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| In [[algebraic topology]], a mathematical discipline, the '''Betti numbers''' can be used to distinguish [[topological space]]s. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces.
| | 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><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>In case you loved this information and you want to receive much more information about [http://www.youtube.com/watch?v=90z1mmiwNS8 Dentists in DC] i implore you to visit the webpage. |
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| Each Betti number is a [[natural number]] or [[Extended real number line|+∞]]. For the most reasonable finite-dimensional spaces (such as [[Compact space|compact]] [[manifold]]s, finite [[simplicial complex]]es or [[CW complex]]es), the sequence of Betti numbers is 0 from some points onwards (Betti numbers vanish above the dimension of a space), and they are all finite.
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| The term "Betti numbers" was coined by [[Henri Poincaré]] after [[Enrico Betti]].
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| ==Informal Definition==
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| [[File:Torus.png|thumb|alt=A torus.|A torus has one connected component, two circular holes (the one in the center and the one in the middle of the "tube"), and one two-dimensional void (the inside of the "tube") yielding Betti numbers of 1,2,1.]]
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| Informally, the ''k''th Betti number refers to the number of unconnected ''k''-dimensional surfaces.<ref>{{cite journal |last=Carlsson |first=G. |authorlink=Gunnar Carlsson |title=Topology and data |journal=[[Bulletin of the American Mathematical Society]] |volume=46 |issue=2 |year=2009 |pages=255–308 |doi= |url=http://www.ams.org/images/carlsson-notes.pdf }}</ref> The first few Betti numbers have the following intuitive definitions:
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| * b<sub>0</sub> is the number of connected components
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| * b<sub>1</sub> is the number of one-dimensional or "circular" holes
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| * b<sub>2</sub> is the number of two-dimensional holes or "voids"
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| ==Definition==
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| For a non-negative [[integer]] ''k'', the ''k''th Betti number ''b''<sub>''k''</sub>(''X'') of the space ''X'' is defined as the [[rank of an abelian group|rank]] of the [[abelian group]] ''H''<sub>''k''</sub>(''X''), the ''k''th [[homology group]] of ''X''. Equivalently, one can define it as the [[vector space dimension]] of ''H''<sub>''k''</sub>(''X''; '''Q'''), since the homology group in this case is a vector space over '''Q''', the field of [[rational number]]s. The [[universal coefficient theorem]], in a very simple case, shows that these definitions are the same.
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| More generally, given a [[Field (mathematics)|field]] ''F'' one can define ''b''<sub>''k''</sub>(''X'', ''F''), the ''k''th Betti number with coefficients in ''F'', as the vector space dimension of ''H''<sub>''k''</sub>(''X'', ''F'').
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| ==Example: the first Betti number in graph theory==
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| In [[topological graph theory]] the first Betti number of a graph ''G'' with ''n'' vertices, ''m'' edges and ''k'' connected components equals | |
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| :<math>m - n + k.\ </math>
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| This may be proved straightforwardly by [[mathematical induction]] on the number of edges. A new edge either increments the number of 1-cycles or decrements the number of connected components.
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| See [[cyclomatic complexity]] for an application of the first Betti number in [[software engineering]].
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| ==Properties==
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| The (rational) Betti numbers ''b''<sub>''k''</sub>(''X'') do not take into account any [[torsion subgroup|torsion]] in the homology groups, but they are very useful basic topological invariants. In the most intuitive terms, they allow one to count the number of ''holes'' of different dimensions. For a [[circle]], the first Betti number is 1. For a general pretzel, the first Betti number is twice the number of holes.
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| In the case of a finite simplicial complex the homology groups ''H''<sub>''k''</sub>(''X'', '''Z''') are finitely-generated, and so have a finite rank. The homology group is 0 when ''k'' exceeds the top dimension of a simplex of ''X''.
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| For a finite CW-complex ''K'' we have
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| :<math>\chi(K)=\sum_{i=0}^\infty(-1)^ib_i(K,F), \,</math>
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| where <math>\chi(K)</math> denotes [[Euler characteristic]] of ''K'' and any field ''F''.
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| For any two spaces ''X'' and ''Y'' we have
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| :<math>P_{X\times Y}=P_X P_Y , \, </math>
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| where ''P''<sub>''X''</sub> denotes the '''Poincaré polynomial''' of ''X'', (more generally, the [[Poincaré series (modular form)|Poincaré series]], for infinite-dimensional spaces), i.e. the
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| [[generating function]] of the Betti numbers of ''X'':
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| :<math>P_X(z)=b_0(X)+b_1(X)z+b_2(X)z^2+\cdots , \,\!</math>
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| see [[Künneth theorem]].
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| If ''X'' is ''n''-dimensional manifold, there is symmetry interchanging ''k'' and ''n'' − ''k'', for any ''k'':
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| :<math>b_k(X)=b_{n-k}(X) , \,\!</math>
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| under conditions (a ''closed'' and ''oriented'' manifold); see [[Poincaré duality]].
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| The dependence on the field ''F'' is only through its [[characteristic (field)|characteristic]]. If the homology groups are [[torsion (algebra)|torsion-free]], the Betti numbers are independent of ''F''. The connection of ''p''-torsion and the Betti number for [[characteristic p|characteristic ''p'']], for ''p'' a prime number, is given in detail by the [[universal coefficient theorem]] (based on [[Tor functor]]s, but in a simple case).
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| ==Examples==
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| #The Betti number sequence for a circle is 1, 1, 0, 0, 0, ...;
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| #:the Poincaré polynomial is
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| #::<math>1+x \,</math>.
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| #The Betti number sequence for a two-[[torus]] is 1, 2, 1, 0, 0, 0, ...;
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| #:the Poincaré polynomial is
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| #::<math>(1+x)^2=1+2x+x^2 \,</math>.
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| #The Betti number sequence for a three-[[torus]] is 1, 3, 3, 1, 0, 0, 0, ... .
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| #:the Poincaré polynomial is
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| #::<math>(1+x)^3=1+3x+3x^2+x^3 \,</math>.
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| #Similarly, for an ''n''-[[torus]],
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| #:the Poincaré polynomial is
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| #::<math>(1+x)^n \,</math> (by the [[Künneth theorem]]), so the Betti numbers are the [[binomial coefficient]]s.
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| It is possible for spaces that are infinite-dimensional in an essential way to have an infinite sequence of non-zero Betti numbers. An example is the infinite-dimensional [[complex projective space]], with sequence 1, 0, 1, 0, 1, ... that is periodic, with [[period length]] 2.
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| In this case the Poincaré function is not a polynomial but rather an infinite series
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| :<math>1+x^2+x^4+\dotsb</math>,
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| which, being a geometric series, can be expressed as the rational function
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| : <math>\frac{1}{1-x^2}=1+x^2+(x^2)^2+(x^2)^3+\dotsb.</math>
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| More generally, any sequence that is periodic can be expressed as a sum of geometric series, generalizing the above (e.g., <math>a,b,c,a,b,c,\dots,</math> has generating function
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| :<math>(a+bx+cx^2)/(1-x^3) \,</math>),
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| and more generally [[linear recursive sequence]]s are exactly the sequences generated by [[rational functions]]; thus the Poincaré series is expressible as a rational function if and only if the sequence of Betti numbers is a linear recursive sequence. | |
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| The Poincaré polynomials of the compact simple Lie groups are: | |
| :<math>P_{SU(n+1)_{}}(x) = (1+x^3)(1+x^5)...(1+x^{2n+1})</math>
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| :<math>P_{SO(2n+1)_{}}(x) = (1+x^3)(1+x^7)...(1+x^{4n-1})</math>
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| :<math>P_{Sp(n)_{}}(x) = (1+x^3)(1+x^7)...(1+x^{4n-1})</math>
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| :<math>P_{SO(2n)_{}}(x) = (1+x^{2n-1})(1+x^3)(1+x^7)...(1+x^{4n-5})</math>
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| :<math>P_{G_{2}}(x) = (1+x^3)(1+x^{11})</math>
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| :<math>P_{F_{4}}(x) = (1+x^3)(1+x^{11})(1+x^{15})(1+x^{23})</math>
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| :<math>P_{E_{6}}(x) = (1+x^3)(1+x^{9})(1+x^{11})(1+x^{15})(1+x^{17})(1+x^{23})</math>
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| :<math>P_{E_{7}}(x) = (1+x^3)(1+x^{11})(1+x^{15})(1+x^{19})(1+x^{23})(1+x^{27})(1+x^{35})</math>
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| :<math>P_{E_{8}}(x) = (1+x^3)(1+x^{15})(1+x^{23})(1+x^{27})(1+x^{35})(1+x^{39})(1+x^{47})(1+x^{59})</math>
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| ==Relationship with dimensions of spaces of differential forms==
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| In geometric situations when <math>X</math> is a [[closed manifold]], the importance of the Betti numbers may arise from a different direction, namely that they predict the dimensions of vector spaces of [[closed differential form]]s ''[[Modular arithmetic|modulo]]'' [[exact differential form]]s. The connection with the definition given above is via three basic results, [[de Rham's theorem]] and [[Poincaré duality]] (when those apply), and the [[universal coefficient theorem]] of [[homology theory]]. | |
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| There is an alternate reading, namely that the Betti numbers give the dimensions of spaces of [[harmonic form]]s. This requires also the use of some of the results of [[Hodge theory]], about the [[Hodge Laplacian]].
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| In this setting, [[Morse theory]] gives a set of inequalities for alternating sums of Betti numbers in terms of a corresponding alternating sum of the number of [[critical point (mathematics)|critical points]] <math>N_i</math> of a [[Morse function]] of a given [[Morse theory|index]]:
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| :<math> b_i(X) - b_{i-1} (X) + \cdots \le N _i - N_{i-1} + \cdots. </math>
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| [[Edward Witten|Witten]] gave an explanation of these inequalities by using the Morse function to modify the [[exterior derivative]] in the [[de Rham complex]].<ref>Witten, Edward (1982). ''Supersymmetry and Morse theory.'' J. Differential Geom. 17 (1982), no. 4, 661–692.</ref>
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| == References ==
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| <references /> | |
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| *{{Citation |first=Frank Wilson |last=Warner |title=Foundations of differentiable manifolds and Lie groups |location=New York |publisher=Springer |year=1983 |isbn=0-387-90894-3 }}.
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| *{{Citation |first=John |last=Roe |title=Elliptic Operators, Topology, and Asymptotic Methods |edition=Second |series=Research Notes in Mathematics Series |volume=395 |location=Boca Raton, FL |publisher=Chapman and Hall |year=1998 |isbn=0-582-32502-1 }}.
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| {{DEFAULTSORT:Betti Number}}
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| [[Category:Algebraic topology]]
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| [[Category:Graph invariants]]
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| [[Category:Topological graph theory]]
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| [[Category:Generating functions]]
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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.
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