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User talk:Eizarazua

 Hello Eizarazua, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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title: User talk:Eizarazua


text:
User:Peter Buch

My user page in the German Wikipedia

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Peter Buch


text:
Pearson's chi-square test

A Pearson's chi-square test can be used as an inferential test of the independence of two nominal variables.

The value of the test-statistic is

 * $$ X^2 = \sum {(O - E)^2 \over E} ,$$

where

 * $$X^2$$ = the test statistic that approaches a χ2 distribution.
 * $$O$$ = frequencies observed;
 * $$E$$ = frequencies expected (asserted by the null hypothesis).

Pearson’s Chi Square (χ^2)
In-Class Notes; CGU PSYCH 308D - 'Categorical Data Analysis' - Dale Berger, taken on 2014-04-01 by Josh Penman = Pearson’s Chi Square - (<- Redirect to) Pearson’s χ^2

χ^2 Rules of Thumb

 * 1) The expected frequencies are at least 10, if you want to be conservative about it.

If you were to take true normal distribution; you would have a Z score - if you randomly took a sample Z score from the distribution and squared it, what bottom value would you have?

A: Zero. (0)

If this distribution is Z^2, what shape do you think it would have? Would it be a nice normal distribution? No: it would be a rapidly descending graph.

Some formulas that need explaining!
Z^2 = χ^2(df=1)

Z^2+Z^2=χ^2(df=2)

Degrees of Freedom

F(Z^2)

E ((Xi-[μ])/(σx) = 1

The expected value of Chi Squared is equal to the degrees of freedom. If someone says, “oh, I found a Chi Square of 15. ..

Some statisticians use a notation “ν” (pronounced “nu” as in “new”) - but there are other beautiful relationships here:

χ^2ν = [∑]*Zi

Mean -

Variance

Kurtosis =

Mode =
 * 1) If ν ≤ 2 then Mode = 0
 * 2) If ν > 2 then Mode = ν-2

Example
If you wanted to test a die to see whether it’s a fair die, what would you do?

Class Answers: Roll it over and over and see what kind of distribution you get

Roll it over and over and see what kind of distribution you get
 * 1) If we roll this die 60 times, how many times would you expect to get a roll of 1? A: 6 (1 out of 6 possible combinations = 1/6; 1/6*60=10.
 * 2) Prior to the Pearson’ Chi Squared era, we’d say “this doesn’t look too bad. . . “
 * 3) If you randomly rolled a die 60 times and got exactly 10 of each. . . that would be extremely improbable!
 * 4) Let’s start by taking the observed frequency minus the expected frequency: (See the Google Example Sheet
 * 5) There are 5 degrees of freedom here (J: How do you calculate degree’s of freedom?)
 * 6) So here, you have Chi Square with 5 degrees of freedom; χf,05^2 at the .05 α level, that zero-value is 11.07
 * So, our value is 9.6: have we proven that the die is fair? A: __
 * 1) In general, if we collect more information, we’ll get a better answer. Here, we don’t have statistical significance at the [α]=.05 level.
 * 1) In general, if we collect more information, we’ll get a better answer. Here, we don’t have statistical significance at the [α]=.05 level.

Now, in gambling; having a 1 and 6 being; if you shave a little bit off the 1 side, and a little bit off the six side; - we want to test whether getting a 1 or a 6; now; if this is a theory not driven by the data, but driven by what we know about our friend and the die, we might run a separate test. ..

 * 1) What is the observed frequency of getting a 1 or a 6?  A: 16 + 16 = 31.
 * 2) What is the observed frequencey of everything else? . . . . A: 29.
 * 3) What is the expected frequencey of a 1 or a 6, and everyting else? A: 20; 40 for everything else.
 * 4) Observed minus expected . . . let’s get the Chi Squared:
 * 5) = (31 - 20)^2/20 + (29-40)^2/40
 * 6) = 9.07
 * 7) Now, how many degrees of freedom do we have here?
 * 8) Two observed frequencies
 * 9) But we have a constraint on here: it’s out of 60; there’s only one piece of information, because if we know the frequency of getting a 1 or a 6, we also know the frequency of getting everything else.
 * So, Chi Squared with one degree of freedom at α=.005 the Chi Squared cutoff is 7.88. If you were to calculate the actual p value for this, it would be __
 * So, Chi Squared with one degree of freedom at α=.005 the Chi Squared cutoff is 7.88. If you were to calculate the actual p value for this, it would be __

The first test we had, you could call a “blob test - 5 degrees of freedom. . . but if you have a test with 1 degree of freedom, that’s more of a laser test.

S^2 = ([∑](Xi-x̅)^2)/(n-1)

Application of Chi Square
[w:Fisher Fisher] had 3 degrees of freedom - with 3, the mode is at one, and so forth. He took what [w:/Mendel Mendel] predicted, and what Mendel observed;

He expected a 3; but got a chi square that was very close: he got a closer model than what one would expect by chance. Fisher had a charitable explanation: that Mendel had over-enthusiastic assistants.

Others have analyzed this too, and made arguments that maybe it wasn’t as bad as Fisher made it out to be, but if the data fits the model too well, that may imply that something fishy (i.e., suspicious) is going on here.

Here’s an issue: If you did a χ^2 test with 1 degree of freedom, is this a 1-Tailed test or a 2-Tailed test?
 * 1) If you get a big χ^2 with one degree of freedom, it’s actually a 2-tailed test: if you want to do a 1-tailed test, you would need to

Question: With Mendel’s work, in genetics, do we really expect a random distribution? A: if parents, each with one blue eyed gene and one brown-eyed gene have children; what’s the probability that they’ll have blue eyes? 1/4th (J: I’m not sure if this is biologically accurate . . .) - same thing with the peas.
 * 1) You have a null hypothesis that you’re dealing with here: each plant is a cross of two particular peas, and 1/4th of them should turn out one way or the other.

Q: How did you get the 11.07 as the cutoff value above? A: That comes from a table.

Other Applications
Now we have a little two-by-two table: you can find this in /Packet 5/

 * 1) See Step 2 in the Google Spreadsheet
 * 2) What % of people with BA are on Salary? A: 85%
 * 3) What % of people without a BA are on Salary? A: 60%

If there is no relationship between salary and education level, what would we expect those frequencies to be? (Refer to Step 2.1 Independence Hypothesis here
 * 1) Null hypothesis is “Independence
 * 2) You would expect 25% of 40; 10, as the expected frequency;

The df = (# of Rows-1) * (# of columns -1)
 * 1) In this case, a 2-by-2 table; it’s (2-1)*(2-1) = 1*1 = 1; df (<- Redirect to Degrees of Freedom) = 1

There’s a neat little hand calculation you can do here:
 * 1) For a 2x2 table:
 * 2) χ^2 = (N(a*d-B*c)^2)/(a+b)*(c+d)*(a+c)*(b+d)
 * 3) Where:
 * 4) a is top left cell;
 * 5) b is top right cell;
 * 6) c is bottom left cell;
 * 7) d is bottom right cell

Now, if you wanted to do __; sometimes called Yate’s Correction for continuity; people have talked about how important it is to apply this. (See https://docs.google.com/spreadsheets/d/12s4TcLMNEvfKl_rVmAmXA5IWqJ2EcxTbzH_PFZtZJ-E/edit#gid=1502721222 Example 3)

There’ only one circumstance where you apply this that is appropriate; the situation is called Fixed Marginals - The Median Test is like this:
 * 1) You have 80 people; and you’re going to runs samples 20 people, and divide them into two groups.
 * 2) You decide a ahead of time you’re going to have 40 people in each group
 * 3) You expect marginal values of 40 for each; if you know the margins before collecting the data, that is called fixed margins.
 * 4) That is the only case in which you will use the Continuity correction (J: in the Behavioral Sciences as of 2014-04-01)
 * 5) Now, your expected frequencies, if you had independent samples, you would expect to find 20 people in each cell.
 * 6) Reduce |fo-fe| by .5  (“fo”= Frequency Observed; “”fe” = Frequency Expected; “|__|” = Absolute Values.

Non-Parametric Chi Square and Fisher’s Exact Test
For cases where you have very small frequencies, you can’t apply this test of independence: Let’s take an example, where we’ve got a program where people start, then drop out.

Now, for this you can calculate the Multinomial outcome;
 * 1) the probability of this exact outcome; give the observed marginals =
 * 2) ((a+b)!*(c+d)!*(a+c)!*(b+d)!)/(n!*A!*b!*c!*d!)
 * 3) (5!*6!*4!*7!)/(11!*0!*5!*4!*2!)
 * 4) = (6*5*4*3*2*1*7!)/11*10*9*8(7!*2!)
 * 5) = 6/(11*2*3*2)
 * 6) = 1/(11*2)
 * 7) = .045

McNemar’s Test
Now, let’s look at McNemar’s Test (Pronounced “Mac-no-mar” by professor) (See Example 5

 * 1) p(+|Judge1) = p(+|Judge 2)
 * 2) The null hypothesis says that the given data is a random outcome of a 50-50 spit; binomial distribution; people have various formulas for this to approximate: You can use a:
 * 3) Chi Square formula: χ^2 = (ABS(b-c)-1)^2/(b+c)
 * 4) = (|15-5|-1)^2/(15+5)
 * 5) = 9^2/20 = 81/20 = ‘’4.05’’
 * 6) We would say that Judge 2 is ___ (J: QUESTION 1!). This is an ‘’’approximation’’’.
 * 7) Z formula for Approximation of the Binomial Distribution (with Yate’s Correction) This is what you actually want to apply: the  binomial distribution (also called, “binomial test, with
 * 8) Z = ‘’(ABS’’(x-Np)’’-.5)’’/ SQRT(n*p*q) where:
 * 9) x = Number of hits you got; the number observed in that cell: (use cell b)
 * 10) P = Likelihood of a hit for cell b
 * 11) Q = Likelihood of a miss for cell b
 * 12) N = b+c = 20
 * 13) p = .05
 * 14) Resulting p = .021 < .05, therefore we have statistical significance (note: current best practice uses 95% Confidence Interval, not just P values

With a small set of data, the McNemar’s Chi Squared Test is not accurate. I would suggest you never use it. In SPSS’s table, for a Chi Square tests, we had a superscript that indicated that a binomial distribution was used for the McNemar Test - that could be similar to SPSS telling you not to use McNemar’s test as well:)

Example 6: Other applications: How people would vote from one year to the next.

Example 7:

title: Pearson's chi-square test


text:
User:Leinad

Meta:user:Leinad

title: User:Leinad


text:
User:Aiacciu9

 * co-wiki
 * en-wiki
 * fr-wiki

title: User:Aiacciu9


text: Template:User co-1

title: Template:User co-1


text:
Portal:Corsican

{| cellpadding="10" cellspacing="5" style="width: 99%; background-color:inherit ; margin-left: auto; margin-right: auto" =Corscian Department=
 * style="background-color: #ffffe0; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px; height: 60px;" colspan="2" |
 * style="width: 100%; background-color: #ffffe0; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" |
 * style="width: 100%; background-color: #ffffe0; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" |

Welcome to the Corscian Department at Wikiversity, part of the Center for Foreign Language Learning and the School of Language and Literature.

Introduction to Corsican
Corsican or Corsu (Corsu or Lingua Corsa) is a romance language spoken and written on the islands of Corsica (France) and northern Sardinia (Italy). Corsu is the traditional native language of the Corsican people, and was long the sole language of the island, which was acquired by France in 1768. In 1990, more than two centuries later, nearly all Corsicans were fluent in French, usually as a first language. In addition, an estimated 50% of those also had some degree of proficiency in Corsu, and a small minority, perhaps 10%, used Corsu as a first language.

To show your interest in the French department, sign up at the French stream.

Courses, Lessons and Resources
Courses currently being offered by the Corsican Department:
 * Corsican I

Planned

 * Corsican II

Department news

 * 28 February 2010 Department founded
 * }

title: Portal:Corsican


text: Corsican I

title: Corsican I


text:
User:Egm6341.s10.Team4.roni/HW4/4.4

= Problem 4: Proof of Higher Order Error for Trap. rule Theorem =

Given
Higher Order Error for Trap. rule equation (1) on slide [[media:Egm6341.s10.mtg21.djvu|21-1]], which is:

 * {| style="width:100%" border="0" align="left"

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle Eq. (1) \ E_n^1= \sum_{k=0}^{n-1}[ \int_{x_k}^{x_{k+1}}f(x)dx- \frac{h}{2}(f_k(x_k)+f(x_{k+1}))]
 * }
 * }

Show
That eq. (1) can be transformed to the following: equation (5) on slide [[media:Egm6341.s10.mtg21.djvu|21-2]]

 * {| style="width:100%" border="0" align="left"

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle E_n^1=\dfrac{h}{2} \sum_{k=0}^{n-1}[ \int_{-1}^{1}g_k(t)dt-(g_k(-1)+g_k(+1))]

where, equation (2) on slide [[media:Egm6341.s10.mtg21.djvu|21-1]]

h= \frac{(b-a)}{n} \ \ \ x_i=a+ih $$ and, equation (3) on slide [[media:Egm6341.s10.mtg21.djvu|21-1]]

x(t)= t*\frac{h}{2} + \frac{x_i+x_{i+1}}{2} \ \ \ t\in[-1,1] $$
 * style= |
 * }.
 * }.

Solution
Defining: equation (1) on slide [[media:Egm6341.s10.mtg21.djvu|21-2]]

g_i(t):=f(x(t)) \ where \ x\in[x_i,x_{i+1}] $$

For one Panel, we have:

x_i=-1 \ \ and \ \ x_{i+1}=1 $$

f(x)_i=g(-1) \ \ \ f(x)_{i+1}=g(1) $$

Also,

\frac{dx(t)}{dt}=\frac{h}{2} $$

\Rightarrow E_n^1= \sum_{k=0}^{n-1} \int_{-1}^{1}g_k(t)\dfrac{h}{2}dt-\dfrac{h}{2}(g_k(-1)+g_k(+1)) $$

Giving,

 * {| style="width:20%" border="0"

$$\displaystyle E_n^1=\dfrac{h}{2} \sum_{k=0}^{n-1}[ \int_{-1}^{1}g_k(t)dt-(g_k(-1)+g_k(+1))] $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

 * style= |

 * }
 * }

Author
Solved and typed by - Reviewed by - .

title: User:Egm6341.s10.Team4.roni/HW4/4.4


text:
User:Egm6341.s10.Team4.roni/HW4/4.5

= Problem 5: Theorem of Higher Order Error for Trap. rule Step1 =

Given
Higher Order Error for Trap. rule equation (5) on slide [[media:Egm6341.s10.mtg21.djvu|21-1]], which is:

 * {| style="width:100%" border="0" align="left"

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle Eq. (5)\ E_n^1=\dfrac{h}{2} \sum_{k=0}^{n-1}[ \int_{-1}^{1}g_k(t)dt-(g_k(-1)+g_k(+1))]
 * }
 * }

Find
That using eq. (1) on slide [[media:Egm6341.s10.mtg21.djvu|21-2]],

 * {| style="width:100%" border="0" align="left"

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle g_k(t):=f(x(t)) \  such \  that \  x \   \epsilon  \left[ x_k,x_{k+1}\right]

For one panel Step 1 :

Eq.(2) \int_{-1}^{1}(-t) {g}^{(1)}(t)dt = \int_{-1}^{1}g_k(t)dt-[(g_k(-1)+g_k(+1))] $$

 * style= |
 * }
 * }

Solution
Using integration by Parts :

\int u {v}^{'}= uv - \int v {u}^{'}\ \ where \ \ u=t \ \ {v}^{'}=g(t) $$

Writing the left side of Eq. (2):

\int_{-1}^{1}(-t) {g}^{(1)}(t)dt = -\int_{-1}^{1}(t) {g}^{(1)}(t)dt $$

= -\Bigg[(t \, g(t)) \Bigg|^{1}_{-1} -\int_{-1}^{1}g(t)dt \Bigg] $$

= -(1*g(1)-(-1)g(-1)-\int_{-1}^{1}g(t)dt) $$

= \int_{-1}^{1}g(t)dt)-(g(-1)+g(1)) $$

 * {| style="width:20%" border="0"

$$\displaystyle \int_{-1}^{1}(-t) {g}^{(1)}(t)dt = \int_{-1}^{1}g(t)dt)-(g(-1)+g(1)) $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

 * style= |

 * }
 * }

Author
Solved and typed by - Reviewed by -

title: User:Egm6341.s10.Team4.roni/HW4/4.5


text:
University of Florida/Egm6341/s10.team3.aks/HW4

Ref. Lecture notes [[media:Egm6341.s10.mtg20.djvu|p.20-2]]

Problem Statement
Discuss Pros and cons of below methods.

1. Taylor Series 2. Composite Trapezoidal rule 3. Composite Simpson's rule 4. Romberg table (including Richardson's extrapolation) 5. Corrected Trapezoidal rule $$ CT_k (n) $$

Solution
 Taylor Series :

Pros

1. Powerful tool that helps to approximate any type of simple or complex functions into a polynomial form with an error in the form of remainder.

2. For oscillatory functions, a high order Taylor seems a good choice, but can lead to problems of loss of significance in the computation if not programmed carefully.

Cons

1. The function should be differentiable in the given limits in which it is approximated as a polynomial.

2. Number of operations can be large. This is not always a problem these days considering how cheap and fast computers are at  present. In general the step size can be made larger the higher the order of the Taylor scheme. But a computational count will tell you whether it is more effective to compute a lot at each step, and take larger steps, or compute little at each step and make due with smaller step sizes. Usually it is more advantageous to go with good lower order method and small steps, but this depends on the problem.

Composite Trapezoidal rule:

Pros:

1. This methods works well for periodic functions like trigonometric functions (sinx, cosx) and it is very accurate because the error is dependent on the difference between the odd derivatives of the limits.

2. Better accuracy then simple trapezoidal rule and also easy to use.

Cons :

1. It's convergence rate is low and we need to have more number of sub intervals.

Composite Simpson's rule:

Pros:

1. Exact for cubic (less then that) polynomials.

2. More accurate then composite trapezoidal rule.

Cons:

1. This rule can be applied to only even number of finite intervals between the limits. So the value of 'n' in Simpson's rule is always an even number. i.e. n=2i; where (i=1,2,3,4....)

 Romberg table (including Richardson's extrapolation):

Pros:

1. The successive computation of $$\;I(2n)$$ is cheaper and faster when $$\;I(n)$$ is already computed.

2. Better accuracy then other methods.

Cons:

1. There is one limitation with this rule.Romberg integration is successful only when integrand satisfies the hypotheses of the Euler–Maclaurin.

= (5) Prove =
 * $$E:= \int\limits_{1}^{-1}(-t)g^{(1)}(t) dt = [P_2(t)g^{(1)}(t)]_{-1}^{1} - \int\limits_{1}^{-1} P_2(t) g^{(2)}(t) dt         $$

Ref. Lecture notes [[media:Egm6341.s10.mtg20.djvu|p.21-2]]

Problem Statement
Prove that $$E:= \int\limits_{1}^{-1}(-t)g^{(1)}(t) dt = [P_2(t)g^{(1)}(t)]_{-1}^{1} - \int\limits_{1}^{-1} P_2(t) g^{(2)}(t) dt   $$

where

$$ P_1(t)= -t $$

and $$ P_2(t)= \int\limits_{1}^{-1} P_1(t) dt $$

Solution
$$ E:= \int\limits_{1}^{-1} \underbrace{(-t)}_{P_1(t)} g^{(1)}(t) dt $$ $$  = \int\limits_{1}^{-1} P_1(t) g^{(1)}(t) dt  $$

$$  = g^{(1)}(t) \int P_1(t) dt - \int [\frac{d}{dt}g^{(1)}(t) \int P_1(t) dt]dt $$

$$  = [g^{(1)}(t) P_2(t))]_{-1}^{1} - \int [g^{(2)}(t) P_2(t)]dt $$

$$  where, P_2(t) = \int P_1(t) dt                        = \int (-t) dt                        = \frac{-t^2}{2} + {\alpha}\rightarrow_{Integration Constt.} $$


text:
Category:Effects of gravitation

This category contains articles describing observable effects of gravitation, including relativistic effects predicted by general relativity and other theories.

title: Category:Effects of gravitation


text:
User talk:Lilsanto34

 Hello Lilsanto34, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 1 March 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 1 March 2010 (UTC)

title: User talk:Lilsanto34


text:
User:B9 hummingbird hovering/Blog/Learning Tibetan Weblog/Archive1/Choying Dzo

Treasury of Dharmadhatu (Tib. Chöying Dzö; Wyl. chos dbyings mdzod) — one of the Seven Treasuries of Longchenpa. It consists of two texts:
 * a short verse work entitled The Precious Treasury of the Basic Space of Phenomena (Skt. Dharmadhātu ratna koṣa nāma; Wyl. chos dbyings rin po che'i mdzod ces bya ba) dealing principally with the view of trekchö, and
 * Longchena's own commentary on those verses, A Treasure Trove of Scriptural Transmission (Skt. Dharmadhātu ratna koṣa nāma vṛtti; Wyl. chos dbyings rin po che'i mdzod ces bya ba'i 'grel pa). Source:

@#/_/d+harma d+hA tu rat+na ko Sha nA ma/ @#/_/chos dbyings rin po che'i mdzod ces bya ba bzhugs so/_/ @##/_/rgya gar skad du/_d+harma d+hA tu rat+na ko Sha nA ma/_bod skad du/_chos dbyings rin po che'i mdzod ces bya ba/_dpal kun tu bzang po la phyag 'tshal lo/_/gdod nas lhun grub ngo mtshar rmad kyi chos/_/rang byung ye shes 'od gsal byang chub sems/_/snang srid snod bcud 'khor 'das 'byung ba'i mdzod/_/mi g.yo sprarkosa dang bral la phyag 'tshal lo/_/theg pa'i yang rtse ri rgyal nyi zla'i klong /_/'od gsal lhun grub rdo rje snying po'i klong /_/rtsol zhing sgrub med rang bzhin babs kyi klong /_/ye 'byams rmad du byung ba bshad kyis nyon/_/lhun grub klong las thams cad 'byung ba'i gzhi/_/ngo bo stong la rang bzhin ma 'gags pas/_/cir yang ma grub cir yang 'char ba ste/_/sku gsum klong nas 'khor 'das rang shar kyang /_/dbyings las ma g.yos chos nyid bde ba'i zhing /_/sems nyid klong chen 'gyur med nam mkha'i ngang /_/rol pa nges med thugs rjes cho 'phrul klong /_/thams cad dbyings kyi rgyan las logs na med/_/phyi nang 'du 'phro byang chub sems kyi rtsal/_/cir yang ma yin cir yang snang ba'i phyir/_/ngo mtshar cho 'phrul ya mtshan rmad kyi chos/_/phyi nang skye 'gro gzugs su snang ba kun/_/dbyings kyi rgyan te sku yi 'khor lor shar/_/ma lus grags pa'i sgrarka skad ji snyed kun/_/dbyings kyi rgyan te gsung gi 'khor lor shar/_/dran rig 'gyu 'phro mi rtog bsam yas kyang /_/dbyings kyi rgyan te thugs kyi 'khor lor shar/_/'gro ba rigs drug skye gnas bzhi po yang /_/chos dbyings ngang las g.yos pa rdul tsam med/_/snang srid yul drug gzung 'dzin snang ba yang /_/chos dbyings ngang na med snang sgyarku ma'i tshul/_/rten med sang seng ye stong yangs pa che/_/rang gsal ji bzhin chos dbyings rgyan du shar/_/ji ltar snang grags dbyings kyi ngang chen 'dir/_/lhun mnyam mi g.yo chos sku byang chub sems/_/ye babs rang stong 'pho dang 'gyur med pas/_/cir snang chos nyid rang byung ye shes ngang /_/bya dang rtsol med bde klong gcig tu 'khyil/_/rang gsal mi g.yo longs spyarkoda rdzogs pa ni/_/gang snang nyid nas rang bzhin lhun gyis grub/_/bcos shing basgyarkura med lhun mnyam

gdal ba'o/_/sna tshogs ma 'dres rol pa'i 'char tshul las/_/sprarkula pa rang byung ya mtshan 'phrul gyi don/_/bya bral kun tu bzang las g.yos pa med/_/g.yang sa med pa'i byang chub sems nyid la/_/rtsol ba med pa'i sku gsum lhun rdzogs kyang /_/dbyings las ma g.yos lhun grub 'dus ma byas/_/sku dang ye shes phrin las lhun rdzogs kyang /_/tshogs chen ye rdzogs ye shar chen po'i klong /_/ye nas lhun grub 'pho 'gyur med pa'i zhing /_/chos dbyings ngang las chos nyid gzigs pa yang /_/mkhyen pa ma 'gags dbyings kyi rgyan du shar/_/byas shing sgrarkuba med ye nas gnas pa nyid/_/nyi mkha' bzhin te ngo mtshar rmad kyi chos/_/ye nas lhun gyis grub pa'i dbyings rum 'dir/_/'khor ba kun bzang mya ngan 'das pa bzang /_/kun bzang klong na 'khor 'das ye nas med/_/snang ba kun bzang stong pa kun tu bzang /_/kun bzang klong na snang stong ye nas med/_/skye shi kun bzang bde sdug kun tu bzang /_/kun bzang klong na bde sdug skye shi med/_/bdag gzhan kun bzang rtag chad kun tu bzang /_/kun bzang klong na bdag gzhan rtag chad med/_/med la yod par 'dzin pas 'khrul par btags/_/rten med rmi lam 'dra ba'i rang bzhin la/_/'khor 'das rang mtshan zhen pa mtshar re che/_/thams cad kun bzang lhun grub chen po ni/_/ma 'khrul mi 'khrul 'khrul par mi 'gyur bas/_/srid pa ming tsam yod med mtha' las 'das/_/su yang gang du sngar yang ma 'khrul la/_/da lta mi 'khrul slad kyis 'khrul mi 'gyur/_/'di ni srid gsum ka dag dgongs pa'o/_/'khrul pa med pas ma 'khrul chos su med/_/ye nas lhun grub rang byung rig pa che/_/ma grol mi grol grol bar mi 'gyur la/_/'das pa ming tsam su yang grol ma myong /_/grol bar mi 'gyur bcings pa ye nas med/_/mkha' bzhin rnam dag rgya chad phyogs lhung bral/_/'di ni yongs grol ka dag dgongs pa'o/_/mdor na lhun grub yangs pa'i dbyings rum nas/_/rol pa'i rtsal gyi 'khor 'das ci shar yang /_/shar tsam nyid nas 'khor 'das yod ma myong /_/gnyid kyi rtsal las rmi lam ci shar yang /_/don la med de rang rig bde ba'i mal/_/lhun mnyam yangs pa chen por phyam gdal lo/_/chos dbyings rin po che'i mdzod las/_'khor 'das dbyings las ma g.yos par bstan pa'i le'u ste dang po'o// _ //dbyings kyi rang bzhin gdod nas lhun grub la/_/phyi dang nang med kun tu khyab par gdal/_/mtha' yi mu med steng 'og phyogs mtshams 'das/_/yangs dog gnyis med rig pa mkha' ltar dag_/dmigs bsam sprarko bsdus bral ba'i klong nyid du/_/skye med dbyings las skye ba'i cho 'phrul rnams/_/cir yang ma nges gar yang rgya chad med/_/'di zhes mi mtshon dngos po mtshan ma med/_/phyogs 'byams nam mkha' 'dra ba'i rang bzhin la/_/skye med lhun grub snga phyi thog mtha' bral/_/'khor 'das kun gyi ngo bo byang chub sems/_/ma byung ma skyes ma nges lhun grub ni/_/gang nas ma 'ongs gar yang song ba med/_/snga phyi ris med byang chub sems kyi klong /_/'gro dang 'ong med kun tu khyab par gdal/_/thog mtha' dbus med chos nyid de bzhin nyid/_/phyam gdal mkha' mnyam dag pa'i rang bzhin la/_/thog mtha' med de snga phyi'i yul las 'das/_/skye 'gag med de dngos po mtshan ma med/_/'gro 'ong med de 'di zhes mtshon dang bral/_/rtsol zhing sgrarkuba med bya ba'i chos kyi stong /_/phyogs cha dbus med de bzhin nyid kyi gzhi/_/dmigs med rgyun chad med de mnyam pa'i klong /_/thams cad chos nyid mnyam pa'i rang bzhin las/_/mnyam pa'i klong na mi gnas gcig kyang med/_/gcig mnyam kun mnyam byang chub sems kyi ngang /_/ma skyes mkha' mnyam yangs par phyam gdal bas/_/mnyam nyid ngang la rgyun chad med pa'i phyir/_/lhun grub phyogs med kun khyab gdal ba'i rdzong /_/steng 'og bar med ye klong yangs pa'i rdzong /_/phyogs med kun shong skye med chos sku'i rdzong /_/mi 'gyur lhun grub rin chen gsang ba'i rdzong /_/snang srid 'khor 'das ye rdzong phyam gcig rdzogs/_/phyogs med kun khyab gdal ba'i sa gzhi la/_/'khor 'das ris med byang chub sems kyi mkhar/_/chos nyid klong yangs rtse mo lhun mtho zhing /_/ma byas rang bzhin phyogs bzhi yangs pa'i dkyil/_/rim rtsol bral ba'i 'jug sgo shin tu yangs/_/lhun grub 'byor pa'i bkod pas brgyan pa der/_/rang byung ye shes rgyal po gdan la bzhugs/_/'du 'phror snang ba ye shes rtsal rnams kun/_/blon por gyur pas yul la dbang basgyarkura zhing /_/rang gnas bsam gtan btsun mo dam pa dang /_/dgongs pa rang shar sras dang bran g.yog bcas/_/bde chen klong dkyil rang gsal rtog pa med/_/mi g.yo bsam brjod bral ba'i ngang nyid las/_/snang srid snod bcud kun la mnga' dbang basgyarkura/_/chos dbyings yangs pa'i yul khams rgya che'o/_/yul der gnas na thams cad chos kyi sku/_/rang byung ye shes gcig las ma g.yos par/_/ma byas ye zin stsol sgrarkuba 'das pa nyid/_/grwarka zur med pa'i thig le zlum pas na/_/ji bzhin dbye gsal med pa'i klong du 'khyil/_/'gro drug gnas dang sangs rgyas zhing khams kyang /_/gzhan na med de chos nyid nam mkha'i ngang /_/rang gsal byang chub sems su ro gcig pas/_/rig pa'i ngang du 'khor 'das 'ub chub bo/_/thams cad kun 'byung chos dbyings mdzod 'di ni/_/'das pa ma btsal ye nas lhun grub pas/_/chos sku mi 'gyur yul med kun 'byams la/_/phyi nang snod bcud snang ba longs spyarkoda rdzogs/_/gzugs brnyan lta bur rang shar sprarkula pa'i sku/_/sku gsum rgyan du ma rdzogs chos med pas/_/thams cad sku gsung thugs kyi rol par shar/_/bde gshegs zhing khams ma lus grangs med kyang /_/nyid las byung ba'i sems nyid sku gsum klong /_/'khor ba'i rang bzhin 'gro drug grong khyer kyang /_/chos dbyings ngang las gzugs brnyan shar ba tsam/_/skye shi bde sdug sna tshogs snang ba yang /_/sems nyid klong 'dir sprarkula pa'i ltad mo bzhin/_/med la yod la snang la gzhi med pas/_/mkha' la sprarkina bzhin glo bur rkyen byung tsam/_/yod min med min rang bzhin mtha' 'das pas/_/sprarkosa bral thig le'i ngang du 'ub chub bo/_/sems nyid byang chub sems kyi rang bzhin ni/_/mkha' ltar dag pas skye shi bde sdug med/_/dngos po ris med 'khor 'das chos las grol/_/'di zhes mi mtshon mkha' klong rab yangs pas/_/mi 'gyur mi 'pho lhun grub 'dus ma byas/_/'od gsal rdo rje snying por sangs rgyas pas/_/thams cad kun kyang rang byung bde ba'i zhing /_/lhun mnyam byang chub mchog gi ngang nyid do/_/chos dbyings rin po che'i mdzod las/_snang srid zhing khams su shar ba'i le'u ste gnyis pa'o//__ //thams cad kun 'dus byang chub sems su 'dus/_/byang chub sems las ma gtogs chos med pas/_/chos kun byang chub sems kyi rang bzhin no/_/byang chub sems kyi mtshon dpe nam mkha' 'dra/_/sems la rgyu med skye ba'i yul med pas/_/mi gnas brjod 'das bsam yul 'das pa nyid/_/nam mkha'i dbyings zhes dpe ru mtshon pa tsam/_/mtshon dpe nyid kyang 'di zhes byar med na/_/mtshon dpe'i don la bsam brjod ga la srid/_/'di ni rang bzhin dag pa'i dper shes bya/_/don ni rang rig mkha' mnyam byang chub sems/_/bsam pa'i yul min mtshon brjod 'das pa ste/_/rang gsal mi g.yo 'od gsal yangs pa'i klong /_/ma byas lhun grub yangs dog mtho dman med/_/chos sku byang chub snying po'i dkyil yangs so/_/rtags ni rtsal las cir yang 'char ba ste/_/shar ba'i dus na shar sa shar mkhan med/_/shar zhes ming tsam dpyad na nam mkha' 'dra/_/ris med mnyam pa chen por 'ub chub pas/_/phyam gdal gzung 'dzin med pa'i klong nyid do/_/rang byung ye shes chos nyid phyogs yan la/_/dpe don rtags kyi nges pa'i mtshon dpe bstan/_/mkha' mnyam gzer chen gsum gyi bdag nyid du/_/thams cad kun 'dus rang bzhin dbye bsal med/_/phyam mnyam yangs pa chen po'i dbyings rum na/_/thams cad ye mnyam snga phyi bar bsam med/_/kun bzang rdo rje sems dpa'i dgongs pa'o/_/byang chub sems ni nyi ma'i snying po 'dra/_/ngang gis 'od gsal ye nas 'dus ma byas/_/sgrarkiba pa'i chos med zang thal lhun gyis grub/_/sprarkosa pa'i tshogs med mi rtog chos nyid ngang /_/stong pas choos sku gsal bas longs spyarkoda rdzogs/_/zer ldan sprarkula pa sku gsum 'du 'bral med/_/ye nas yon tan lhun gyis grub zin pas/_/skyon dang nyes cha'i mun pas basgrarkibasa pa med/_/snga phyi dus gsum 'pho 'gyur med par gcig_/sangs rgyas sems can kun la khyab par gcig_/'di ni rang byung byang chub sems zhes bya/_/de yi rtsal ni cir yang 'char ba ste/_/rtog dang mi rtog snang srid snod bcud dang /_/skye 'gro'i snang ba sna tshogs ji snyed do/_/'di kun shar yang rang bzhin 'ga' med de/_/smig rgyu'i chu dang rmi lam sgrarka brnyan bzhin/_/sprarkula pa gzugs brnyan dri za'i grong khyer dang /_/mig yor ji bzhin med pa gsal snang du/_/gzhi med rten med glo bur snang ba tsam/_/bar skabs re 'ga'i chos su rtogs par bya/_/lhun grub byang chub sems kyi rang bzhin las/_/rol pa ma 'gags 'khor 'das cho 'phrul 'byung /_/cho 'phrul de kun dbyings su 'ub chub pas/_/gdod ma'i ngang las g.yos pa med shes bya/_/'dir ni thams cad byang chub sems kyi ngang /_/gcig rdzogs kun rdzogs ma byas don kun rdzogs/_/rang bzhin lhun rdzogs rang byung ye shes so/_/byang chub sems ni snang dang mi snang las/_/'khor 'das phyi nang chos su med na yang /_/de yi rtsal las g.yos pa'i rang bzhin gyis/_/snang srid 'khor 'das sna tshogs rol par shar/_/shar tsam nyid nas rang bzhin stong pa'i gzugs/_/skye ba med las skye bar snang ba ste/_/snang dus nyid nas skyes pa 'ga' yang med/_/'gag pa med pa 'gag par snang na yang /_/'gag pa med de sgyarku ma stong pa'i gzugs/_/gnas pa nyid nas gnas pa'i chos med de/_/gnas mkhan gzhi med 'gro 'ong med pa'i ngang /_/ji ltar snang ba de ltar ma grub pas/_/rang bzhin med ces btags pa tsam du zad/_/snang ba de yang rtsal las rang shar bas/_/rten 'brel nyid ces rang bzhin brda tsam brdzod/_/rtsal las shar bar snang ba'i rang dus nas/_/shar dang ma shar phyogs dang ris med pas/_/rtsal yang brda tsam ngo bo 'ga' med pas/_/thams cad rtag tu 'pho 'gyur med pa'i ngang /_/byang chub sems las g.yos pa rdul tsam med/_/chos dbyings rin po che'i mdzod las/_byang chub sems kyi mtshon dpe bstan pa'i le'u ste gsum pa'o//__ // kun 'dus byang chub sems kyi rang bzhin ni/_/snang ba ma yin snang ba'i chos las 'das/_/stong pa ma yin stong pa'i chos las 'das/_/yod pa ma yin dngos po mtshan ma med/_/med pa ma yin 'khor 'das kun la khyab/_/yod med ma yin lhun mnyam gdod ma'i dbyings/_/phyogs dang ris med gzhi rtsa dngos po med/_/rgyun chad med pas rig pa byang chub klong /_/'pho dang 'gyur med mkha' dbyings ye nas gdal/_/rang byung ye shes ye zla med pa'i don/_/mi skye mi 'gag thig le gcig tu 'dus/_/ma nges kun khyab phyogs mtha' yongs kyis med/_/lhun mnyam mi g.yo rdo rje snying po'i gdung /_|'du 'bral med pa'i dbyings mchog rab 'byams 'di/_/tshig gis mtshon pa'i spyarkoda yul ma yin te/_/shes rab klong rdol so sor rang rig yul/_/bsam brjod sprarkosa dang bral ba'i rnal 'byor pas/_/mtshon dang mi mtshon med par thag bcad de/_/sgom dang bsgom par bya ba ma rnyed pas/_/bying rgod rnam rtog dgra bo bsad ma dgos/_/ye nas spyarki blugs gnas pa'i chos nyid la/_/bdag dang gzhan du rtog pa mi mnga' bas/_/khams gsum 'di nyid rang bzhin mnyam pa'i zhing /_/dus gsum rgyal ba rang snang dag pa ste/_/spang blang med par thams cad phyam gcig pas/_/gzhan nas thob par bya ba rdul tsam med/_/chos kun sems nyid klong chen der gsal la/_/mnyam pa'i don la g.yos pa cung zad med/_/phyi dang nang med 'char nub rnyog pa med/_/mtha' yi mun sel rtsa ba byang chub sems/_/gang yang ma spangs gol sa shugs la chod/_/'gro ba'i snang tshul sna tshogs 'jig rten dang /_/dag pa'i sangs rgyas sku dang ye shes kyang /_/rtogs dang ma rtogs rtsal las shar ba yi/_/rol pa ma 'gags nam mkha'i dbyings khyab kyang /_/chos dbyings ngang la rtogs dang ma rtogs tsam/_/rtogs pas bde gshegs dag pa'i snang ba dang /_/ma rtogs ma rig gzung 'dzin bag chags las/_/sna tshogs snang yang dbyings las g.yos pa med/_/byang chub sems ni kun gyi dngos gzhi ste/_/mtshan nyid ma 'gags sna tshogs cir shar yang /_/rang gsal chos nyid dag pa'i dbyings su gsal/_/dbye gsal med do rgya yan rig pa'i 'gros/_/zang thal ye shes rang byung klong yangs shing /_/ma basgrarkibasa phyi nang med par 'od gsal bas/_/rang rig sems kyi me long 'od po che/_/'dod 'byung nor bu rin chen chos kyi dbyings/_/btsal ba med pa thams cad rang byung bas/_/rang byung ye shes 'dod dgur 'byung ba'i dpal/_/che ba'i yon tan rnam grangs ji snyed pa/_/dbyings las dbyings byung thabs mchog ma 'gags shar/_/thams cad skye med dbyings su lhun rdzogs pas/_/dngos po zil gnon stong nyid byang chub klong /_/stong pa zil gnon byang chub rang rig klong /_/byang chub sems la snang stong ye nas med/_/gnyis med ma zhen bsam yas cho 'phrul 'byung /_/dus gsum dus med skye med chos kyi dbyings/_/mi 'gyur mi phyed 'dus ma byas pa'i klong /_/dus gsum sangs rgyas rig pa'i ye shes dbyings/_/gzung 'dzin zil gnon rang rig byang chub klong /_/phyi dang nang med chos nyid lhun yangs so/_/chos dbyings rin po che'i mdzod las/_byang chub sems kyi rang bzhin bstan pa'i le'u ste bzhi pa'o//______//sems nyid byang chub sems kyi ngo bo la/_/lta ba bsgom med spyarkoda pa spyarkada du med/_/'bras bu basgrarkuba med sa lam bgrod du med/_/dkyil 'khor bskyed med bzlas brjod rdzogs rim med/_/dbang la bskur med dam tshig bsrung du med/_/ye nas lhun grub dag pa'i chos nyid la/_/glo bur rim rtsol rgyu 'bras chos las 'das/_/'di dag byang chub sems kyi ngo bo ste/_/nyi ma sprarkina dang mun pas ma basgrarkibasa la/_/glo bur ma byas dbyings na ngang gis gsal/_/rtsol zhing sgrarkuba pa'i chos bcu gang bstan pa/_/rtsal las glo bur 'khrul ngor gsungs pa nyid/_/rim rtsol dbang po'i rim pas 'jug pa'i thabs/_/a ti yo ga rdo rje snying po'i don/_/ji bzhin rnal du 'byor la bstan pa min/_/rim 'jug rtsol ba can gyi gang zag rnams/_/chos nyid gdod ma'i dbyings su drang ba'i phyir/_/nyan thos rang rgyal byang chub sems dpa'i theg_/chung ngu gsum la bstan pa'i rim pa ste/_/krI ya u pa yo ga rnam pa gsum/_/'bring po gsum la rang bzhin babs kyis grub/_/ma hA a nu a ti rnam pa gsum/_/chen po gsum la gdod nas snang ba ste/_/rgyu 'bras theg pa'i chos kyi sgo dbye bas/_/skal ldan 'gro ba byang chub gsum la 'dren/_/kun kyang mthar thug rdo rje snying po'i don/_/gsang chen rmad du byung 'dir 'jug dgos pas/_/kun gyi rtse mo 'od gsal mchog mi 'gyur/_/mngon par byang chub snying po'i theg par grags/_/chos kyang gnyis las blang dor bya rtsol can/_/rtsal las rol par shar ba'i rang bzhin gyi/_/sems dang sems byung bag chags sbyarkanga phyir bstan/_/de dag sems las ye shes dag par 'dod/_/blang dor bya rtsol med pa'i chos chen ni/_/rang byung ye shes byang chub sems nyid kyi/_/ngo bo thad drang ngang las ma g.yos par/_/mngon du byed pas gzhan du rtsol mi dgos/_/rang la bzhag nas gzhan du 'tshol mi byed/_/'di ni nyi ma'i ngo bo de nyid don/_/rang gzhag 'od gsal mi g.yo gnas par 'dod/_/gzhan ni sprarkina dang mun sel rtsol sgrarkuba kyi/_/nyi ma gdod sgrarkuba byed dang mtshungs par bstan/_/des na 'di gnyis khyad par gnam sa bzhin/_/ding sang a tir rlom pa'i glang chen dag_/'gyu 'phro'i rtog tshogs byang chub sems yin lo/_/rmongs pa 'di kun mun pa'i klong nyid du/_/rang bzhin rdzogs pa chen po'i don la ring /_/rtsal dang rtsal las shar ba'ang mi shes na/_/byang chub sems kyi ngo bo smos ci dgos/_/'dir ni gdod nas dag pa'i byang chub sems/_/don dam dbyings kyi chos nyid bden pa ni/_/smrarka bsam 'das pas shes rab pha rol phyin/_/ngang gis mi g.yo rang bzhin 'od gsal zhing /_/'gyu 'phro'i sprarkosa dang ye nas bral ba la/_/ngo bo zhes brjod nyi ma'i snying po 'dra/_/de yi rtsal ni 'char tshul ma 'gags pa'i/_/rig pa zang thal rtog dpyod gnyis dang bral/_/sal gyi gsal yang gzung 'dzin med pa yin/_/rtsal las shar ba'i rig pa sprarkosa pa'i blo/_/des bskyed gzung 'dzin bag chags sna tshogs can/_/yul med yul du gzung ba'i yul lnga dang /_/bdag med bdag du 'dzin pa'i nyon mongs lnga/_/phyi nang snod bcud 'khrul snang ji snyed de/_/'khor bar snang ba rtsal las yang shar ba/_/ma rtogs log par gzung ba'i snang ba nyid/_/gang nas ma 'ongs gang du ma song la/_/gar yang mi gnas chos nyid klong chen du/_/rtogs pas khams gsum yongs grol dgongs pa zhes/_/a ti lhun grub rdo rje snying po'i lung /_/kun bzang yangs pa chen pa'i klong nas shar/_/rnam dag byang chub sems kyi ngo bo la/_/lta ba'i yul med lta ba'i chos su med/_/blta bar bya dang byed pa rdul tsam med/_/sgom pa'i blo med bsgom bya'i chos kyang lhad/_/spyarkada dang spyarkoda pa gnyis med lhun grub pas/_/basgrarkuba par bya ba'i 'bras bu rdul tsam med/_/med pa'i chos la bgrod pa'i sa med pas/_/phyin par bya ba'i lam yang ye nas med/_/'od gsal thig le chen por grub zin pas/_/rnam rtog 'phro 'dus bskyed pa'i dkyil 'khor dang /_/sngags dang kha ton dbang dang dam tshig med/_/rim sdud la sogs mi dmigs rdzogs rim med/_/ye nas grub zin sku dang ye shes la/_/'dus byas glo bur rkyen byung rgyu 'bras med/_/'di dag yod na rang byung ye shes min/_/'dus byas nyid phyir 'jig pa nyid dang ni/_/lhun grub 'dus ma byas zhes gang slad mtshon/_/'di phyir don dam dbyings kyi ngo bo la/_/rgyu 'bras las 'das rang bzhin rnam bcu med/_/rtsol dang basgrarkuba med sems nyid rnal ma'i don/_/yod med sprarkosa kun zhi bar mkhyen 'tshal lo/_/chos dbyings rin po che'i mdzod las/_rtsol sgrarkuba rgyu 'bras las 'das pa'i le'u ste lnga pa'o//_ ___// nyi ma'i snying po 'od rnams 'dus pa ltar/_/chos kun rtsa ba byang chub sems su 'dus/_/snang srid snod bcud ma dag 'khrul pa yang /_/gang byung rten dang gnas pa'i dbyings brtags pas/_/gzhi med ye grol sems kyi ngang du 'dus/_/chos nyid ye klong yangs pa chen po'i ngang /_/'khrul dang ma 'khrul ming don 'das par 'dus/_/dag pa'i rang snang sku dang zhing khams dang /_/ye shes phrin las ngo mtshar rol pa yang /_/rang byung ngang la 'du 'bral med par 'dus/_/snang srid 'khor 'das kun 'dus byang chub sems/_/nyi mkha' bzhin du stong gsal 'dus ma byas/_/gdod nas rang byung ye klong yangs pa yin/_/sems nyid klong chen 'gyur med nam mkha'i ngang /_/rol pa nges med byang chub sems kyi rtsal/_/'khor 'das theg pa kun la dbang basgyarkura ba'i/_/byar med gcig gis thams cad zil gyis mnan/_/mtha' ru gyur pa'i yul gzhan logs na med/_/chos nyid byang chub sems las gar ma g.yos/_/thams cad kun bzang lhun grub gcig shar bas/_/ma lus kun 'dus 'gran zla bral ba'i mchog_/che ba'i che ba kun bzang chos kyi dbyings/_/rgyal po lta bur thams cad kun 'dus pas/_/'khor 'das kun la dbang basgyarkura gar ma g.yos/_/thams cad kun bzang mi bzang gcig med pas/_/bzang ngan med par kun tu bzang por gcig_/grub dang ma grub thams cad dbyings gcig pas/_/thams cad lhun grub mi g.yo mnyam pa'i gcig_/gcig las kun shar ma lus chos kyi dbyings/_/byar med ngang la sgrarkuba med btsal du med/_/rtsol sgrarkuba rang gi dbyings las gzhan med pas/_/gang las rtsol zhing gang du sgrarkuba par byed/_/btsal bas yul med bsgoms pas mthong ba med/_/basgrarkuba pa'i gnas med gzhan nas 'ong ba med/_/'gro 'ong med pas mnyam nyid chos kyi sku/_/lhun rdzogs thig le chen po'i dbyings su 'dus/_/nyan thos rang rgyal byang chub sems dpa'i lung /_/bdag dang bdag gi med par thag bcad nas/_/sprarkosa bral nam mkha' 'dra bar dgongs don gcig_/a ti mchog gsang rnal 'byor chen po'i lung /_/bdag gzhan dbyer med yangs pa'i nam mkha' la/_/rang byung ye shes ji bzhin rnal 'jog pas/_/dgongs don de kun snying po'i mchog 'dir 'dus/_/krI ya u pa yo ga rigs gsum yang /_/bdag dang lha dang ting 'dzin mchod sprarkina las/_/sgo gsum rnam dag dngos grub 'dod par gcig_/rdo rje rtse mo lung rgyal gsang ba yang /_/snang grags rig pa rnam dag ye nas lha/_/sgo gsum rnam dag dngos grub mngon gyur pas/_/dgongs pa de kun snying po'i mchog 'dir 'dus/_/ma hA a nu a ti rnam gsum yang /_/snang srid snod bcud lha dang lha mo'i zhing /_/dbyings dang ye shes rnam dag dbyer med pas/_/chos nyid mi g.yo rang byung ye shes 'dod/_/mchog gsang rab 'dir thams cad rnam dag pas/_/ma byas gzhal yas ye klong bde ba'i zhing /_/phyi dang nang med kun khyab gdal ba las/_/blang dor bya rtsol mtshan ma'i chos med par/_/thams cad ye 'byams chos sku'i klong du grol/_/dgongs pa de kun gsang chen snying por 'dus/_/gcig rdzogs kun rdzogs chos kun 'dus pa'i klong /_/ye babs rang gsal lhun grub chen por 'dus/_/chos dbyings rin po'i mdzod las/_thams cad byang chub sems su 'dus par bstan pa'i le'u ste drug pa'o// ___// rang bzhin lhun grub byang chub sems kyi lung /_/ma byas don grub ri rgyal rtse mo ni/_/kun las 'phags so theg mchog rgyal po che/_/ji ltar ri rgyal rtse mor phyin pa na/_/dma' ba'i lung rnams dus gcig mthong ba ste/_/lung gi rtse mo'i rang bzhin mthong dang bral/_/de bzhin a ti rdo rje snying po ni/_/theg pa'i yang rtse don kun gsal bar mthong /_/'og ma'i theg pas 'di don mthong ba med/_/de phyir lhun grub rtse mor gyur pa'i tshe/_/ji ltar yid bzhin nor bu chen po la/_/gsol ba btab na 'dod dgu ngang gis 'byung /_/phal pa'i dngos la de lta ma yin no/_/rdo rje snying po sku gsum lhun grub pas/_/rang bzhag dbyings las sangs rgyas rang la 'grub/_/rtsol zhing sgrarkuba med che ba de nyid do/_/'og ma'i theg pas blang dor 'bad byas kyang /_/bskal par mi 'grub tshi chad nad du che/_/ye nas lhun mnyam rig pa byang chub sems/_/ji bzhin rang babs chos nyid yangs pa ni/_/rang bzhin chos sku mnyam nyid gdod ma'i klong /_/kun la yod de skal ldan 'ga' yi yul/_/ji bzhin bzhag na ngang der babs kyis 'grub/_/khyab gdal rang gsal lhun grub longs spyarkoda rdzogs/_/kun la yod kyang mthong ba 'ga' yi yul/_/gang snang rang gzhag bya rtsol bral na mngon/_/ma 'gags rol pa sprarkula sku gdal ba'i klong /_/kun la yod de 'char ba'i dus na gsal/_/yid bzhin yon tan phrin las cho 'phrul yang /_/gzhan na med de rang rig dag pa'i klong /_/chu dang rnyog bzhin rang dwarkangasa bzhag na gsal/_/btsal bas mi rnyed gdod nas dag pa'i chos/_/sangs rgyas byang chub rang byung klong na gsal/_/sngar grub zin pas da gzod basgrarkuba mi dgos/_/che ba rang gnas dgongs pa chos nyid klong /_/mi 'gyur lhun gyis grub la rtsol mi byed/_/ye gzhi babs gzhi byang chub snying po'i gzhi/_/rang bzhin ngang las g.yos pa 'ga' med pas/_/klong gsal rig pa'i don las ma g.yo zhig_/thams cad bzhag pas 'grub pa'i rgyu mtshan yang /_/mi 'gyur kun 'gro khyab bdag ye shes lnga/_/sku lnga gsung lnga thugs lnga yon tan lnga/_/phrin las lnga ste dang po'i sangs rgyas kyang /_/thog mtha' med pa'i klong 'dir lhun grub pas/_/gzhan du ma 'tshol rang bzhin ye nas grub/_/sangs rgyas kun gyi chos sku byang chub kyang /_/mi 'gyur mnyam pa'i don las gzhan du med/_/de yang rang byung ngang 'dir lhun grub pas/_/ma tshol ma sgrarkubasa re dogs lhun gyis zhog_/sems can kun gyi rang byung ye shes kyang /_/ma byas ma btsal chos skur lhun grub pas/_/spang blang ma 'dzin chos dbyings ngang 'dir zhog_/mi g.yo mi bsam lhun mnyam ngo bo la/_/ma byas don grub gzhi klong yangs pa yin/_/mi 'gyur kun 'gro sku dang ye shes bdag_/rgyal thabs spyarki blugs rang byung chen po'i dbang /_/snang srid snod bcud ye grol lhun grub pas/_/bya rtsol mi dgos rang bzhin lhun gyis grub/_/thams cad kun grub lhun grub chen por rgyas/_/chos dbyings rin po che'i mdzod las/_thams cad byang chub kyi sems su ye nas lhun gyis 'grub par bstan pa'i le'i ste bdun pa'o// ___//thams cad dbyings gcig rang byung ye shes la/_/gnyis su med pa ngo bo'i 'dug tshul te/_/gnyis snang ma 'gags rtsal las rol par shar/_/snang btags gnyis med byang chub sems zhes bya/_/'pho 'gyur med pa'i rig pa byang chub la/_/spangs thob med pa'i snang srid 'khor 'das shar/_/gzung 'dzin med pa'i rnal 'byor ngo kha la/_/med la snang 'di ya mtshan dgod po 'chor/_/snang bar med la sna tshogs snang bar shar/_/stong par med la mtha' dbus khyab par gdal/_/gzung 'dzin med la nga bdag so sor zhen/_/gzhi rtsa med la tshe rabs brgyud mar snang /_/dgag sgrarkuba med la bde sdug blang dor byed/_/phar bltas skye 'gro'i snang ba mtshang re che/_/mi bden bden par zhen pas bden bden 'dra/_/ma 'khrul 'khrul par zhen pas 'khrul 'khrul 'dra/_/nges med nges par bzung bas nges nges 'dra/_/yin min yin par bzung bas yin yin 'dra/_/mi 'thad 'thad par bzung bas 'thad 'thad 'dra/_/sna tshogs col chung yul gyis sems khrid nas/_/don med rig pa skad cig brgyud mar mthud/_/nyin zhag zla ba lo dang mi tshe 'das/_/gnyis med gnyis su bzung bas 'gro ba bslus/_/rnal 'byor dag pa'i sems la tshur bltas pas/_/rten gzhi med pa'i rig pa ming dang bral/_/mtshon brjod ma mthong lta sgom khrigs chags bud/_/phyal ba lhug pa yangs pa phyam gdal bas/_/nyams len ma shes thun mtshams phyogs ris med/_/thams cad rgya yan phyam phyam bar mtshams bral/_/lus dang yul dang snang ba'i gtad med par/_/nam mkha'i klong mnyam yangs par phyam gdal bas/_/nang gi chos zhes bdag tu 'dzin pa med/_/phyi rol snang ba'i yul la phar bltas pas/_/thams cad sang seng 'al 'ol zang ma thal/_/ra ri zab zeb bza' gtad chos dang bral/_/snang grags dran rig myong tshor sngar bzhin med/_/'di ci rang bzhin smyarkona pa'i snang ba'am/_/rmi lam nang bzhin rang la dgod po 'chor/_/dgra gnyen chags sdang nye ring 'du shes bral/_/nyin mtshan ris med mnyam par phyam gcig pas/_/dmigs gtad mtshan mar 'dzin pa'i 'khor ba sangs/_/rang byung ye shes ngang zhes mi rtog pas/_/blang dor spang gnyen gzeb las 'das pa yin/_/'di ltar rtogs na gnyis med ye shes te/_/rang byung kun tu bzang po'i dgongs par phyin/_/ldog pa'i gnas med zad pa'i sar phyin te/_/rang byung ngang las mnyam nyid ma rtogs par/_/gnyis med nyid ces tshig la mngon zhen nas/_/ci yang mi dmigs yid dpyod gdeng 'cha' ba/_/log rtogs nyid de ma rig mun pa'i klong /_/de phyir rang byung 'pho 'gyur med pa la/_/bsam rdzogs rgyal po gnyis su med par sbyarkanga /_/khams gsum yongs grol 'khor 'das gnyis med don/_/rang bzhin khong nas rang shar chos sku'i rdzong /_/mkha' ltar rnam dag dpe las 'das pa 'byung /_/'di dang 'di zhes so sor zhen pa'i bar/_/gnyis su gnas par rang gzhan 'khrul pa'i gzeb/_/gang tshe 'di zhes tha dad ris med cing /_/thams cad phyam mnyam dmigs gtad med pa na/_/gnyis med rtogs zhes rdo rje sems dpas gsungs/_/chos dbyings rin po che'i mdzod las/_byang chub kyi sems la gnyis su med par bstan pa'i le'u ste brgyad pa'o//______//rang bzhin yangs pa chen po'i klong gcig la/_/mkha' mnyam byang chub sems kyi gnam gzer ni/_/gnad du dril te bcud du phyung ba ni/_/che ba'i che ba kun bzang yangs pa'i thugs/_/rang gi ngo bos spyarki rgya rlabs kyis bcad/_/klong chen gcig la rtogs dang ma rtogs dang /_/grol dang ma grol gnyis med mnyam pa che/_/sgo nga'i nang nas 'dab gshog rgyas pa'i bya/_/rgya dang bral bas nam mkha'i klong na gnas/_/klu rnams zil gnon g.yang sa shugs kyis chod/_/theg pa'i yang rtse rdo rje snying po yang /_/ji bzhin rtogs pa'i rnal 'byor skal ba can/_/theg dman zil gnon 'khor ba'i g.yang sa chod/_/kun grol mnyam pa chen por gnas pa de/_/rgyu 'bras rtsol sgrarkuba can la mi rigs kyang /_/theg mchog mi g.yo mnyam pa'i don la 'thad/_/thams cad bde chen mkha' mnyam chos sku'i klong /_/chos sku'i klong du ma grol 'ga' yang med/_/chos nyid rang bzhengs rdo rje snying po'i sku/_/bag chags lus la snying po'i rtsal rdzogs te/_/skye shi bar do srid pa'i las bor nas/_/rig pa gcig pu kun dang dbyer med cing /_/lhun grub sa la rgyal ba'i srid zin nas/_/rgya chad med par sprarkula pa 'byung ba dang /_/thogs pa med par kun la 'jug pa ni/_/byar med rlung zhon rnal 'byor spyarkoda yul te/_/dman pa'i theg pa kun la mi rigs kyang /_/a tis rigs par ston pa 'bras bu'i gnad/_/skye med skye ba'i cho 'phrul 'byung ba la/_/rgyu 'bras mtshan mar 'dzin pa 'khrul pa'i blo/_/a tis rgyu rkyen med par bstan pa ste/_/'og ma rnams la mi rigs rigs pa'i gnad/_/sangs rgyas sems can dgongs spyarkoda dbyer med la/_/'khor 'das gnyis su 'dzin pa 'khrul pa'i blo/_/a tis gnyis su med par bstan pa de/_/'og ma rnams la mi rigs rigs pa'i gnad/_/rtogs dang ma rtogs med par grol ba la/_/rtogs nas grol bar 'dod pa nyams pa'i dgra/_/a tis mnyam nyid gcig tu bstan pa de/_/'og ma rnams la mi rigs rigs pa'i gnad/_/mtshon byed thabs kyi khyad par ma bsten par/_/brjod med rtogs par mi 'dod blun po'i blo/_/a tis don dam dbyer med bstan pa de/_/'og ma rnams la mi rigs rigs pa'i gnad/_/rdzogs chen ye gdal gting mtha' med pa la/_/thug pa med ces zer ba blun po'i blo/_/a tis mu med chig chod bstan pa de/_/'og ma'i blo yul mi rigs rigs pa'i gnad/_/thig le gcig la rgyu mtshan go bzlog pas/_/'bras bu re dogs chod de mkha' dang mnyam/_/yangs so che'o mkha' mnyam rgyal ba'i thugs/_/spangs thob med de thig le gcig gi klong /_/ye nas grol lo rtogs dang ma rtogs med/_/rnal 'byor mkha' mnyam bya bral lam du bde/_/ye sangs rgyas pa'i rig pa yul med 'di/_/'khor bar mi 'khyam 'khrul gzhi kun las 'das/_/su yang ma 'khrul 'khrul pa'i gnas med de/_/thams cad chos dbyings klong gsal gcig gi ngang /_/snga phyi ris med mkha' mnyam yangs pa nyid/_/ye babs lhun grub 'khor ba gdod nas dag_/grol bar mi 'jug mya ngan 'das mi len/_/mi 'gyur klong chen 'khor 'das yod ma myong /_/'di la spangs thob re dogs mi dmigs shing /_/gdod nas byang chub gzhi klong yangs pa che/_/thams cad ming tsam don la mtshon brjod 'das/_/grol dang 'khrul med 'khor 'das la zlos pas/_/su yang ma rtsol bcos basgyarkura ma byed cig_/yangs dog mtho dman med pa'i rig pa la/_/rgya chad phyogs lhung med kyis dmigs gtad shol/_/bya byed 'gro 'ong med pa'i rig pa la/_/dus dang gnyen po med kyis 'dzin rtsol zhog_/ched du dmigs pa yod na 'ching ba'i rgyu/_/gang la'ang gtad 'dzin ma 'cha' phyam la thong /_/chos kun ye nas grol rung ma grol rung /_/gnas lugs rang bzhin dag rung ma dag rung /_/sems nyid sprarkosa dang bral rung ma bral rung /_/gnyug ma'i gshis la grub rung ma grub rung /_/'khor 'das rang bzhin gnyis rung ma gnyis rung /_/bsam brjod kun las 'das rung ma 'das rung /_/dgag sgrarkuba 'khrul pa zhig rung ma zhig rung /_/rtogs pa'i lta ba rtogs rung ma rtogs rung /_/chos nyid don la bsgoms rung ma bsgoms rung /_/blang dor med pas spyarkada rung ma spyarkada rung /_/gnas lugs 'bras bu grub rung ma grub rung /_/sa dang lam rnams bgrod rung ma bgrod rung /_/sgrarkiba pa kun dang bral rung ma bral rung /_/bskyed rdzogs chos nyid rdzogs rung ma rdzogs rung /_/thar pa'i 'bras bu thob rung ma thob rung /_/'gro drug 'khor bar 'khyams rung ma 'khyams rung /_/rang bzhin lhun gyis grub rung ma grub rung /_/rtag chad gnyis 'dzin bcings rung ma bcings rung /_/chos nyid dgongs par sleb rung ma sleb rung /_/gong ma'i rjes su snyogs rung ma snyogs rung /_/gnam sa 'dre ldog snang ba ci shar yang /_/phyal ba lhug pa gzhi med zang ka ma/_/gtad med zang zing ban bun chal ma chol/_/re dogs gnyis med smyarkona pa'i ngang tshul can/_/lta sgom ris med ched 'dzin 'dod blo zhig_/zhe 'dod 'khris med 'di zhes rtsol sgrarkuba med/_/gang byung byung la gang snang snang du chug_/gang shar shar la gang yin yin du chug_/gang yang yin la gang yang min du chug_/kun spyarkoda nges med rig pa thod rgal ngang /_/chos dang chos min rtsis gzhi 'ga' med pas/_/gtad med zang ka grub mtha'i gzeb las 'das/_/za 'chag nyal 'dug nyin zhag phyam gdal bas/_/rang bzhin chos nyid mnyam pa'i ngang nyid de/_/mchod pa'i lha med brdung ba'i 'dre yang med/_/sgom pa'i chos med tha mal rang dga'i ngang /_/ma bcos rgyal po snyems med phyam gcig pas/_/phyal pa lhug pa lhun grub gcig pu nyid/_/ma byas ye zin rtsol sgrarkuba bral bas bde/_/lta ba'i gzhi med sgom pa'i ngang med la/_/spyarkada pa'i chos med basgrarkuba 'dod 'bras bu med/_/thams cad ris med mnyam par phyam gdal bas/_/bya rtsol ma dgos yangs dog med par bde/_/smon pa med pas sgrarkuba blo'i chos zad de/_/spang bya med pas gnyen po'i 'ching zhen 'das/_/gang yin kun yin yin min 'ga' med pas/_/gang snang gang shar 'dem ka med par grol/_/ma grol ye grol rang grol chos med pas/_/phyam gcig gtad med la bzla'i chos las 'das/_/klong yangs klong yangs klong chen yangs pa'i ngang /_/klong chen rab 'byams klong gsal klong 'byams pas/_/klong gcig gnyis med bde klong 'khyil ba ni/_/sna tshogs rang grol chos nyid zad sar phyin/_/mi 'gyur lhun grub 'dun ma legs pa'i rtse/_/bdag gzhan rjes su 'jug pa'i 'gro rnams kyang /_/'di bzhin ye 'byams klong chen gcig tu dril/_/kun bzang sa la gtan srid zin pa yin/_/chos dbyings rin po che'i mdzod las/_chos thams cad byang chub sems kyi klong du la bzla bar bstan pa'i le'u ste dgu pa'o//_ //rang bzhin gdod nas dag pa'i byang chub sems/_/btang gzhag 'gro 'ong med pa'i chos nyid la/_/btsal bas mi 'grub chos nyid nam mkha'i klong /_/rang bzhin bzhag pas 'od gsal nyi zla 'char/_/yul kyang mi dgag sems kyang mi gzung bar/_/rang bzhin lhun grub ngang las mi g.yo na/_/kun bzang yangs pa chen po'i dgongs par phyin/_/mi sprarko mi bsdu rang dwarkangasa sang nge ba/_/dwarkangasa pa'i rgya mtsho mi g.yo mnyam pa bzhin/_/chos nyid gting gsal rang byung ye shes ngang /_/'byung 'jug re dogs bral bar gnas pa yin/_/tshig gis mi mtshon 'jur bu'i sems med par/_/rang babs gyin 'da' bcos bslad med pa ni/_/klong thim chos nyid mtshan ma med pa ste/_/sgom dang bsgom par bya ba'i chos med pas/_/bying rgod rang yan rang byung dgongs par shar/_/spangs pas mi spong kun rtog rig pa'i rtsal/_/chos nyid ngang du dbye bsal ris med pas/_/basgrarkubasa pas mi 'grub chos nyid dbyings su shar/_/'khor ba ma spangs rang byung ye shes su/_/klong chen rtsal gyi rnal 'byor dag pas mthong /_/ye nas snang sems rang babs chos nyid ngang /_/ting 'dzin mi g.yo chu bo'i rgyun shar bas/_/rdo rje rtse mo kun bzang thugs kyi mchog_/yangs pa'i chos mchog nam mkha'i mtha' dang mnyam/_/dbye bsal med par thams cad sgom pa'i mchog_/ye 'byams rmad byung rgyal por lhun gyis grub/_/ye nas spyarki blugs 'od gsal chu bo'i rgyun/_/btang gzhag med pa'i ngang 'dir lhun grub pas/_/'khor 'das rang bzhin chos dbyings dgongs pa'i mchog_/mi g.yo brjod 'das mkha' mnyam klong chen nyid/_/'gro ba kun la ye nas babs kyis grub/_/bdag las gzhan du snang ba 'khrul pa'i sems/_/bsgom dang rtsol bar 'dod pa 'khrul pa'i sems/_/'khrul pa chos nyid ngang bzhag mnyam pa'i zhing /_/mi g.yo gdod nas dag pa'i rang bzhin klong /_/bya dang rtsol med gzhag dang ma bzhag med/_/mi 'gyur lhun gyis grub pa'i chos nyid la/_/dmigs bsam tshol khro bral ba'i rang rig gis/_/yang yang bltas na blta ru med pa mthong /_/bltar med rig pa spyarki blugs lta ba yin/_/ma bsgoms rig pa btang gzhag bral ba la/_/yang yang bsgoms na bsgom du med pa mthong /_/sgom med rig pa spyarki blugs sgom pa yin/_/gnyis med blang dor bral ba'i gnas lugs la/_/yang yang spyarkada na spyarkada du med par mthong /_/spyarkada med rig pa spyarki blugs spyarkoda pa yin/_/ye zin re dogs bral ba'i lhun grub la/_/yang yang basgrarkubasa na basgrarkuba tu med pa mthong /_/basgrarkuba med rig pa spyarki blugs 'bras bu yin/_/mnyam nyid ngang las yul du mi rtog cing /_/sems su mi 'dzin re dogs 'byung 'jug gzhi/_/yul sems mnyam pa'i ngang der gnas pa ni/_/chos nyid klong las ngang gis g.yos pa med/_/mtshan ma'i yul la yul med spyarki blugs gnas/_/ye nas gnyis med rig pa sbyarki blugs pas/_/'khor 'das dbyer med rdzogs pa chen po'i ngang /_/thams cad blang dor med par phyam gdal lo/_/dngos dang dngos med dbyings su mnyam pa dang /_/sangs rgyas sems can dbyings su mnyam pa dang /_/kun rdzob don dam dbyings su mnyam pa dang /_/skyon dang yon tan dbyings su mnyam pa dang /_/mtho dman phyogs mtshams dbyings su mnyam pa'i phyir/_/rang byung ngang las rol pa ci shar yang /_/shar ba'i dus na mnyam shar bzang ngan med/_/de la blang dor gnyen pos bcos ci dgos/_/gnas pa'i tshe na mnyam gnas bzang ngan med/_/da lta sems las gang byung rang zhir glod/_/grol ba'i tshe na mnyam grol bzang ngan med/_/dran pa'i rjes la dgag sgrarkuba 'phro ma mthud/_/thams cad gzhi klong byang chub sems nyid las/_/rtsal dang rol pa'i 'char tshul ma nges pas/_/mnyam par shar yang gdod ma'i klong nas shar/_/mi mnyam shar yang mnyam pa'i dbyings nas shar/_/mnyam par gnas kyang rang gzhag chos nyid ngang /_/mi mnyam gnas kyang mnyam pa'i dbyings na gnas/_/mnyam par grol yang rang byung ye shes klong /_/mi mnyam grol yang mnyam pa'i dbyings su grol/_/thams cad ye mnyam rang byung rig pa la/_/shar dang ma shar dbyings la ye nas med/_/gnas dang mi gnas dbyings la ye nas med/_/grol dang ma grol dbyings la ye nas med/_/mi g.yo mnyam pa chen po'i rig pa la/_/shar ba'i dus na rang shar rang sa zin/_/gnas pa'i dus na rang gnas rang sa zin/_/grol ba'i dus na rang grol rang sa zin/_/mi 'gyur sprarkosa dang bras ba'i rig pa la/_/shar ba ye shar gnas pa ye gnas la/_/grol bar ye grol nam mkha'i rang bzhin no/_/shar gnas grol gsum shar grol rgyun chad med/_/rgyun chad med pas rgyu 'bras bar ma chod/_/rgya 'bras med pas 'khor ba'i g.yang sa chod/_/g.yang sa med pas gol sa ga la yod/_/ye nas mi 'gyur kun tu bzang po'i klong /_/'pho 'gyur med pa rdo rje sems dpa'i klong /_/gnas lugs rang ngo shes pa tsam nyid la/_/sangs rgyas zhes su ming 'di btags par zad/_/'di nyid rtogs nas blang dor chos med pas/_/thams cad chos nyid gcig tu phyam gdal ba/_/gser gling lta bur dbye bsal med pa yin/_/mtha' yis ma reg gol sgrarkiba gdar sha chod/_/g.yang sa med pa'i byang chub sems nyid la/_/'bad rtsol med pa'i sku gsum lhun rdzogs kyang /_/bsam brjod 'das shes ming tsam brjod par zad/_/snang ba rgya yan rig par rang byung gsal/_/ma basgrarkibasa phyi nang med par zang thal bas/_/ma bcos rnal gzhag chos nyid chen por gsal/_/blo bde'i mal na lus sems khong glod de/_/shes pa bag yangs byar med skyes bu bzhin/_/sgrarkimasa glod med par lus sems gang bder gzhag_/ji ltar 'dug kyang rang gi ngang la 'dug_/ji ltar gnas kyang rang gi ngang la gnas/_/ji ltar 'gro yang rang gi ngang la 'gro/_/byang chub dbyings la 'gro 'ong ngang gis med/_/'gro 'ong med pa rgyal ba rnams kyi sku/_/ji ltar smrarkasa kyang rang gi ngang la smrarka/_/ji ltar brjod kyang rang gi ngang la brjod/_/byang chub sems la smrarka brjod ngang gis med/_/smrarka brjod med pa dus gsum rgyal ba'i gsung /_/ji ltar bsams kyang rang gi ngang la bsam/_/ji ltar rtog kyang rang gi ngang la rtog_/byang chub sems la bsam rtog ye nas med/_/bsam rtog bral ba dus gsum rgyal ba'i thugs/_/med la cir yang 'byung ba sprarkula pa'i sku/_/nyid la nyid longs spyarkoda pa longs spyarkoda rdzogs/_/de la dngos gzhi med pas chos sku ste/_/'bras bu sku gsum lhun gyis grub pa'i klong /_/byang chub sems kyi klong chen ngang nyid las/_/dran pa'i rnam par rtog pa mi 'byung ste/_/shes pa'i mtshan ma yid la mi g.yo na/_/de nyid sangs rgyas nyag gcig dgongs pa yin/_/byang chub rang bzhin nam mkha'i dkyil yangs 'dra/_/dran dang rtog pa med pa sgom pa'i mchog_/rang gi rang bzhin ma g.yos bcos pa med/_/mi bsam yid la byed pa rnam bral ba/_/rang babs chos nyid dus gsum 'pho 'gyur med/_/'gyu 'phro'i kun rtog med pa sgom pa'i mchog_/de bzhin nyid der gang gnas dam pa'i sems/_/sangs rgyas nyag gcig mtshon pa kun dang bral/_/mi g.yo chos dbyings 'dzin rtog phyam 'das pa/_/rgyal ba'i dgongs klong rang bzhin yangs pa'i mchog_/lus sems bcos pa'i 'ching ba rnams spangs shing /_/gyin 'dar lhod chags dran bsam ci 'gyus kyang /_/gzhi gzhag chos nyid ngang las mi g.yo na/_thams cad kun bzang dgongs klong yangs pa yin/_/ma bzang ma btang 'jur bu'i sgrarkimasa glod med/_/ji ltar rang babs rgya yan ngang gis zin/_/mi g.yo phyam gdal yangs dog med pa'i klong /_/dran bsam thams cad rang byung rang zhi na/_/rdo rje sems dpa' nam mkha'i dgongs pa yin/_/ma bcos klong du ma yengs ngang ldan na/_/dran rtog yul la 'jug pa'ang chos nyid ngang /_/ched du 'jur bus bcos na chos nyid kyang /_/mi rtog mkha' ltar yangs kyang mtshan ma'i gzeb/_/nyin mtshan sgom pas 'da' yang 'ching zhen nyid/_/bsam gtan lha dang mtshungs par rgyal bas gsungs/_/de bas ma yengs 'bad rtsol zhig pa'i sems/_/rang babs 'dzin rtsol 'das pa shin tu gces/_/rang byung ye shes phyogs dang ris med pas/_/'di zhes mi mtshon rang bzhin sprarkosa kun zhi/_/de bas yid la byed pa rnams spangs te/_/gzhi bral yangs pa chen po'i don la bslab/_/chos nyid nyag gcig rang byung ye shes te/_/lta ba nyag gcig sprarkosa pa'i mtha' dang bral/_/sgom pa nyag gcig btang gzhag 'gro 'ong med/_/spyarkoda pa nyag gcig blang dor byar med cing /_/'bras bu nyag gcig spangs thob gnyis dang bral/_/'di ni rang byung lhun grub dgongs pa yin/_/snang srid snod bcud 'khor dang mya ngan 'das/_/chos so cog kun chos nyid gdod ma'i ngang /_/rang byung ye shes nyid las ma g.yos pas/_/gang yang gzhi gzhag dgongs par shes par bya/_/sna tshogs yul du snang ba'i chos nyid la/_/gang du 'di ltar 'jog ces mi rtog par/_/rang babs sprarko bsdu bral bar lhun gyis zhog_/chos nyid mnyam pa'i klong du ngang gi gnas/_/dkar dmar yul du snang ba'i rnam pa la/_/dbang po mi bsdu mig kyang mi 'gul bas/_/bdag la mi bsam gzhan la mi rtog par/_/phyam phyal yangs pa chen por ngang gis zhog_/rgya bskyed dpangs bstod sprarko bsdu bral ba'i sems/_/kun mnyam rang byung ye shes dgongs pa la/_/phyi nang bar med nam mkha' 'dres pa'i nyams/_/bde gsal sprarkosa dang bral ba'i ting 'dzin 'char/_/gzhi gzhag mi g.yo mnyam pa'i dgongs pa la/_/phyi dang nang med gzung 'dzin sprarkosa dang bral/_/yul zhes gzhan du zhen pa'i sems med pas/_/gzung ba'i chos med snod bcud snang zhen bral/_/'khor bar skye ba'i yul med nam mkha' 'dra/_/rang zhes sems la nang du mi rtog pas/_/'dzin pa'i chos med srid pa'i kun rtog zhi/_/'khor bar skye ba'i mkhan po rtsad nas chod/_/de tshe mkha' 'dra phyi nang 'khrul pa'i chos/_/gang yang mi dmigs chos sku'i dgongs par phyin/_/zad pa'i sar thug 'gro dang 'ong med pas/_/thams cad klong 'byams kun tu bzang po'i zhing /_/chos sku'i pho brang mchog tu phyin pa yin/_/da lta'i rig pa gzhi las ma g.yos na/_/de nyid goms chos phyi ma'i srid pa stongs/_/yang srid len pa'i las dang bag chags bral/_/rgyu 'bras la zlos 'khor 'das mnyam par brjod/_/srid zhir mi gnas byang chub snying por phyin/_/'dir yang zhi gnas rtse gcig phyed pa gces/_/rang bzhin rdzogs pa chen po'i dgongs pa yin/_/ngang las g.yos na yid dpyod 'khor ba nyid/_/de la rgyu 'bras nyid de la ma zlos/_/nor ba'i skye bo 'og nas 'og tu 'gro/_/dos na mchog gsang rdzogs pa chen po ni/_/dbyings las ma g.yos rtsal rnams gzhi la 'khyol/_/dgongs pa mi g.yo mnyam par gnas pa yin/_/'di yi ngang la rgyu 'bras bya rtsol med/_/lta ba bsgom du med pa la sogs te/_/mtha' dbus gnyis med 'gog pa'i tshul brjod kyi/_/gzhan du ngang las phyir 'khyal rtsal nyid las/_/rol pa sna tshogs snang srid dgu 'char bas/_/rgyu 'bras med ces nam yang ma brjod cig_/rten 'brel rkyen 'byung 'dus byas grangs bsam 'das/_/'khor ba'i 'khrul snang zhi bde'ang grangs bsam 'das/_/de kun rgyu rkyen tshogs pa'i rten 'brel nyid/_/gshis la gzhal na gang yang ma grub ltar/_/lam du byed pas gshis thog mi g.yo'i dus/_/gang yang mi dmigs dgongs pa'i dus na rtsi/_/gshis thog mthar phyin gang yang gos mi 'gyur/_/nyon mongs las dang bag chags klong chen 'di/_/rten med sgyarku mas sprarkula pa'i rtsed mo byed/_/'di las thar dgos rgyu 'bras la zlos 'tshal/_/de yi thabs ni 'di las mchog gzhan med/_/de phyir chos dbyings dgongs pa ma g.yos gces/_/'di nyid kho bo'i snying gtam zab mo'i klong /_/kun yin kun min yin min 'das pa gces/_/chos dbyings rin po che'i mdzod las/_dgongs pa chos nyid las mi g.yo bar bstan pa'i le'u ste bcu pa'o//_ //thams cad mkha' mnyam byang chub sems gcig la/_/gnyis su bzung bas rgyu 'bras srid par 'khrul/_/'khrul snang rten med sgyarku ma'i snang ba la/_/thug phrad rjes med rtsis gdab bral bar skyong /_/mi 'dod thog tu bab la dor bya'i sems/_/khro dang mi dga' phrag dog 'khrug dang 'tshig_/skyo bas yid dbyung na tsha sdug bsngal sems/_/'chi dang skye bas 'jigs la sogs pa ste/_/rtsal las rol par shar dus ngos bzung la/_/mi spong mi spyarkoda mi sgyarkura mi len zhing /_/mi lta mi sgom rang babs phyam gcig tu/_/dmigs bsam sprarko bsdu bral bar lhun gyis zhog_/rjes med rang yal mkha' klong yangs pa'i sems/_/gsal dwarkangasa ngar dang bcas pa khong nas 'char/_/yin min med pa'i rig pa phyogs yan la/_/'dir gtad med pa'i snang ba thug phrad kyis/_/'dir gzung med pa'i ngang der rang bzhag pas/_/'dir grol med pa'i dgag sgrarkuba rjes med yal/_/'dzin zhen med pa'i nyams myongs khong rdol 'byung /_/'di nyid ji bzhin dgongs klong ye nas yangs/_/de bzhin 'dod cing yid la dga' ba'i sems/_/grub bde gnyen dang gtam snyan longs spyarkoda dang /_/gnas dang phyogs ni yid du 'ong pa der/_/rang bzhin dga' bas brgyan pa'i sems shar ba/_/de nyid ngos gzung rang babs cog gzhag gis/_/gdod ma'i dbyings su ma bcos lhun gyis grub/_/rang bzhin bar ma 'gro 'dug gyin 'dar gnas/_/dga' dang mi dga' gnyis min ci shar yang /_/shar dus ngos gzung spang blang ma byas pas/_/rang babs chos nyid dbye bsal med pa ni/_/gti mug 'od gsal chen por grol zhes bya/_/mtshan mo la sogs gnyid kyis myos pa na'ang /_/rang babs sprarko bsdu bral ba'i ngang nyal bas/_/rags par snang ba nub pas der 'dzin nub/_/phra ba ches phra 'dzin par bcas nub pas/_/cha mnyam mi rtog ngang du rig pa'i sems/_/'byung 'jug re dogs bral ba rang bzhin gnas/_/kun rtog chos dbyings grol ba'i dus yin pas/_/'khor ba myang 'das grol zhes brjod pa yin/_/gnyid kyang rang byung gdod ma'i klong chen te/_/rtsal rnams ngo bo'i dbyings su gzhir thim pas/_/rol par 'dzin pa'i sprarkosa kun ngang gis zhi/_/byar med rang byung ye shes dgongs pa'o/_/de ltar 'dod dang mi 'dod bar ma'i sems/_/dug gsum rtsal las rol par shar to cog_/dbyings las byung zhing dbyings kyi ngang du shar/_/dbyings su ma gtogs g.yos pa 'ga' med pas/_/gzeb dang bcos basgyarkura gang yang mi byed par/_/dbyings nyid ngos gzung ngang der gzhag ma thag_/rang zhi rang yal rang sar grol ba gnad/_/nyon mongs las dang bag chags thams cad kyang /_/rtsal las rol par shar ba'i cho 'phrul la/_/gnyen po bzang byas thar pa'i lam nyid kyang /_/rtsal las rol pa shar ba'i cho 'phrul te/_/gnyis ka rtsal las rol par ye shar nas/_/ngo shes ngang du ma bcos 'jog pa gnad/_/bang mnyam 'gros mnyam gzhi nas g.yos par mnyam/_/rkyen byung 'dus byas rgyu 'bras ma 'das pas/_/rang gzhag cog gzhag rgyu 'bras zlos pa gces/_/'di ni mchog gsang theg pa'i yang rtse ste/_/blo dman rnams la mi smrarka shin tu gsang /_/sgrarko skur dbang gis snying po'i bstan pa 'chal/_/sgrarko 'debs nyid dang dgongs pa log par zhugs/_/gsang sgo 'chol rnams mtha' med ngan 'gror ltung /_/de bas rab gsang theg pa rgyal po'i gdung /_/skal bzang dam pa rnams la bstan zhing gtad/_/mdor na gang snang yul sems rkyen rnams la/_/gnyen po mi 'jug rtsol bas ma spangs par/_/rang zhag cog gzhag rang babs rig pa'i gnad/_/bde sdug thams cad rig pa'i 'char tshul la/_/blang dor gnyis su bzung bas srid par 'ching /_/gang snang yul mnyam dbang po'i ngor gsal tsam/_/gang shar sems mnyam dran rig rjes med tsam/_/gnyis ka 'phral mnyam dgag sgrarkuba 'ching ba tsam/_/don la phug mnyam gzhi med snang ba tsam/_/yul rnams ris mnyam gzhig na rjes med tsam/_/blo rnams ngor mnyam dpyad na nam mkha' tsam/_/yul sems gnyis med bar snang dag pa tsam/_/de ltar sus shes kun tu bzang po'i gdung /_/rgyal ba'i sras mchog sa rab rig pa 'dzin/_/'di ltar chos rnams yod mnyam med mnyam la/_/snang mnyam stong mnyam bden mnyam rdzun mnyam pas/_/spang gnyen 'bad rtsol 'ching zhen kun thong la/_/yul med mnyam pa chen por phyam gdol cig_/sems med rig pa chen por phyam gdol cig_/skyon med dag mnyam chen por phyam gdol cig_/chos dbyings rin po che'i mdzod las/_rkyen snang mkha' mnyam dag par bstan pa'i le'u ste bcu gcig pa'o// _//chos kun byang chub sems su ye grol bas/_/ma grol ba yi chos ni yod ma yin/_/'khor ba ye grol gdod nas dag par grol/_/'das pa ye grol lhun gyis rdzogs par grol/_/snang ba ye grol gzhi rtsa med par grol/_/srid pa ye grol byang chub snying por grol/_/sprarkosa pa ye grol mu mtha' med par grol/_/sprarkosa med ye grol ma skyes dag par grol/_/bde ba ye grol chos nyid phyam du grol/_/sdug bsngal ye grol gzhi mnyam yangs par grol/_/bar ma ye grol mkha' mnyam chos skur grol/_/dag pa ye grol dag gzhis stong par grol/_/ma dag ye grol yongs grol chen por grol/_/sa lam ye grol bskyed rdzogs bral bar grol/_/lta sgom ye grol spang blang med par grol/_/spyarkoda pa ye grol kun tu bzang por grol/_/'bras bu ye grol re dogs med par grol/_/dam tshig ye grol chos nyid chen por grol/_/bzlas brjod ye grol gleng brjod bral bar grol/_/ting 'dzin ye grol bsam yul med par grol/_/yod med ye grol mtha' las 'das par grol/_/rtag chad ye grol gzhi rtsa med par grol/_/yang dag ye grol dmigs bsam 'das par grol/_/yang dag min grol phyogs bsam 'das par grol/_/las rnams ye grol gos pa med par grol/_/nyon mongs ye grol 'ching grol med par grol/_/bag chags ye grol rten gzhi med par grol/_/rnam smin ye grol myong gzhi med gar grol/_/gnyen po ye grol spang bya med par grol/_/spang blang gnyis med mkha' mnyam yangs par grol/_/grol ba ye grol bcings pa med par grol/_/ma grol ye grol bcing grol med par grol/_/glod pa ye grol glod rgyu med par grol/_/cog gzhag ye grol 'jog rgyu med par grol/_/mdor na snang zhing srid pa'i chos rnams dang /_/mi snang mi srid chos las 'das so cog_/thams cad ye nas dbyings su grol zin pas/_/da gzod 'bad pas sus kyang grol mi dgos/_/de la 'bad rtsol byas kyang don med pas/_/ma byed ma byed rtsol zhing sgrarkuba ma byed/_/ma lta ma lta yid kyi chos ma lta/_/ma bsgoms ma bsgoms blo yi chos ma bsgoms/_/ma dpyad ma dpyad yul sems rjes ma dpyad/_/ma basgrarkubasa ma basgrarkubasa re dogs 'bras ma basgrarkubasa/_/ma spangs ma spangs nyon mongs las ma spangs/_/ma len ma len yang dag chos ma len/_/ma 'chings ma 'chings rang gi rgyud ma 'chings/_/thams cad phyam log gang la'ang yul med pas/_/khrigs dang chos med dmigs gtad ngos gzung med/_/gzhi log lam log 'bras bu'i chos log pas/_/legs nyes shor god rdul tsam dmigs su med/_/phyam phyam phyang chad ye chad snang srid chad/_/'khor 'das ru log dbyings kyi so na med/_/gang yin ci yin 'di yin gtad so med/_/khyed cag ci bya nga ni gang na 'dug_/sngar shul da med 'di la sus ci bya/_/ha ha 'di 'dra mtshar che dgod re bro/_/snang srid snod bcud 'khrul pa ru log pas/_/nyin mtshan ye sangs rang sangs nam mkha' sangs/_/zhag dang tshes sangs lo zla bskal pa sangs/_/gcig sangs kun sangs chos dang chos min sangs/_/'khor 'das 'khrul gzhi gdod ma'i dbyings su sangs/_/dbyings zhes tha snyad blo yi chos sangs pas/_/gang sgrarkuba ci 'bad da ni ci zhig gnyer/_/'dod blo'i 'khris zad nam mkha' ya mtshan che/_/chos med sprarkanga po'i rang bzhin de 'drar zad/_/sa gzhi rin chen nam mkha' bar snang rdzong /_/rten med ye grol dgongs par lhun grub pas/_/srid gsum snod bcud yul med chen por grol/_/phyogs med phyogs su 'dzin pas bcings pa rnams/_/rang bzhin ma shes rang gis rang bslad pas/_/rang la rang rmongs 'khrul pa a re 'khrul/_/'khrul pa med la g.yang sar 'dzin pas 'khrul/_/'khrul dang ma 'khrul byang chub sems kyi klong /_/byang chub sems la 'khrul grol ye nas med/_/de las rol par shar la 'dzin pas bcings/_/don la 'ching grol gnyis med yul sems med/_/med la yod par 'dzin pas ma bslu zhig_/ye sangs rgyas pas grol ba'i rig pa de/_/gtad 'dzin chos kyi gzeb tu ma 'chings shig_/ye nas yul med rnam kun dag pa'i klong /_/mkha' mnyam bde chen gzhi rtsa byang chub klong /_/de la 'khor ba mi srid gdod ma'i babs/_/thig le nyag gcig grwarka zur med pa la/_/gcig dang tha dad 'dzin pa 'khrul pa'i sems/_/rang byung ye shes rgyu rkyen med pa la/_/'khor ba'i lam du 'dzin pa byang chub gegs/_/lhun grub phyogs med mtha' dang bral ba la/_/phyogs lta'i mtha' la zhen pa snyem byed bdud/_/dngos mtshan med pa'i stong pa 'gag med la/_/yod med snang stong 'dogs pa log pa'i blo/_/des na gang 'dod phyogs ris gzeb bor la/_/lhun grub phyogs med nam mkha' ltar shes byos/_/snang grags mthong thos tshogs drug ci shar yang /_/thams cad rang gsal dbye bsal med pa'i klong /_/ye grol mnyam pa'i klong du la zlos shing /_/mnyam nyid gcig la cir snang 'byung bas dbyings/_/yon tan thams cad skyed par byed pas gzhi/_/thams cad bya byed med par grol bas klong /_/kun 'byung snying po shar bas byang chub sems/_/mkha' 'dra gdod nas dag par shes par bya/_/rang byung ye shes gzhi klong yangs pa la/_/ye nas dri med 'khor bas ma gos kyang /_/yon tan lhun grub rgyu 'bras 'das par chub/_/rang rig snying po 'od gsal dag pas sems/_/byang chub sems su kun 'dus rnam par dag_/rtsal las rol par shar ba'i rang ngo la/_/rtogs pas glo bur yang sangs rgyas pa dang /_/ma rtogs ma rig 'khrul pa'i tshogs shar bas/_/kun gzhi las mched tshogs brgyad yul dang bcas/_/snang srid snod gcud rol par ci shar yang /_/byang chub sems kyi klong las g.yos pa med/_/sems klong ma g.yos mnyam pa'i ngang gnas na/_/'khor 'das 'ub chub dgongs klong yangs par grol/_/gshis la 'khor 'das mi srid rang bzhin babs/_/bzang ngan blang dor mi srid rang bzhin babs/_/spangs thob gzung 'dzin mi srid rang bzhin babs/_/dug lnga'i nyon mongs mi srid rang bzhin babs/_/rgya chad phyogs lhung mi srid rang bzhin babs/_/rtsal dang 'char ba mi srid rang bzhin babs/_/phyogs tsam btags pa mi 'gog rang bzhin babs/_/rang byung ye shes chos zad ming med pa/_/rtsal dang rol pa ci shar gzhi med nyid/_/'ching grol med pa gnas lugs rang bzhin babs/_/grol zhes brdar btags rang yal rjes med tsam/_/kun yin kun min btags par mi 'gal bas/_/ye nas grol zhes tshig tu brjod pa yin/_/dbye bsal med do lhun grub klong du grol/_/'du 'bral med do thig le'i klong du grol/_/cir yang 'char ro nges med klong du grol/_/gzugs su snang ngo snang ba rang sar grol/_/sgrarka ru grags so grags par rang sar grol/_/dri ru tshor ba tshor ba'i dbyings su grol/_/ro myong reg pa rang sa'i ngang der grol/_/dran rig byung tshor gzhi rtsa rten med grol/_/gcig tu grol lo chos nyid klong du grol/_/gnyis su med do yul sems mnyam par grol/_/rang byung grol lo ye shes klong du grol/_/lhun grub grol lo gzhi dbyings dag par grol/_/sna tshogs grol lo nyag gcig klong du grol/_/phyogs med grol lo lhun grub klong du grol/_/thams cad grol lo snying po'i klong du grol/_/'od gsal grol lo nyi zla'i klong du grol/_/chos nyid grol lo nam mkha'i klong du grol/_/chos can grol lo rgya mtsho'i klong du grol/_/mi 'gyur grol lo ri rgyal klong du grol/_/gdod nas grol lo skye med klong du grol/_/phyam gcig grol lo ye sangs klong du grol/_/yongs grol grol lo ye rgyas klong du grol/_/chos dbyings rin po che'i mdzod las/_chos thams cad byang chub sems su ye nas grol ba'i rang bzhin bstan pa'i le'u ste bcu gnyis pa'o//_ _ //chos rnams lhun grub byang chub snying po la/_/'bad rtsol med pa'i gnad kyi goms byas na/_/ye sangs rgyas la yang sangs rgyas 'byung ste/_/'di ni bla med rdo rje snying po'i don/_/rim dgu'i snying po byang chub klong chen yin/_/mkha' dkyil nyi zla'i dkyil 'khor 'od gsal yang /_/ma rtogs sprarkina chen snying pos yongs basgrarkibasa pas/_/mi snang byang chub rang la yod pa'i tshul/_/sprarkina chen dbyings su bzhag pas rang dengs ltar/_/'bad rtsol med pas rgyu 'bras sprarkina bral nas/_/mkha' dkyil byang chub snying po rang la 'char/_/dbang po'i rim pas theg pa tha dad la/_/ngo bo nyi bzhin chos dbyings klong na gsal/_/rtsal las zer bzhin kun shar ris med pas/_/sa dang chu mtshor drod kyis khyab pa na/_/rlangs las sprarkina gyi rol par shar ba yis/_/ngo bo nyid dang rtsal yang basgrarkibasa pa bzhin/_/ngo bo nyid las rang rtsal ma dag pa'i/_/rol pas snying po'i de nyid rang ngor sgrarkiba/_/snang srid snod bcud 'khrul snang bsam mi khyab/_/nyi zer rtsal las rlung bskyod sprarkina dengs ltar/_/rang ngo rtogs las rol pa rgyan du shar/_/'khrul pa ye grol rang sar rang grol bas/_/'khrul snang 'khrul 'dzin ma spangs dbyings su zhi/_/gar song cha med dwarkangasa pa'i nam mkha' la/_/sku dang ye shes lhun grub nyi ma shar/_/gzhan nas ma 'ongs rang snang dag pa tsam/_/gnyis skyes sgo nga'i nang nas 'dab rgyas pa/_/sgong rgyas 'thums ltar da ltar mi snang yang /_/sgong rgya ral bas mkha' dkyil lding ba ltar/_/gzung 'dzin 'khrul rtog zag pa sngar zad kyang /_/zag 'bras lhag bcas sgong rgya ral ma thag_/lhun grub rig pa rang gsal rang la 'char/_/sku dang ye shes snang bas mkha' dbyings khyab/_/rang ngo shes pas kun bzang klong du grol/_/phyogs bcur thugs rje'i rol pa tshad med las/_/sprarkula pa 'phro bas 'gro ba'i don kun byed/_/'khor ba ji srid mdzad pa nye bar ston/_/'di ni rang bzhin babs kyi ngo bo las/_/rtsal gyi thugs rje phyogs med shar ba ste/_/rol pas gzhan don phun sum tshogs pa yin/_/ma dag cha bcas rol pa nyer zhi yang /_/ma dag 'gro la sprarkula pa snang ba ni/_/ston pa'i ji bzhin shugs kyi thugs rje dang /_/'gro sems gtsang ma'i las smon dag pas 'char/_/de tshe zhing kun sprarkula pa tshad med cing /_/mtha' yas 'gro ba byang chub 'dren mdzad kyang /_/ston pa'i chos sku dbyings las mi g.yo ste/_/rang byung ye shes rtag chad phyogs lhung med/_/dbyings las rang shar stug po bkod par ni/_/rig 'dzin mkha' 'gro sa bcu'i sems dpa' la/_/longs spyarkoda rdzogs pa'i bkod pa bsam yas snang /_/de yang dbyings las ston pa'i thugs rje dang /_/gdul bya'i mos dges lhun grub rang ngor snang /_/chos sku'i ngo bo rang byung ye shes te/_/rol par thams cad mkhyen pa'i ye shes mtsho/_/gdod ma'i dbyings na thig le gcig tu bzhugs/_/longs sku'i ngo bo rang bzhin lhun grub ste/_/rol par rigs lnga ye shes rnam pa lnga/_/nam mkha'i dbyings kun gang bar snang ba yin/_/sprarkula sku'i ngo bo thugs rje'i 'char gzhi ste/_/rol par gang la gang 'dul der snang zhing /_/phrin las chen pos mnga' dbang 'byor pa yin/_/'di dag rgyu 'bras rtsol bas ma basgrarkubasa te/_/ye nas lhun grub cog gzhag ngang la snang /_/mchog gsang rab la tshe 'dir snang ba ste/_/de las gzhan du bar dor mi bslu bas/_/rdo rje snying po rtse mo'i theg pa ni/_/rgyu 'bras theg pa kun las khyad par 'phags/_/chos dbyings rin po che'i mdzod las/_chos thams cad byang chub kyi sems su ye sangs rgyas pa la rtsol ba dang sgrarkuba pa med par yang sangs rgyas par bstan pa'i le'u ste bcu gsum pa'o//____ _//de ltar chos nyid rdo rje snying po'i glu/_/mkha' mnyam gdod nas dag pa'i rang bzhin 'di/_/mi 'gyur gzhi rtsa bral ba'i gnas nyid du/_/rang shar 'pho 'gyur med pa'i rol par shar/_/ye mnyam phyam gdal klong chen yangs pa'i don/_/gang yang ma phyin gdod ma'i rang bzhin ngang /_/mi g.yo lhun gyis grub pa'i chos nyid la/_/rgya chad med cing phyogs lhung bral bar gyur/_/don rnams ji bzhin mkha' mnyam yangs pa'i dkyil/_/gang du rang byung klong chen rgyal po ni/_/rtag tu mi g.yo sna tshogs rang sar grol/_/'di zhes mi mtshon dbyings rum yangs par phyin/_/rtogs pa'i dus des mkha' 'dra'i rnal 'byor pas/_/rang nyams phyogs gcig lung don mthun pa ni/_/rtsa ba'i sems lung nyi shu rtsa gcig dang /_/klong gsum man ngag sde bzhi mthun par bkod/_/dge ba 'di yis ma lus 'gro ba kun/_/ma 'bad bzhin du gdod ma'i sar phyin nas/_/kun bzang sa la 'pho 'gyur med pa yi/_/don gnyis lhun grub chos kyi rgyal por shog_/phyogs rnams kun tu bde dpal 'byor pa ni/_/dag pa'i zhing bzhin 'dod dgu lhun grub cing /_/chos rnga sgrarkogasa pas thar pa'i rgyal mtshan 'dzugs/_/dam chos mi nub bstan pa rgyas par shog_/chos dbyings rin po che'i mdzod cesabya ba/_theg pa mchog gi rnal 'byor pa klong chen rab 'byams kyis gangs ri thod dkar gyi mgul du legs par bkod pa rdzogs so//____//_!_na maHsarba dz+nyag+g+hA na gu ru we/_kun bzang dgongs klo~Mng dri~M me~Md mkha' dbyings las/_/rdzogs che~Mn ka lhun dgongs gsang ra~Mb 'abyama~Msa gnad/_/cig char 'gyed 'khas 'o~Md ze~Mr bsa~Mm ya~Ms pa~M/_/rdo~M rje~M gzi~M brji~Md zhing kun nyin mor byed/_/sku gsum dgongs par rang grol klong rdol gter/_/gsang chen nA da'i sgrarka dbyangs mi zad rgyan/_/sku dang ye shes rang byung rig pa'i rgyud/_/rdo rje 'chang gi bka' dang mi gnyis pa/_/zab las ches zab cho~Ms dabyinga~Msa rtogs pa'i mdzod/_/dus gsum rgyal dgongs chos sku'i ring bsrel mchog_/kun mkhyen chos rje'i rtogs pa'i 'dra zhal dngos/_/mngon sum mjal du yod 'di skal pa bzang /_/rgyal bstan yongs rdzogs dgongs don gcig bsdus pa/_/gzhung bzang 'di 'dra srid pa gsum na med/_/dus gsum 'byon dka' zab gsang nor bu'i tog_/mthong thos dran grol srid zhi'i rgyan gcig pu/_/rnam dag par du baskrarkuna pa'i dmigs med dges/_/snying po'i bstan dar de 'dzin bshad sgrarkuba rgyas/_/'gro kun theg rtse'i myur lam gdod ma'i gzhir/_/mngon par byang chub don gnyis lhun grub shog_/ces pa'ang chos sbyarkina gsar baskrarkuna dang smon tshig chab gcig kun mkhyen yab sras kyi bka' 'bangs su smon pa grub rigs gdung grol can tshe dbang rig 'dzin dpal gyis lha ldan gtsug lag khang gi nye 'dabs chu srin shar du rab byung bcu drug pa chu bya sa gas nya ba'i dus tshes dge bar bris pa'i dge bas kun tu bzang po ru gyur cig gu/_/dge'o/_/dge'o/_/dge'o//____//_!_mkhan slob chos gsum ring lugs che/_/'dzam gling sa gsum khyab par 'phel/_/'gro rgyud mchog gsum snang ba dang /_/mi 'bral dus gsum dge legs shog/__//ces pa'ang dz+nyag+g+hA nas so/_/sarba mang+galaM//___//

Source:


text:
File talk:MPSoCPlatformModel.gif

The image is a snapshot of a research tool, and was uploaded to Wikiversity by one of the creators of the tool (and the person who did the snapshot).

title: File talk:MPSoCPlatformModel.gif


text:
Boundary Value Problems/BVP-Exercise 1-1:Problems and Solutions

Review ../Ordinary-Differential-Equations/


text:
User:Winterstein

I am an Edinburgh-based mathematics and Artificial Intelligence consultant. The website for my work is at http://www.winterwell.com/.

I am the lead architect for Sodash, an AI-powered social media dashboard: http://soda.sh/

I keep a personal web-page at http://winterstein.me.uk and a blog at http://www.platypusinnovation.com. Visitors welcome.

title: User:Winterstein


text:
User talk:Winterstein

 Hello Winterstein, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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title: User talk:Winterstein


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User talk:Vickyagarwal

 Hello Vickyagarwal, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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User talk:T.thanos

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User:Egm6341.s10.team3.ynahn81/HW4-6(c)

(c) Compare Comp Trap, Comp Simp, Romberg for $$\displaystyle f(x) = \frac{e^{x}-1}{x}$$
Ref: Lecture notes [[media:Egm6341.s10.mtg25.djvu|p.25-1]]

Problem Statement
For $$\displaystyle I = \int_{0}^{1}\frac{e^{x}-1}{x}dx$$, compare $$\displaystyle I_{n}$$ to $$\displaystyle O(10^{-10})$$ and computational time for - Composite Trap rule, - Composite Simpson's rule and - Romberg table.

Solution
In what follows, we are using the follow matlab function as a given function $$\displaystyle f(x) = \frac{e^{x}-1}{x}$$.

i) Composit Trap. rule

Recall the formula of comp. Trap. rule.

 * {| style="width:100%" border="0" align="left"

T_{0}(n) = h\left[\frac{1}{2}f(x_{0}) + f(x_{1}) + f(x_{2}) + \cdot\cdot\cdot + f(x_{n-1}) + \frac{1}{2}f(x_{n}) \right] $$ where $$\displaystyle h = \frac{b-a}{n} $$ and $$\displaystyle x_{0} = a,\quad x_{n} = b $$.
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Also, realize the fact that if we want to double the number of sub-intervals, we don't have to calculate the functions at even point, i.e. $$\displaystyle f(x_{0}), f(x_{2}), \cdot\cdot\cdot, f(x_{n-2}), f(x_{n}) $$ because we already calculated the functions at these even point at the previous step. Therefore, all we have to do is evaluating the functions only at the odd points and combine them with the calculated values at the previous step. As a result, we can use the following inductive formula.
 * {| style="width:100%" border="0" align="left"

T_{0}(2n) = T_{0}(n) + h\sum_{i=1}^{n}f(x_{2i-1}). $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Matlab code

Result

$$\displaystyle I_{n,trap} = T_{0}(n) = 1.317902151493210, \quad n = 2^{15} = 32768 $$ $$\displaystyle E_{n,trap} = I - I_{n,trap} = 3.880606946893295e-11 $$

ii) Composite Simpson's rule

Recall the formula of comp. Simpson's rule.

 * {| style="width:100%" border="0" align="left"

T_{1}(n) = \frac{h}{3}\left[f(x_{0}) + 4f(x_{1}) + 2f(x_{2}) + 4f(x_{3}) + \cdot\cdot\cdot + 4f(x_{n-3}) + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_{n}) \right] $$ where $$\displaystyle h = \frac{b-a}{n} $$ and $$\displaystyle x_{0} = a,\quad x_{n} = b $$.
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Also, realize the fact that if we want to double the number of sub-intervals, we don't have to calculate the functions at even point, i.e. $$\displaystyle f(x_{0}), f(x_{2}), \cdot\cdot\cdot, f(x_{n-2}), f(x_{n}) $$ because we already calculated the functions at these even point at the previous step. Therefore, all we have to do is evaluating the functions only at the odd points and combine them with the calculated values at the previous step. As a result, we can use the following inductive formula.
 * {| style="width:100%" border="0" align="left"

T_{1}(2n) = T_{1}(n) + h\sum_{i=1}^{n}4f(x_{2i-1}) - \underbrace{h\sum_{i=1}^{\frac{n}{4}}2f(x_{4i-2})}_{Already\,\, obtained\,\, when\,\, T_{1}(n) \,\, was\,\, calculated.}. $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Matlab code

Result

$$\displaystyle I_{n,Simpson} = T_{1}(n) = 1.317902151460891, \quad n = 2^{7} = 128 $$ $$\displaystyle E_{n,Simpson} = I - I_{n,Simpson} = 6.486811088279865e-12 $$

iii) Romberg Table

Since the first column of Romberg table corresponds to Trap. rule, we can use the below formulas in order to obtain the first column of Romberg talbe.

 * {| style="width:100%" border="0" align="left"

T_{0}(n) = h\left[\frac{1}{2}f(x_{0}) + f(x_{1}) + f(x_{2}) + \cdot\cdot\cdot + f(x_{n-1}) + \frac{1}{2}f(x_{n}) \right] $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

 * {| style="width:100%" border="0" align="left"

T_{0}(2n) = T_{0}(n) + h\sum_{i=1}^{n}f(x_{2i-1}). $$ where $$\displaystyle h = \frac{b-a}{n} $$ and $$\displaystyle x_{0} = a,\quad x_{n} = b $$.
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Also, we gradually set up the Romberg table in parallel, as soon as the result for $$\displaystyle T_0(2^j) $$ becomes available using the below formula.

 * {| style="width:100%" border="0" align="left"

T_{k}(2n) = \frac{2^{2k}T_{k-1}(2n) - T_{k-1}(n)}{2^{2k}-1}. $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Matlab code

Result

$$\displaystyle I_{n,Romberg} = T_{3}(n) = 1.317902151489847, \quad n = 2^{3} = 8 $$ $$\displaystyle E_{n,Romberg} = I - I_{n,Romberg} = 3.544298188273842e-11 $$

iv) Compare computational time

We can compare computational times of above three methods using tic/toc commands in Matlab.

Matlab code

As above table shown, Romberg Table has the shortest computational time. Note that Simpson's rule also has the same order of magnitude of computational time with Romberg Table even though Romberg Table has slightly shorter computational time than Simpson's rule. Therefore,  both Romberg table and Simpson's rule are efficient ways to get a numerical integration value for the given function.

title: User:Egm6341.s10.team3.ynahn81/HW4-6(c)


text:
Russian/Lesson 1

Describing language skills
Read the dialogue of two well-acquainted people - Ivan (male) and Olga (female):

 Russian has a variety of ways to address a person. Personal names have at least 3 forms: full (Константин), short (Костя) and pet (Костик, Костенька). Other possible forms are vulgar (Костян) and familiar (Костька). Note that Russians almost never use full forms of personal names.

Иван: Привет, Ольга.
 * Ivan: Hi, Olga.

Ольга: Привет.
 * Olga: Hi.

Иван: Слушай, ты хорошо говоришь по-китайски?
 * Ivan: Listen, do you speak Chinese well?

Ольга: Да, а что?
 * Olga: Yes; why?

Иван: К нам партнёры из Китая завтра приезжают, а переводчик заболел. Не могла бы ты его заменить? Мы заплатим.
 * Ivan: We have an appointment with partners from China tomorrow, but our interpreter has fallen ill. Could you please replace him? We'll pay.

Ольга: Да, конечно.
 * Olga: Sure.

 Привет is the most popular informal greeting. Expression "Привет. Как дела?" (Hi, How are you?) is often used to begin conversation.

title: Russian/Lesson 1


text:
User:Egm6341.s10.team3.ynahn81/HW4-7(c)

(c) Compare Comp Trap, Comp Simp, Romberg for $$\displaystyle f(x) = \frac{1}{1+x^{2}}$$
Ref: Lecture notes [[media:Egm6341.s10.mtg25.djvu|p.25-2]]

Problem Statement
For $$\displaystyle I = \int_{-5}^{5}\frac{1}{1+x^2}dx$$, compare $$\displaystyle I_{n}$$ to $$\displaystyle O(10^{-10})$$ and computational time for - Composite Trap rule, - Composite Simpson's rule and - Romberg table.

Solution
In what follows, we are using the follow matlab function as a given function $$\displaystyle f(x) = \frac{1}{1+x^{2}}$$.

i) Composit Trap. rule

Recall the formula of comp. Trap. rule.

 * {| style="width:100%" border="0" align="left"

T_{0}(n) = h\left[\frac{1}{2}f(x_{0}) + f(x_{1}) + f(x_{2}) + \cdot\cdot\cdot + f(x_{n-1}) + \frac{1}{2}f(x_{n}) \right] $$ where $$\displaystyle h = \frac{b-a}{n} $$ and $$\displaystyle x_{0} = a,\quad x_{n} = b $$.
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Also, realize the fact that if we want to double the number of sub-intervals, we don't have to calculate the functions at even point, i.e. $$\displaystyle f(x_{0}), f(x_{2}), \cdot\cdot\cdot, f(x_{n-2}), f(x_{n}) $$ because we already calculated the functions at these even point at the previous step. Therefore, all we have to do is evaluating the functions only at the odd points and combine them with the calculated values at the previous step. As a result, we can use the following inductive formula.
 * {| style="width:100%" border="0" align="left"

T_{0}(2n) = T_{0}(n) + h\sum_{i=1}^{n}f(x_{2i-1}). $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Matlab code

Result

$$\displaystyle I_{n,trap} = T_{0}(n) = 2.746801533832637, \quad n = 2^{16} = 65536 $$ $$\displaystyle E_{n,trap} = I - I_{n,trap} = 5.739320130260239e-11 $$

ii) Composite Simpson's rule

Recall the formula of comp. Simpson's rule.

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T_{1}(n) = \frac{h}{3}\left[f(x_{0}) + 4f(x_{1}) + 2f(x_{2}) + 4f(x_{3}) + \cdot\cdot\cdot + 4f(x_{n-3}) + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_{n}) \right] $$ where $$\displaystyle h = \frac{b-a}{n} $$ and $$\displaystyle x_{0} = a,\quad x_{n} = b $$.
 * $$\displaystyle
 * $$\displaystyle
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Also, realize the fact that if we want to double the number of sub-intervals, we don't have to calculate the functions at even point, i.e. $$\displaystyle f(x_{0}), f(x_{2}), \cdot\cdot\cdot, f(x_{n-2}), f(x_{n}) $$ because we already calculated the functions at these even point at the previous step. Therefore, all we have to do is evaluating the functions only at the odd points and combine them with the calculated values at the previous step. As a result, we can use the following inductive formula.
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T_{1}(2n) = T_{1}(n) + h\sum_{i=1}^{n}4f(x_{2i-1}) - \underbrace{h\sum_{i=1}^{\frac{n}{4}}2f(x_{4i-2})}_{Already\,\, obtained\,\, when\,\, T_{1}(n) \,\, was\,\, calculated.}. $$
 * $$\displaystyle
 * $$\displaystyle
 * }
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Matlab code

Result

$$\displaystyle I_{n,Simpson} = T_{1}(n) = 2.746801533879843, \quad n = 2^{9} = 512 $$ $$\displaystyle E_{n,Simpson} = I - I_{n,Simpson} = 1.018696238475059e-11 $$

iii) Romberg Table

Since the first column of Romberg table corresponds to Trap. rule, we can use the below formulas in order to obtain the first column of Romberg talbe.

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T_{0}(n) = h\left[\frac{1}{2}f(x_{0}) + f(x_{1}) + f(x_{2}) + \cdot\cdot\cdot + f(x_{n-1}) + \frac{1}{2}f(x_{n}) \right] $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

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T_{0}(2n) = T_{0}(n) + h\sum_{i=1}^{n}f(x_{2i-1}). $$ where $$\displaystyle h = \frac{b-a}{n} $$ and $$\displaystyle x_{0} = a,\quad x_{n} = b $$.
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Also, we gradually set up the Romberg table in parallel, as soon as the result for $$\displaystyle T_0(2^j) $$ becomes available using the below formula.

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T_{k}(2n) = \frac{2^{2k}T_{k-1}(2n) - T_{k-1}(n)}{2^{2k}-1}. $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Matlab code

Result

$$\displaystyle I_{n,Romberg} = T_{2}(n) = 2.746801533889937, \quad n = 2^{8} = 256 $$ $$\displaystyle E_{n,Romberg} = I - I_{n,Romberg} = 9.281464485866309e-14 $$

iv) Compare computational time

We can compare computational times of above three methods using tic/toc commands in Matlab.

Matlab code

As above table shown, Simpson's rule has the shortest computational time. However, note that Romberg Table has almost same computational time with Simpson's rule even though Simpson's rule has slightly shorter computational time than Romberg Table. Therefore,  both Simpson's rule and Romberg Table are efficient ways to get a numerical integration value for the given function.

title: User:Egm6341.s10.team3.ynahn81/HW4-7(c)