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In the mathematical field of [[numerical ordinary differential equations]], a '''geometric integrator''' is a numerical method that preserves geometric properties of the exact [[Vector field#Flow curves|flow]] of a differential equation.
It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you have any kind of inquiries concerning where and the best ways to make use of [http://www.youtube.com/watch?v=90z1mmiwNS8 dentist DC], you can contact us at the page.
 
==Pendulum example==
We can motivate the study of geometric integrators by considering the motion of a [[simple pendulum|pendulum]].
 
Assume that we have a pendulum whose bob has mass <math>m=1</math> and
whose rod is massless of length <math>\ell=1</math>.  Take the
acceleration due to gravity to be <math>g=1</math>. Denote by
<math>q(t)</math> the angular displacement of the rod from the vertical,
and by <math>p(t)</math> the pendulum's momentum. The [[Hamiltonian mechanics|Hamiltonian]] of
the system, the sum of its [[kinetic energy|kinetic]] and [[potential energy|potential]] energies, is
 
:<math>H(q,p) = T(p)+U(q) = \frac{1}{2}p^2 - \cos q, </math>
 
which gives [[Hamilton's equations]]
 
:<math>(\dot q,\dot p) = (p,-\sin q). \, </math>
 
It is natural to take the [[configuration space]] <math>Q</math> of all <math>q</math> to be the unit
circle <math>\mathbb S^1</math>, so that <math>(q,p)</math> lies on the
cylinder <math>\mathbb S^1\times\mathbb R</math>. However, we will take
<math>(q,p)\in\mathbb R^2</math>, simply because <math>(q,p)</math>-space is
then easier to plot. Define <math>z(t) = (q(t),p(t))^{\mathrm T}</math>
and <math>f(z) = (p,-\sin q)^{\mathrm T}</math>. Let us experiment by
using some simple numerical methods to integrate this system. As usual,
we select a constant step size, <math>h</math>, and for an aribtrary non-negative integer <math>k</math> we write
<math>z_k:=z(kh)</math>.
We use the following methods.
 
: <math> z_{k+1} = z_k + hf(z_k) \, </math> ([[Euler method|explicit Euler]]),
 
: <math> z_{k+1} = z_k + hf(z_{k+1}) \, </math> ([[implicit Euler method|implicit Euler]]),
 
: <math> z_{k+1} = z_k + hf(q_k,p_{k+1}) \, </math> ([[Euler–Cromer algorithm|symplectic Euler]]),
 
: <math> z_{k+1} = z_k + hf((z_{k+1}+z_k)/2) \, </math> ([[implicit midpoint rule]]).
 
(Note that the symplectic Euler method treats ''q'' by the explicit and <math>p</math> by the implicit Euler method.)
 
The observation that <math>H</math> is constant along the solution
curves of the Hamilton's equations allows us to describe the exact
trajectories of the system: they are the [[level set|level curves]] of <math>p^2/2 -
\cos q</math>. We plot, in <math>\mathbb R^2</math>, the exact
trajectories and the numerical solutions of the system. For the explicit
and implicit Euler methods we take <math>h=0.2</math>, and ''z''<sub>0</sub>&nbsp;=&nbsp;(0.5,&nbsp;0) and (1.5,&nbsp;0) respectively; for the other two methods we take <math>h=0.3</math>, and ''z''<sub>0</sub>&nbsp;=&nbsp;(0,&nbsp;0.7), (0,&nbsp;1.4) and (0,&nbsp;2.1).
[[Image:pendulumtrajectories.png|thumb|300px|Simple pendulum: trajectories]]
The explicit (resp. implicit) Euler method spirals out from (resp. in to) the origin. The other two methods show the correct qualitative behaviour, with the implicit midpoint rule agreeing with the exact solution to a greater degree than the symplectic Euler method.
 
Recall that the exact flow <math>\phi_t</math> of a Hamiltonian system with one degree of freedom is
area-preserving, in the sense that
:<math>\det\frac{\partial\phi_t}{\partial (q_0,p_0)} = 1</math> for all <math>t</math>.
This formula is easily verified by hand. For our pendulum
example we see that the numerical flow <math>\Phi_{{\mathrm{eE}},h}:z_k\mapsto z_{k+1}</math> of the explicit Euler method is '''not''' area-preserving; viz.,
 
:<math>\det\frac{\partial}{\partial (q_0,p_0)}\Phi_{{\mathrm{eE}},h}(z_0)
    = \begin{vmatrix}1&h\\-h\cos q_0&1\end{vmatrix}
  = 1+h^2\cos q_0.</math>
 
A similar calculation can be carried out for the implicit Euler method,
where the determinant is
 
:<math>\det\frac{\partial}{\partial (q_0,p_0)}\Phi_{{\mathrm{iE}},h}(z_0)
    = (1+h^2\cos q_1)^{-1}.</math>
 
However, the symplectic Euler method '''is''' area-preserving:
 
:<math>
    \begin{pmatrix}1&-h\\0&1\end{pmatrix}\frac{\partial}{\partial (q_0,p_0)}\Phi_{{\mathrm{sE}},h}(z_0)
    = \begin{pmatrix}1&0\\-h\cos q_0&1\end{pmatrix},</math>
 
thus <math>\det(\partial\Phi_{{\mathrm{sE}},h}/\partial (q_0,p_0)) = 1</math>. The implicit midpoint rule has similar geometric properties.
 
To summarize: the pendulum example shows that, besides the explicit and
implicit Euler methods not being good choices of method to solve the
problem, the symplectic Euler method and implicit midpoint rule agree
well with the exact flow of the system, with the midpoint rule agreeing
more closely. Furthermore, these latter two methods are area-preserving,
just as the exact flow is; they are two examples of geometric (in fact, [[symplectic integrator|symplectic]]) integrators.
 
==Moving frame method==
 
The [[moving frame]] method can be used to construct numerical methods which preserve [[Lie group|Lie]] [[Symmetry group|symmetries]] of the ODE. Existing methods such as [[Runge-Kutta]] can be modified using moving frame method to produce invariant versions.<ref>[http://www.math.ohio-state.edu/~pwkim/invode.pdf Invariantization of Numerical Schemes Using Moving Frames], Pilwon Kim, 2006, BIT Numerical Mathematics, 2007 &ndash; Springer</ref>
 
==See also==
* [[Energy drift]]
 
==References==
<references/>
*Ernst Hairer, Christian Lubich and Gerhard Wanner, ''Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations.'' Springer, Berlin, 2002. ISBN 3-540-43003-2.
*Ben Leimkuhler and Sebastian Reich, ''Simulating Hamiltonian Dynamics.''  Cambridge University Press, 2005. ISBN 0-521-77290-7.
 
{{DEFAULTSORT:Geometric Integrator}}
[[Category:Numerical differential equations]]

Latest revision as of 19:08, 29 December 2014

It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.

Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.

At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.

Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.

Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.

Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.

Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.

Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.

In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.

If you have any kind of inquiries concerning where and the best ways to make use of dentist DC, you can contact us at the page.

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