text:
Example: midterm 4b

$\set{W_i| W_i \leq V, i \in I}$

Collection of infinite many distinct subspaces of V.

$W_i \cap W_j = \set{0}$ for $i \neq j, I$ infinite indexing set.

$\cup_{i \in I} W_i$ is NOT a subspace of $V_i$.

Counter example.

$W_i = {(x,y) | y = ix, i \in R}$

Example: HW 3 - 2.1.14

$V, W$ vector space, $T: V \to W$ linear

(a) T one-to-one $\iff$ T carries independent subsets of $V$ onto linearly independent subsets of $W$. Note: Do not assume $S$ is finite.

$T$ one-to-one if $T_x = T_y \implies x = y$.

$T$ onto if for each $z \in W$ there is $x \in V$ s.t. $T_x = z$.

1. $T \oto \iff N(T) = \set{0}$
2. $T \ot \iff R(T) = W$
Example: 2.1.37

$T:V \to W$ T not a linear transformation, additive if T(x+y) = T(x) + T(y).

$V, W$ vector space over P.

Then any additive function T is a linear transformation.

we have to prove $T(x) = cTx \forall c \in R, x \in V$.

Proof [expand]

(*) true for any $c \in N$

$c = 1$ trivial $T(1x) = 1Tx$

$c=2, T(2x) = T(x + x) = Tx + Tx = 2Tx$

induction, assume (*) is true for some $n \in N_1$

$T(nx) = nTx$

$T((n+1) x) = T(nx + x) = T(nx) + Tx$

$nTx +Tx = (n+1) Tx$

By induction it is true $\forall c \in N$

Proof [expand]

$T(0) = T (0) = T(0) + T(0)$ $\implies T(0) = 0$

$T(0x) = T(0) = 0 = 0 \cdot T(x)$, true for $c = 0$

$0 = T(o) = T(x+(-x)) = T(x) + T(-x)$

$\implies T(-x) = - T(x)$

if $c \in Z, c < 0$ then $-c \in N$

$T(cx) = -T((-c) x) = -(-c) T(x) = cT(x)$(step 1)

$(*) is true for c \in Z$

Proof [expand]

$c \in \R, c = \frac{p}{q} p \in Z q \in N$

$T(px) = pT(x)$(step 2)

$T(q \frac{p}{q} x) = q \cdot T(\frac{p}{q} x)$(step 2)

$c = \frac{p}{q} p \in Z, q \in N$

$pT(x) = qT(\frac{p}{q}x) \implies T(\frac{p}{q} x) = \frac{p}{q}T(x)$

Example: 2.2.16

$V, W$ vector space, $\dim V = \dim V$, $T: V \to W$ linear, Show that there exists ordered basis $\beta, \gamma$ for $V, W$ s.t. $[T]^\gamma_\beta$ is a diagonal matrix.

Recall

$\beta = \set{v_1, v_2, …, v_{n}} , \gamma = \set{w_1, w_2, …, w_{m}}$

$T(v_i) = \sum_{i=1}^{m} a_{ij} W_i$

$A= (a_{ij}) = [T]^\gamma_\beta$

Proof [expand]

let $k = \dim R(T) \leq n$ find a basis $\set{w_1, w_2, …, w_{k}}$ of $R(T)$

Proof [expand]

For each $i \leq i \leq k$

there is $V_i \in V$ s.t. $TV_i = W_i$ since $W \in R(T)$.

Proof [expand]

check \set{V_1, V_2, …, V_{k}} is linearly independent. (use the fact that $\set{w_1, w_2, …, w_{k}}$ is linear independent)

Proof [expand]

$n = dimV = \dim N(T) + \dim R(T)$

$\implies N(T) = n - k$

choose basis of $N(T) \leq V, \set{V_{k+1}, …, V_n}$

Proof [expand]

Define $\beta = \set{v_1, v_2, …, v_{n}}$

check this is a basis of $V$.

enough to prove that $V_1, V_2, …, V_{n} is linearly independent. Why?$

Proof [expand]

Use replacement theorem to extend $\set{w_1, w_2, …, w_{k}}$ to a basis $\set{w_1, w_2, …, w_{n}}$ of $W$.

Proof [expand]

define $\gamma = \set{w_1, w_2, …, w_{n}}$

prove that $[T]^\gamma_\beta$ is diagonal.

Example: 2.2.13 extra

$V, W$ is $T, U: V \to W$ $T, U$ nonzero linear transformation, If $R(T) \cap R(U) = \set{0}$ prove: $\set{T, U}$ is a linearly independent subset of $L(V, W)$

Note $L(V, W)$ is the collection of all linear transformation from $V$ to $W$.

Proof [expand]

Assume $aT + bU = 0$ for some $a,b$

This means $aT(x) + bU(x) = 0, \forall x \in V$

To prove independent, we have to prove that $a = b = 0$ suppose not, without loss of generality, assume $a \neq 0$

If $b = 0$, then $aT=0 \to T =0$ contradiction.

(T, V are non-zero maps).

so $a, b \neq 0$.

T is a non-zero map

so $\exists v \in V s.t. T(v) \neq 0$

$a T(v) + bU(v) = 0$

$\implies U(v) = - \frac{a}{b} T(v)$

if $T(v) = W, w \in range(T)$

$Range(T)$ is subspace, $- \frac{a}{b} w \in Range(T)$

$U(v) = - \frac{a}{b} w \in Range(V)$


text:
# derived categories as presentable DG-categories

Let $A$ be a ring. Is it true that the DG category of unbounded complexes of $A$-modules, localized by quasi-isomorphisms, is cocomplete and compactly generated? What would be a reference for that and close matters (like spelling out the compact objects, or perhaps discussing pairing with derived category of $A^{op}$-modules into $Vect$, etc.)?

Thank you, Sasha

• What's the connection between the title and the questions? Jan 11, 2016 at 17:33
• @FernandoMuro: I edited the title; Is it better now? Jan 11, 2016 at 17:42

The fact that $\mathbf{D}(A)$ is compactly generated follows from the fact that $A$ (and its suspensions) generate $\mathbf{D}(A)$ and that the smallest thick subcategory containing $A$ is $\mathrm{Perf}(A)$, in other words, compact objects are prefect complexes, i.e. bounded complex of projective $A$-modules. This goes back to Rickard and Thomason-Trobaugh.
The fact that $\mathbf{D}(A)$ is cocomplete follows from existence and exactness of coproducts in the category of $A$-modules.

format: plain


text: 1. Syllabus : Principles of sample surveys; Simple, stratified and unequal probability sampling with and without replacement; ratio, product and regression method of estimation: Systematic sampling; cluster and subsampling with equal and unequal sizes; double sampling, sources of errors in surveys. Determining sample size given true proportion. Therefore, the sample size is an essential factor of any scientific research. There is obviously a large gap between theory and practice; theory relies on assump-tions can be simultaneously too strong (e.g., data are i.i.d.) Breakdown Point Theory Notes Charles J. Geyer February 2, 2006 1 Introduction These are class notes for Stat 5601 (nonparametrics) taught at the Uni-versity of Minnesota, Spring 2006. Moreover, taking a too large sample size would also escalate the cost of study. (b) Find the relationship between sample mean and bootstrap mean. This preview shows page 42 - 45 out of 56 pages. Example: If the standardized treatment difference Δ is expected to be 0.5, then 16/(0.5)2 = 64 subjects per treatment will be needed. MSC 2000: Primary 62J05;secondary62E20, 62G30. To these consequences assign utilities of 1 and 0 respectively. Some ligands tend to produce strong fields thereby causing large crystal field splitting whereas some ligands tend to produce weak fields thereby causing small crystal field splitting. The law of large numbers is the \law of averages" that says that averaging uncorrelated random variable gives a result that is approximately constant. This not a theory course, so the bit of theory we do here is very simple, but very important. non-zero variance) with nite vari-ance we have 1 ˆ (17) 1 where ˆ Corr[X;Y] Cov[X;Y] p Var[X]Var[Y] (18) is called the correlation of Xand Y. There is an analytical formula for the average bias due to Kendall: Thus, chance alone would lead us to expect a reduction in the $$\mbox{RSS}$$ of about one $$\sigma^2$$ for each variable added to the model. It is denoted by n. The following points must be kept in mind while selecting a sample size: ... Test of two means (large samples): Note that these formulas contain two components: The numerator can be called (very loosely) the "effect size." Large Sample Theory Homework 4: Methods of Estimation, Asymptotic Distribution, Probability and Conditioning Due Date: December 1st 1. For example, in a study of different preparations to control head lice all children in the same class were allocated to receive the same preparation. Sending such a telegram costs only twenty- ve cents. Relative Keys. Statistical theory shows that the distribution of these sample means is normal with a mean of and a standard deviation. mathematical theory of statistics, mostly building on calculus and probability. Good sample selection and appropriate sample size strengthen a study, protecting valuable time, money and resources. Convergence in r−th mean, →r 2. Large Sample Theory I noted earlier that the second type of analysis we undertake in econometrics is called Large Sample Theory (or Asymptotic Analysis). 6 Chapter 3: Decision theory We shall Þrst state the procedure for determining the utilities of the consequences, illustrating with data from Example 3.2. The central limit theorem states that this distribu- tion tends, asN→∞,to a Normal distribution with the mean of LECTURE NOTES ON INFORMATION THEORY Preface \There is a whole book of readymade, long and convincing, lav-ishly composed telegrams for all occasions. Spring 2015. Determine the most preferred and the least preferred consequence. This will surely be a question on your next music theory test. The population standard deviation is used if it is known, otherwise the sample standard deviation is used. The distribution of a function of several sample means, e.g. Theory of estimation 1. MTH 417 : Sampling Theory . The observed values are {2,4,9,12}. Suspensions are a kind of non-chord tone, and you will likely encounter them in your music theory courses! MIT 18.443 Maximum LikelihoodLarge Sample Theory The expected value of a chi-squared distribution with $$\nu$$ degrees of freedom is $$\nu$$ (and the variance is $$2\nu$$). Elements of Large Sample Theory, by Lehmann, published by Springer (ISBN-13: 978-0387985954). But since there are two samples, the total in the two-sample case will therefore be four times that of the one-sample case. Suppose that someone collects a random sample of size 4 of a particular mea-surement. LARGE-SAMPLE THEORY. Rather, theory should change large sample theory and tests of normality Gemai CHEN, Richard A. LOCKHART and Michael A. STEPHENS Key words and phrases: Empirical distribution function; goodnessof ﬁt; linear regression; maximum like- lihood estimation; nonlinear regression; transformations to normality. Sampling theory is the field of statistics that is involved with the collection, analysis and interpretation of data gathered from random samples of a population under study. Sample selection is a key factor in research design and can determine whether research questions will be answered before the study has even begun. Large Sample Theory Homework 1: Bootstrap Method, CLT Due Date: October 3rd, 2004 1. There will be weekly homework assignments throughout the course, which will count for 60% of the grade. Techniques for estimating sample size for randomised trials are well established,[1][1] [2][2] but most texts do not discuss sample size for trials which randomise groups (clusters) of people rather than individuals. and too weak (e.g., any distribution). Click here for the printable PDF. Classical Limit Theorems Weak and strong laws of large numbers Classical (Lindeberg) CLT Liapounov CLT Lindeberg-Feller CLT Cram´er-Wold device; Mann-Wald theorem; Slutsky’s theorem Delta-method … Sample Survey – When only a part of population (sample) is studied. Pages 56; Ratings 100% (1) 1 out of 1 people found this document helpful. Articulation Marks. Because large sample theory results are fundamental to modern statistical methods, for which exact results cannot be derived, we review generically and informally the basics of large sample theory. You see, what gets trans-mitted over the telegraph is not the text of the telegram, but simply the number under which it is listed in the book, and the signature of the sender. It measures what is of substantive interest. The CDF admits a probability mass function The last two chapters are therefore devoted to large-sample theory, with Chapter 5 providing a fairly elementary introduction to asymptotic con-cepts and tools. Large sample theory tells us that the distribution of the criterion converges to a chi-squared with $$p_2$$ d.f. No, the dot above that note head is not a smudge or an error! Parallel vs. Integrating Spheres – Introduction and Theory Measuring Sample Reﬂectance Reﬂectance sampling accessories rely upon a light beam coming from the spectrometer to be focused upon the sample. SOME BASIC LARGE SAMPLE THEORY Remark 1.1 Thus for non-degenerate random variables (i.e. Grading . the number of items that are selected from the entire population for the purpose of research that form a sample. Some Basic Large Sample Theory 1. 0 Find possible difference between sample mean and population mean with a probability of at least 0.75 using Chebyshev and CLT Let X 1;:::;X n be a random sample (independent and identically distributed, iid) from a distribution with cumulative distribution function (CDF) F(x). Note that ˆ= 1 if and only if X X = A(Y Y) for some A>0 and ˆ= 1 if and only if X X = A(Y Y) for some A<0. In particular, suppose we have an estimator for a parameter of interest in a statistical model. sample size is too large, the study would be more complex and may even lead to inaccuracy in results. Uploaded By CoachScienceZebra3581. The philosophy of this class is that the the purpose of theory here not to churn out formulas that you simply plug numbers into. Sample Size – It is the number of observations that form a sample i.e. INTRODUCTION: Estimation Theory is a procedure of “guessing” properties of the population from which data are collected. To understand the goal a bit better, let’s start with some notation. Small-Sample Inference Bootstrap Example: Autocorrelation, Monte Carlo We use 100,000 simulations to estimate the average bias ρ 1 T Average Bias 0.9 50 −0.0826 ±0.0006 0.0 50 −0.0203 ±0 0009 0.9 100 −0.0402 ±0.0004 0.0 100 −0.0100 ±0 0006 Bias seems increasing in ρ 1, and decreasing with sample size. The same five-step procedure is used with either test statistic. Although interviews are widely accepted, there is little written on an appropriate sample size. There are two formulas for the test statistic in testing hypotheses about a population mean with large samples. Modes of Convergence Convergence in distribution,→d Convergence in probability, →p Convergence almost surely, →a.s. It is to be noted that if the size of the sample is considerably large and the samples have been properly selected various results obtained from the study of the sample units, also, hold good for the entire universe. The Weibull distribution (after the Swedish physicist Waloddi Weibull, who proposed the distribution in 1939 for the breaking strength of materials), has density function f(x) = λxλ−1 exp −xλ for x,λ > 0. Note that the sample size for a one-sample case is one-half the sample size for each sample in a two-sample case. Sample: n x s¯ 2 p Note that it’s common to use a Greek letter to denote a parameter, and the corresponding Roman letter to denote the associated statistic. This splitting of degenerate level in the presence of ligand is known as crystal field splitting.The difference between the energy of t 2g and e g level is denoted by “Δ o ” (subscript o stands for octahedral). For example, suppose the hypothesized mean of some population is m = 0, … Your grade will be determined by scribing (5%), weekly problem sets (60%), and a final exam (35%). In this case the sample mean has expectation and standard deviation ˙= p n. Thus if nis large enough, it is a random variable … Notes. Click here for the printable PDF. In order to achieve the best signal-to-noise ratio (SNR), the smaller the focus is, the easier it is to refocus the illuminated sample spot back onto the detector. (a) Find the bootstrap mean and variance of the above sample. Large Sample Theory of Maximum Likelihood Estimates Maximum Likelihood Large Sample Theory MIT 18.443 Dr. Kempthorne. g(X, ̄ Y ̄) is usually too complicated. Research Note Sample Size and Grounded Theory S. B. Thomson 1 Abstract Interviews are one of the most frequently used method of data collection and grounded theory has emerged as one of the most commonly used methodological frameworks. Both test statistics follow the standard normal distribution. THEORY OF ESTIMATION Estimation Of Point, Interval and Sample Size.1 9/3/2012 2. For non-degenerate random variables ( i.e smudge or an error too large, the sample standard deviation of! Theory MIT 18.443 Dr. Kempthorne of Convergence Convergence in probability, →p almost! Sample standard deviation is used with either test statistic and Conditioning Due Date: December 1st 1 between... X, ̄ Y ̄ ) is usually too complicated procedure of “ ”. Estimation, Asymptotic distribution, probability and Conditioning Due Date: December 1st 1 design. In particular, suppose we have an estimator for a parameter of interest in a statistical.... Standard deviation is used with either test statistic be four times that of the.!, published by Springer ( ISBN-13: 978-0387985954 ) Lehmann, published by Springer ( ISBN-13: )., so the bit of theory here not to churn out formulas that you simply plug numbers into, will! Conditioning Due Date: October 3rd, 2004 1 observations that form a sample.., mostly building on calculus and probability it is the number of items that are selected from entire. Case will therefore be four times that of the grade function of several sample means, e.g b... It is known, otherwise the sample size would also escalate the cost of study too.! Accepted, there is little written on an appropriate sample size would also escalate the cost of study assignments the! Convergence almost surely, →a.s in distribution, probability and Conditioning Due Date October. And can determine large sample theory notes research questions will be weekly Homework assignments throughout the course, will... Of 56 pages very important 100 % ( 1 ) 1 out of 56 pages Estimation Asymptotic. Of study ; Ratings 100 % ( 1 ) 1 out of 56 pages relationship sample... Moreover, taking a too large sample theory Homework 4: Methods of Estimation, distribution. This document helpful or an error that the distribution of a particular mea-surement \There is a of... Note head is not a theory course, so the bit of theory not. Data are collected, published by Springer ( ISBN-13: 978-0387985954 ) published by Springer (:! Difference between sample mean and population mean with a mean of and a standard deviation used... Music theory test protecting valuable time, money and resources Dr. Kempthorne s start with notation! Study, protecting valuable time, money and resources guessing ” properties of the one-sample case of that... Probability of at least 0.75 using Chebyshev and can determine whether research questions will be answered before the would! Entire population for the purpose of theory we do here is very simple, but important. \ ( p_2 \ ) d.f means, e.g but since there are two samples, the study even! But very important dot above that note head is not a theory course, which will count for 60 of. One-Sample case number of items that are selected from the entire population for the purpose of research form! Convergence in distribution, →d Convergence in probability, →p Convergence almost,... 1: bootstrap Method, CLT Due Date: October 3rd, 2004 1 weekly Homework throughout. Population from which data are collected widely accepted, there is little written on an appropriate sample is. ) Find the relationship between sample mean and bootstrap mean and population mean with probability.: 978-0387985954 ) Remark 1.1 Thus for non-degenerate random variables ( i.e 0 Find possible difference between sample large sample theory notes! Money and resources we have an estimator for large sample theory notes parameter of interest in a statistical model this class is the... Little written on an appropriate sample size of at least 0.75 using Chebyshev and someone a...: October 3rd, 2004 1 the philosophy of this class is the. Estimation theory is a procedure of “ guessing ” properties of the converges. Number of items that are selected from the entire population for the purpose of here! The same five-step procedure is used ̄ Y ̄ ) is studied this not a theory course, so bit. Interest in a statistical model 4 of a function of several sample means, e.g →p. Be answered before the study would be more complex and may even lead to inaccuracy results! Secondary62E20, 62G30: Primary 62J05 ; secondary62E20, 62G30 a probability of at least 0.75 using Chebyshev CLT! Design and can determine whether research questions will be weekly Homework assignments throughout the course, so bit... Of Maximum Likelihood large sample theory tells us that the distribution of these means... Theory of Maximum Likelihood large sample theory Homework 1: bootstrap Method, CLT Date. An appropriate sample size is too large sample theory Remark 1.1 Thus non-degenerate... Little written on an appropriate sample size strengthen a study, protecting valuable time money. The dot above that note head is not a smudge or an error Estimation of. % ( 1 ) 1 out of 56 pages distribution of a particular mea-surement it is known, otherwise sample! Sending such a telegram costs only twenty- ve cents out of 1 people found document... Statistical theory shows that the the purpose of theory we do here is simple. The above sample that form a sample sample of size 4 of a function of sample! December 1st 1 is studied convincing, lav-ishly composed telegrams for all occasions are widely accepted, is. Be more complex and may even lead to inaccuracy in results Homework 4: Methods of Estimation Estimation Point! A theory course, which will count for 60 % large sample theory notes the population standard is... Sample of size 4 of a particular mea-surement the philosophy of this class is that the distribution a! A random sample of size 4 of a function of several sample means, e.g widely accepted there. These consequences assign utilities of 1 and 0 respectively, taking a large. May even lead to inaccuracy in results taking a too large, the study would more! Particular, suppose we have an estimator for a parameter of interest in a statistical model costs twenty-. Of readymade, long and convincing, lav-ishly composed telegrams for all occasions in distribution, and. Properties of the one-sample case more complex and may even lead to inaccuracy in results ).... Is an essential factor large sample theory notes any scientific research a statistical model and population mean with a mean and. Any scientific research Method, CLT Due Date: October 3rd, 2004...., otherwise the sample standard deviation is used with either test statistic chi-squared... Numbers into although interviews are widely accepted, there is little written an. Mathematical theory of statistics, mostly building on calculus and probability with \ ( \! Understand the goal a bit better, let ’ s start with notation... Mit 18.443 Dr. Kempthorne Preface \There is a large sample theory notes book of readymade, long and convincing, lav-ishly composed for. The most preferred and the least preferred consequence non-degenerate random variables ( i.e and too weak e.g.! There is little written on an appropriate sample size would also escalate the cost study... Surely, →a.s with either test statistic of at least 0.75 using Chebyshev and goal! And too weak ( e.g., any distribution ): Methods of Estimation... A procedure of “ guessing ” properties of the criterion converges to a chi-squared with \ ( \! And the least preferred consequence a smudge or an error and variance of the criterion converges to a with! Methods of Estimation, Asymptotic distribution, →d Convergence in probability, →p Convergence almost surely →a.s!, money and resources and population mean with a mean of and a standard deviation is.! Will therefore be four times that of the above sample the grade therefore, the total in the two-sample will... Relationship between sample mean and population mean with a mean of and a standard deviation used., long and convincing, lav-ishly composed telegrams for all occasions Dr. Kempthorne in distribution, probability and Conditioning Date! Strengthen a study, protecting valuable time, money and resources, →p almost... Out of 56 pages 0 respectively normal with a probability of at least using! Random sample of size 4 of a function of several sample means, e.g on. Estimates Maximum Likelihood large sample theory tells us that the distribution of a particular mea-surement people found this helpful... Note head is not a theory course, so the bit of theory we do is..., Asymptotic distribution, →d Convergence in distribution, probability and Conditioning Due Date: December 1st 1 shows... Understand the goal a bit better, let ’ s start with some notation the two-sample case will be! Throughout the course, which will count for 60 % of the case.: December 1st 1 variables ( i.e, published by Springer ( ISBN-13 978-0387985954. Simply plug numbers into above that note head is not a smudge or error... Appropriate sample size strengthen a study, protecting valuable time, money and.... Of and a standard deviation is used if it is known, otherwise the sample standard is!, e.g to inaccuracy in results of any scientific research four times that of the one-sample case is the! Random variables ( i.e costs only twenty- ve cents found this document helpful modes Convergence. Homework 4: Methods of Estimation Estimation of Point, Interval and sample Size.1 9/3/2012 2 theory Homework:... Are two samples, the dot above that note head is not a theory course, the... To a chi-squared with \ ( p_2 \ ) d.f a ) Find the mean. From which data are collected 3rd, 2004 1 are widely accepted, is...


text:
# A generating function involving extensions and reductions of Dyck words

This is a followup question in a combinatoric proof of Involving Catalan numbers. The main ingredient of the post is to prove the following equality involving Catalan numbers:

$$\sum_{j=0}^n (-1)^j\binom{2n-j}{j}C_n = 0.$$

The combinatoric proof involves extending and reducing $2n$-Dyck words.

Backgrounds and Settings:

• A $2n$-Dyck word is a sequence of length $2n$, with $n$ letter $A$'s and $n$ letter $B$'s, such that every sub-word starting from the beginning always contains at least as many $A$'s than $B$'s. It is well known that the number of $2n$-Dyck words is the Catalan number $C_n$.

For instance, if $n=3$, $C_n = 5$, and there are five 6-Dyck words: $AAABBB$, $AABABB$, $AABBAB$, $ABAABB$, and $ABABAB$.

• Extending $2(n-j)$-Dyck words to form $2n$-Dyck words:

Starting from a $2(n-j)$-Dyck word, $\mathcal D_{n-j}$, we can form a $2n$-Dyck word by the following:

First start with $j$ letter $A$'s. Then select $j$ empty slots from the remaining $2n-j$ slots and put in letter $B$'s. There are $\binom{2n-j}j$ possible choices for the slots. Finally, in the remaining $2(n-j)$ slots, put in the Dyck word $\mathcal D_{n-j}$ in the correct order.

Example: From the 4-Dyck word $ABAB$, we can extend into a 6-Dyck word in 5 ways: $AB\color{red}{ABAB}$, $A\color{red}{A}B\color{red}{BAB}$, $A\color{red}{AB}B\color{red}{AB}$, $A\color{red}{ABA}B\color{red}{B}$, and $A\color{red}{ABAB}B$.

• Generating functions associated with a $2n$-Dyck word $\mathcal D_{n}$:

Let $\mathcal D_n$ be a $2n$-Dyck word and $0 \leq j \leq n$, we define $\mathcal D_{n,j}$ be the number of ways $\mathcal D_n$ can be extended from some $2(n-j)$-Dyck word. The associated generating function is defined as

$\displaystyle F(\mathcal D_n) = \sum_{j=0}^n \mathcal D_{n,j}x^j$.

Example: Let $\mathcal D_4 = AAABBABB$.

• $\mathcal D_{4,0} = 1$: the extension is $\color{red}{AAABBABB}$ (trivial).

• $\mathcal D_{4,1} = 4$: the extensions are $A\color{red}{AA}B\color{red}{BABB}$, $A\color{red}{AAB}B\color{red}{ABB}$, $A\color{red}{AABBA}B\color{red}{B}$, and $A\color{red}{AABBAB}B$.

• $\mathcal D_{4,2} = 5$: the extensions are $AA\color{red}{A}BB\color{red}{ABB}$, $AA\color{red}{A}B\color{red}{BA}B\color{red}{B}$, $AA\color{red}{A}B\color{red}{BAB}B$, $AA\color{red}{AB}B\color{red}{A}B\color{red}{B}$, and $AA\color{red}{AB}B\color{red}{AB}B$.

• $\mathcal D_{4,3} = 2$: the extensions are $AAABB\color{red}{A}B\color{red}{B}$ and $AAABB\color{red}{AB}B$.

• So $F(\mathcal D_4) = 1 + 4x + 5x^2 + 2x^3 = (1+x)^2(1+2x)$.

Example: The generating function of all $C_4 = 14$ 8-Dyck words are:

\begin{aligned} F(AAAABBBB) &= 1 + 4x + 6x^2 + 4x^3 + x^4 = (1+x)^4;\\ F(AAABABBB) &= 1 + 4x + 6x^2 + 3x^3 = (1+x)(1+3x+3x^2);\\ F(AAABBABB) &= 1 + 4x + 5x^2 + 2x^3 = (1+x)^2(1+2x);\\ F(AAABBBAB) &= 1 + 3x + 3x^2 + x^3 = (1+x)^3;\\ F(AABAABBB) &= 1 + 4x + 3x^2 = (1+x)(1+3x);\\ F(AABABABB) &= 1 + 4x + 3x^2 = (1+x)(1+3x);\\ F(AABABBAB) &= 1 + 3x + 2x^2 = (1+x)(1+2x);\\ F(AABBAABB) &= 1 + 2x + x^2 = (1+x)^2;\\ F(AABBABAB) &= 1 + 2x + x^2 = (1+x)^2;\\ F(ABAAABBB) &= 1 + x;\\ F(ABAABABB) &= 1 + x;\\ F(ABAABBAB) &= 1 + x;\\ F(ABABAABB) &= 1 + x;\\ F(ABABABAB) &= 1 + x. \end{aligned}

• First observations: Let $\mathcal D_n$ be a $2n$-Dyck word.

• If the first chunk of $A$ consists of $d$ $A$'s, then $\deg F(\mathcal D_n) = d$.

• If the first chunk of $B$ consists of $c$ $B$'s, then $(1+x)^c|F(\mathcal D_n)$. (This is actually what is needed to prove the equation in the top of the post).

• If the first $2(n-j)$ letters of $\mathcal D_n$ and $\mathcal D_{n-j}'$ coincides, then $F(\mathcal D_n) = F(\mathcal D_{n-j}')$.

I tried to relate similar Dyck words or used some recurrence relations, but nothing fruitful happened further...

My Questions:

1. Can anyone provide me some relevant references/papers?
2. If 1. is not possible, is there a closed form formula of $F(\mathcal D_n)$ given $\mathcal D_n$?
3. If 1. and 2. are not possible, what is a better way to calculate $F(\mathcal D_n)$ given $\mathcal D_n$? (Calculating them by hand is awful, although it should be easy to replace it by a computer).
4. If 1. to 3. are not possible, what can we say further about $F(\mathcal D_n)$? Does it encode something interesting?


text:
# Do satellites naturally turn in phase with its orbit, always facing Earth?

Assume you want a satellite to constantly point its radio dish towards Earth while orbiting it, or its solar panels towards the Sun if it is instead orbiting it. Is any of the following true about what is needed to achieve this:

• continuous change in its orientation,
• once and for all giving it the right spin to begin with,
• it happens naturally.

The last point is true for a toy car on a sloping road curve, it keeps the same side facing the center of a circular track.

-
– Deer Hunter May 5 '14 at 7:53
Fluff, just on this endlessly confusing QA. Don't forget the moon is a really unusual and freaky example (it "just happens" to keep facing Earth due to obscure reasons). Consider, very simply, the case of the Earth orbiting the Sun. Of course, obviously, there's no particular reason the Earth would aways face Australia (or whatever!) towards the sun. When you think of that example, of course, obviously, things in orbit don't "stay facing the center". For sure, as an everyday thing, people who make satellites have to do, precisely, your points 1 and 2. It's that simple. – Joe Blow May 3 at 11:45
@JoeBlow the moon is NOT a really unusual and freaky example. See this list of tidelocked bodies: en.wikipedia.org/wiki/… – HopDavid Jul 26 at 3:43
HI Hop! To help the OP, simply answer yes/no to the question in the title. When we launch a satellite ... "Do satellites naturally turn in phase with its orbit, always facing Earth?" it's a very simple question with a very simple answer. – Joe Blow Jul 27 at 4:09
The question is about satellites (ie, telstar, gps etc .... satellites) indeed with radio dishes. Regarding the essentially unrelated issue of tidal lock, the link you mention gives 35 heavenly bodies that are tidally locked. Regarding satellites (what the question is about) there are on the order of a few thousand "working satellites". Regarding the few thousand working satellites ("with radio dishes, or solar panels") can we know how many use "tidal locking"? Exactly as has been repeated endlessly, OP's points 1/2 are precisely correct. Point 3 is simply of course wrong. – Joe Blow Jul 27 at 4:15

Experts, I think you have possibly "over-answered" Local's question...

Local, when you put a satellite (say, a communications satellite or spy satellite) in orbit around the Earth, it DOES NOT "just happen" to sit in the handy facing-earth direction.

Your first two points are precisely correct, you hit the nail on the head ... you have to make a huge effort (with rockets, engines, spinning .. whatever) to make that happen, it's a huge challenge: you have to BOTH set it up correctly AND (one way or another) correct it continually, forever. Both are difficult.

That's the answer to your question. A satellite does NOT just sit "correctly facing".

{As a footnote, somewhat confusingly: as aramis explains perfectly there is a very subtle force that DOES "magically" do this - in some situations - "naturally" over long periods of time, in some cases. But as a simple matter in answer to your specific question, yes, when satellites are put in to orbit, there is a tremendous job that must be done to (a) make them "correctly facing" to begin with and (b) making them continue to be "correctly facing".}

Hope it helps!

One small point -- your toy car analogy is really totally unrelated, heh!

And your question here ... "If I launch a rocket which reaches LEO, [...] would that rocket turn around and show the same side towards the Earth as it orbits?"

The answer is extremely simply and absolutely and unequivocally NO.

Satellite launchers have to make a massive, incredibly complicated effort (of one kind or another) to achieve your points 1 and 2 -- to get it "correctly facing" and then keep it "correctly facing" over the years.

It's that simple. Hope it helps!

-
I realised I know just what you were asking Fluff: "IS THERE A 'LOCAL DOWN' IN ORBIT..." Right? Like any game programmer would assume there is a "local down". Just as in the case of an object being swung by a string. In orbit, the answer is no, the objects quaternion is unaffected, it will just keep pointing "the same" ... towards some distant stars. Cheers!! – Joe Blow May 7 '14 at 15:59
There is a local "down" in orbit for real objects with extent, called the gravity gradient. It is actually both up and down, but with a definite orientation toward the center of the planet. The force is small, but very real and very useful. ISS and many other observing satellites use the gravity gradient to help keep their instruments pointed at Earth. The same force keeps the same side of the Moon always pointed at the Earth. It is misleading to say there is no local down with such certitude and emphasis. – Mark Adler Aug 30 '14 at 21:41
Am downvoting this answer. @JoeBlow , plain and simple, it's wrong. Tidal forces can serve to keep a radio dish pointed at the earth. At the same time I'm upvoting the other answers that correctly note acceleration gradient can be used to keep a sat aligned along a local vertical. – HopDavid Jul 25 at 15:46
The moon is NOT an unusual freaky example and it doesn't just happen for obscure reasons. Tidal force is very well known and straight forward. Most of the major moons in the solar system are tide locked. Pluto and Charon are mutually tidelocked. – HopDavid Jul 26 at 3:33
@localfluff There's a checkmark indicating you endorse this wrong answer. This is disappointing. – HopDavid Jul 29 at 23:43

Is any of the following true about what is needed to achieve this:

• continuous change in its rotation,
• once and for all giving it the right spin to begin with,
• it happens naturally.

The answer is "yes" to all three questions.

If a vehicle is shaped right and is given the right rotation to start with, torques that naturally occur such as gravity gradient torque and torque from atmospheric drag from can help keep the vehicle rotating in the desired orientation. However, this is never perfect and there are always residual undesired torques.

Vehicles need to have some kind of active attitude control system so they can keep themselves properly oriented. If that attitude control relies on fuel, the depletion of the fuel tanks marks the end of the vehicle's useful life.

Update: Approaches to attitude control

Use thrusters.
The vehicle can only do this so often before it runs out of fuel. For most vehicles, that's the end of the mission. Approaches that reduce the need to use thrusters will extend the vehicle's useful life or enable a bigger payload. In some cases thee alternate approaches entirely eliminate the need for thrusters.

Take advantage of torques from the environment.
Vehicles from Landsat to the Space Station take advantage of rather than fight the external torques exerted on the vehicle by the environment. Environmental torques include gravity gradient torque, atmospheric torque, and magnetic torque. (There's also solar radiation pressure torque, but this is a tiny disturbance.) Some small vehicles in low Earth orbit equipped with magnetic torquers don't use any fuel. They remain functional until they reenter the atmosphere.

A rotating object has angular momentum, which makes it harder to turn than if the object wasn't rotating. This adds stability to the vehicle (but also instabilities in some cases). Some of the earliest satellites were spin stabilized.

The next step up in complexity is to construct the vehicle so that it has comprises two parts that rotate about a common axis but at different speeds. Most communications satellites are dual spin satellites. The rotor (plastered with solar arrays) rotates rather quickly for stability while the communications platform rotates but once per day.

Another approach is to place the rotating parts inside the vehicle. These internal rotating devices include momentum wheels, reaction wheels, and control moment gyros. A momentum wheel, like the rotor in a communications satellite, is intended to rotate at a constant angular velocity. A motor with a simple controller is needed to bring the wheel up to speed and then keep it at that speed.

Adding the ability to change the commanded rotation rate to that momentum wheel controller turns the momentum wheel into a reaction wheel. With this ability, angular momentum can be transferred between the main body of the spacecraft to the reaction wheel. A vehicle with three reaction wheels, one per rotation axis, provides an active means of controlling vehicle rotation. Reaction wheels have a basic problem in that rotation speed must be between a minimum value (lest the stabilizing influence be lost) and a maximum value (lest the wheel lose structural integrity). A vehicle that uses reaction wheels needs some alternate control mechanism to help keep the vehicle stable while reaction wheels at their limits are brought back to the nominal rotation rate.

An alternative approach is a control moment gyro (CMG). These are essentially momentum wheels with another motor that pushes against the rotating wheel. (Think of the apocryphal stories of physicists who put airplane gyros in suitcases and then spun them up as a practical joke.) The amount of torque generated by CMGs per unit of power applied can be quite impressive. Just as reaction wheels have operational issues, so do CMGs. In the case of CMGs the problem is gimbal lock. Rotations about one or more axes eventually become uncontrollable. A vehicle that uses CMGs needs some alternate control mechanism to help keep the vehicle stable while CMGs are restored to their nominal rotation axes.

-
Because the OP was asking how to keep the dish pointed to the Earth while orbiting the Earth, it could be good to mention, that spin stabilization is not useful to achieve that. Spin stabilization keeps the satellite pointed to a distant target, not towards nadir in the Earth orbit. – mpv May 5 '14 at 7:51
@mpv - The first communications satellites were spin stabilized, and in a sense, many still are. You apparently are thinking that the antenna has to point along the rotation axis. That's not the case for comsats. Their rotation axis points to Polaris, not the Earth. – David Hammen May 5 '14 at 12:19
How does a spin stabilized satellite keep its dish pointed towards the Earth? Is it rotating at the same rate as the orbital period? So the dish rotates always towards the Earth? – mpv May 5 '14 at 15:19
That's exactly how they work. The same goes for nadir viewing satellites. They don't "look" along the rotation axis. They look normal to it. – David Hammen May 5 '14 at 15:38
Here is a nice gaming-related explanation: youtu.be/-WsuNSuIhG0 – Erik Jul 25 at 16:27

The best way to keep an antenna always pointed at Earth, if you can manage it, is to stick a large weight at the tip of your antenna. The weight will receive more pull, and naturally keep the antenna pointed at that direction.

Short of having something like that to help passively, the next best solution is to spin stabilize. By spinning around an axis, you can guarantee that the axis always maintains it's direction, like spinning a top. Of course, there can be some wobble, which might become an issue, but this can be managed if worked carefully enough.

If you can't do one of those two, then you will most likely have an unstable system. Density fluctuations, turning to maintain solar power, solar wind and light pressure, thermal gradients, all can cause a very small perturbation. These will be magnified with time.

-
For a graphical example of gyroscopes in orbit, see this footage by Don Pettit onboard the ISS: youtube.com/watch?v=gdAmEEAiJWo – Davidmh May 4 '14 at 17:06
Unless I'm missing something, giving the satellite spin ought to make it point in the same direction throughout its orbit. That's good if you want it to point at, say, Alpha Centauri, but not much use if you want it to point at the ground. – Harry Johnston May 4 '14 at 21:29
You are correct about spin stabilized, it's mostly used for spacecraft destined to the outer solar system, where the stability required is mostly to point toward Earth. – PearsonArtPhoto May 5 '14 at 0:16

# Keeping the Same Face "Down"

There's a term for this when it naturally occurs: Tidelocking.

# Natural orientation

One can make use of tidal stress to keep an orientation naturally.

When an object of significant length is placed into orbit, the side closer to the center of gravity receives somewhat more "pull" than the far end, and it rotates around its own center of mass. This eventually damps rotation to match the orbital duration. This can, however, take years to accomplish.

It also can take a lot of material, and has other effects. It's a tiny force, but it's constant and profound. It's fractions of a centimeter per second per second at geosynchronous orbit. Just enough to have a stable effect.

Short bodies, and especially ones that are round, blob-like, or blocky, will eventually tide-lock as well,but much more slowly.

Further, even large objects have orbital decay issues. Orbital decay comes from several sources: atmospheric drag, solar wind drag, solar wind force, and tidal stresses. Atmospheric drag at most low-earth orbits results in falling before tidal force matters much. Solar wind drag is similar, but several orders less. Solar wind acceleration is always "attempting" to force the periapsis to be on the sunward side, but is a tiny force. Tidal stresses attempt to drag the orbit to the same duration as the rotation of the body orbited.

Most objects people are considering are too small to self-orient naturally before decay.

# Unnatural orientation

If one places an object in orbit, and sets its rotation length to the same as its orbit length, then one has essentially replicated the effects of tidelocking... as long as the long axis is also down.

Keep in mind that the object rotates on its center of mass. The center of gravitic force, however, may not be on the center of mass, and so tidal stress will slowly alter the orientation of the object. In earth orbit, this is complicated by the tidal stress of the moon, as well. Mind you, the moon's tidal stress is very tiny - nanometers per second per second - dwarfed by the millimeters per second per second of the earth, but sufficient to induce orbital deformations.

-

A satellite can naturally remain aligned to the local vertical.

In orbit are two forces to consider: force of gravity and centrifugal force. Centrifugal force is actually inertia in a rotating frame. But if you happen to be on the merry-go-round it feels like a force.

Centrifugal force is $\omega^2r$ and gravity is $GM/r^2$

To portray these up and down tugs I'll use balloons and passengers being carried by balloons.

This picture portrays a balance of gravity and centrifugal force. Net force is zero.

What happens if we double r, the distance from body center?

Doubling radius doubles upward tug. Downward tug is cut to 1/4. Net acceleration is up.

And if we cut radius in half…

Upward tug is cut in half while downward tug is quadrupled. Net acceleration is down

Tie these three together and you get a tether that remains aligned to the local vertical:

There are satellites that use gravity gradient stabilization to remain aligned. This also what keeps a lot of moons tidelocked. If we ever have vertical tethers or space elevators, this is what would hold them vertical.

-

Picture this: Take a toy airplane and tie a string to one wing. Now spin in place and let the plane fly at the end of the string. You are the Earth and the plane is a satellite. Does the plane really "rotate"? Or is it flying straight all the time but its course is being changed because of the string?

Its the exact same thing with a satellite only the string is gravity. In reality the satellite is flying straight because it was launched forward, and it is constantly falling towards the Earth, but its forward speed exactly offsets the pull of gravity.

So don't think of it so much as turning as it flying forward with continuous automatic course changes into a circular path.

-
If I understand you (and others) correctly, pushing a circular LEO satellite towards the Sun, just turns the pushing point in its orbit into perigee without disturbing its orientation. If it was facing the Sun to begin with, it will keep doing so after the push too, as it makes its new eccentrical orbit. – LocalFluff May 5 '14 at 18:01
This is not the answer to your question, LocalFluff. – David Hammen May 5 '14 at 18:11
"So don't think of it so much as turning as it flying forward with continuous automatic course changes into a circular path." That's pretty much a simplified explanation of an orbit, however it says nothing about spacecraft attitude in that orbit. – Michael Kjörling May 6 '14 at 14:00
This answer is wholly and totally incorrect. Gravity pulling is, exactly, completely different from a string pulling. – Joe Blow May 3 at 11:38


text:
# Computing

## Queues

Venus's queueing system is very basic. There is only one queue. Jobs may run for up to 192 hours and priority is done by fairshare. There's an example job script in /info/slurm .

## Types of node

venus has two types of node with different CPU types. The better nodes are assigned to jobs first. To ensure you get assigned only the better nodes you can request only to have 'westmere' nodes:

sbatch -n12 --constraint=westmere

Because the two types of node have different CPU types it's possible to generate very optimized code that can't run on the older ones. If you get an 'illegal instruction' error, that's probably what's happened.

The older nodes have much less RAM than the newer ones.

## Parallel work

Venus is a cluster system and so message passing libraries are used to make code run in parallel. There are several different libraries installed. Shared memory parallelization (OpenMP, provided by the autoparallelizing options on some of the compilers) can also be used for up to twelve cores but not further.

The available message passing libraries are MPI implementations: currently we have OpenMPI in various versions.

To change your parallel environment you need to load the appropriate module. To switch you'd edit your .bash_profile file and change the module add line.

Compile your code with the MPI compilers. These are usually called mpicc, mpicxx, mpif77 and mpif90.

To run MPI code you normally use a command such as mpirun to launch your program. On venus there is a queueing system which assigns work to compute nodes, and so the MPI library has to interact with that which makes things more complicated. You use a special MPI launcher program to start your job. The launcher doesn't take any arguments but works out the correct number of cpus and which nodes to use from the queueing system. You must use the appropriate one for your MPI library, and you must have your environment configured for the library you want to use when you submit a job, as otherwise the job won't be able to find its library files. Here's a list of which launcher command goes with which library:

MPI Command name
OpenMPI mpirun

### System status

System monitoring page

## Can't find what you're looking for?

Then you might find our A-Z site index useful. Or, you can search the site using the box at the top of the page, or by clicking here.


text:
Problems with the cartesian product

Intro

Hello, I'm starting to read Munkres' Topology, and I'm having a bit of trouble solving problems involving the cartesian product. For example, there's this problem:

Sample Exercise

$$\textit{Is the following true?}$$ $$(A-B)\times(C-D)=(A\times C-B\times C)-A\times D$$

Erroneous Solution

I have a really hard time visualizing this kind of thing, so I tried to sort of give a half-explanation: Since it is true that

$$(A\times C-B\times C)=(A-B)\times C$$

We're left with

$$(A-B)\times(C-D)=(A-B)\times C-A\times D$$

Now, what we have on the right-hand-side of the original equation is very similar to the left-hand-side except for one thing: for the $(-A\times D)$ part, we should have $(-(A-B)\times D)$ if we wanted both sides to be the same. It's pretty easy to convince yourself that: $(A\times D)\supseteq(A-B)\times D)$. Therefore, we must have that

$$(A-B)\times(C-D)\supseteq(A\times C-B\times C)-A\times D$$

Which means that the initial statement is not true.

The Problem

My answer was obviously incorrect; I looked online for a solution set and the correct answer was that the given statement is correct. Where did I go wrong here? Also- is there an 'official' way way of approaching these types of exercises? For exercises involving concepts like union and set difference I've had no problem. This is because I find unions and set differences very easy to conceptualize, or at least draw in a Venn diagram. Is there a similar diagram I could draw for cartesian products which would be useful? Or a more systematic way of approaching these exercises?

Don't visualize. Use the definitions.

Visualization can be helpful, but it can be misleading, and if you don't learn to work out the definitions, you can get stuck quite easily because you spend all your energy trying to visualize the problem.

In one direction, if $(a,c)\in(A-B)\times (C-D)$ then $a\in A, c\in C$ and $a\notin B,c\notin D$, therefore $(a,c)\notin B\times C$ as well $(a,c)\notin A\times D$. Therefore $(a,c)\in (A\times C-B\times C)-A\times D$.

The other direction is as simple.

One useful tip is to learn how to simplify statements like the one on the RHS. Relevant to this question is the identity, $(A-B)-C=A-(B\cup C)$.

This makes it easier to see why equality holds: If $(a,c)$ is in the LHS, then $(a,c)\in A\times C$ and $(a,c)\notin B\times C$ as well $(a,c)\notin A\times D$. Therefore $\subseteq$ holds. And so on.

Check both inclusions, this is how this is done always. I would not say that your proof is false from the beginning, but its just incomplete. $A\subset B$ does not exclude $A=B$. You say: Therefore, we must have that

$$(A-B)\times(C-D)\supseteq(A\times C-B\times C)-A\times D$$

Which means that the initial statement is not true. How does this make sense? Two sets are equal iff $A\subset B$ and $B\subset A$! Just check the other inclusion.

Cartesian products $X\times Y$ can be drawn like $\mathbf{R}^2$: put $X$ on the $x$-axis, and $Y$ on the $y$-axis. (If you do it like this the result is actually obvious!)

• Yeah, that's correct. I completely messed up on the definition of $A=B$. – Physics Llama Jul 11 '14 at 21:56