Maple Document: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=1337
en-us2020 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemSun, 29 Mar 2020 03:58:28 GMTSun, 29 Mar 2020 03:58:28 GMTNew applications in the Maple Document categoryhttps://www.maplesoft.com/images/Application_center_hp.jpgMaple Document: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=1337
Quantum Mechanics of Infrared Spectroscopy
https://www.maplesoft.com/applications/view.aspx?SID=154616&ref=Feed
Infrared Spectroscopy is an incredibly versatile experimental technique that allows us to gather structural information about gaseous, liquid, or solid samples. While interpreting IR spectra is a basic practice in any undergraduate chemistry curriculum, understanding the theory requires a deeper grasp of the principles of quantum mechanics. Here we will explore the fundamental quantum mechanics that allows us to understand how IR spectroscopy works, and we will then take a deeper dive into more advanced topics that connect the theory and practice of IR spectroscopy.<A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.<img src="https://www.maplesoft.com/view.aspx?si=154616/harm_osc.JPG" alt="Quantum Mechanics of Infrared Spectroscopy" style="max-width: 25%;" align="left"/>Infrared Spectroscopy is an incredibly versatile experimental technique that allows us to gather structural information about gaseous, liquid, or solid samples. While interpreting IR spectra is a basic practice in any undergraduate chemistry curriculum, understanding the theory requires a deeper grasp of the principles of quantum mechanics. Here we will explore the fundamental quantum mechanics that allows us to understand how IR spectroscopy works, and we will then take a deeper dive into more advanced topics that connect the theory and practice of IR spectroscopy.<A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.https://www.maplesoft.com/applications/view.aspx?SID=154616&ref=FeedMon, 16 Mar 2020 04:00:00 ZMiah TurkeMiah TurkeCoupled Cluster Theory
https://www.maplesoft.com/applications/view.aspx?SID=154609&ref=Feed
Hartree Fock does not account for electron correlation, so many post-Hartree-Fock methods have been developed including coupled cluster theory which excels at treating smaller molecules. In the worksheet, we present the definitions and ideas behind coupled cluster theory. Comparisons will be made between coupled cluster and the configuration interaction method including comments on computational complexity. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.<img src="https://www.maplesoft.com/view.aspx?si=154609/Screen_Shot_2020-03-16_at_9.26.35_AM.png" alt="Coupled Cluster Theory" style="max-width: 25%;" align="left"/>Hartree Fock does not account for electron correlation, so many post-Hartree-Fock methods have been developed including coupled cluster theory which excels at treating smaller molecules. In the worksheet, we present the definitions and ideas behind coupled cluster theory. Comparisons will be made between coupled cluster and the configuration interaction method including comments on computational complexity. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.https://www.maplesoft.com/applications/view.aspx?SID=154609&ref=FeedMon, 16 Mar 2020 04:00:00 ZJoshua WagnerJoshua WagnerUse of DFT in ab initio Molecular Dynamics
https://www.maplesoft.com/applications/view.aspx?SID=154613&ref=Feed
Molecular dynamics (MD) simulations work by generating trajectories for particles calculated from the equations of motion that describe the system. In classical MD, we treat these systems with the Newtonian equations of motion along a potential energy landscape. Within this approximation various methods have arisen, namely from the use different approximations to the potential function that describes the potential landscape. However, this treatment does not allow for the inclusion of more complex electronic effects.
<BR><BR>
A way to address this problem is by the use of the Schrodinger equation of quantum mechanics instead of Newtonian mechanics as is done in ab initio Molecular Dynamics (AIMD). Using many of the approximations familiar to quantum mechanics, we are able to simulate phenomena that are much more difficult to describe using classical MD such as hydrogen bonding and chemical reactions. Using density functional theory (DFT) to help define the potential is one way that allows a more accurate description of these. Here we will explore the differences in the two approaches to Molecular Dynamics (MD and AIMD) by comparison of a typical force field used in classical MD and a DFT calculation that could be used in AIMD, both for water.
<BR><BR>
<A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.<img src="https://www.maplesoft.com/view.aspx?si=154613/Water_e_density.png" alt="Use of DFT in ab initio Molecular Dynamics" style="max-width: 25%;" align="left"/>Molecular dynamics (MD) simulations work by generating trajectories for particles calculated from the equations of motion that describe the system. In classical MD, we treat these systems with the Newtonian equations of motion along a potential energy landscape. Within this approximation various methods have arisen, namely from the use different approximations to the potential function that describes the potential landscape. However, this treatment does not allow for the inclusion of more complex electronic effects.
<BR><BR>
A way to address this problem is by the use of the Schrodinger equation of quantum mechanics instead of Newtonian mechanics as is done in ab initio Molecular Dynamics (AIMD). Using many of the approximations familiar to quantum mechanics, we are able to simulate phenomena that are much more difficult to describe using classical MD such as hydrogen bonding and chemical reactions. Using density functional theory (DFT) to help define the potential is one way that allows a more accurate description of these. Here we will explore the differences in the two approaches to Molecular Dynamics (MD and AIMD) by comparison of a typical force field used in classical MD and a DFT calculation that could be used in AIMD, both for water.
<BR><BR>
<A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.https://www.maplesoft.com/applications/view.aspx?SID=154613&ref=FeedMon, 16 Mar 2020 04:00:00 ZJesus AlvarezJesus AlvarezAn Introduction to Relativistic Quantum Mechanics
https://www.maplesoft.com/applications/view.aspx?SID=154614&ref=Feed
The influence of special relativity on quantum chemical phenomena can be understood entirely from the Dirac equation. The Dirac equation is the fully relativistic wave equation for the electron, which holds amongst its predictions spin and its coupling to magnetic fields. The Dirac equation will be examined in the low-speed limit to return a Schrodinger equation with relativistic perturbations. These scalar and vector perturbations, which explain s/p orbital contraction and d/f orbital destabilization, will be derived. The effect of special relativity on chemical bonding will also be discussed through an example. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.<img src="https://www.maplesoft.com/view.aspx?si=154614/image.png" alt="An Introduction to Relativistic Quantum Mechanics" style="max-width: 25%;" align="left"/>The influence of special relativity on quantum chemical phenomena can be understood entirely from the Dirac equation. The Dirac equation is the fully relativistic wave equation for the electron, which holds amongst its predictions spin and its coupling to magnetic fields. The Dirac equation will be examined in the low-speed limit to return a Schrodinger equation with relativistic perturbations. These scalar and vector perturbations, which explain s/p orbital contraction and d/f orbital destabilization, will be derived. The effect of special relativity on chemical bonding will also be discussed through an example. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.https://www.maplesoft.com/applications/view.aspx?SID=154614&ref=FeedMon, 16 Mar 2020 04:00:00 ZPatrick SahrmannPatrick SahrmannPath Integral Formulation of Quantum Mechanics
https://www.maplesoft.com/applications/view.aspx?SID=154615&ref=Feed
There are many mutually equivalent physical formulations of quantum mechanics. The two earliest and most well known are Schrodinger's wavefunction formulation and Heisenberg's matrix formulation. After that, there were attempts to generalize classical mechanics concepts into quantum mechanics, which saw the introduction of Bohm formulation and Wigner formulation, etc. However, it was until Richard P. Feynman, that a rigid connection between quantum mechanics and one of the most important quantity in classical mechanics, the action, was established.
<BR><BR>
In this worksheet, we started from the motivations of Feynman to introduce path integral quantum mechanics formulation, its physical interpretation and how it goes back to Schrodinger's picture, and an application on harmonic oscillators.
<BR><BR>
<A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.<img src="https://www.maplesoft.com/view.aspx?si=154615/WeChat_Image_20200316135409.jpg" alt="Path Integral Formulation of Quantum Mechanics" style="max-width: 25%;" align="left"/>There are many mutually equivalent physical formulations of quantum mechanics. The two earliest and most well known are Schrodinger's wavefunction formulation and Heisenberg's matrix formulation. After that, there were attempts to generalize classical mechanics concepts into quantum mechanics, which saw the introduction of Bohm formulation and Wigner formulation, etc. However, it was until Richard P. Feynman, that a rigid connection between quantum mechanics and one of the most important quantity in classical mechanics, the action, was established.
<BR><BR>
In this worksheet, we started from the motivations of Feynman to introduce path integral quantum mechanics formulation, its physical interpretation and how it goes back to Schrodinger's picture, and an application on harmonic oscillators.
<BR><BR>
<A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.https://www.maplesoft.com/applications/view.aspx?SID=154615&ref=FeedMon, 16 Mar 2020 04:00:00 ZDa TengDa TengQuantum Phase Transitions
https://www.maplesoft.com/applications/view.aspx?SID=154617&ref=Feed
While classical phase transitions are common phenomena widely understood on a basic level, the subject of quantum phase transitions is much less frequently broached at a pedagogical level. Here, our discussion will start with a basic introduction of the fundamental differences (and similarities) between classical and quantum phase transitions. We will introduce the Ising model as a quantitative model of phase transitions, and major results of the 1D Ising model will be briefly reviewed. Moving into applications of quantum phase transitions using the Ising model, the magnetization phase diagram of LiHoF4 will be presented, along with some of the electronic structure properties of the compound to demonstrate why it is a suitable example. Electronic structure properties, such as dipole moment, will be calculated in the worksheet using Maple. Overall, students should come away with an appreciation of using the Ising model to understand quantum phase transitions, and how this model can be applied to real systems. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.<img src="https://www.maplesoft.com/view.aspx?si=154617/qpt.png" alt="Quantum Phase Transitions" style="max-width: 25%;" align="left"/>While classical phase transitions are common phenomena widely understood on a basic level, the subject of quantum phase transitions is much less frequently broached at a pedagogical level. Here, our discussion will start with a basic introduction of the fundamental differences (and similarities) between classical and quantum phase transitions. We will introduce the Ising model as a quantitative model of phase transitions, and major results of the 1D Ising model will be briefly reviewed. Moving into applications of quantum phase transitions using the Ising model, the magnetization phase diagram of LiHoF4 will be presented, along with some of the electronic structure properties of the compound to demonstrate why it is a suitable example. Electronic structure properties, such as dipole moment, will be calculated in the worksheet using Maple. Overall, students should come away with an appreciation of using the Ising model to understand quantum phase transitions, and how this model can be applied to real systems. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.https://www.maplesoft.com/applications/view.aspx?SID=154617&ref=FeedMon, 16 Mar 2020 04:00:00 ZJeriann BeiterJeriann BeiterVariational Quantum Monte Carlo
https://www.maplesoft.com/applications/view.aspx?SID=154619&ref=Feed
The many electron system is one that quantum mechanics have no exact solution for because of the correlation between each electron with every other electron. Many methods have been developed to account for this correlation. One family of methods known as quantum Monte Carlo methods takes a numerical approach to solving the many-body problem. By employing a stochastic process to estimate the energy of a system, this method can be used to calculate the ground state energy of a system using the variational principle. The core of this method involves choosing an ansatz with a set of variational parameters that can be optimized to minimize the energy of the system. Recent research in this field is focused on developing methods to faster and more efficiently optimize this trial ansatz and also to make the ansatz more flexible by allowing more parameters. These topics along with the theory behind variational Monte Carlo (VMC) will be the subject of this Maple worksheet. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.<img src="https://www.maplesoft.com/view.aspx?si=154619/pimc.png" alt="Variational Quantum Monte Carlo" style="max-width: 25%;" align="left"/>The many electron system is one that quantum mechanics have no exact solution for because of the correlation between each electron with every other electron. Many methods have been developed to account for this correlation. One family of methods known as quantum Monte Carlo methods takes a numerical approach to solving the many-body problem. By employing a stochastic process to estimate the energy of a system, this method can be used to calculate the ground state energy of a system using the variational principle. The core of this method involves choosing an ansatz with a set of variational parameters that can be optimized to minimize the energy of the system. Recent research in this field is focused on developing methods to faster and more efficiently optimize this trial ansatz and also to make the ansatz more flexible by allowing more parameters. These topics along with the theory behind variational Monte Carlo (VMC) will be the subject of this Maple worksheet. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.https://www.maplesoft.com/applications/view.aspx?SID=154619&ref=FeedMon, 16 Mar 2020 04:00:00 ZZhuoxun JiangZhuoxun JiangPath Integrals and Interferometry
https://www.maplesoft.com/applications/view.aspx?SID=154620&ref=Feed
This worksheet attempts to motivate and derive the path-integral propagator of quantum mechanics. As an example, the results from a single photon Mach-Zehnder interferometer experiment are derived, demonstrating the necessity of superposition. The example is extended to the Elitzur-Vaiman bomb tester to provide an example of interaction-free measurement. Finally, the use of the propagator in dynamics is motivated using the potential energy surface of molecular nitrogen. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.<img src="https://www.maplesoft.com/view.aspx?si=154620/Mach-Zehnder_Diagram.png" alt="Path Integrals and Interferometry" style="max-width: 25%;" align="left"/>This worksheet attempts to motivate and derive the path-integral propagator of quantum mechanics. As an example, the results from a single photon Mach-Zehnder interferometer experiment are derived, demonstrating the necessity of superposition. The example is extended to the Elitzur-Vaiman bomb tester to provide an example of interaction-free measurement. Finally, the use of the propagator in dynamics is motivated using the potential energy surface of molecular nitrogen. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.https://www.maplesoft.com/applications/view.aspx?SID=154620&ref=FeedMon, 16 Mar 2020 04:00:00 ZJohn PetersonJohn PetersonExplicitly correlated wavefunctions
https://www.maplesoft.com/applications/view.aspx?SID=154622&ref=Feed
Orbital based electronic structure methods are widely used to compute properties of chemical systems and yield accurate results in systems without strong correlations, yet with the basis sets commonly used today, these exhibit slow convergence towards the complete basis set limit. This is partially due to the inability of commonly used basis sets, such as cc-pVXZ, to properly represent the correlation cusp condition with a finite number of contracted Gaussian basis elements. By including an explicitly addressing correlation cusp, these techniques can exhibit better convergence properties, but sometimes at the cost of additional computational resources required. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Explicitly correlated wavefunctions" style="max-width: 25%;" align="left"/>Orbital based electronic structure methods are widely used to compute properties of chemical systems and yield accurate results in systems without strong correlations, yet with the basis sets commonly used today, these exhibit slow convergence towards the complete basis set limit. This is partially due to the inability of commonly used basis sets, such as cc-pVXZ, to properly represent the correlation cusp condition with a finite number of contracted Gaussian basis elements. By including an explicitly addressing correlation cusp, these techniques can exhibit better convergence properties, but sometimes at the cost of additional computational resources required. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.https://www.maplesoft.com/applications/view.aspx?SID=154622&ref=FeedMon, 16 Mar 2020 04:00:00 ZChris ChiChris ChiPath Integral Method in Quantum Mechanics
https://www.maplesoft.com/applications/view.aspx?SID=154623&ref=Feed
This worksheet first introduces and derives the basic concepts and functional form of the path integral in Quantum Mechanics, then it explores two types of Path Integrals (Statistical and Time-dependent) and with example programs written for the latter one. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Path Integral Method in Quantum Mechanics" style="max-width: 25%;" align="left"/>This worksheet first introduces and derives the basic concepts and functional form of the path integral in Quantum Mechanics, then it explores two types of Path Integrals (Statistical and Time-dependent) and with example programs written for the latter one. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.https://www.maplesoft.com/applications/view.aspx?SID=154623&ref=FeedMon, 16 Mar 2020 04:00:00 ZGuan WangGuan WangGroup Theory and Vibrational Spectroscopy
https://www.maplesoft.com/applications/view.aspx?SID=154607&ref=Feed
In this worksheet, we will explore the topic of symmetry and Group Theory. After a self-contained introduction of some formalisms, we will learn how to classify molecules depending on their structure, and how to account for their symmetry elements using the so-called irreducible representations. We will work on specific examples to show how this approach can convey information regarding the vibrational modes of molecules, which of these will be active in Infrared and Raman spectroscopy, and even the specific type of vibration that will be observed. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.<img src="https://www.maplesoft.com/view.aspx?si=154607/gtvs.png" alt="Group Theory and Vibrational Spectroscopy" style="max-width: 25%;" align="left"/>In this worksheet, we will explore the topic of symmetry and Group Theory. After a self-contained introduction of some formalisms, we will learn how to classify molecules depending on their structure, and how to account for their symmetry elements using the so-called irreducible representations. We will work on specific examples to show how this approach can convey information regarding the vibrational modes of molecules, which of these will be active in Infrared and Raman spectroscopy, and even the specific type of vibration that will be observed. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.https://www.maplesoft.com/applications/view.aspx?SID=154607&ref=FeedSun, 15 Mar 2020 04:00:00 ZLuis Busto de MonerLuis Busto de MonerTwo and four component Hamiltonians in Relativistic Quantum Chemistry
https://www.maplesoft.com/applications/view.aspx?SID=154608&ref=Feed
In this paper, we will first go over the theory of Relativistic quantum mechanics applied to chemistry, then talk about the Influence of relativity on properties of chemical molecules, and use Maple together with ORCA quantum chemistry packages to calculate energies of Helium and Mercury atoms as an example to illustrate the theory. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.<img src="https://www.maplesoft.com/view.aspx?si=154608/project_36.gif" alt="Two and four component Hamiltonians in Relativistic Quantum Chemistry" style="max-width: 25%;" align="left"/>In this paper, we will first go over the theory of Relativistic quantum mechanics applied to chemistry, then talk about the Influence of relativity on properties of chemical molecules, and use Maple together with ORCA quantum chemistry packages to calculate energies of Helium and Mercury atoms as an example to illustrate the theory. <A HREF="/products/toolboxes/quantumchemistry/">This worksheet uses the Maple Quantum Chemistry Toolbox</A>.https://www.maplesoft.com/applications/view.aspx?SID=154608&ref=FeedSun, 15 Mar 2020 04:00:00 ZNan ShengNan ShengAdd Reverb to Audio with Convolution
https://www.maplesoft.com/applications/view.aspx?SID=154606&ref=Feed
You can add special effects (such as echo and reverb) to audio with a technique known as convolution.
<BR><BR>
In this application, we:
<UL>
<LI>import a recording of a voice, recorded on a microphone near to the speaker.
<LI>import an impulse response (a single clap of the hands recorded in an enclosed space that has hard walls).
<LI>convolve the audio with the impulse response.
</UL>
After convolution, the human voice now has echo and reverb.<img src="https://www.maplesoft.com/view.aspx?si=154606/reverb_thumb.png" alt="Add Reverb to Audio with Convolution" style="max-width: 25%;" align="left"/>You can add special effects (such as echo and reverb) to audio with a technique known as convolution.
<BR><BR>
In this application, we:
<UL>
<LI>import a recording of a voice, recorded on a microphone near to the speaker.
<LI>import an impulse response (a single clap of the hands recorded in an enclosed space that has hard walls).
<LI>convolve the audio with the impulse response.
</UL>
After convolution, the human voice now has echo and reverb.https://www.maplesoft.com/applications/view.aspx?SID=154606&ref=FeedThu, 12 Mar 2020 04:00:00 ZSamir KhanSamir KhanGraph Theory and MaplePrimes
https://www.maplesoft.com/applications/view.aspx?SID=154605&ref=Feed
<A HREF="https://www.mapleprimes.com">MaplePrimes</A> is a discussion forum for Maple users. Users interact with each other by asking and answering questions, or posting interesting content that others comment on. Some users are motivated to post often, while others take a more relaxed attitude to the forum.
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Naturally, this means that some users create a bigger ripple than others. This application identifies the most influential contributors on MaplePrimes using techniques from graph theory.<img src="https://www.maplesoft.com/view.aspx?si=154605/MaplePrimesGraphTheory_thumb.png" alt="Graph Theory and MaplePrimes" style="max-width: 25%;" align="left"/><A HREF="https://www.mapleprimes.com">MaplePrimes</A> is a discussion forum for Maple users. Users interact with each other by asking and answering questions, or posting interesting content that others comment on. Some users are motivated to post often, while others take a more relaxed attitude to the forum.
<BR><BR>
Naturally, this means that some users create a bigger ripple than others. This application identifies the most influential contributors on MaplePrimes using techniques from graph theory.https://www.maplesoft.com/applications/view.aspx?SID=154605&ref=FeedFri, 06 Mar 2020 05:00:00 ZSamir KhanSamir KhanEtoile fractale et flocon de Koch
https://www.maplesoft.com/applications/view.aspx?SID=154599&ref=Feed
Cette Maplet utilise une methode fractale pour générer à partir d'un polygone régulier,à sa première itération une étoile et d'autres formes pour les itérations suivantes.
Une variante permet de générer le flocon de Koch.
L'image présente montre la deuxième itération sur un pentagone.<img src="https://www.maplesoft.com/view.aspx?si=154599/etoile_fractale.gif" alt="Etoile fractale et flocon de Koch" style="max-width: 25%;" align="left"/>Cette Maplet utilise une methode fractale pour générer à partir d'un polygone régulier,à sa première itération une étoile et d'autres formes pour les itérations suivantes.
Une variante permet de générer le flocon de Koch.
L'image présente montre la deuxième itération sur un pentagone.https://www.maplesoft.com/applications/view.aspx?SID=154599&ref=FeedWed, 19 Feb 2020 05:00:00 Zxavier cormierxavier cormierMaplets etoile de neige et étoiles imbriquées
https://www.maplesoft.com/applications/view.aspx?SID=154588&ref=Feed
Cette Maplet permet de genérer des étoiles de neige,des sapins ou des fougères.
On peut sauvegarder le dessin en .gif<img src="https://www.maplesoft.com/view.aspx?si=154588/flocon_de_neige.jpg" alt="Maplets etoile de neige et étoiles imbriquées" style="max-width: 25%;" align="left"/>Cette Maplet permet de genérer des étoiles de neige,des sapins ou des fougères.
On peut sauvegarder le dessin en .gifhttps://www.maplesoft.com/applications/view.aspx?SID=154588&ref=FeedFri, 14 Feb 2020 06:08:09 Zxavier cormierxavier cormierSpeed-up calculation of nextprime
https://www.maplesoft.com/applications/view.aspx?SID=5729&ref=Feed
<p>A speed-up calculation of the functions nextprime and prevprime is intended. In some distributions used it was observed similarities to "Prime Number Races" (primes of the form q<i>n</i>+a).</p><img src="https://www.maplesoft.com/view.aspx?si=5729/nextprime_19_sm.gif" alt="Speed-up calculation of nextprime" style="max-width: 25%;" align="left"/><p>A speed-up calculation of the functions nextprime and prevprime is intended. In some distributions used it was observed similarities to "Prime Number Races" (primes of the form q<i>n</i>+a).</p>https://www.maplesoft.com/applications/view.aspx?SID=5729&ref=FeedFri, 14 Feb 2020 05:00:00 ZGiulio BonfissutoGiulio BonfissutoBee-Cell Structure.mw
https://www.maplesoft.com/applications/view.aspx?SID=154603&ref=Feed
Nature, in general, affords many examples of economy of space and time for anyone who is curious enough to stop for a while and think.
Pythagoras (6th Century BC) was reported to have known the fact that the circle is the figure that has the greatest surface area among all plane figures having the same perimeter which is obviously an example of economy of space.
Heron (2d Century BC) deduced the fact that light after reflection follows the shortest path, hence an example of economy of space.
Fermat (1601-1665) was seeking a way to put the law of light refraction under a form similar to that given by Heron for the light reflection but this time he was looking for an economy of time rather than space.
Huygens (1629-1695), building on Fermat finding, considered the path of light not as a straight line but rather as a curve when passing through mediums where its velocity is variable from one point to an other. Hence economy of time.
Jean Bernoulli (1667-1748) he too was building on Huygens concept when he solved the problem of the brachystochron which is based on economy of time.
This is not to say that nature economy is concerned with only physical phenomena but examples taken from living creatures abound around us. To take one example that many researches have examined very carefully and in many details in the past is that of the honeycomb building in a bee hive.
It turns out, as we shall soon prove, that the bottom of any bee-cell has the form of a trihedron with 3 equal rhombi (rhombotrihedron) which, once added to the hexagonal right prism, will make the total surface area smaller resulting in economy on the precious wax that is secreted and used by worker bee in the construction of the entire cell.
Our plan in this article has a double purpose:
1- to prove the minimal surface we referred to above using a classical proof.
2- To start with no preconceived idea about the bee-cell then
A- to consider a trihedron having 90 degrees dihedral angle between all 3 planes.
B- To get an equation relating dihedral angle with the larger angle in a rhombus that, once
solved, gives exactly the dihedral angle of 120 degrees along with the larger angle in each
rhombus as 109.47122 degrees which are the exact data one can find at the bottom end of a
honey-cell.
This configuration which is that of a minimal surface of the cell is the only one that our
equation can give for all 3 planes to have in common these two angles . All others have
different dihedral and larger angles.
I believe that the neat and simple equation I arrived at is somehow original and so far I have no idea if anyone else has found it before me.<img src="https://www.maplesoft.com/view.aspx?si=154603/3b4238cc9e7aeb803a42872bde31a350.gif" alt="Bee-Cell Structure.mw" style="max-width: 25%;" align="left"/>Nature, in general, affords many examples of economy of space and time for anyone who is curious enough to stop for a while and think.
Pythagoras (6th Century BC) was reported to have known the fact that the circle is the figure that has the greatest surface area among all plane figures having the same perimeter which is obviously an example of economy of space.
Heron (2d Century BC) deduced the fact that light after reflection follows the shortest path, hence an example of economy of space.
Fermat (1601-1665) was seeking a way to put the law of light refraction under a form similar to that given by Heron for the light reflection but this time he was looking for an economy of time rather than space.
Huygens (1629-1695), building on Fermat finding, considered the path of light not as a straight line but rather as a curve when passing through mediums where its velocity is variable from one point to an other. Hence economy of time.
Jean Bernoulli (1667-1748) he too was building on Huygens concept when he solved the problem of the brachystochron which is based on economy of time.
This is not to say that nature economy is concerned with only physical phenomena but examples taken from living creatures abound around us. To take one example that many researches have examined very carefully and in many details in the past is that of the honeycomb building in a bee hive.
It turns out, as we shall soon prove, that the bottom of any bee-cell has the form of a trihedron with 3 equal rhombi (rhombotrihedron) which, once added to the hexagonal right prism, will make the total surface area smaller resulting in economy on the precious wax that is secreted and used by worker bee in the construction of the entire cell.
Our plan in this article has a double purpose:
1- to prove the minimal surface we referred to above using a classical proof.
2- To start with no preconceived idea about the bee-cell then
A- to consider a trihedron having 90 degrees dihedral angle between all 3 planes.
B- To get an equation relating dihedral angle with the larger angle in a rhombus that, once
solved, gives exactly the dihedral angle of 120 degrees along with the larger angle in each
rhombus as 109.47122 degrees which are the exact data one can find at the bottom end of a
honey-cell.
This configuration which is that of a minimal surface of the cell is the only one that our
equation can give for all 3 planes to have in common these two angles . All others have
different dihedral and larger angles.
I believe that the neat and simple equation I arrived at is somehow original and so far I have no idea if anyone else has found it before me.https://www.maplesoft.com/applications/view.aspx?SID=154603&ref=FeedThu, 13 Feb 2020 17:34:36 ZAhmed BaroudyAhmed BaroudyThe Dirac Equation in Robertson-Walker spacetime
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The simplest example of m=0, k=0 Dirac equation in Robertson-Walker spacetime. This is the first step for more general examples of the Dirac equation coupled to Einstein's, supergravity and so on. The computation takes out components and so could be improved,<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="The Dirac Equation in Robertson-Walker spacetime" style="max-width: 25%;" align="left"/>The simplest example of m=0, k=0 Dirac equation in Robertson-Walker spacetime. This is the first step for more general examples of the Dirac equation coupled to Einstein's, supergravity and so on. The computation takes out components and so could be improved,https://www.maplesoft.com/applications/view.aspx?SID=154604&ref=FeedThu, 13 Feb 2020 05:00:00 ZDr. Mark RobertsDr. Mark RobertsAtwood Machine
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The following is a detailed study of the motion of an unconventional Atwood Machine where one mass is constrained to move along a fixed vertical axis.
The differences with the regular Atwood Machine are :
1- the tension T on the string on either side of the pulley though it is the same, however it is not constant in the present case because of the obliquity of the 2d part of the string.
2- the unique and constant acceleration (a) in the simple machine is replaced in here with two different and variable accelerations whose ratio is however constant.
3- In the simple machine the constant acceleration makes plotting and animation of the system a straightforward procedure according to
s = (1/2)*at^2.However in the modified Atwood machine that we present in here the accelerations being variable there is no way to get the displacement as a direct function of time. This seems to make plotting & animation an impossible task. However we were able to devise a trick to overcome this difficulty.<img src="https://www.maplesoft.com/view.aspx?si=154598/Modified_Atwood_Machine.jpg" alt="Atwood Machine" style="max-width: 25%;" align="left"/>The following is a detailed study of the motion of an unconventional Atwood Machine where one mass is constrained to move along a fixed vertical axis.
The differences with the regular Atwood Machine are :
1- the tension T on the string on either side of the pulley though it is the same, however it is not constant in the present case because of the obliquity of the 2d part of the string.
2- the unique and constant acceleration (a) in the simple machine is replaced in here with two different and variable accelerations whose ratio is however constant.
3- In the simple machine the constant acceleration makes plotting and animation of the system a straightforward procedure according to
s = (1/2)*at^2.However in the modified Atwood machine that we present in here the accelerations being variable there is no way to get the displacement as a direct function of time. This seems to make plotting & animation an impossible task. However we were able to devise a trick to overcome this difficulty.https://www.maplesoft.com/applications/view.aspx?SID=154598&ref=FeedSat, 25 Jan 2020 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed Baroudy