Transverse Index Research Articles

Changes in Quality of Carya illinoinensis at Different Harvest Periods.

In order to determine the appropriate harvesting period of Carya illinoinensis planted in Nanjing city of China, this study analyzed the phenotypic characteristics and inclusions, including single fruit quality, fruit transverse and vertical diameter, kernel rate, water content, color, respiratory strength, crude fat, soluble sugar, soluble protein, and total phenols, of two cultivars 'Pawnee' and 'Wichita' during September and October, respectively. Results showed that the respiration intensity and IAD values of pecan fruits decreased as the harvest date was delayed. 'Pawnee' fruits exhibited the highest seed kernel fullness, ∆E value, fruit transverse diameter, shape index, kernel yield, crude fat, and total phenolic content in late September and early October, while the quality of 'Wichita' fruits reached its peak in late October. The appropriate harvest period is conducive to the material accumulation of Carya illinoinensis, which is of great practical significance for improving the commodity value of pecans. The optimal harvesting period for 'Pawnee' in Nanjing is from the end of September to the beginning of October, and the optimal harvesting period for 'Wichita' is from mid- to late-October to the end of October.

A transverse index theorem in the calculus of filtered manifolds

We use filtrations of the tangent bundle of a manifold starting with an integrable subbundle to define transverse symbols to the corresponding foliation, define a condition of transversally Rockland, and prove that transversally Rockland operators yield a K-homology class. We construct an equivariant KK-class for transversally Rockland transverse symbols, and show a Poincaré duality type result linking the class of an operator and its symbol.

Relationship between Preoperative Maxillomandibular Transverse Discrepancy and Post-Surgical Stability in Class II Malocclusion

The aim of this study was to examine the relationship between the presurgical maxillomandibular transverse index and post-surgical stability one year after mandibular advancement. For the material and methods, twenty-two subjects who were treated with mandibular advancement were enrolled in this study. Postsurgical stability was defined as the horizontal mandibular position change of <2 mm in lateral cephalogram 1 year after surgery. Subjects were divided into two groups according to the maintenance of postsurgical stability: a stable group (group S) and a less stable group (group LS). Presurgical maxillomandibular transverse index was determined as Yonsei transverse index (YTI) one month before surgery. A logistic analysis was performed on the postsurgical stability according to the YTI value. The presurgical, post-expansion target YTI value was obtained using receiver operating characteristic (ROC) curve. There were no notable differences in the baseline characteristics of the two groups except for vertical positions of point A, B, and gender distribution. Before surgery, however, there was a significant difference in YTI at both the fossa and CR level between the groups. The amount of mandibular advancement did not show a significant difference. The odds ratio for YTI was 0.35 (p = 0.024). The prediction of stability of presurgical YTI yielded an area under the ROC curve of 0.88. The cut-off value for YTI was 1.45 mm. It can thus be concluded that presurgical transverse index showed a correlation with postsurgical stability, and correcting it in the presurgical phase to a certain level appears to aid in securing postsurgical stability.

Efficient mechanical modeling of sandwich panels with functionally graded lattice cores under thermal environments

3D-printed sandwich panels with functionally graded (FG) lattice cores have garnered widespread attention owing to their outstanding mechanical performance. However, the straightforward interplay between the service environment, strut structure, and their mechanical behavior has not been systematically explored. In this study, an analytical two-step prediction modeling is developed to efficiently obtain the mechanical response of 3D printed sandwich panels with FG lattice cores. Initially, the effective elastic moduli of FG lattice cores are identified based on finite element (FE) analysis. The stress and deformation of the core-homogenized sandwiches in thermal environments are predicted using the higher-order shear deformation theory. Subsequently, the de-homogenization is employed to calculate strut stresses. The prediction shows excellent agreement with the FE results in the literature. Furthermore, parametric analysis is performed to uncover the impact of crucial variables on the strut stresses within the FG lattice cores. The results show that the strut stress decreases with the increased transverse layer number and power index, while increasing with the higher ambient temperature, length-to-thickness ratio, and thermal expansion of the sandwich panels. The proposed model can serve as an efficient tool for the design and printing of high-performance lattice-core sandwich panels.

Evaluation of morphological traits using multivariate techniques in Kotdhar goat native to Shivalik hill region of Himachal Pradesh.

The objective of this study was to analyze the diverse morphological attributes of Kotdhar goat native to Shivalik hill region of Himachal Pradesh. A total number of 180 goats, comprising 150 females and 30 males aged between 1 and 3years, were randomly selected to estimate15 distinct morphological traits and ten body indices. Mean values of 17.66cm, 11.68cm, 8.73cm, 71.94cm, 69.83cm, 74.48cm, 38.10cm, 73.38cm, 20.55cm, 16.48cm, 14.89cm, 13.57cm, 19.20cm, 7.71cm and 26.12kg were estimated for head length, head width, face length, body length, wither height, heart girth, chest depth, rump height, rump length, rump width, haunch width, bicoastal width, shoulder width, cannon bone and body weight respectively. In addition, morphological indices viz. body index, length index, body ratio, thoracic development, pelvic index, longitudinal pelvic index, transverse pelvic index, proportionality index, dactyl thorax index and relative depth of thorax were estimated from morphological traits. Results of morphological indices indicated that Kotdhar goat are medium bodied, longilinear, tall animals and well-suited for meat production. Phenotypic correlation among most of traits was positive and significant. Among all correlations, correlation between head length and head width was found lowest (- 0.270), while the correlation between heart girth and chest depth was observed highest (0.861). Canonical discriminant function analysis showed that there was no considerable level of intermixing for morphological traits in Kotdhar goat. Employing principal component analysis with varimax rotation, three principal components were observed, accounting for 68.47% of the total variation. First principal component (PC1) encapsulated the highest proportion of variance (44.39), revealing the high loadings on head length (0.777), body length (0.747), wither height (0.782), heart girth (0.785), chest depth (0.706), rump height (0.762), haunch width (0.734) and bicoastal width (0.793). Hence, it is possible to utilize these eight variables out of the 15 morphological traits for the characterization of the Kotdhar goat.

Correlation between Foot Transverse Arch Index and Foot Posture Index in Identifying Foot Arches in Healthy Population

Background: Medial longitudinal arch of foot plays a crucial role on maintaining ankle during weight bearing activities. In previous studies arch index derived from foot print shows a negative correlation with the navicular height. There are no other studies have correlated mathematically derived foot print values with standing arch height measurements with an acceptable equation. This study correlates arch index and foot posture index to identify foot arches. Methods: subjects with high arch have taken for the study. The study type is observational study was done on sample size n=60,foot arch index and foot posture index was evaluated with the samples. Findings: a r- value of 0.333, there is a Negative correlation between Arch index and Foot posture index in low arched foot. There is no significant correlation, and the a r- value of 0.173, there is a low positive correlation between arch index and foot posture index in high arched foot. Interpretation: Among high arched foot and low arched foot, Foot Posture Index and Arch Index is correlated, according to Pearson's correlation, only high arched foot shows low positive correlation between Foot Posture Index and Arch Index. Flatfoot shows negative correlation.

Phaneroptic characterization and zoometric indices of Creole goats in the Ayacucho Region, Peru: First step for breeding programs, selection, and conservation.

The purpose of this study is to evaluate the phenotypic characteristics and typification of Creole goats in five localities of the Ayacucho region in south-central Peru. Data from 149 goats (25 males and 124 females) were collected, excluding animals under 2 years of age, pregnant, and sick. Seven qualitative characteristics and 11 zometric measurements were evaluated, and then 9 zometric indices were estimated. In the region, goats with composite colors predominated (76.50%). Additionally, supernumerary (24.20%) and divergent teats (22.60%) were observed in females, while males exhibited a scrotal bifurcation (32.00%). The origin of the flock had a significant effect (p < 0.05) on body measurements, as well as on pelvic index, transverse pelvic index, longitudinal pelvic index, compactness index, and load cannon bone index, except for body index (BOI), proportionality index, dactyl thorax index (DTI), and relative cannon bone thickness index (<p > 0.05). In the Ayacucho region, there is a predominance of light animals (57.72%) with a significant DTI (<p < 0.05) among the populations, including goats with good balance and capable of walking long distances. Furthermore, there are compact animals (47.65%) with a significant BOI (<p < 0.05) among the flock populations, which animals are of the biotype of meat. Conversely, the strongest positive correlation (<r = 0.89) was discovered between chest girth and body weight. In the Ayacucho region, there are various creole goat biotypes range from light to very compact heavy goats, with a predominance of meat biotype animals, as well as a marked size dimorphism between localities of origin.

Quantitative proton radiography and shadowgraphy for arbitrary intensities

Charged-particle radiography and shadowgraphy data can be directly inverted to obtain a line-integrated transverse Lorentz force or a line-integrated transverse refractive index gradient if intensity modulations due to scattering and absorption are negligible, and angular deflections are small. We develop a new direct-inversion algorithm based on plasma physics and compare it to a new Monge–Ampère code and an existing power diagram code (Kasim et al., 2017). The measured or source intensity is represented by electrons subject to drag, and the other intensity by fixed ions. The decrease in kinetic plus electrostatic energy determines convergence. The displacement of the electrons from their initial to their equilibrium positions determines the line-integrated force or refractive index gradient. We have implemented two approaches: PIC (particle in cell) and Lagrangian fluid, in 1-D and 2-D. The PIC code works for arbitrary intensities, can work efficiently in parallel, and can make use of existing codes. The Lagrangian code requires less memory and is faster than the PIC code without massively parallel processing, but fails in 2-D for large intensity modulations. The Monge–Ampère code is by far the fastest in 2-D, without massively parallel processing, but fails for intensities with large voids, high contrast ratios and large deflections across the boundaries, and could not obtain the degree of convergence possible with the PIC code. The power diagram code was by far the slowest and most memory intensive, and failed for large peaks in the measured intensity.

Forecast of pain degree of lumbar disc herniation based on back propagation neural network.

To further explore the pathogenic mechanism of lumbar disc herniation (LDH) pain, this study screens important imaging features that are significantly correlated with the pain score of LDH. The features with significant correlation imaging were included into a back propagation (BP) neural network model for training, including Pfirrmann classification, Michigan State University (MSU) regional localization (MSU protrusion size classification and MSU protrusion location classification), sagittal diameter index, sagittal diameter/transverse diameter index, transverse diameter index, and AN angle (angle between nerve root and protrusion). The BP neural network training model results showed that the specificity was 95 ± 2%, sensitivity was 91 ± 2%, and accuracy was 91 ± 2% of the model. The results show that the degree of intraspinal occupation of the intervertebral disc herniation and the degree of intervertebral disc degeneration are related to LDH pain. The innovation of this study is that the BP neural network model constructed in this study shows good performance in the accuracy experiment and receiver operating characteristic experiment, which completes the prediction task of lumbar Magnetic Resonance Imaging features for the pain degree of LDH for the first time, and provides a basis for subsequent clinical diagnosis.

Light wave propagation and diffraction inhibition in three-dimensional photonic quasicrystal lattices

We have investigated the linear propagation of light wave in three-dimensional photonic quasicrystal lattices by numerical simulation. The diffraction characteristics of the light waves in the longitudinal non-modulated, in-phase modulated and out-of-phase modulated quasicrystal lattices have been compared and discussed. We have demonstrated the diffraction inhibition of light wave in three-dimensional photonic quasicrystals. Linear transverse light localization has been realized in the longitudinal out-of-phase modulated three-dimensional quasicrystal lattices by adjusting the structural parameters appropriately. Furthermore, the results also show that the longitudinal out-of-phase modulated photonic quasicrystals are easier to produce diffraction inhibition than the longitudinal out-of-phase modulated periodic lattices under low transverse refractive index contrast. Distortionless dot image transmission has numerically demonstrated in out-of-phase modulated photonic quasicrystals array. It provides a new way for distortion-free image transmission, which has promising applications in optical information transmission and all-optical signal processing.

Quasi-soliton control in optical lattices with longitudinal exponentially modulation

The dynamical evolution of one-dimensional quasi-soliton in Kerr-type linear-nonlinear optical lattices with the exponential asymptotically longitudinal modulation is investigated. The optical lattice features both the transverse linear refractive index modulation and the nonlinear coefficient periodically modulation. The analytical solutions of the integral beam center, the mean square of the beam width, and the critical incident angle are obtained by adopting the effective particle approach. The dynamical trapping and switching phenomena have been observed. Moreover, the introduction of the longitudinal modulation improves greatly the mobility of the soliton. The propagation states of the quasi-soliton can be controlled by varying a variety of parameters, such as the depths of the linear nonlinear lattices, the form factor and the incident angle of soliton, as well as the asymptotic rate of longitudinal modulation. Thus our results offer a new opportunity for the all optical manipulation of light beam.

Failure Mechanism of the Corrugated Medium under Simulated Cold Chain Logistics.

It is necessary to develop corrugated medium food packaging, which is suitable for highly humid environments, to meet the demands of cold chain logistics. In this paper, we investigated the influence of the transverse ring crush index of different environmental factors of corrugated medium and the failure mechanisms during cold chain transportation. After freeze-thaw treatment of corrugated medium, XRD and DP showed a decrease in crystallinity and polymerization of 3.47 and 7.83%, respectively. Also, the FT-IR spectra of the paper showed that the number of intermolecular hydrogen bonds decreased by about 3.00% after freezing. SEM and XRD showed CaCO3 precipitation on the paper surface and a 26.01% increase in pore size. This study would be beneficial in further expanding the applications of cellulose-based paperboard cold chain transportation.

Predicting Fundamental Transverse Electric Mode of Slab Waveguide Based on Physics-Informed Neural Networks

Over the past few years, deep learning has proven to be an effective and fast tool in many areas, especially in the field of photonics. The design of integrated optical devices relies on optical waveguides which requires a reliable and fast methods for determining the waveguide’s characteristics before the fabrication process. In this work, the newly emerging paradigm of physics-informed neural networks (PINNs) is employed for analyzing and predicting the fundamental transverse electric (TE) mode and effective refractive index ( ) of a slab waveguide. PINNs is particularly useful as it is a data and mesh-free method, which solve the most critical problems of computational modeling which are the speed and computational hardware cost. The suggested model has a prediction accuracy of up to 99% and effective refractive index relative error ranges between 10-5 and 10-6. Model results are validated against finite difference time domain method using Lumerical software and variational method.

Evaluation of neotectonic activity using watershed geomorphic analysis: A case study in the west of Dokan Lake, Kurdistan Region, Iraq

The Iraqi Kurdistan Region's landscapes are the product of a complex merging of the effects of the collision between the Arabian and Eurasian plates and various geomorphic processes. A study of the Khrmallan drainage basin in the west of Dokan Lake, based on a morphotectonic study is a significant contribution to our understanding of the Neotectonic activity in the High Folded Zone. This study investigated an integrated method of detail morphotectonic mapping and geomorphic indices' analysis for determining the signal of Neotectonic activity, using the digital elevation model (DEM) and satellite images. The detailed morphotectonic map, along with extensive field data, revealed considerable variation in the relief and morphology of the study area, and eight morphotectonic zones were recognized. The presence of a high anomalous value in stream length gradient (SL) ranges from 19 to 769, increasing the channel sinuosity index (SI) reaching 1.5, and the tendency of basin shifting through the transverse topographic index (T) ranging from 0.02 to 0.5, conclude that the study area is tectonically active. The strong relationship between the growth of the Khalakan anticline and the activation of faulting is concurrent with the collision of the Arabian and Eurasian plates. A hypothesis of an antecedent can be applied to the Khrmallan valley.

Revealing the geomorphic parameters of neotectonics based on geological and seismic data at Wadi Beda basin, NW Gulf of Suez, Egypt

The current study used remote sensing, GIS, and seismic data to determine the geological and morphometric parameters of the Wadi Beda Basin area, which is located in the northwestern part of the Gulf of Suez Province. The geomorphic parameters related with the development of neotectonics are deduced using the Shuttle Radar Topography Mission (SRTM), DEM-derived catchments, and drainage networks of the research area. The most important geomorphic parameters applied to the current study area are valley floor width to height ratio (Vf), stream length gradient index (Sl), transverse topographic symmetry index (T), hypsometric curve and integral (HI), and drainage basin shape (Bs). Based on the obtained geomorphic parameter results, the Wadi El-Noqra and Wadi Umm Athala basins exhibit moderate tectonic activity and moderate symmetry and are considered to be at their mature stage. The main basins of Wadi Umm Elda and Wadi Beda are moderately symmetrical and reflect the youth stage. However, the Bs index suggests that the Beda main basin has moderate tectonic activity, whereas Wadi Umm Elda has strong tectonic activity. By integrating the geomorphic and geological data, it is possible to determine that the majority of the study area (83.4%) is categorized by a moderate risk. Low-risk and high-risk zones, on the other hand, make up 11.1% and 5.5%, respectively, of the study area. The majority of the wadis in the Wadi Beda Main Basin are controlled by NW-trending faults, and the perturbation sites are concentrated there where these faults cross with E-W-trending faults, making them high-risk areas.

Investigating the impact of typical bridge transverse dislocations on the running performance of high-speed trains

Bridge dislocations are hard to be voided in the operation of high-speed trains, which will lead to abnormal vibration of trains. In this study, the train–track–bridge interaction theory was adopted to develop and validate a three-dimensional coupled train–track–bridge vibration model using the finite element method. Thereafter, the influence of the dislocation amplitude on the dynamic performance of high-speed trains under five typical bridge additional transverse dislocation modes was investigated utilizing the overrun probability analysis approach. The results showed that the dislocation amplitude significantly affects the transverse dynamic behavior of high-speed trains, but it has a considerably smaller impact in the vertical direction when exposed to five typical bridge transverse dislocation modes. Moreover, under constant dislocation amplitude, the transverse parallel misalignment of the girders has the greatest influence on the running performance of high-speed trains. Although all the running performance indicators did not exceed the limit in the transverse unilateral cornering condition of the girder, speed control was suggested to guarantee operational stability. At various vehicle speeds, dislocation thresholds for the transverse symmetrical cornering, transverse parallel cornering, transverse misalignment, and transverse parallel misalignment were provided based on probabilistic guarantee rates utilizing the transverse Sperling index as the control index.

Modeling of functioning of a functionally gradient transversally isotropic hydroacoustic coating on a rigid vibrating body wall

A numerical and analytical technique is presented for the computational analysis of the performance indicators of a functionally gradient transversally isotropic flat single-layer hydroacoustic coating with a variable exponential transverse inhomogeneity index, fixed along one of the boundary surfaces without slippage with an ideal contact on a rigid vibrating body wall and contacting on the opposite surface with a half-space of an ideal slightly compressible fluid. In addition to the vibrations of the housing surface, the model takes into account the impact of a plane hydroacoustic pressure wave normally incident on the contact boundary of the coating and the liquid, and the evaluation of the effectiveness of the coating is based on the analysis of the cumulative intensity of the pressure wave emitted by the surface of the coating contact with the liquid under these complex influences. An example of calculations using the described technique is considered, within the framework of which estimates are obtained for the values of the inhomogeneity parameter that ensure the most effective functioning of the coating in various frequency ranges of the effects under consideration.

Cellulose nanofibers/polyvinyl alcohol blends as an efficient coating to improve the hydrophobic and oleophobic properties of paper

The effect of cellulose nanofibers (CNFs)/polyvinyl alcohol (PVA) coating on the hydrophobic, oleophobic, and strength properties of paper were investigated. The results showed that the size of bamboo fibers (BFs) decreased significantly and the crystallinity increased significantly after biological enzyme treatment. The average length of CNFs obtained by high pressure homogenization was 2.4 µm, the diameter was 28.7 nm, and the crystallinity was 63.63%. When the coating weight of PVA/CNF was 2.0 g/m2 and the CNF dosage was increased from 0.0% to 3.0%, the paper grease resistance grade was increased from 7 to 9, the Cobb value was decreased from 22.68 ± 0.29 g/m2 to 18.37 ± 0.63 g/m2, the contact angle was increased from 67.82° to 93.56°, and the longitudinal and transverse tensile index were increased from 67.72 ± 0.21 N m/g and 37.63 ± 0.25 N m/g to 68.61 ± 0.55 N m/g and 40.71 ± 0.78 N m/g, respectively. When the CNF dosage was 3.0% and the coating weight of PVA/CNF was 4.0 g/m2, the grease resistance grade of the paper was 12, the Cobb value was 21.80 ± 0.39 g/m2, and the longitudinal and transverse tensile indices were 72.11 ± 0.43 N m/g and 42.58 ± 0.48 N m/g, respectively. In summary, the increase of CNFs can effectively improve the lipophobicity, hydrophobicity and tensile strength of the PVA coated paper.

Feature selection approach for failure mode detection of reinforced concrete bridge columns

Selecting optimal input variables for machine learning (ML) algorithms is essential for any model outputs. This study presented a feature selection-based approach for determining the optimal input parameters for classifying reinforced concrete columns failure modes. The comprehensive datasets of 311 reinforced columns involving different parameters were collected from the previous studies. The Pearson correlation (PC) and mutual information (MI) techniques were used to test input variables' linear and nonlinear relevance to the outputs. In addition, minimum redundancy maximum relevance (mRMR) algorithms were employed to select and rank the relevance of eleven input variables for the model outputs. i.e., flexural (F), flexural-shear (F-S), and shear (S) failure modes using predictor importance score. Three different classification algorithms, artificial neural networks (ANN), Decision Tree (DT), and Naïve Bayes (NB), were used to analyze five different models, M1 to M5, developed using different combinations of the selected input variables. The aspect ratio, longitudinal rebar index, transverse rebar index, and axial load ratio are the optimal input parameters that classify the failure mode reinforced concrete column.

Quark and gluon helicity evolution at small x: revised and updated

We revisit the problem of small Bjorken-x evolution of the gluon and flavor-singlet quark helicity distributions in the shock wave (s-channel) formalism. Earlier works on the subject in the same framework [1–3] resulted in an evolution equation for the gluon field-strength F12 and quark “axial current” ψ¯γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\\psi}\\gamma $$\\end{document}+γ5ψ operators (sandwiched between the appropriate light-cone Wilson lines) in the double-logarithmic approximation (summing powers of αs ln2(1/x) with αs the strong coupling constant). In this work, we observe that an important mixing of the above operators with another gluon operator, D←i\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {}_D{}^{\\leftarrow i} $$\\end{document}Di, also sandwiched between the light-cone Wilson lines (with the repeated transverse index i = 1, 2 summed over), was missing in the previous works [1–3]. This operator has the physical meaning of the sub-eikonal (covariant) phase: its contribution to helicity evolution is shown to be proportional to another sub-eikonal operator, Di − D←i\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {}_D{}^{\\leftarrow i} $$\\end{document} , which is related to the Jaffe-Manohar polarized gluon distribution [4]. In this work we include this new operator into small-x helicity evolution, and construct novel evolution equations mixing all three operators (Di − D←i\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {}_D{}^{\\leftarrow i} $$\\end{document}, F12, and ψ¯γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\\psi}\\gamma $$\\end{document}+γ5ψ), generalizing the results of [1–3]. We also construct closed double-logarithmic evolution equations in the large-Nc and large-Nc&Nf limits, with Nc and Nf the numbers of quark colors and flavors, respectively. Solving the large-Nc equations numerically we obtain the following small-x asymptotics of the quark and gluon helicity distributions ∆Σ and ∆G, along with the g1 structure function,∆ΣxQ2∼∆GxQ2∼g1xQ2∼1x3.66αsNc2π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Delta \\Sigma \\left(x,{Q}^2\\right)\\sim \\Delta G\\left(x,{Q}^2\\right)\\sim {g}_1\\left(x,{Q}^2\\right)\\sim {\\left(\\frac{1}{x}\\right)}^{3.66\\sqrt{\\frac{\\alpha_s{N}_c}{2\\pi }}} $$\\end{document}in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin [5].