scholarly journals SOLITON DEFORMATION OF INVERTED CATENOID

2021 ◽  
Vol 2 (336) ◽  
pp. 24-32
Author(s):  
D. Kurmanbayev ◽  
K. Yesmakhanova
Keyword(s):  
Dirac Operator ◽  
Minimal Surface ◽  
Kdv Equation ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Weierstrass Representation ◽  
Blowing Up ◽  
Novikov Equation ◽  
Particular Solution ◽  
Definition Of

The minimal surface (see [1]) is determined using the Weierstrass representation in three-dimensional space. The solution of the Dirac equation [2] in terms of spinors coincides with the representations of this surface with conservation of isothermal coordinates. The equation represented through the Dirac operator, which is included in the Manakov’s L, A, B triple [3] as equivalent to the modified Veselov-Novikov equation (mVN) [4]. The potential 𝑈 of the Dirac operator is the potential of representing a minimal surface. New solutions of the mVN equation are constructed using the pre-known potentials of the Dirac operator and this algorithm is said to be Moutard transformations [5]. Firstly, the geometric meaning of these transformations which found in [6], [7], gives us the definition of the inversion of the minimal surface, further after finding the exact solutions of the mVN equation, we can represent the inverted surfaces. And these representations of the new potential determine the soliton deformation [8], [9]. In 2014, blowing-up solutions to the mVN equation were obtained using a rigid translation of the initial Enneper surface in [6]. Further results were obtained for the second-order Enneper surface [10]. Now the soliton deformation of an inverted catenoid is found by smooth translation along the second coordinate axis. In this paper, in order to determine catenoid inversions, it is proposed to find holomorphic objects as Gauss maps and height differential [11]; the soliton deformation of the inverted catenoid is obtained; particular solution of modified Karteweg-de Vries (KdV) equation is found that give some representation of KdV surface [12],[13].

scholarly journals Proposing 3D Thermal Technology for Heritage Building Energy Monitoring

2021 ◽  
Vol 13 (8) ◽  
pp. 1537
Author(s):  
Antonio Adán ◽  
Víctor Pérez ◽  
José-Luis Vivancos ◽  
Carolina Aparicio-Fernández ◽  
Samuel A. Prieto
Keyword(s):  
Computer Vision ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Building Energy ◽  
Energy Monitoring ◽  
3D Computer Vision ◽  
Building Monitoring ◽  
Heritage Buildings ◽  
Heritage Building ◽  
Definition Of

The energy monitoring of heritage buildings has, to date, been governed by methodologies and standards that have been defined in terms of sensors that record scalar magnitudes and that are placed in specific positions in the scene, thus recording only some of the values sampled in that space. In this paper, however, we present an alternative to the aforementioned technologies in the form of new sensors based on 3D computer vision that are able to record dense thermal information in a three-dimensional space. These thermal computer vision-based technologies (3D-TCV) entail a revision and updating of the current building energy monitoring methodologies. This paper provides a detailed definition of the most significant aspects of this new extended methodology and presents a case study showing the potential of 3D-TCV techniques and how they may complement current techniques. The results obtained lead us to believe that 3D computer vision can provide the field of building monitoring with a decisive boost, particularly in the case of heritage buildings.

scholarly journals Dirac operators with Lorentz scalar shell interactions

2018 ◽  
Vol 30 (05) ◽  
pp. 1850013 ◽  
Author(s):  
Markus Holzmann ◽  
Thomas Ourmières-Bonafos ◽  
Konstantin Pankrashkin
Keyword(s):  
Dirac Operator ◽  
Spectral Properties ◽  
Smooth Surface ◽  
Three Dimensional ◽  
The Self ◽  
Eigenvalue Asymptotics ◽  
Yang Mills ◽  
Lorentz Scalar ◽  
Definition Of ◽  
The Individual

This paper deals with the massive three-dimensional Dirac operator coupled with a Lorentz scalar shell interaction supported on a compact smooth surface. The rigorous definition of the operator involves suitable transmission conditions along the surface. After showing the self-adjointness of the resulting operator, we switch to the investigation of its spectral properties, in particular, to the existence and non-existence of eigenvalues. In the case of an attractive coupling, we study the eigenvalue asymptotics as the mass becomes large and show that the behavior of the individual eigenvalues and their total number are governed by an effective Schrödinger operator on the boundary with an external Yang–Mills potential and a curvature-induced potential.

scholarly journals Microfabrication of Nonplanar Polymeric Microfluidics

Micromachines ◽  
2018 ◽  
Vol 9 (10) ◽  
pp. 491
Author(s):  
Pin-Chuan Chen ◽  
Chung-Ying Lee ◽  
Lynh Duong
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Microlens Array ◽  
Microfluidic Chips ◽  
Pmma Substrate ◽  
In Vitro Characterization ◽  
First Case ◽  
Definition Of ◽  
Solvent Bonding

For four decades, microfluidics technology has been used in exciting, state-of-the-art applications. This paper reports on a novel fabrication approach in which micromachining is used to create nonplanar, three-dimensional microfluidic chips for experiments. Several parameters of micromachining were examined to enhance the smoothness and definition of surface contours in the nonplanar poly(methyl methacrylate) (PMMA) mold inserts. A nonplanar PMMA/PMMA chip and a nonplanar polydimethylsiloxane (PDMS)/PMMA chip were fabricated to demonstrate the efficacy of the proposed approach. In the first case, a S-shape microchannel was fabricated on the nonplanar PMMA substrate and sealed with another nonplanar PMMA via solvent bonding. In the second case, a PDMS membrane was casted from two nonplanar PMMA substrates and bonded on hemispherical PMMA substrate via solvent bonding for use as a microlens array (MLAs). These examples demonstrate the effectiveness of micromachining in the fabrication of nonplanar microfluidic chips directly on a polymeric substrate, as well as in the manufacture of nonplanar mold inserts for use in creating PDMS/PMMA microfluidic chips. This technique facilitates the creation of nonplanar microfluidic chips for applications requiring a three-dimensional space for in vitro characterization.

scholarly journals Vector Products As A Tool For The Investigation Of The Isometric Transformation Of Objects

2015 ◽  
Vol 98 (1) ◽  
pp. 60-71
Author(s):  
Ryszard Józef Grabowski
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Necessary Condition ◽  
Vector Product ◽  
Isometric Transformation ◽  
Check Points ◽  
Normal Vectors ◽  
Definition Of ◽  
Three Dimensional Space

Abstract The identification of isometric displacements of studied objects with utilization of the vector product is the aim of the analysis conducted in this paper. Isometric transformations involve translation and rotation. The behaviour of distances between check points on the object in the first and second measurements is a necessary condition for the determination of such displacements. For every three check points about the measured coordinate, one can determine the vector orthogonal to the two neighbouring sides of the triangle that are treated as vectors, using the definition of the vector product in three-dimensional space. If vectors for these points in the first and second measurements are parallel to the studied object has not changed its position or experienced translation. If the termini of vectors formed from vector products treated as the vectors are orthogonal to certain axis, then the object has experienced rotation. The determination of planes symmetric to these vectors allows the axis of rotation of the object and the angle of rotation to be found. The changes of the value of the angle between the normal vectors obtained from the first and second measurements, by exclusion of the isometric transformation, are connected to the size of the changes of the coordinates of check points, that is, deformation of the object. This paper focuses mainly on the description of the procedure for determining the translation and rotation. The main attention was paid to the rotation, due to the new and unusual way in which it is determined. Mean errors of the determined parameters are often treated briefly, and this subject requires separate consideration.

Dynamic Simulation of Multibody Mechanical Systems Using the Vector-Network Model

1988 ◽  
Vol 12 (1) ◽  
pp. 21-30 ◽  
Author(s):  
M.J. Richard
Keyword(s):  
Dynamic Models ◽  
Analytical Procedure ◽  
Dimensional Space ◽  
Digital Simulation ◽  
Three Dimensional ◽  
General Purpose ◽  
New Approach ◽  
Network Method ◽  
Definition Of ◽  
Three Dimensional Space

Pressing technological problems have created a growing interest in the development of dynamic models for the digital simulation of multibody systems. This paper describes a new approach to the problem of motion prediction. An extension of the “vector-network” method to rigid body systems in three-dimensional space is introduced. The entire procedure is a basic application of concepts of graph theory in which laws of vector dynamics are combined. The analytical procedure was successfully implemented within a general-purpose digital simulation program since, from a minimal definition of the mechanism, it will automatically predict the behavior of the system as output, thereby giving the impression that the equations governing the motion of the mechanical system have been completely formulated and solved by the computer. Simulations of the response of a rail vehicle which demonstrate the validity, applicability and self-formulating aspect of the automated model are provided.

scholarly journals Embedding the Cartesian Space Principles in a Smile, an Inclusive Learning Method for K12.

10.29007/9n6r ◽  
2020 ◽  
Author(s):  
Stefano Vuga ◽  
Eleonora Vuga
Keyword(s):  
Modular Forms ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Emotional Memory ◽  
Free Play ◽  
Two Dimensional ◽  
Real Representation ◽  
Cognitive Intelligence ◽  
Cartesian Space ◽  
Definition Of

It is now well established that the negative emotions the child experiences for not understanding a mathematical topic mark their emotional memory associated with that topic. We’ve been investigating which tangible and accessible tools prevent the development of a pathological allergy to a fundamental concept as it is the Cartesian space, seeking for kid-friendly gates to the subject. When fear and pain for not understanding traces an escape pattern from this topic at a young age the child’s ability to relate to all its didactic applications can be seriously jeopardized, marking (when not identified) the school career and sub-sequent attitudes towards all the fields of theoretical and practical application of it.The elementary approach in explaining the Cartesian space principles to the children remains mainly linked to traditional visualization models of three-dimensional images on two-dimensional space, e.g., paper, blackboard, and screens. Only recently, augmented reality has been used as a teaching aid for visualizing objects in the actual three-dimensional space. Those systems are suitable for children naturally predisposed to mathematical and/or visual-cognitive intelligence, who are not suffering from any visual impairment. This is a non-inclusive system of access to understanding such fundamental topic as the Cartesian space. Topic which is later essential to an extended comprehension of geometry, mathematics, representation of objects, and concepts. The aim of the research was to find and test a support system to complement the standard two-dimensional and visual-only approach and to guarantee a complete and consistent sensorial experience of the definition of the Cartesian space through physical, material, and modular forms. We sought to create a bond between the concept and its real representation. This system should be extended to different ages of development and types of intelligence and backgrounds, transversal to environments and contexts of usage (family/school), also for visually impaired children. The developed tools pro-vide the child an early and positive emotional bond, prior to any traditional scholastic approach, with the fundamental principles of the Cartesian space through methods such as free play, trial and error, experimentation and share of the emotions while engaging in cooperative activities.

scholarly journals Ideals of Herzog–Northcott type

2011 ◽  
Vol 54 (1) ◽  
pp. 161-186 ◽  
Author(s):  
Liam O'Carroll ◽  
Francesc Planas-Vilanova
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Regular Sequence ◽  
Point Of View ◽  
Prime Ideals ◽  
Monomial Curves ◽  
Definition Of ◽  
Minimal Prime ◽  
Special Case ◽  
Three Dimensional Space

AbstractThis paper takes a new look at ideals generated by 2×2 minors of 2×3 matrices whose entries are powers of three elements not necessarily forming a regular sequence. A special case of this is the ideals determining monomial curves in three-dimensional space, which were studied by Herzog. In the broader context studied here, these ideals are identified as Northcott ideals in the sense of Vasconcelos, and so their liaison properties are displayed. It is shown that they are set-theoretically complete intersections, revisiting the work of Bresinsky and of Valla. Even when the three elements are taken to be variables in a polynomial ring in three variables over a field, this point of view gives a larger class of ideals than just the defining ideals of monomial curves. We then characterize when the ideals in this larger class are prime, we show that they are usually radical and, using the theory of multiplicities, we give upper bounds on the number of their minimal prime ideals, one of these primes being a uniquely determined prime ideal of definition of a monomial curve. Finally, we provide examples of characteristic-dependent minimal prime and primary structures for these ideals.

scholarly journals Un modelo para visualizar objetos en 4D con el Mathematica

2018 ◽  
pp. 51-58
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Two Dimensional ◽  
3D Objects ◽  
El Sistema ◽  
Definition Of ◽  
Three Dimensional Space

Un modelo para visualizar objetos en 4D con el Mathematica A model to visualize objects in 4D with Mathematica Ricardo Velezmoro y Robert Ipanaqué Universidad Nacional de Piura, Urb. Miraflores s/n, Castilla, Piura, Perú.  DOI: https://doi.org/10.33017/RevECIPeru2014.0008/ Resumen Una variedad de técnicas de gráficos por computadora han permitido la visualización de objetos, que existen en dimensiones más altas, en una pantalla 2D. En este artículo se propone un nuevo modelo a partir de la extensión de una técnica útil en la visualización de objetos en 3D en una pantalla 2D para realizar algo similar con objetos en 4D. Dicha técnica se basa en la definición de una inmersión, en primera instancia, del espacio tridimensional en el espacio bidimensional que luego se toma como referencia para definir otra inmersión, que constituye el modelo propuesto en este artículo, del espacio tetra dimensional en el espacio tridimensional. En teoría la visualización de objetos en 4D en una pantalla 2D se consigue mediante la composición de las dos inmersiones mencionadas, pero en la práctica se aprovechan los comandos incorporados en el sistema de cálculo simbólico Mathematica para tal fin. Descriptores: objetos 4D, modelo, inmersión Abstract A variety of computer graphics techniques have enabled the display of objects, which exist in higher dimensions, on a 2D screen. In this paper a new model from the extension of a technique useful in visualizing 3D objects on a 2D screen to make something similar with 4D objects is proposed. This technique is based on the definition of a immersion, in the first instance, from the three-dimensional space in two-dimensional space which is then taken as a reference to define another immersion, which is the model proposed in this paper, from the fourdimensional space in three dimensional space. Theoretically the visualization of objects in 4D on a 2D screen is achieved by the composition of the two immersions mentioned, but in practice the incorporated commands into the computer algebra system Mathematica for this purpose are utilized. Keywords: objects 4D, model, immersion.

New Mathematical Definition and Calculation of Axial Rotation of Anatomical Joints

1991 ◽  
Vol 113 (3) ◽  
pp. 270-275 ◽  
Author(s):  
Shinji Miyazaki ◽  
Akimasa Ishida
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Axial Rotation ◽  
The Body ◽  
Body Segment ◽  
Angular Velocity Vector ◽  
Mathematical Definition ◽  
Stationary Coordinate System ◽  
The Difference ◽  
Definition Of

In the field of joint kinematics, clinical terms such as internal-external, or medical-lateral, rotations are commonly used to express the rotation of a body segment about its own long axis. However, these terms are not defined in a strict mathematical sense. In this paper, a new mathematical definition of axial rotation is proposed and methods to calculate it from the measured Euler angles are given. The definition is based on the integration of the component of the angular velocity vector projected onto the long axis of the body segment. First, the absolute axial rotation of a body segment with respect to the stationary coordinate system is defined. This definition is then generalized to give the relative axial rotation of one body segment with respect to the other body segment where the two segments are moving in the three-dimensional space. The well-known Codman’s paradox is cited as an example to make clear the difference between the definition so far proposed by other researchers and the new one.

Improvements to the Convected Co-Ordinates Method for Predicting Large Deflection Extreme Riser Response

2002 ◽  
Author(s):  
Patrick J. O’Brien ◽  
Michael Lane ◽  
John F. McNamara
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Mathematical Expression ◽  
Equilibrium Equations ◽  
Solution Vector ◽  
Analysis Technique ◽  
Rotational Deformation ◽  
Precise Measure ◽  
Definition Of ◽  
Three Dimensional Space

The geometric nonlinearity associated with large motions of compliant riser structures is catered for by defining an orthogonal triad of axes, known as the convected co-ordinate axes, for each beam element. This triad rigidly translates and rotates in three-dimensional space so as to closely follow the motions of the deforming element. The method enables equilibrium equations to be written by expressing the total nodal solution vector as the sum of rigid body and deformation terms. This paper presents and develops newly defined measures of beam deformation that provide a precise measure of beam rotational deformation relative to the convected axes. The use of these new terms no longer depends on the assumption that the deformation rotations behave as true vectors. A mathematical expression that accurately predicts the nonlinear coupling between torque and bending that occurs along the convected axes is also developed. This upgraded definition of rotational deformation also increases the robustness of the analysis technique, enabling larger time-steps during a dynamic simulation and with rapid convergence being achieved at each solution step.

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