron microscopy, Arabidopsis leaves and fern Adiantum capillus-veneris gametophytes were frozen using a high-pressure freezer, and observed under electron microscopy. However, no bundled microfilaments were found, suggesting that the cp-actin filaments were unstable even under high-pressure freezing.

Interactive design of intelligent wearable products based on network communication technology

At this stage, network communication technology is increasingly mature, and intelligent wearable products are also widely used in human daily life. Wearable products are popular with users because of their numerous types, complete functions and convenient services. Wearable products integrate interaction technology, and users can interact with products. However, how to improve the user’s interaction experience and reduce the user’s cognitive burden on the interaction interface is an urgent problem in the current product interaction design. Therefore, based on the analysis of the types and related technologies of wearable products, this paper made a specific analysis of the interaction design of wearable products, and established an interaction design model. At the same time, the wearable fall detection system was also tested by machine learning algorithm. The experimental results showed that the average test result of the algorithm in this paper was 87.39%, while the average test result of the traditional algorithm was 83.79%. In terms of the missed alarm rate of fall detection, the average test result of this algorithm was 6.4%, while the average test result of the traditional algorithm was 12.33%. In terms of fall detection sensitivity, the average test result of this algorithm was 92.50%, while the average test result of the traditional algorithm was 88.24%. Compared with traditional algorithms, this method performs better, with lower missed detection rate and higher sensitivity. Innovative combination of machine learning algorithm, through three-dimensional coordinate system, differentiation and vector sum formula, improves the accuracy and reliability of fall detection. In conclusion, the algorithm in this paper can effectively optimize the relevant performance of the system, thus improving the accuracy of the system’s fall detection.

Determination of the magnetization direction via correlation between reduced‐to‐the‐pole magnetic anomalies and total gradient of the magnetic potential with vertical magnetization

AbstractThe total magnetization of an underground magnetic source is the vector sum of the induced magnetization and the natural remanent magnetization. The direction of the total magnetization serves as important a priori information in the inversion and processing of magnetic data. We demonstrated that the total gradient of the magnetic potential with vertical magnetization constitutes the envelope of the vertical component of the magnetic field for all directions of the Earth's field and source magnetization. The total gradient of the magnetic potential with vertical magnetization and the reduction‐to‐the‐pole field simultaneously tend to achieve maximum symmetry near the correct total magnetization direction. As a result, the total magnetization direction can be estimated by computing the correlations between the reduction‐to‐the‐pole and the total gradient of the magnetic potential with vertical magnetization. The proposed method yields accurate magnetization directions in synthetic model examples. The total gradient of the magnetic potential with vertical magnetization is less susceptible to data noise than transforms which are derived from the high‐order magnetic field derivatives or tensors. The estimation results are slightly affected by changes in the source magnetization direction. In a field example in the Weilasito region (North China), the reduction‐to‐the‐pole fields calculated using the estimated magnetization directions are well centred over the source. The proposed method obtained a more focused magnetization direction than that of a three‐dimensional magnetization vector inversion. The total gradient of the magnetic potential with vertical magnetization therefore provides a novel and accurate approach to determine the total magnetization direction from the total field anomaly in a variety of situations.

A Blockwise Bootstrap-Based Two-Sample Test for High-Dimensional Time Series.

We propose a two-sample testing procedure for high-dimensional time series. To obtain the asymptotic distribution of our ℓ∞-type test statistic under the null hypothesis, we establish high-dimensional central limit theorems (HCLTs) for an α-mixing sequence. Specifically, we derive two HCLTs for the maximum of a sum of high-dimensional α-mixing random vectors under the assumptions of bounded finite moments and exponential tails, respectively. The proposed HCLT for α-mixing sequence under bounded finite moments assumption is novel, and in comparison with existing results, we improve the convergence rate of the HCLT under the exponential tails assumption. To compute the critical value, we employ the blockwise bootstrap method. Importantly, our approach does not require the independence of the two samples, making it applicable for detecting change points in high-dimensional time series. Numerical results emphasize the effectiveness and advantages of our method.

A wind speed forcasting model based on rime optimization based VMD and multi-headed self-attention-LSTM

Due to the nonlinearity, fluctuation, and intermittency of wind speed, its accurate prediction is essential for improving efficiency in wind power operation systems. In this regard, a hybrid model that combines the rime optimization algorithm (RIME), variational mode decomposition (VMD), multi-headed self-attention (MSA) mechanism and long short-term memory (LSTM) is proposed for wind speed prediction. First, the number of modes and VMD penalty parameter are optimized with RIME, the optimized parameters are brought into the VMD to decompose the raw wind speeds, and a Lagrange multiplier and quadratic penalty function are introduced to obtain the input series. Then, a LSTM short-term wind speed prediction model is constructed based on the MSA mechanism and solved for the hidden states and weights of each layer of attention in the model. Finally, a ReLU activation function is used to activate the hidden states of the LSTM model, and a weighted sum vector is used as the final sequence representation, which is inputted to the output layer for specific prediction to obtain the short-term wind speed prediction results. To verify the effectiveness of the proposed model, wind speed data from four wind farms in Ningxia, China, and two sets of wind speed data from an M2 tower in the USA are selected, and 19 models are built to compare the performance of the proposed model. The results show that the proposed model outperforms other models on all datasets in terms of all five performance metrics, with smaller errors and higher prediction accuracy.

Electrochemical etching of n-type GaN in different electrolytes

Nanosized pores were introduced into GaN epitaxial films via electrochemical (EC) etching in strong electrolytes including NaOH, HNO3, NaNO3, and NaCl. The growth behaviors of GaN nanopores show a strong correlation with the electrolyte types under the same applied bias and electrolyte concentration. Specifically, the resulting morphologies (e.g. pore growth direction and pore size) were mainly pertinent to the anions instead of cations. Pore sizes as small as a few nanometers and pore directions between the horizontal and vertical directions can be adjusted by selecting the electrolytes and etching voltages. The depletion model was employed to interpret the carrier flow pathways, and the pore propagation direction can be regarded as the sum of etch rate vectors of intended etching direction and preferential oxidation direction. Finally, the near-infrared nanoporous (NP) GaN distributed Bragg reflectors (DBRs) were manufactured by the EC etching in different electrolytes. Our study points out that the type of anions in electrolyte greatly impacts pore morphologies of EC etched GaN:Si, thereby affecting the future applications.

The Canonical component of the nilfibre for parabolic adjoint action in type A

This work is a continuation of [Y. Fittouhi and A. Joseph, Parabolic adjoint action, Weierstrass Sections and components of the nilfibre in type A]. Let P be a parabolic subgroup of an irreducible simple algebraic group G. Let P′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P'$$\\end{document} be the derived group of P, and let m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {m}}$$\\end{document} be the Lie algebra of the nilradical of P. A theorem of Richardson implies that the subalgebra C[m]P′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}[{\\mathfrak {m}}]^{P'}$$\\end{document}, spanned by the P semi-invariants in C[m]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}[{\\mathfrak {m}}]$$\\end{document}, is polynomial. A linear subvariety e+V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e+V$$\\end{document} of m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {m}}$$\\end{document} is called a Weierstrass section for the action of P′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P'$$\\end{document} on m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {m}}$$\\end{document}, if the restriction map induces an isomorphism of C[m]P′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}[{\\mathfrak {m}}]^{P'}$$\\end{document} onto C[e+V]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}[e+V]$$\\end{document}. Thus, a Weierstrass section can exist only if the latter is polynomial, but even when this holds its existence is far from assured. Let N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}$$\\end{document} be zero locus of the augmentation C[m]+P′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}[{\\mathfrak {m}}]^{P'}_+$$\\end{document}. It is called the nilfibre relative to this action. Suppose G=SL(n,C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G=\ extrm{SL}(n,{\\mathbb {C}})$$\\end{document}, and let P be a parabolic subgroup. In [Y. Fittouhi and A. Joseph, loc. cit.], the existence of a Weierstrass section e+V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e+V$$\\end{document} in m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {m}}$$\\end{document} was established by a general combinatorial construction. Notably, e∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e \\in {\\mathscr {N}}$$\\end{document} and is a sum of root vectors with linearly independent roots. The Weierstrass section e+V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e+V$$\\end{document} looks very different for different choices of parabolics but nevertheless has a uniform construction and exists in all cases. It is called the “canonical Weierstrass section”. Through [Y. Fittouhi and A. Joseph, loc. cit. Prop. 6.9.2, Cor. 6.9.8], there is always a “canonical” component Ne\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}^e$$\\end{document} of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}$$\\end{document} containing e. It was announced in [Y. Fittouhi and A. Joseph, loc. cit., Prop. 6.10.4] that one may augment e to an element eVS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e_\ extrm{VS}$$\\end{document} by adjoining root vectors. Then the linear span EVS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_\ extrm{VS}$$\\end{document} of these root vectors lies in Ne\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {N}^e$$\\end{document} and its closure is just Ne\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}^e$$\\end{document}. Yet, this same result shows that Ne\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}^e$$\\end{document} need not admit a dense P orbit [Y. Fittouhi and A. Joseph, loc. cit., Lemma 6.10.7]. For the above [Y. Fittouhi and A. Joseph, loc. cit., Theorem 6.10.3] was needed. However, this theorem was only verified in the special case needed to obtain the example showing that Ne\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}^e$$\\end{document} may fail to admit a dense P orbit. Here a general proof is given (Theorem 4.4.5). Finally, a map from compositions to the set of distinct non-negative integers is defined. Its image is shown to determine the canonical Weierstrass section. One may anticipate that the remaining components of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}$$\\end{document} can be similarly described. However, this is a long story and will be postponed for a subsequent paper. These results should form a template for general type.