Computer Algebra Systems have been extensively used in higher education. The reasons are many e.g., visualize mathematical problems, correlate real-world problems on a conceptual level, are flexible, simple to use, ac...Computer Algebra Systems have been extensively used in higher education. The reasons are many e.g., visualize mathematical problems, correlate real-world problems on a conceptual level, are flexible, simple to use, accessible from anywhere, etc. However, there is still room for improvement. Computer algebra system (CAS) optimization is the set of best practices and techniques to keep the CAS running optimally. Best practices are related to how to carry out a mathematical task or configure your system. In this paper, we are going to examine these techniques. The documentation sheets of CASs are the source of data that we used to compare them and examine their characteristics. The research results reveal that there are many tips that we can follow to accelerate performance.展开更多
This report shows how starting from classic electric circuits embodying commonly electric components we have reached semi-complicated circuits embodying the same components that analyzing the signal characteristics re...This report shows how starting from classic electric circuits embodying commonly electric components we have reached semi-complicated circuits embodying the same components that analyzing the signal characteristics requires a Computer Algebra System. Our approach distinguishes itself from the electrical engineers’ (EE) approach that relies on utilizing commercially available software. Our approach step-by-step shows how Kirchhoff’s rules are applied conducive to the needed circuit information. It is shown for the case at hand the characteristic information is a set of coupled differential equations and that with the help of Mathematica numeric solutions are sought. Our report paves the research road for unlimited creative similar circuits with any degree of complications. Occasionally, by tweaking the circuits we have addressed the “what if” scenarios widening the scope of the investigation. Justification of the accuracy of our analysis for the generalized circuits is cross-checked by arranging the components symmetrizing the circuit leading to an intuitively predictable reasonable result. Mathematica codes are embedded assisting the interested reader in producing and extending our results.展开更多
With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV-Burgers equation and KP-B equation are chosen to illustrate this approach. As a res...With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV-Burgers equation and KP-B equation are chosen to illustrate this approach. As a result, their abundant new soliton-like solutions and period form solutions are found.展开更多
The method of Riccati equation is extended for constructing travelling wave solutions of nonlinear partial differential equations. It is applied to solve the Karamoto-Sivashinsky equation and then its more new explici...The method of Riccati equation is extended for constructing travelling wave solutions of nonlinear partial differential equations. It is applied to solve the Karamoto-Sivashinsky equation and then its more new explicit solutions have been obtained. From the results given in this paper, one can see the computer algebra plays an important role in this procedure.展开更多
For the purpose of overcoming the difficulty of the so-called 'intermediate expression swell' in applying computer algebra, a semi-inverse algorithm is proposed. The or del of seeking solutions for various pro...For the purpose of overcoming the difficulty of the so-called 'intermediate expression swell' in applying computer algebra, a semi-inverse algorithm is proposed. The or del of seeking solutions for various problems is partly inverted, i. e., the intermediate expressions appearing in computation are 'frozen' in the symbolic form at first, and 'unfrozen' till the formal expressions of final solutions are found out. In this rca!, the overflow due to the shortage of saving space is avoided. The applications of the algorithm in the problems on nonlinear oscillation, dynamical optimization and interfacial solitary waves are described, which show the effectiveness of the semi-inverse algorithm.展开更多
The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations . T...The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations . The asymptotic expansions of solution were constructed. The remainders were estimated. And an example was analysed. It provides a new foreground for the application of the method of boundary layer with multiple scales .展开更多
In the EasyDyn multibody open source project, computer algebra has been used from the beginning to generate the expressions of velocities and accelerations of the bodies, by symbolic differentiation of their position....In the EasyDyn multibody open source project, computer algebra has been used from the beginning to generate the expressions of velocities and accelerations of the bodies, by symbolic differentiation of their position. Originally, the MuPAD computer algebra system had been retained because it was freely available for non commercial purposes and showed very good technical features.Unfortunately, MuPAD is nowadays only available through commercial channels and needs to be replaced to keep EasyDyn publicly available. This paper presents why Xcas/Giac is finally selected,among other long-term promising projects like Axiom, Maxima, Sage or Yacas. Among the choice criteria, the accessibility, the portability, the ease of use, the automatic export to C language, and the similarity with the -MuPAD language are all considered. The performances of the MuPAD and Xcas/Giac implementations are also compared on some examples.C 2013 The Chinese Society of Theoretical and Applied Mechanics. [doi：10.1063/2.13013012]展开更多
<span style="font-size:10.0pt;font-family:;" "="">Nucleons are fermions with intrinsic spins exhibiting dipole character. Dipole-dipole interaction via their dipole moments is the key fe...<span style="font-size:10.0pt;font-family:;" "="">Nucleons are fermions with intrinsic spins exhibiting dipole character. Dipole-dipole interaction via their dipole moments is the key feature quantifying the short-range nucleonics interaction in two-body physics. For a pair of interacting dipoles, the energy of a pair is the quantity of interest. The same is true for chemical polar molecules. For both cases, derivation of energy almost exclusively is carried out vectorially </span><span style="font-size:10.0pt;font-family:" color:#943634;"=""><a href="#ref1">[1]</a></span><span style="font-size:10.0pt;font-family:;" "=""></span><span style="font-size:10.0pt;font-family:" times="" new="" roman",serif;"=""><span></span></span><span style="font-size:10.0pt;font-family:;" "="">. Although uncommon the interacting energy can be derived algebraically too. For the latter Taylor, expansion is applied </span><span style="font-size:10.0pt;font-family:" color:#943634;"=""><a href="#ref2">[2]</a></span><span style="font-size:10.0pt;font-family:" background:lime;"=""></span><span style="font-size:10.0pt;font-family:" minion="" pro="" capt",serif;background:lime;"=""><span></span></span><span style="font-size:10.0pt;font-family:;" "="">.</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">The given expression although appears to be correct it is incomplete.</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">In our report,</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">by applying Taylor</span><span style="font-size:10.0pt;font-family:;" "="">’</span><span style="font-size:10.0pt;font-family:;" "="">s expansion up to the 4th order and utilizing a Computer Algebra System we formulate the missing terms.</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">Our report highlights the impact of correcting missing terms by giving two explicit examples.</span><span style="font-family:;" "=""></span>展开更多
A new algorithm for symbolic computation of polynomial-type conserved densities for nonlinear evolution systems is presented. The algorithm is implemented in Maple. The improved algorithm is more efficient not only in...A new algorithm for symbolic computation of polynomial-type conserved densities for nonlinear evolution systems is presented. The algorithm is implemented in Maple. The improved algorithm is more efficient not only in removing the redundant terms of the genera/form of the conserved densities but also in solving the conserved densities with the associated flux synchronously without using Euler operator. Furthermore, the program conslaw.mpl can be used to determine the preferences for a given parameterized nonlinear evolution systems. The code is tested on several well-known nonlinear evolution equations from the soliton theory.展开更多
An improved algorithm for symbolic computations of polynomial-type conservation laws (PCLaws) of ageneral polynomial nonlinear system is presented.The algorithm is implemented in Maple and can be successfully usedfor ...An improved algorithm for symbolic computations of polynomial-type conservation laws (PCLaws) of ageneral polynomial nonlinear system is presented.The algorithm is implemented in Maple and can be successfully usedfor high-dimensional models.Furthermore,the algorithm discards the restriction to evolution equations.The programcan also be used to determine the preferences for a given parameterized nonlinear systems.The code is tested on severalknown nonlinear equations from the soliton theory.展开更多
The inversion of a non-singular square matrix applying a Computer Algebra System (CAS) is straightforward. The CASs make the numeric computation efficient but mock the mathematical characteristics. The algorithms cond...The inversion of a non-singular square matrix applying a Computer Algebra System (CAS) is straightforward. The CASs make the numeric computation efficient but mock the mathematical characteristics. The algorithms conducive to the output are sealed and inaccessible. In practice, other than the CPU timing, the applied inversion method is irrelevant. This research-oriented article discusses one such process, the Cayley-Hamilton (C.H.) [1]. Pursuing the process symbolically reveals its unpublished hidden mathematical characteristics even in the original article [1]. This article expands the general vision of the original named method without altering its practical applications. We have used the famous CAS Mathematica [2]. We have briefed the theory behind the method and applied it to different-sized symbolic and numeric matrices. The results are compared to the named CAS’s sealed, packaged library commands. The codes are given, and the algorithms are unsealed.展开更多
As a general feature, the electric field of a localized electric charge distribution diminishes as the distance from the distribution increases;there are exceptions to this feature. For instance, the electric field of...As a general feature, the electric field of a localized electric charge distribution diminishes as the distance from the distribution increases;there are exceptions to this feature. For instance, the electric field of a charged ring (being a localized charge distribution) along its symmetry axis perpendicular to the ring through its center rather than as expected being a diminishing field encounters a local maximum bump. It is the objective of this research-oriented study to analyze the impact of this bump on the characteristics of a massive point-like charged particle oscillating along the symmetry axis. Two scenarios with and without gravity along the symmetry axis are considered. In addition to standard kinematic diagrams, various phase diagrams conducive to a better understanding are constructed. Applying Computer Algebra System (CAS), [1] [2] most calculations are carried out symbolically. Finally, by assigning a set of reasonable numeric parameters to the symbolic quantities various 3D animations are crafted. All the CAS codes are included.展开更多
It is a common misconception that electric “resistance” always is a positive defined electric element. <em>i.e.</em>, the plot of the voltage across the resistor, V vs. its current, i is a slanted straig...It is a common misconception that electric “resistance” always is a positive defined electric element. <em>i.e.</em>, the plot of the voltage across the resistor, V vs. its current, i is a slanted straight line with a positive slope. Esaki diode also known as tunnel diode is an exception to this character. For a certain voltage range, the current recedes resulting in a line with a negative slope;it is interpreted as negative resistance. In this research flavored report, we investigate the impact of the negative resistance in a typical classic electric circuit. E.g., a tunnel diode, D is inserted in a classic electric circuit that is composed of an ohmic resistor, R and a capacitor, C which are all in series with a DC power supply. The circuit equation for the RCD circuit is a nonlinear ordinary differential equation (NLODE). In line with the ever-growing popular Computer Algebra System (CAS), this is solved numerically utilizing two distinctly different CASs. The consistency of the solutions confidently leads to the understanding of the impact of the negative resistance. The circuit characteristics are compared to the classic analogous RC circuit. The report embodies an atlas of characteristics of the circuits making the analysis visually comprehensible.展开更多
Calculation of the interactive force between two horizontally stacked circular uniformly charged rings placed along the common vertical axis conducive to nonlinear oscillations under gravity has been addressed [1]. Al...Calculation of the interactive force between two horizontally stacked circular uniformly charged rings placed along the common vertical axis conducive to nonlinear oscillations under gravity has been addressed [1]. Although challenging, nonetheless the scope of the study limited to uniform charge distributions of the rings. Here we extend the analysis considering a charged ellipse with a nonuniform, curvature-dependent elliptic charge distribution exerting a force on a point-like charge placed on the vertical symmetry axis. Nonuniform charge distribution and its impact on various practical scenarios are not a common theme addressed in literature. Applying Computer Algebra System (CAS) particularly <em>Mathematica</em> [2], we analyze the issue on hand augmenting the traditional scope of interest. We overcome the CPU expensive symbolic computation following our newly developed numeric/symbolic method [1]. For comprehensive understanding, we simulate the nonlinear oscillations.展开更多
Utilizing Mathematica this report shows how from a practitioner’s point of view useful quantities some known, and some unknown and fresh properties about the Maxwell-Boltzmann distribution are calculated. We shortcut...Utilizing Mathematica this report shows how from a practitioner’s point of view useful quantities some known, and some unknown and fresh properties about the Maxwell-Boltzmann distribution are calculated. We shortcut circling the usage of antiquated incomplete tabulated error functions given in the textbooks and professional literature replacing them with efficient upgrades. And, utilizing the animation features of Mathematica displaying the temperature-dependence of the distribution function assists in visualizing the character of the distribution.展开更多
We envision utilizing the versatility of a Computer Algebra System, specifically Mathematica to explore designing physics problems. As a focused project, we consider for instance a thermo-mechanical-physics problem sh...We envision utilizing the versatility of a Computer Algebra System, specifically Mathematica to explore designing physics problems. As a focused project, we consider for instance a thermo-mechanical-physics problem showing its development from the ground up. Following the objectives of this investigation first by applying the fundamentals of physics principles we solve the problem symbolically. Applying the solution we investigate the sensitivities of the quantities of interest for various scenarios generating feasible numeric parameters. Although a physics problem is investigated, the proposed methodology may as well be applied to other scientific fields. The codes needed for this particular project are included enabling the interested reader to duplicate the results, extend and modify them as needed to explore various extended scenarios.展开更多
The I-V diagram of a tunnel diode inherits a voltage range corresponding to a specific current domain with a negative slope. Within this range, the electric resistance is negatively impacting the characteristics of th...The I-V diagram of a tunnel diode inherits a voltage range corresponding to a specific current domain with a negative slope. Within this range, the electric resistance is negatively impacting the characteristics of the electric circuits. One such circuit containing a tunnel diode in series with an inductor driven by a DC source is considered. The negative resistance significantly alters the characteristics of the circuit. In this research-oriented project, we unveil these characteristics comparing them to the classic inductive circuit with an ohmic resistor. This project stems from our previous work [1] and may be considered an application of the tunnel diode embodying unseen surprises. The circuit analysis is entirely based on utilizing a Computer Algebra System (CAS) specifically Mathematica. Without a CAS, the completion of the project wouldn’t have been possible otherwise.展开更多
We consider the motion of a massive point-like projectile thrown with initial velocity with respect to horizontal in a two-dimensional vertical plane under the influence of gravity in a viscose media. Two different ve...We consider the motion of a massive point-like projectile thrown with initial velocity with respect to horizontal in a two-dimensional vertical plane under the influence of gravity in a viscose media. Two different velocity-dependent resistive media models are considered—linear and quadratic. With an objective to utilizing a Computer Algebra System (CAS), specifically <em>Mathematica</em> [1] numerically we solve the corresponding equations of motions. For a set of compatible parameters characterizing viscose forces graphically we display comparing the trajectories explicitly showing the impact of the models. Utilizing the model-dependent trajectory equations numerically we evaluate their associated arc-lengths. What distinguishes our approach vs. the existing body of work is the notion of the “reverse engineering”. Meaning, utilizing our numeric data we establish their corresponding analytic counter parts. Ultimately, utilizing both outputs numerically and analytically we determine the matching initial projectile angles maximizing their respective arc-lengths.展开更多
Two electrically charged rings of different sizes are assembled along their common vertical symmetry axis through their centers. The bottom ring is secured on a horizontal support while the top one is loose. For a set...Two electrically charged rings of different sizes are assembled along their common vertical symmetry axis through their centers. The bottom ring is secured on a horizontal support while the top one is loose. For a set of practical values characterizing the charged rings we envision a scenario where the mutual electric repulsion between the rings and the weight of the top ring results in stable nonlinear oscillations. To quantify the characteristics of the oscillations, we utilize a Computer Algebra System specifically <em>Mathematica</em> [1]. We accompany the analysis with a simulation for a comprehensive visual understanding.展开更多
In search of nonlinear oscillations, we envision a 3D elliptic curva-ture-dependent nonuniform charge distribution to creating an electric field along the symmetry axis causing a massive point-like charged particle pl...In search of nonlinear oscillations, we envision a 3D elliptic curva-ture-dependent nonuniform charge distribution to creating an electric field along the symmetry axis causing a massive point-like charged particle placed on the symmetry axis to oscillate in a delayed/hesitant nonlinear mode. The charge distribution is a 3D twisted line creating nontrivial electric field causing an unexpected oscillation that is non-orthodox defying the common sense. Calculation of this research flavored investigation is entirely based on utilities accompanied with Computer Algebra Systems (CAS) especially <em>Mathematic</em> [1]. The characteristics of the delayed oscillations in addition to embodying classic graphics displaying the time-dependent kinematic quantities are augmented including various phase diagrams signifying the nonlinear oscillations. The output of our investigation is compared to nonlinear non-delayed oscillations revealing fresh insight. For comprehensive understanding of the hesitant oscillator a simulation program is crafted clarifying visually the scenario on hand.展开更多
文摘Computer Algebra Systems have been extensively used in higher education. The reasons are many e.g., visualize mathematical problems, correlate real-world problems on a conceptual level, are flexible, simple to use, accessible from anywhere, etc. However, there is still room for improvement. Computer algebra system (CAS) optimization is the set of best practices and techniques to keep the CAS running optimally. Best practices are related to how to carry out a mathematical task or configure your system. In this paper, we are going to examine these techniques. The documentation sheets of CASs are the source of data that we used to compare them and examine their characteristics. The research results reveal that there are many tips that we can follow to accelerate performance.
文摘This report shows how starting from classic electric circuits embodying commonly electric components we have reached semi-complicated circuits embodying the same components that analyzing the signal characteristics requires a Computer Algebra System. Our approach distinguishes itself from the electrical engineers’ (EE) approach that relies on utilizing commercially available software. Our approach step-by-step shows how Kirchhoff’s rules are applied conducive to the needed circuit information. It is shown for the case at hand the characteristic information is a set of coupled differential equations and that with the help of Mathematica numeric solutions are sought. Our report paves the research road for unlimited creative similar circuits with any degree of complications. Occasionally, by tweaking the circuits we have addressed the “what if” scenarios widening the scope of the investigation. Justification of the accuracy of our analysis for the generalized circuits is cross-checked by arranging the components symmetrizing the circuit leading to an intuitively predictable reasonable result. Mathematica codes are embedded assisting the interested reader in producing and extending our results.
基金The project supported by the National Key Basic Research Development Project Program under Grant No.G1998030600the Foundation of Liaoning Normal University
文摘With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV-Burgers equation and KP-B equation are chosen to illustrate this approach. As a result, their abundant new soliton-like solutions and period form solutions are found.
文摘The method of Riccati equation is extended for constructing travelling wave solutions of nonlinear partial differential equations. It is applied to solve the Karamoto-Sivashinsky equation and then its more new explicit solutions have been obtained. From the results given in this paper, one can see the computer algebra plays an important role in this procedure.
文摘For the purpose of overcoming the difficulty of the so-called 'intermediate expression swell' in applying computer algebra, a semi-inverse algorithm is proposed. The or del of seeking solutions for various problems is partly inverted, i. e., the intermediate expressions appearing in computation are 'frozen' in the symbolic form at first, and 'unfrozen' till the formal expressions of final solutions are found out. In this rca!, the overflow due to the shortage of saving space is avoided. The applications of the algorithm in the problems on nonlinear oscillation, dynamical optimization and interfacial solitary waves are described, which show the effectiveness of the semi-inverse algorithm.
文摘The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations . The asymptotic expansions of solution were constructed. The remainders were estimated. And an example was analysed. It provides a new foreground for the application of the method of boundary layer with multiple scales .
文摘In the EasyDyn multibody open source project, computer algebra has been used from the beginning to generate the expressions of velocities and accelerations of the bodies, by symbolic differentiation of their position. Originally, the MuPAD computer algebra system had been retained because it was freely available for non commercial purposes and showed very good technical features.Unfortunately, MuPAD is nowadays only available through commercial channels and needs to be replaced to keep EasyDyn publicly available. This paper presents why Xcas/Giac is finally selected,among other long-term promising projects like Axiom, Maxima, Sage or Yacas. Among the choice criteria, the accessibility, the portability, the ease of use, the automatic export to C language, and the similarity with the -MuPAD language are all considered. The performances of the MuPAD and Xcas/Giac implementations are also compared on some examples.C 2013 The Chinese Society of Theoretical and Applied Mechanics. [doi：10.1063/2.13013012]
文摘<span style="font-size:10.0pt;font-family:;" "="">Nucleons are fermions with intrinsic spins exhibiting dipole character. Dipole-dipole interaction via their dipole moments is the key feature quantifying the short-range nucleonics interaction in two-body physics. For a pair of interacting dipoles, the energy of a pair is the quantity of interest. The same is true for chemical polar molecules. For both cases, derivation of energy almost exclusively is carried out vectorially </span><span style="font-size:10.0pt;font-family:" color:#943634;"=""><a href="#ref1">[1]</a></span><span style="font-size:10.0pt;font-family:;" "=""></span><span style="font-size:10.0pt;font-family:" times="" new="" roman",serif;"=""><span></span></span><span style="font-size:10.0pt;font-family:;" "="">. Although uncommon the interacting energy can be derived algebraically too. For the latter Taylor, expansion is applied </span><span style="font-size:10.0pt;font-family:" color:#943634;"=""><a href="#ref2">[2]</a></span><span style="font-size:10.0pt;font-family:" background:lime;"=""></span><span style="font-size:10.0pt;font-family:" minion="" pro="" capt",serif;background:lime;"=""><span></span></span><span style="font-size:10.0pt;font-family:;" "="">.</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">The given expression although appears to be correct it is incomplete.</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">In our report,</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">by applying Taylor</span><span style="font-size:10.0pt;font-family:;" "="">’</span><span style="font-size:10.0pt;font-family:;" "="">s expansion up to the 4th order and utilizing a Computer Algebra System we formulate the missing terms.</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">Our report highlights the impact of correcting missing terms by giving two explicit examples.</span><span style="font-family:;" "=""></span>
文摘A new algorithm for symbolic computation of polynomial-type conserved densities for nonlinear evolution systems is presented. The algorithm is implemented in Maple. The improved algorithm is more efficient not only in removing the redundant terms of the genera/form of the conserved densities but also in solving the conserved densities with the associated flux synchronously without using Euler operator. Furthermore, the program conslaw.mpl can be used to determine the preferences for a given parameterized nonlinear evolution systems. The code is tested on several well-known nonlinear evolution equations from the soliton theory.
基金the Scientific Fund of Education Department of Zhejiang Province of China under Grant No.20070979the National Natural Science Foundations of China under Grant Nos.10675065,90503006,and 10735030+1 种基金the State Basic Research Program of China (973 Program) under Grant No.2007CB814800the K.C.Wong Magna Fund in Ningbo University
文摘An improved algorithm for symbolic computations of polynomial-type conservation laws (PCLaws) of ageneral polynomial nonlinear system is presented.The algorithm is implemented in Maple and can be successfully usedfor high-dimensional models.Furthermore,the algorithm discards the restriction to evolution equations.The programcan also be used to determine the preferences for a given parameterized nonlinear systems.The code is tested on severalknown nonlinear equations from the soliton theory.
文摘The inversion of a non-singular square matrix applying a Computer Algebra System (CAS) is straightforward. The CASs make the numeric computation efficient but mock the mathematical characteristics. The algorithms conducive to the output are sealed and inaccessible. In practice, other than the CPU timing, the applied inversion method is irrelevant. This research-oriented article discusses one such process, the Cayley-Hamilton (C.H.) [1]. Pursuing the process symbolically reveals its unpublished hidden mathematical characteristics even in the original article [1]. This article expands the general vision of the original named method without altering its practical applications. We have used the famous CAS Mathematica [2]. We have briefed the theory behind the method and applied it to different-sized symbolic and numeric matrices. The results are compared to the named CAS’s sealed, packaged library commands. The codes are given, and the algorithms are unsealed.
文摘As a general feature, the electric field of a localized electric charge distribution diminishes as the distance from the distribution increases;there are exceptions to this feature. For instance, the electric field of a charged ring (being a localized charge distribution) along its symmetry axis perpendicular to the ring through its center rather than as expected being a diminishing field encounters a local maximum bump. It is the objective of this research-oriented study to analyze the impact of this bump on the characteristics of a massive point-like charged particle oscillating along the symmetry axis. Two scenarios with and without gravity along the symmetry axis are considered. In addition to standard kinematic diagrams, various phase diagrams conducive to a better understanding are constructed. Applying Computer Algebra System (CAS), [1] [2] most calculations are carried out symbolically. Finally, by assigning a set of reasonable numeric parameters to the symbolic quantities various 3D animations are crafted. All the CAS codes are included.
文摘It is a common misconception that electric “resistance” always is a positive defined electric element. <em>i.e.</em>, the plot of the voltage across the resistor, V vs. its current, i is a slanted straight line with a positive slope. Esaki diode also known as tunnel diode is an exception to this character. For a certain voltage range, the current recedes resulting in a line with a negative slope;it is interpreted as negative resistance. In this research flavored report, we investigate the impact of the negative resistance in a typical classic electric circuit. E.g., a tunnel diode, D is inserted in a classic electric circuit that is composed of an ohmic resistor, R and a capacitor, C which are all in series with a DC power supply. The circuit equation for the RCD circuit is a nonlinear ordinary differential equation (NLODE). In line with the ever-growing popular Computer Algebra System (CAS), this is solved numerically utilizing two distinctly different CASs. The consistency of the solutions confidently leads to the understanding of the impact of the negative resistance. The circuit characteristics are compared to the classic analogous RC circuit. The report embodies an atlas of characteristics of the circuits making the analysis visually comprehensible.
文摘Calculation of the interactive force between two horizontally stacked circular uniformly charged rings placed along the common vertical axis conducive to nonlinear oscillations under gravity has been addressed [1]. Although challenging, nonetheless the scope of the study limited to uniform charge distributions of the rings. Here we extend the analysis considering a charged ellipse with a nonuniform, curvature-dependent elliptic charge distribution exerting a force on a point-like charge placed on the vertical symmetry axis. Nonuniform charge distribution and its impact on various practical scenarios are not a common theme addressed in literature. Applying Computer Algebra System (CAS) particularly <em>Mathematica</em> [2], we analyze the issue on hand augmenting the traditional scope of interest. We overcome the CPU expensive symbolic computation following our newly developed numeric/symbolic method [1]. For comprehensive understanding, we simulate the nonlinear oscillations.
文摘Utilizing Mathematica this report shows how from a practitioner’s point of view useful quantities some known, and some unknown and fresh properties about the Maxwell-Boltzmann distribution are calculated. We shortcut circling the usage of antiquated incomplete tabulated error functions given in the textbooks and professional literature replacing them with efficient upgrades. And, utilizing the animation features of Mathematica displaying the temperature-dependence of the distribution function assists in visualizing the character of the distribution.
文摘We envision utilizing the versatility of a Computer Algebra System, specifically Mathematica to explore designing physics problems. As a focused project, we consider for instance a thermo-mechanical-physics problem showing its development from the ground up. Following the objectives of this investigation first by applying the fundamentals of physics principles we solve the problem symbolically. Applying the solution we investigate the sensitivities of the quantities of interest for various scenarios generating feasible numeric parameters. Although a physics problem is investigated, the proposed methodology may as well be applied to other scientific fields. The codes needed for this particular project are included enabling the interested reader to duplicate the results, extend and modify them as needed to explore various extended scenarios.
文摘The I-V diagram of a tunnel diode inherits a voltage range corresponding to a specific current domain with a negative slope. Within this range, the electric resistance is negatively impacting the characteristics of the electric circuits. One such circuit containing a tunnel diode in series with an inductor driven by a DC source is considered. The negative resistance significantly alters the characteristics of the circuit. In this research-oriented project, we unveil these characteristics comparing them to the classic inductive circuit with an ohmic resistor. This project stems from our previous work [1] and may be considered an application of the tunnel diode embodying unseen surprises. The circuit analysis is entirely based on utilizing a Computer Algebra System (CAS) specifically Mathematica. Without a CAS, the completion of the project wouldn’t have been possible otherwise.
文摘We consider the motion of a massive point-like projectile thrown with initial velocity with respect to horizontal in a two-dimensional vertical plane under the influence of gravity in a viscose media. Two different velocity-dependent resistive media models are considered—linear and quadratic. With an objective to utilizing a Computer Algebra System (CAS), specifically <em>Mathematica</em> [1] numerically we solve the corresponding equations of motions. For a set of compatible parameters characterizing viscose forces graphically we display comparing the trajectories explicitly showing the impact of the models. Utilizing the model-dependent trajectory equations numerically we evaluate their associated arc-lengths. What distinguishes our approach vs. the existing body of work is the notion of the “reverse engineering”. Meaning, utilizing our numeric data we establish their corresponding analytic counter parts. Ultimately, utilizing both outputs numerically and analytically we determine the matching initial projectile angles maximizing their respective arc-lengths.
文摘Two electrically charged rings of different sizes are assembled along their common vertical symmetry axis through their centers. The bottom ring is secured on a horizontal support while the top one is loose. For a set of practical values characterizing the charged rings we envision a scenario where the mutual electric repulsion between the rings and the weight of the top ring results in stable nonlinear oscillations. To quantify the characteristics of the oscillations, we utilize a Computer Algebra System specifically <em>Mathematica</em> [1]. We accompany the analysis with a simulation for a comprehensive visual understanding.
文摘In search of nonlinear oscillations, we envision a 3D elliptic curva-ture-dependent nonuniform charge distribution to creating an electric field along the symmetry axis causing a massive point-like charged particle placed on the symmetry axis to oscillate in a delayed/hesitant nonlinear mode. The charge distribution is a 3D twisted line creating nontrivial electric field causing an unexpected oscillation that is non-orthodox defying the common sense. Calculation of this research flavored investigation is entirely based on utilities accompanied with Computer Algebra Systems (CAS) especially <em>Mathematic</em> [1]. The characteristics of the delayed oscillations in addition to embodying classic graphics displaying the time-dependent kinematic quantities are augmented including various phase diagrams signifying the nonlinear oscillations. The output of our investigation is compared to nonlinear non-delayed oscillations revealing fresh insight. For comprehensive understanding of the hesitant oscillator a simulation program is crafted clarifying visually the scenario on hand.
