The most significant assumptions in the subdomain technique (i.e., based on the formal resolution of Maxwell's equations applied in subdomain) is defined by: “The iron parts (i.e., the teeth and the back-iron) are considered to be infinitely permeable so that the saturation effect is neglected”. In this paper, the author presents a new scientific contribution on improving of this method in two-dimensional (2-D) and in Cartesian coordinates by focusing on the consideration of iron. The subdomains connection is carried out in the two directions (i.e., x- and y-edges). For example, the improvement was performed by solving magnetostatic Maxwell's equations for an air- or iron-cored coil supplied by a direct current. To evaluate the efficacy of the proposed technique, the magnetic flux density distributions have been compared with those obtained by the 2-D finite-element analysis (FEA). The semi-analytical results are in quite satisfying agreement with those obtained by the 2-D FEA, considering both amplitude and waveform.

The vibrations of a Cartesian cutting machine caused by the pneumatic tool are studied with a sub-system approach. The cutting head is modeled as an equivalent robot arm which is able to mimic the measured resonances, The Cartesian structure is modeled according to the mode superposition approach. The full model is able to predict the variations in the response of the machine to tool excitation that are caused by the motion of the head along the rails of the Cartesian structure. Comparisons with experimental results are made. Finally, the mathematical model is used for assessing the effectiveness of a vibration absorber tuned to main resonance frequency of the cutting head.

In this paper we introduce the hypo-q-norms on a Cartesian product of inner product spaces. A representation of these norms in terms of inner products, the equivalence with the q-norms on a Cartesian product and some reverse inequalities obtained via the scalar Shisha-Mond, Birnacki et al., Grüss type inequalities, Boas-Bellman and Bombieri type inequalities are also given.

Quantifying an imaging modality’s ability to reproduce results is important for establishing its utility. In Magnetic Resonance Spectroscopic Imaging (MRSI), new acquisition protocols are regularly introduced, which improve upon their precursors with respect to signal-to-noise ratio (SNR), total acquisition duration, and nominal voxel resolution. This study has quantified the within-subject and between-subject reproducibility of one such new protocol (reduced-Field-of-View-Density-Weighted-Concentric-Ring-Trajectory (rFOV-DW-CRT) MRSI) by calculating the Coefficient of Variance of data acquired from a test-retest experiment. The Posterior Cingulate Cortex (PCC) and the Right Superior Corona Radiata (SCR) were selected as the Regions of Interest (ROIs) for Grey Matter (GM) and White Matter (WM), respectively. CVs for between-subject and within-subject are consistently around or below 15% for Glx, tCho, and Myo-Ins and below 5% for tNAA and tCr.

Since the launch of Protein Data Bank (PDB) in 1971, Cartesian coordinate system (CCS) has been the default approach to specify atomic positions in biomolecular experimental structures with X, Y and Z. In 2020, a local spherical coordinate system (LSCS) approach was proposed as an alternative to CCS, i.e., ρ, θ and φ. Recently, the continued application of deep learning technology in protein structure prediction (PSP) saw a leap forward in the accuracy of PSP, as evidenced by AlphaFold of Google’s DeepMind. However, there still is room for the improvement of the performances of PSP algorithms to date. Given that geometrically, CCS and LSCS are like the two sides of a coin, this short article puts forward a hypothesis that the time is now ripe to end the half-a-century burial of ρ, θ and φ in PDB, and use them as LSCS features for the design of novel PSP algorithms in future.

Due to the vast conformational space proteins can adopt, accurate and efficient prediction of protein structure remains still a challenging task, coupled with the intricacies of interatomic interactions and the limitations of current computational models in effectively navigating this complex molecular landscape. Additionally, the lack of comprehensive experimental data for all protein structures further exacerbates the difficulty in reliable machine learning-based prediction of the three-dimensional conformations of the proteios building block of life. Geometrically, Cartesian coordinate system (CCS, X, Y and Z) and spherical coordinate system (SCS, ρ, θ and φ) are two interconvertible coordinate systems, and are like two sides of one coin. Since the beginning of Protein Data Bank (PDB) in 1971, CCS has been the default approach to specify atomic positions with X, Y and Z in PDB. In this manuscript, therefore, I present a novel method for the reversible spherical geometric conversion of protein backbone structure coordinate matrices to three independent vectors: ρ, θ and φ. This reversible conversion facilitates lossless extraction of essential structural features from protein backbone structural data, enabling the development of advanced novel algorithms for protein structure prediction in future. In short, this inter-atomic SCS approach offers a comprehensive yet efficient means of representing protein backbone geometry, leveraging spherical coordinates to capture spatial relationships in a compact and intuitive inter-atomic manner, and to provide a robust framework for reversible feature extraction for the ongoing efforts in advancing the field of protein structure prediction, the holy grail of computational structural biology.

In this work, the manufacture of thin walls with sharp corners has been optimized by adjusting the limits of a 3-axis cartesian kinematics through data recorded and analyzed off-line, such as axis speed, acceleration and the positioning of the X and Y axes. The study was carried out with two powder materials (SS316L and IN718) using the directed energy deposition process with laser. 1 mm thick walls were obtained with only one bead per layer and straight/sharp corners at 90º. After adjusting the in-position parameter G502 for positioning precision on the FAGOR 8070 CNC system, it has been possible to obtain walls with minimal accumulation of material in the corner, and with practically constant layer thickness and height, with a radii of internal curvature between 0.11 and 0.24 mm for two different precision configuration. The best results have been obtained by identifying the correct balance between the decrease in programmed speed and the precision in the positioning to reach the point defined as wall corner, with speed reductions of 29% for a programmed speed of 20 mm/s and 61% for a speed of 40 mm/s. The walls show minimal defects such as residual porosities, and the microstructure is adequate.

In this paper we introduce the hypo-q-operator norm and hypo-q-numerical radius on a Cartesian product of algebras of bounded linear operators on Banach spaces. A representation of these norms in terms of semi-inner products, the equivalence with the q-norms on a Cartesian product and some reverse inequalities obtained via the scalar reverses of Cauchy-Buniakowski-Schwarz inequality are also given.

Delta parallel robot has been widely used in precision parts processing, handling, sorting, and precision assembly, with high efficiency and motion stability being its important performance indexes. Any small line segment corner in trajectory planning causes tangential discontinuous trajectory abrupt change, and the process vibration and shock caused by it seriously affect its high-speed and high-precision performance. In this paper, a trajectory planning method combining Cartesian space and joint space is proposed. Firstly, the vector method and micro-element integration method are used to establish the complete kinematics and dynamics equations of the Delta parallel robot, and the inverse kinematics/dynamics model-solving program is written based on Matlab software. Secondly, the end-effector trajectory of the Delta parallel robot is planned in Cartesian space, and the data points and inverse control points of the end-effector trajectory are obtained by the normalization method. Finally, the data points and control points are mapped to the joint space through the inverse kinematics equation, and the fifth-order B-spline curve is adopted for quadratic trajectory planning, which realizes the high-order continuous smoothing of trajectory planning. The simulation and experiment results show that the trajectory smoothing performance of continuous high-order curvature change can be improved using the proposed method, the tracking error of the end-effector is reduced by 21.12%, and the peak torques change of three joints is reduced by 3.5%, 11.6% and 1.6%, respectively.

To date, Cartesian (x, y, z) coordinate system (CCS) has been the default approach to geometrically specify atomic spatial positions in protein structures since the launch of Protein Data Bank (PDB) in 1971. To this end, this paper proposes a local spherical coordinate system (SCS) approach as an alternative to the default approach and a previously reported global SCS approach. The local SCS approach here requires only two parameters (θ and φ), instead of x, y and z as required by the default CCS approach. Essentially, CCS and SCS are like the two sides of one coin, i.e., geometric coordinate system for three-dimensional position specification. Therefore, this paper furthermore argues that it is time to flip the coin over, and have a look at the other side of the coin, e.g., the local SCS approach, which possesses an intrinsically lower degree of descriptional complexity than that of the default CCS approach, and constitutes a potentially useful alternative perspective for all protein structural research field.