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https://cran.hafro.is/web/packages/crunch/vignettes/variables.html
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Crunch Variables
Previous: Getting Started
Once you’ve uploaded data to Crunch, you’ll likely want to spend some time cleaning up the metadata associated with your dataset. Legacy statistical programs offer only limited support for variable metadata, both in type and quality. Because Crunch facilitates data visualization, collaboration, and sharing results with others, making your dataset presentation-quality is worthwhile, and this often requires additional work after uploading a data file.
Most of the operations described below can also be accomplished in the web client. However, users comfortable with scripting may find it faster or easier to automate these actions with crunch. Such edits made within R are carried out on the remote Crunch dataset, thereby keeping data in sync across all client applications.
A rose by any other moniker: “names” and “aliases”
Crunch takes the principled stand that working with data in the 21st Century should not be constrained by legacies of the punch-card era. Variables should have “names” that are human-readable and searchable by their meaning—there is no reason to constrain variable names to be eight characters, all caps, etc. “Aided awareness: coffee roasters” is much nicer and more presentable than “Q2B_V1”.
At the same time, shorter, user-defined, unique identifiers for variables do have their uses. For one, it’s what most any legacy statistical software uses for its identifiers, so retaining them on import will help us line up variables when appending a subsequent batch of imported data, for example. For another, when interacting with a dataset from the command line, it can be useful to have shorter, machine-friendlier references.
So, Crunch stores two user-settable identifiers for variables. What you may have thought of as a variable “label”, Crunch elevates to the status of “name”. What you may be used to thinking of as a variable “name”, Crunch calls “alias”.
Aliases and names have slightly different validation constraints. Aliases must be unique across the entire dataset, including among array subvariables. “Names”, however, only must be unique within variable order groups. Any string is valid for either alias or name, though you may want more machine-friendly strings as aliases. In most cases, you probably won’t even set aliases, though: they’ll be set when you import your dataset and will be whatever the names were in your source data.
In sum, name is crunch alias, label is crunch name.
Except in one place in crunch: referencing variables within a dataset.
Accessing variables within a dataset
When dealing with variables within a dataset, the alias is used to identify variables. This is because (1) aliases are typically what were used to identify variables in whatever format from which the dataset was imported, and consequently, (2) aliases are typically more machine-friendly as names, less likely to contain characters that are not valid as variable names in R.
Because the names attribute is used for indexing elements in R, if we want to extract variables based on alias, it means that the names attribute of dataset must actually expose aliases. This may be dissonant, but it has some nice properties. For one, comparing our Crunch dataset with the data.frame from which it was created, their names attribute have the same contents:
identical(names(ds), names(economist))
## [1] TRUE
You can reference and extract variables from a dataset as if it were a data.frame, using the $, [, and [[ methods. track.var <- ds$track
track.var
## track (categorical)
##
## Count
## Off on the wrong track 137
## Generally headed in the right direction 80
## Not sure 33
Like datasets, variables have various attributes like name and description that can be set naturally.
name(track.var) <- "Direction of country"
description(track.var) <- "In your opinon, is the country going in the right direction, or is it on the wrong track?"
Two caveats. First, because we first extracted the variable from the dataset before making edits, the dataset object has stale metadata for this variable.
name(track.var) == name(ds$track) ## [1] FALSE If we had instead modified track within ds, like ## Not run name(ds$track) <- "Direction of country"
ds would be up to date.
This can be remedied one of two ways. We could either assign track.var back to ds, as in
ds$track <- track.var or we can just refresh the dataset and fetch data from the server again: ds <- refresh(ds) Now, ds has our edits: name(track.var) == name(ds$track)
## [1] TRUE
It is not always convenient that the names attribute of the dataset actually yields aliases. Moreover, if we want to edit the Crunch names of many variables, we need a way of accessing the Crunch metadata more directly. It will be very slow to edit each variable in the dataset individually, referencing them with $, because each edit would send a request to the server. Instead, we’d rather bundle those into a single request. To do this, we can access the variables attribute of the dataset, which is a “variable catalog”: class(variables(ds)) ## [1] "VariableCatalog" ## attr(,"package") ## [1] "crunch" In the variable catalog, Crunch names are names, and aliases are aliases. Hence, identical(names(ds), aliases(variables(ds))) ## [1] TRUE but identical(names(ds), names(variables(ds))) ## [1] FALSE because “Direction of country” is the name for ds$track
head(names(variables(ds)), 10)
## [1] "perc_skipped" "newsint2" "Direction of country" "snowdenfav" "snowdenleakapp"
## [6] "snowdenpros" "snowdenpenalty" "manningknowledge" "manningfavorability" "manningguilt"
These attributes all allow assignment with <-. The methods names and aliases yield character vectors, and they take characters in assignment. Hence, you can use any vectorized string manipulation tools available in R, such as regular expressions, to edit variable names efficiently. You can also just supply a replacement vector, like
names(variables(ds))[4:7] <- c("Favorability of Edward Snowden",
"Approval of Snowden's Leak",
"Support for Prosecution of Snowden",
"Penalty for Snowden")
head(names(variables(ds)), 10)
## [1] "perc_skipped" "newsint2" "Direction of country"
## [4] "Favorability of Edward Snowden" "Approval of Snowden's Leak" "Support for Prosecution of Snowden"
## [7] "Penalty for Snowden" "manningknowledge" "manningfavorability"
## [10] "manningguilt"
Categorical variables
Many variables in survey data are categorical: respondents have a finite set of answers to the survey question, and the answers are first and foremost of a nominal, not quantitative nature. In R, this data type is represented as a factor. The response options, are contained in the factor’s “levels” as a character vector. Manipulation of these levels is limited and often challenging.
In Crunch, categorical variables’ “categories” are objects with richer metadata.
is.Categorical(track.var)
## [1] TRUE
categories(track.var)
## id name value missing
## 1 1 Generally headed in the right direction 1 FALSE
## 2 2 Off on the wrong track 2 FALSE
## 3 3 Not sure 3 FALSE
## 4 -1 No Data NA TRUE
Category attributes
Categories have names, the factor’s levels; numeric values which can be used when interpreting the categorical variable as numeric; and ids, which are analogous to the integer values that underlie an R factor. Categories also have their own “missing” status. Indeed, because Crunch supports more complex missing value support than does R, multiple categories can be marked as missing: there’s not a single “NA” value.
names(categories(track.var))
## [1] "Generally headed in the right direction" "Off on the wrong track"
## [3] "Not sure" "No Data"
values(categories(track.var))
## [1] 1 2 3 NA
ids(categories(track.var))
## [1] 1 2 3 -1
is.na(categories(track.var))
## Generally headed in the right direction Off on the wrong track Not sure
## FALSE FALSE FALSE
## No Data
## TRUE
Names and values can be assigned into categories, but ids cannot: they are immutable references to values within the column of data on the server. Missingness can be set with is.na. Character values assigned will mark those categories as missing, leaving other categories unchanged. Logical values assigned will set the missing TRUE/FALSE accordingly.
names(categories(track.var))[1:2] <- c("Right track", "Wrong track")
values(categories(track.var)) <- c(1, -1, 0)
is.na(categories(track.var)) <- "Not sure"
categories(track.var)
## id name value missing
## 1 1 Generally headed in the right direction 1 FALSE
## 2 3 Not sure 0 TRUE
## 3 2 Wrong track -1 FALSE
## 4 -1 No Data NA TRUE
ids(categories(track.var)) <- sample(ids(categories(track.var)), replace=FALSE)
## Error : Cannot modify category ids
Reordering categories
Categories can also be reordered by index, like any list object
categories(track.var) <- categories(track.var)[c(1,3,2)]
categories(track.var)
## id name value missing
## 1 1 Generally headed in the right direction 1 FALSE
## 2 2 Wrong track -1 FALSE
## 3 3 Not sure 0 TRUE
As with all other metadata edits discussed, updating with these methods automatically sends the changes to the server, so your local edits are reflected in the cloud.
Hiding variables
Datasets often contain variables that you may want to use – perhaps through a derived variable, a transformation, or a recode – or that may simply not be relevant for the analytic tasks at hand. In short, you want to hide them. They aren’t deleted, so you can restore them if you need them later, but they no longer clutter the dataset “workspace”.
As when working with a data.frame, you typically assign the return of a dataset-level function back to the variable representing the dataset in your R script or session.
In our example dataset, we have two copies of a voter-registration variable, “votereg_new” and “votereg_old”. Let’s hide the old version:
ds <- hideVariables(ds, "votereg_old")
hiddenVariables(ds)
## [1] "votereg_old"
As with the is.na function, you can update a variable by assigning it to the hidden variables list.
hiddenVariables(ds) <- "pid7others"
hiddenVariables(ds)
## [1] "votereg_old" "pid7others"
These variables are now hidden, both locally in your R session and remotely on the server, which you can see in the web application. And, just as you could restore them there, you can also restore them from R:
ds <- unhideVariables(ds, "pid7others")
hiddenVariables(ds)
## [1] "votereg_old"
Deleting variables
Sometimes you do want to delete variables permanently. Doing so is easy, but we have some protections in place to keep you from accidentally deleting things from a dataset that may be shared with many people on the server.
To delete, you can assign NULL in to the dataset for that variable, just like you were removing a column from a data.frame. Let’s kill the “votereg_old” variable permanently:
ds$votereg_old <- NULL ## Really delete "votereg_old"? The delete function requires confirmation when you’re running from an interactive session, just to make sure you aren’t accidentally assigning something in that is NULL and deleting your variable. If you know that you want to delete the variable, you can give your approval in advance by wrapping it in a with statement, using the consent context manager: with(consent(), ds$votereg_old <- NULL)
"votereg_old" %in% names(ds)
## [1] FALSE
The with(consent(), ...) pattern works everywhere in crunch that requires confirmation to do an action, such as deleting datasets.
Next: create and manipulate array variables
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2022-12-09 02:01:34
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https://zenodo.org/record/3383947/export/xd
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Journal article Open Access
# Hitastigsbreytingar við vinnslu, flutning og geymslu á frosinni síld / Temperature profiles during processing, transportation and storage of frozen herring products
Karlsdóttir, Magnea; Jónasson, Finnur; Jónsson, Ásbjörn; Arason, Sigurjón
### Dublin Core Export
<?xml version='1.0' encoding='utf-8'?>
<oai_dc:dc xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:oai_dc="http://www.openarchives.org/OAI/2.0/oai_dc/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd">
<dc:creator>Karlsdóttir, Magnea</dc:creator>
<dc:creator>Jónasson, Finnur</dc:creator>
<dc:creator>Jónsson, Ásbjörn</dc:creator>
<dc:creator>Arason, Sigurjón</dc:creator>
<dc:date>2019-08-30</dc:date>
<dc:description>The main objective of the project was to maximize the qualiry and value of herring products. Quality and stability of frocen herring products seasonally and impact of pre-cooling, freezing and storage condition were explored. The results will not only lead to a less decrease in quality due to storage and transportation, but also increase understanding on connection between product defect and their influence on the raw material from catch to market.
The results showed thar frozen herring products didn‘t get enough cooling through the process, from production to export, and in some cases the variation was too much. The freezing planti n Iceland was good, and all the pallets was kept at right temperature. The problem is, when they were taken out of the freezing storage in Iceland, the cooling was not sufficient, which was necessary to maintain low temperature. It is difficult to prevent heat stress when exporting frozen product.
The results of the project indicated that the temprature in freezing trawls was stable during the sailing. Also the results indicated that it is a need for improvements in the freezing plant in Poland, where the temperature in freezing storage is higher compared to Iceland. Transportation in containers was much better than in freezing trawlers, but much more expensive.</dc:description>
<dc:description>Funding: AVS R&D Fund</dc:description>
<dc:identifier>https://zenodo.org/record/3383947</dc:identifier>
<dc:identifier>10.5281/zenodo.3383947</dc:identifier>
<dc:identifier>oai:zenodo.org:3383947</dc:identifier>
<dc:language>eng</dc:language>
<dc:relation>doi:10.5281/zenodo.3383946</dc:relation>
<dc:rights>info:eu-repo/semantics/openAccess</dc:rights>
<dc:subject>temperature variation</dc:subject>
<dc:subject>transport</dc:subject>
<dc:subject>storage</dc:subject>
<dc:subject>containers</dc:subject>
<dc:subject>freezing trawlers</dc:subject>
<dc:title>Hitastigsbreytingar við vinnslu, flutning og geymslu á frosinni síld / Temperature profiles during processing, transportation and storage of frozen herring products</dc:title>
<dc:type>info:eu-repo/semantics/article</dc:type>
<dc:type>publication-article</dc:type>
</oai_dc:dc>
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2019-12-15 01:41:37
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https://www.physicsforums.com/threads/vector-subspaces.633403/
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# Vector Subspaces
1. Sep 4, 2012
### markovchain
Show that T = (x, y, z) : -1 ≤ x + y + z ≤ 1
is not a vector subspace of R3
Thanks!
2. Sep 4, 2012
### DonAntonio
$$(1,0,0)\in T\,\,\,but\,\,\,2(1,0,0)=(2,0,0)\notin T$$
Thus T cannot be v. subspace as it isn't closed under scalar multiplication
DonAntonio
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2017-11-21 14:37:48
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https://math.stackexchange.com/questions/1806478/codes-attaining-the-hamming-bound
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# Codes attaining the Hamming bound
I am interested in codes with the following property. Given any codeword $x^n$ of a $(n,k,d)_2$ code passed through a binary symmetric channel $y^n = x^n + e^n$, if $|e^n| = d$, then $x^n+e^n \in C$. In other words, if we flip any $d$ bits of any codeword, we always get a new codeword.
My first guess was that only codes attaining the Singleton bound $d-1 \leq n-k$ could attain this property. But on the other hand, codes attaining the Hamming bound are said "to fill out the entire space" with spheres of radius $d$.
With that phrashing, it seems to me that flipping an $d$ bits of any codeword should result in a new codeword. But that's not true, is it? If not, is there some other class of codes with this property?
• Looks like you are asking an awful lot. Start with a valid codeword $x$. Flip the first $d$ bits, you require that the resulting word $x'$ is another codeword. Flip the bits at positions $1,2,3,\ldots,d-1$ and $d+1$ you get yet another codeword $x''$. But $x'$ and $x''$ only differ from each other at positions $d$ and $d+1$. In other words, the minimum distance of your code is $\le 2$. Doesn't look like such codes would be very interesting. But, there are codes achieving the Hamming bound. Not too many, but some. Kodlu's answer has a comprehensice list. This was proved by Tietäväinen. – Jyrki Lahtonen Jul 1 '16 at 8:16
• Aha! But that's exactly true for binary MDS codes! :) – Benjamin Lindqvist Jul 1 '16 at 8:17
Let $n=2w$ be the length of the code. Consider the set $E=\{a: w_H(a)=w\}$ of all $n-$vectors of weight $w.$ These are the allowed error patterns, and $|E|=\binom{2w}{w}.$ Now let $$C=E \cup \{\mathbf{1},\mathbf{0}\}$$ where $\mathbf{1}$ is the all 1 vector of length $n$ and $\mathbf{0}$ is the all 0 vector of length $n$. Clearly, for all errors $e \in E$ (which are all errors of weight $w$) an errored codeword $c$ maps to another codeword $c'.$
Let the support of a vector $(x_1,\ldots,x_n)$ be the subset$S_x$ of $\{1,\ldots,n\}$ where, $$i\in S_x \Leftrightarrow x_i=1.$$ There are two special cases: if $e$ is the complement of $c$ then $c+e=\mathbf{1}$ and a if $e=c$ then $e+c=\mathbf{0}.$ In general, if the overlap in the supports of $c$ and $e$ is $t \in \{1,2,\ldots,w-1\},$ i.e., $|S_{c+e}\cap S_c|=t,$ and $c'=c+e$ we have that the size of the support of $c'$ is $$|S_{c+e}|=|S_{c}\setminus (S_c \cap S_e)|+|S_{e}\setminus (S_c \cap S_e)|+|=(w-t)+[n-w-(w-t)]=n-w$$ which equals $w$ when $n=2w$ as required.
End of Edit
If a binary code is perfect (with $d=2t+1$) then the $t-$spheres of radius $t$ around codewords fill the space, not those of radius $d.$
The only binary perfect codes are the repetition code of odd length, the Hamming code and the Golay Code, and these are the only possible parameters.
Consider the $[n,k,d]=[2^m-1,2^m-m-1,3]$ Hamming code, flipping any $d=3$ bits of the Hamming code would give you $\binom{2^m-1}{3}=\frac{(2^m-1)(2^m-2)(2^m-3)}{6}$ distinct vectors but some of those will have pairwise distance $1$ and $2$, surely they are not codewords in the Hamming code.
See wikipedia for more.
Another code with pairwise equal distances is the Simplex code which is a $[2^m-1,m, 2^{m-1}-1]$ code, closely related to the Hadamard code, but these codewords don't have that property either. It can only be that for some set of $d$ flips you get another codeword.
For the trivial repetition code with $d=n$ your property holds, of course.
Edit: So, the only binary MDS codes are
1. the repetition codes $[n,1,n]$, thus with the two codewords of all zeroes and all ones. When $n$ is odd, these are perfect, and have your property. They achieve the Hamming bound with $t=0.$
2. the universal code $[n,n,1]$ containing all vectors of length $n$ and have your property. For odd $n$, they achieve the Hamming bound with $t=(n-1)/2,$ but not for $n$ even.
3. the even weight code $[n,n-1,2]$ made up of codewords of even weight only. These have your property. These don't achieve the Hamming bound.
• Good answer, but I can't accept since it does not address which codes have the property. I suspect we require the code to be MDS, but I'll leave it up until I have an answer – Benjamin Lindqvist Jun 4 '16 at 20:15
• So the MDS property is sufficient, but we haven't established that it's necessary. – Benjamin Lindqvist Jun 30 '16 at 16:07
• I'm going to attempt it tomorrow, will post if I find it – Benjamin Lindqvist Jun 30 '16 at 16:08
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2019-08-23 04:20:42
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https://openreview.net/forum?id=Ske9VANKDH
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## An Optimization Principle Of Deep Learning?
Sep 25, 2019 ICLR 2020 Conference Blind Submission readers: everyone Show Bibtex
• Abstract: Training deep neural networks (DNNs) has achieved great success in recent years. Modern DNN trainings utilize various types of training techniques that are developed in different aspects, e.g., activation functions for neurons, batch normalization for hidden layers, skip connections for network architecture and stochastic algorithms for optimization. Despite the effectiveness of these techniques, it is still mysterious how they help accelerate DNN trainings in practice. In this paper, we propose an optimization principle that is parameterized by $\gamma>0$ for stochastic algorithms in nonconvex and over-parameterized optimization. The principle guarantees the convergence of stochastic algorithms to a global minimum with a monotonically diminishing parameter distance to the minimizer and leads to a $\mathcal{O}(1/\gamma K)$ sub-linear convergence rate, where $K$ is the number of iterations. Through extensive experiments, we show that DNN trainings consistently obey the $\gamma$-optimization principle and its theoretical implications. In particular, we observe that the trainings that apply the training techniques achieve accelerated convergence and obey the principle with a large $\gamma$, which is consistent with the $\mathcal{O}(1/\gamma K)$ convergence rate result under the optimization principle. We think the $\gamma$-optimization principle captures and quantifies the impacts of various DNN training techniques and can be of independent interest from a theoretical perspective.
0 Replies
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2019-12-09 08:52:25
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http://fivethirtyeight.com/features/cutting-off-emergency-unemployment-benefits-hasnt-pushed-people-back-to-work/
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Cutting Off Emergency Unemployment Benefits Hasn’t Pushed People Back to Work
Helene Laurusevage still gets up at 6 a.m. every day and packs lunch for her husband. She still sits down at her computer every morning to hunt for a job, and still updates the meticulous spreadsheet that she uses to document that search. Just one thing has changed: In January, the biweekly checks for $1,126 from the Pennsylvania Department of Labor and Industry stopped showing up in her bank account. Laurusevage, 52, is one of more than a million Americans who lost payments when Congress allowed the Emergency Unemployment Compensation program to expire at the end of last year. The program, which Congress created in 2008, extended jobless benefits beyond the standard 26 weeks provided by most states; at its peak, the federal government provided an unprecedented 6 million workers with up to 73 weeks of benefits.1 The Senate earlier this year voted to renew the program, but House Speaker John Boehner hasn’t allowed the measure to come to a vote in the House. The case against extending unemployment benefits essentially boils down to two arguments. First, the economy has improved, so the unemployed should no longer need extra time to find a new job. Second, extended benefits could lead job seekers either to not search as hard or to become choosier about the kind of job they will accept, ultimately delaying their return to the workforce.2 But the evidence doesn’t support either of those arguments. The economy has indeed improved, but not for the long-term unemployed, whose odds of finding a job are barely higher today than when the recession ended nearly five years ago. And the end of extended benefits hasn’t spurred the unemployed back to work; if anything, it has pushed them out of the labor force altogether. Of the roughly 1.3 million Americans whose benefits disappeared with the end of the program, only about a quarter had found jobs as of March, about the same success rate as when the program was still in effect; roughly another quarter had given up searching. The rest, like Laurusevage, were still looking.3 Laurusevage didn’t expect it to be this hard. She had been her family’s primary breadwinner, earning roughly$60,000 as a health and safety officer for a Philadelphia-area heating and air conditioning company. Her husband, David, earns less than $35,000 a year selling truck parts. When her position was outsourced in April of last year, she thought that as a college graduate with a three-plus-decade history of steady work, she would find a job relatively quickly. But in many ways, her experience is typical. The long-term unemployed — typically defined as those out of work more than six months — are slightly more educated on average than the broader population of job seekers. And older workers like Laurusevage face a particularly tough time: The typical job seeker in her 50s has been out of work 26 weeks, versus 17 weeks for the typical 20-something. Laurusevage also lost her job at a time when the economy, though improving, was still far from fully healed. The unemployment rate in her home state of Pennsylvania was 7.6 percent in April 2013, down from 8.7 percent in the worst of the economic downturn, but still three percentage points above its pre-recession level. That made her much less likely to find a job quickly: The unemployment rate at the time someone loses a job, more than demographics or any other factor, is by far the strongest predictor of how long it will take that person to find a new job.4 Once Laurusevage’s search crossed the six-month mark, her chances of finding a job dropped sharply. Only about 10 percent of the long-term unemployed find jobs each month, a metric known as the job-finding rate. Among those unemployed six months or less, the finding rate is nearly 25 percent. Moreover, the short-term unemployed have seen their chances improve steadily during the recovery, while the finding rate for the long-term unemployed has been nearly flat. In other words, the long-term unemployed have hardly experienced a recovery at all. Some economists have argued that the end of the emergency unemployment benefits program could be a harsh but necessary medicine, prodding the unemployed back to work. But Laurusevage, at least, needed no prodding. The carefully formatted spreadsheet she uses to track her job search is a record of frustration: 246 applications, 24 interviews, no offers. In most entries, the “Outcome” field is blank; many companies these days don’t even bother to send a formal rejection notice. “I can’t even find that crap job. I’ve applied to Wawa and Wawa won’t take me, or I haven’t heard back from them,” she said, referring to the regional convenience store chain. “I will take whatever you have. Temporary, part-time, seasonal, anything.” The government doesn’t directly track what happens to people when their benefits run out. But the Current Population Survey, the monthly survey used to calculate the unemployment rate and related data, does ask people what state they live in and how long they’ve been unemployed. That means we can identify people who are likely to have lost benefits when the emergency program ended and compare their success at finding jobs to people who likely received benefits when the program was still in effect.5 This approach isn’t perfect — for one thing, we can only tell whether people likely qualified for benefits, not whether they actually received them — but past research suggests this methodology has a strong correlation with actual receipt of benefits.6 This analysis suggests the end of benefits has had at most a small impact on the rate at which workers find jobs. In 2013, people who likely qualified for emergency benefits had a monthly job-finding rate of 12 percent. In the four months since the program ended, the job-finding rate for likely cutoff victims — people who would likely have qualified for extended benefits if the program had been renewed — was slightly higher, at 14 percent. But the difference isn’t statistically significant. Even if it holds up as more data comes in, it could be the result of the improving economy rather than the direct impact of the end of the emergency program. The end of the program might have had another effect, however: It may have made people less likely to keep trying to find jobs. Unemployment programs usually require recipients to show that they’re actively applying for jobs; once benefits expire, that incentive goes away. About 19 percent of cutoff victims have dropped out of the labor force each month this year, meaning they stopped actively looking for work. That’s a bit higher than the 16 percent who dropped out each month last year; that difference, too, is at most marginally statistically significant.7 We don’t yet have enough evidence to draw a firm conclusion about what impact the end of emergency benefits has or hasn’t had on the prospects of the long-term unemployed. The Current Population Survey is large, covering about 60,000 households each month, but only a small fraction of the people interviewed each month are unemployed, and even fewer were affected by the benefits cutoff. The total sample for likely cutoff victims over four months is a bit under 1,200 households, enough to draw only preliminary conclusions about a group that totals well over 1 million people in the overall U.S. population. This much is clear, however: There has been no sudden surge of former benefits recipients into jobs. Nor have they abandoned the labor force in droves. Most have done what Laurusevage has done: continued looking for work, but without the lifeline that benefits provided. For Laurusevage, the cutoff has been wrenching. Her husband’s salary, combined with the$563 a week she got in unemployment benefits — the equivalent of about \$30,000 a year — was enough to make ends meet. But once her benefits expired in December, life got harder. Their savings depleted, they scraped together this month’s mortgage payment only by borrowing from David’s 79-year-old mother. They don’t know what they’ll do this month.
“We are about to go under,” Laurusevage said. “My entire savings account is gone. Everything I spent years to save is gone.”
## Footnotes
1. These figures combine the Emergency Unemployment Compensation (EUC) program and another federal program known as Extended Benefits. ^
2. Boehner has also argued against renewing the program on the grounds that a retroactive reinstatement would be difficult to implement, a claim the U.S. Labor Department rejects. ^
3. These figures, based on longitudinally linked Current Population Survey microdata, look at the labor force status in March of people who were unemployed in December and were likely recipients of EUC benefits (as determined using the methodology described later in this article). Because the CPS is conducted in the middle of the month, it isn’t possible to know exactly how many of these people actually saw their benefits expire and how many found jobs or stopped looking in late December, when the program was still running. However, my methodology identifies 1.3 million people as likely EUC recipients in December, almost exactly the figure reported by the Department of Labor. ^
4. Based on my earlier research, which looked at national rather than state-level data, someone who lost a job in Pennsylvania in April 2013 had roughly double the chances of becoming long-term unemployed as someone who lost a job before the recession, when the state unemployment rate was 4.6 percent. ^
5. A bit more on the methodology here: To identify people who likely lost benefits, I found those who reported having been unemployed longer than their state’s maximum duration of benefits (26 weeks in most states) but for less time than their state’s maximum in December, when the EUC program was still in place. I also limited the sample to people who reported losing a job rather than having quit or entered the labor force for the first time. For the comparison group, I used the same basic approach to identify people who likely qualified for EUC benefits in their state.
The two groups are demographically very similar, but not identical. Most important, the median duration of unemployment is slightly lower for the cutoff group than for the comparison group. All else equal, this would tend to make the reported job-finding rate for the cutoff group slightly higher, which is another reason to be cautious about the slightly higher finding rate for this group. ^
6. Another problem: This methodology assumes that people file for benefits immediately upon losing a job and receive them weekly until those benefits expire. Neither assumption entirely reflects reality. Many people delay filing for benefits, in some cases for several weeks or even months. And many people stop receiving benefits for some period during their eligibility. That means my sample likely captures at least some people who were still receiving benefits at the time of their interview. To test whether this would have an effect, I re-ran my calculations excluding anyone unemployed for less than 30 weeks, and again for anyone unemployed less than 35 weeks. Both groups had slightly lower job-finding rates, which is consistent with the notion that the cutoff in benefits had little impact on the probability of returning to work. ^
7. These findings are consistent with past research that found that unemployment benefits have a bigger impact on the dropout rate than on the job-finding rate. That is, benefits make people more likely to keep looking (or at least to say they’re looking) for jobs when they might otherwise have given up, but not much less likely to find or accept jobs. ^
Ben Casselman is FiveThirtyEight’s chief economics writer.
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2016-05-01 09:53:33
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https://space.stackexchange.com/questions/35310/what-does-the-core-stage-of-a-launch-vehicle-mean
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# What does the core stage of a launch vehicle mean?
I have seen the use of the term 'core stage' in the context of space launch vehicles. I am not able to clearly understand what it exactly means? I have learnt about launch vehicles having multiple stages: first, second and so on. Is the core stage just a different name for the first stage itself?
I tried to google it but couldn't get a clear meaning for it. I had a look at the SLS core stage fact sheet by NASA which says a few things about core stage.
• It says that the core stage stores cryogenic liquid hydrogen and liquid oxygen. So, this makes me think that this might not be the same as the first stage because you don't generally carry a higher $$I_{sp}$$ fuel like liquid hydrogen in the first stage (however, I may be completely wrong and maybe it is used in first stage here) and so this must stay at least till the stage which actually uses this cryogenic liquid hydrogen separates.
• It mentions that it houses the flight computers and much of the avionics needed to control rocket's flight. This again makes me think that the core stage is not same as first stage because the avionics and flight computers should be necessary for rocket after first stage separation too. Is it that the upper stages have their own flight computers and avionic system?
Thank you.
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2019-10-20 12:00:14
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https://homework.cpm.org/category/CON_FOUND/textbook/mc2/chapter/3/lesson/3.1.2/problem/3-19
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### Home > MC2 > Chapter 3 > Lesson 3.1.2 > Problem3-19
3-19.
Remember, perimeter is the total length of all the sides of a shape.
What are the missing lengths for the sides of the shape? Try to fill these in and sum up each side length to get the perimeter.
2+2+4+4+6+6 = 24in
Be sure to know why this is the answer.
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2021-05-14 20:25:29
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http://openstudy.com/updates/556a4bd0e4b01de5673b4df1
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## anonymous one year ago What is the solution to the inequality? 2(w − 5) < −8 A. w < −21 B. w < −16 C. w < 1 D. w < −11
• This Question is Open
1. anonymous
@Omar_Elboredy can u help me with this one also its the last one i promise
2. shamim
w-5<-8/2 Right?
3. anonymous
i hope so
4. shamim
w-5<-4
5. geerky42
6. anonymous
First , U have to get the "w" in one side an all other stuff on the other side , to do this simply you have to divide both sides by 2 like this : $\frac{ 2(w - 5) }{ 2 } < \frac{ -8}{ 2 }$ nw you get : $(w-5) < -4$ nw get the 5 from left side to the right one nw you get : $w < 1$ Which is what ! "C" Simple, isn't it ! :D
7. anonymous
thx
8. anonymous
Welcomes ! :)
9. UsukiDoll
You can also distribute the 2 first and then add 10 and divide 2 on both sides. 2(w − 5) < −8 2w-10 < -8 2w - 10 + 10 <-8+10 2w < 2 w < 1
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2017-01-19 07:04:46
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https://testbook.com/question-answer/a-change-in-the-value-of-the-emitter-resistance-re--5facf6c571b1b35312850315
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# A change in the value of the emitter resistance RE in a differential amplifier
This question was previously asked in
ESE Electronics 2012 Paper 2: Official Paper
View all UPSC IES Papers >
1. Affects the difference mode gain, Ad
2. Affects the common mode gain, Ac
3. Affects both Ad and Ac
4. Does not affect either Ad or Ac
Option 2 : Affects the common mode gain, Ac
Free
CT 3: Building Materials
2894
10 Questions 20 Marks 12 Mins
## Detailed Solution
Differential amplifier is shown below:
Differential voltage Gain (Ad) is given by:
And the Common-mode gain (Acm) is given by:
$${A_{cm}} = \frac{{ - {R_c}}}{{2{R_E}}}$$
We can observe that with an increase in RE, Acm decreases.
Note:
$$CMRR = \left| {\frac{{{A_d}}}{{{A_{cm}}}}} \right|$$
$$CMRR = \frac{{{g_m}{R_c}}}{{\frac{{{R_c}}}{{2{R_E}}}}}$$
$$= 2{g_m}{R_E}$$
We can observe that with an increase in RE, Acm decreases, and CMRR increases.
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2021-10-15 23:39:13
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https://gmatclub.com/forum/sarah-s-seafood-restaurant-gets-a-delivery-of-fresh-seafood-every-day-89970.html
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# Sarah‟s seafood restaurant gets a delivery of fresh seafood every day,
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01 Feb 2010, 20:01
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Sarah's seafood restaurant gets a delivery of fresh seafood every day, 7 days per week, and her delivery company charges her d dollars per delivery plus c cents per item delivered. If lst week Sarah's seafood restaurant had an average of x items per day delivered, then which of the following is the total cost, in dollars, of last week's deliveries ?
A. 7cdx/100
B. d + 7cx/100
C. 7d + xc/100
D. 7d + 7xc/100
E. 7cdx
[Reveal] Spoiler: OA
Last edited by Bunuel on 28 Dec 2014, 05:29, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: Sarah‟s seafood restaurant gets a delivery of fresh seafood every day, [#permalink]
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01 Feb 2010, 21:43
1
KUDOS
Straightforwar "D"
7d + (c/100)*7x
What is unclear_?
Manager
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Re: Sarah‟s seafood restaurant gets a delivery of fresh seafood every day, [#permalink]
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02 Feb 2010, 10:27
Well, i would say "D" as well.
$= 7 Days *d + Average Items * c * 7 days; cents need to be written in "/100" -->$= 7*d + (XC*7)/100
Micha
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Re: Sarah‟s seafood restaurant gets a delivery of fresh seafood every day, [#permalink]
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02 Feb 2010, 11:27
LM wrote:
Straightforwar "D"
7d + (c/100)*7x
What is unclear_?
yea quite an easy question.
Letme elaborate
Total delivery charges per delivery = d* number of days = 7d
total delivery charges per item = c/100
total number of items per day = x
total number of items for the week = 7x
so total delivery charges (for per item condition) = 7xc/100
so total charges = 7d+ 7xc/100
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Re: Sarah‟s seafood restaurant gets a delivery of fresh seafood every day, [#permalink]
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02 Feb 2010, 21:16
Okay, thanks gurpreetsingh. I wasn't sure the purpose of c/100. I thought the decimals value was already reflected in the variable c ( as in c= .25 cents)
Thanks
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Re: Sarah‟s seafood restaurant gets a delivery of fresh seafood every day, [#permalink]
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04 Feb 2010, 16:59
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Re: Sarah‟s seafood restaurant gets a delivery of fresh seafood every day, [#permalink]
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27 Dec 2014, 04:53
Hello from the GMAT Club BumpBot!
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Re: Sarah‟s seafood restaurant gets a delivery of fresh seafood every day, [#permalink]
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27 Dec 2014, 11:32
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Hi All,
While I agree that this question is relatively straight-forward, whenever Test Takers see a drop in scores on Test Day, the reason is almost always SILLY/LITTLE MISTAKES. As such, it's important to be sure that you have the correct answer to a question that you "think" is easy/straight-forward/gettable. This question can be solved by TESTing VALUES.
We're told to calculate the cost of 7 days of deliveries.
On each day, there's a cost for the delivery itself (D DOLLARS) and a cost per item delivered (C CENTS/item). Note the difference in UNITS. During the week, we're told that an average of X items are delivered each DAY.
Let's TEST VALUES:
D = 2
C = 3
X = 10
So, per day, the Total Cost = 2 dollars + 3 cents(10 items) = $2.30 per day For 7 days, that's 7($2.30) = $16.10 Now, we just have to check the answers and find the match IN DOLLARS... Answer A = 7(3)(2)(10)/100 = 420/100 = 4.2 dollars. NOT a match Answer B = 2 + 7(10(3)/100 = 2 + 210/100 = 4.1 dollars. NOT a match Answer C = 7(2) + 10(3)/100 = 14.3 dollars. NOT a match Answer D = 7(2) + 7(10)(3)/100 = 16.1 dollars. This IS a MATCH Asnwer E = 7(3)(2)(10) = 420 dollars. NOT a match Final Answer: [Reveal] Spoiler: D GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin # Special Offer: Save$75 + GMAT Club Tests
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Re: Sarah‟s seafood restaurant gets a delivery of fresh seafood every day, [#permalink]
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01 Feb 2016, 04:09
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Re: Sarah‟s seafood restaurant gets a delivery of fresh seafood every day, [#permalink] 01 Feb 2016, 04:09
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2017-06-26 14:08:22
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https://mathhelpboards.com/threads/inequality-is-there-an-elegant-way-to-solve-this.5568/
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Inequality--is there an elegant way to solve this?
dwsmith
Well-known member
$x,y,z >0$ and $x^2 + y^2 + z^ = 1$, show that
$$xyz+\sqrt{x^2y^2+y^2z^2+x^2z^2}\ge \dfrac{4}{3}\sqrt{xyz(x+y+z)}$$
MarkFL
Staff member
Re: inequality--is there an elegant way to solve this?
I would try using Lagrange multipliers.
chisigma
Well-known member
$x,y,z >0$ and $x^2 + y^2 + z^ = 1$, show that
$$xyz+\sqrt{x^2y^2+y^2z^2+x^2z^2}\ge \dfrac{4}{3}\sqrt{xyz(x+y+z)}$$
A way that doesn't require high level knowledege [even if non comfortable from the point od view of computation...] is fo find the point $\displaystyle (x_{0},y_{0}, z_{0})$ of minimum of the function...
$\displaystyle f(x,y,z)= x\ y\ z + \sqrt{x^{2}\ y^{2}\ + x^{2}\ z^{2} + y^{2}\ z^{2}} - \frac{4}{3}\ \sqrt{x\ y\ z\ (x + y + z)}\ (1)$
... under the hypothesis that $\displaystyle x_{0}^{2} + y_{0}^{2}+ z_{0}^{2} = 1$ and then to verify that is $\displaystyle f(x_{0},y_{0},z_{0}) \ge 0$...
Kind regards
$\chi$ $\sigma$
Last edited:
chisigma
Well-known member
A way that doesn't require high level knowledege [even if non comfortable from the point od view of computation...] is fo find the point $\displaystyle (x_{0},y_{0}, z_{0})$ of minimum of the function...
$\displaystyle f(x,y,z)= x\ y\ z + \sqrt{x^{2}\ y^{2}\ + x^{2}\ z^{2} + y^{2}\ z^{2}} - \frac{4}{3}\ \sqrt{x\ y\ z\ (x + y + z)}\ (1)$
... under the hypothesis that $\displaystyle x_{0}^{2} + y_{0}^{2}+ z_{0}^{2} = 1$ and then to verify that is $\displaystyle f(x_{0},y_{0},z_{0}) \ge 0$...
Kind regards
$\chi$ $\sigma$
And an 'elegant way' to do that is to use spherical coordinates...
$\displaystyle x= r\ \sin \theta\ \cos \phi$
$\displaystyle y = r\ \sin \theta\ \sin \phi$
$z=r\ \cos \theta\ (1)$
... then evaluate the absolute minimum $\displaystyle (\theta_{0}, \phi_{0})$ of $\displaystyle f(1,\theta,\phi)$ and finally verify that $\displaystyle f(1,\theta_{0},\phi_{0}) \ge 0$...
Kind regards
$\chi$ $\sigma$
Last edited:
dwsmith
Well-known member
And an 'elegant way' to do that is to use spherical coordinates...
$\displaystyle x= r\ \sin \theta\ \cos \phi$
$\displaystyle y = r\ \sin \theta\ \sin \phi$
$z=r\ \cos \theta\ (1)$
... then evaluate the absolute minimum $\displaystyle (\theta_{0}, \phi_{0})$ of $\displaystyle f(1,\theta,\phi)$ and finally verify that $\displaystyle f(1,\theta_{0},\phi_{0}) \ge 0$...
Kind regards
$\chi$ $\sigma$
When we re-write $f$, we get
\begin{align}
f(1,\theta,\phi) &= \sqrt{\sin ^2(\theta ) \left(\sin ^2(\theta ) \sin ^2(\phi ) \cos ^2(\phi )+\cos ^2(\theta )\right)}+\sin ^2(\theta ) \cos (\theta ) \sin (\phi ) \cos (\phi )\\
&-\frac{4}{3} \sqrt{\sin ^2(\theta ) \cos (\theta ) \sin (\phi ) \cos (\phi ) (\sin (\theta ) (\sin (\phi )+\cos (\phi ))+\cos (\theta ))}
\end{align}
Are there some trig identities I need to be utilizing now?
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2022-05-21 05:54:56
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https://www.math-worksheet.org/solving-equations-by-taking-square-roots
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### Solving equations by taking square roots
We have previously learned how to solve an equation for x. But what happens when it’s not just x but $$x^2$$ instead? Don’t get nervous, it’s only one extra step!
$$x^2 = 49$$
Here’s a bit of a tricky question. What is the inverse, or opposite, operation of squaring something? (Hint: It’s in the title of this article)… Taking the square root!
If we take the square root of both sides, we get:
$$x = 7$$
But wait! Of course, it can’t be that simple, can it? There is one more concept that you do have to understand. If we plug seven back into the original equation, it works just fine. But there is ANOTHER number that we can plug in that would also work. Can you figure it out? How about:
$$(-7)^2 = 49$$
Well, that works too. So the actual answer to the above question is:
$$x = +7$$ and $$-7$$
or
$$x = \pm 7$$
(this is said “plus or minus seven”)
In fact, every time you solve an equation using a square root (unless x=0) you will have TWO answers, a positive and a negative number. Let’s try a tougher one:
$$8x^2 - 8 = 328$$
We cannot take the square root until x2 is all by itself. So let’s start by getting rid of the -8. We have to use the opposite operation so we will add 8 to each side.
$$\array{ 8x^2 - 8 =& 328 \cr +8 & +8}$$
$$8x^2 = 336$$
We just have the other 8 now to cancel out. This 8 is being multiplied so we must “undo” it by dividing each side by 8.
$$\frac{8x^2}{8}=\frac{336}{8}$$
$$x^2 = 42$$
$$x = \pm \sqrt{42}$$
Uh oh. We are at the step where we take the square root, but forty-two is not a perfect square! Well, this is where the calculator comes in handy. Forty-two has a square root but it is a decimal. Don’t forget that we need the positive and the negative answer. If we round off to three decimals our answers are:
$$x = \pm 6.481$$
So, the two major things to remember in order to solve these equations are:
• Get $$x^2$$ by itself and then take the square root of both sides
• Don’t forget that you will have TWO answers, a positive and a negative number.
3787 x
Solve each equation by taking square roots.
This free worksheet contains 10 assignments each with 24 questions with answers.
Example of one question:
Watch below how to solve this example:
2649 x
Solve each equation by taking square roots.
This free worksheet contains 10 assignments each with 24 questions with answers.
Example of one question:
Watch below how to solve this example:
2286 x
Solve each equation by taking square roots.
This free worksheet contains 10 assignments each with 24 questions with answers.
Example of one question:
Watch below how to solve this example:
### Geometry
Circles
Congruent Triangles
Constructions
Parallel Lines and the Coordinate Plane
Properties of Triangles
### Algebra and Pre-Algebra
Beginning Algebra
Beginning Trigonometry
Equations
Exponents
Factoring
Linear Equations and Inequalities
Percents
Polynomials
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2021-04-11 18:57:25
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https://techwhiff.com/learn/state-the-null-and-alternative-hypotheses-using/445409
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# State the Null and Alternative Hypotheses using population parameters. A fourth grade class is studying the...
###### Question:
State the Null and Alternative Hypotheses using population parameters.
A fourth grade class is studying the environment. One of their projects is to grow bean plants in 3 different soils. 100 plants each are planted in soil mixed with dryer lint, soil mixed with fertilizer and soil with nothing added. At the end of the growing period, 5 plants from each soil type were measured (in inches) producing the data in the table below. We will test the claim that there is no difference in mean bean plant height among each of the 3 soil types using a 5% significance level. (Assume that independent samples were drawn from normal populations with equal variances.) Soil with Dryer Lint Soil with Fertilizer Plain Soil 23 18 21 29 19 26 25 23 27 24 19 27 19 14 31
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2023-03-28 00:13:08
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https://pypi.org/project/humilis/
|
AWS cloudformation-based deployment framework
## humilis
Helps you deploy AWS infrastructure with Cloudformation.
This project is originally based on the cumulus project. See CUMULUS_LICENSE for license information.
## Installation
Install the AWS CLI:
pip install awscli
Configure the AWS CLI:
aws configure
humilis will use whatever credentials you introduced when configuring your AWS CLI installation.
You can now install the latest “stable” version of humilis:
pip install humilis
or the development version if you prefer that:
pip install git+https://github.com/humilis/humilis
After installation you need to configure humilis. To configure globally for your system:
humilis configure
The command above will store and read the configuration options from ~/.humilis.ini. You can also store the configuration in a .humilis.ini file stored in your current working directory by using:
humilis configure --local
## Development environment
Assuming you have virtualenv installed:
make develop
. .env/bin/activate
## Testing
At the moment, most tests are integration tests with the AWS SDK. This means that you will need to set up your system to access AWS resources if you want to run the test suite.
py.test tests
## Quickstart
Define your infrastructure environment following the examples in the examples directory. Then to create the environment:
humilis create examples/humilis-firehose.yaml
To update the environment after it has been deployed:
humilis update examples/humilis-firehose.yaml
And to delete it:
humilis delete examples/humilis-firehose.yaml
## Humilis environments
A humilis environment is just a collection of cloudformation stacks that are required for an application. Instead of having a monolytic CF template for your complete application, humilis allows you to define infrastructure layers that are combined into an environment. Each humilis layer translates exactly into one CF template (therefore into one CF stack after the layer is deployed).
Breaking a complex infrastructure environment into smaller layers has at least two obvious advantages:
• Easier to maintain. It’s easier to maintain a simple layer that contains just a bunch of CF resources than serve a well-defined purpose.
• Easier to reuse. You should strive to define your infrastructure layers in such a way that you can reuse them across various environments. For instance, many projects may require a base layer that defines a VPC, a few subnets, a gateway and some routing tables, and maybe a (managed) NAT. You can define a humilis layer with those resources and have a set of layer parameters (e.g. the VPC CIDR) that will allow you to easily reuse it across environments.
### Environment anatomy
An environment definition file is a yaml document that specifies the list of layers that form your enviroment. The file should be named as your environment. That is, for environment my-app-environment the environment description file should be called my-app-environment.yaml. The contents of the environment definition should be organized as follows:
---
my-app-environment:
description:
A description of what this environment is for
layers:
# The layers that you environment requires. They will be deployed in the
# same order as you list them. Note that you can also pass parameters
# to a layer (more on that later).
- {layer: name_of_first_layer, layer_param: layer_value}
- {layer: name_of_second_layer}
- {layer: name_of_third_layer}
### Layer anatomy
Anything associated to a given layer must be stored in a directory with the same name as the layer, within the same directory where the environment definition file is located. If we consider the my-app-environment environment we used above then your directory tree should look like this:
.
├── my-app-environment.yaml
├── name_of_first_layer
│ ├── meta.yaml
│ └── resources.yaml
├── name_of_second_layer
│ ├── meta.json
│ └── meta.yaml
└── name_of_third_layer
├── resources.json.j2
└── resources.yaml.j2
A layer must contain at least two files:
• meta.yaml: Meta information about the layer such as a description, and layer parameters.
• resources.yaml: Basically a CF template with the resources that the layer contains.
Those two files can also be in .json format (meta.json and resources.json). Or you can add the extension .j2 if you want the files to be pre-processed with the Jinja2 template compiler.
Below an example of how a layer meta.yaml may look like:
---
meta:
description:
Creates a VPC, that's it
parameters:
vpc_cidr:
description: The CIDR block of the VPC
value: 10.0.0.0/16
Above we declare only one layer parameter: vpc_cidr. humilis will make pass that parameter to Jinja2 when compiling any template contained in the layer. So the resources.yaml.j2 for that same layer may look like this:
---
resources:
VPC:
Type: "AWS::EC2::VPC"
Properties:
CidrBlock: {{ vpc_cidr }}
## References
You can use references in your meta.yaml files to refer to thing other than resources within the same layer (to refer to resources within a layer you can simply use Cloudformation’s Ref or GetAtt functions). Humilis references are used by setting the value of a layer parameter to a dict that has a ref key. Below an a meta.yaml that refers to a resource (with a logical name VPC) that is contained in another layer (called vpc_layer):
---
meta:
description:
Creates an EC2 instance in the vpc created by the vpc layer
dependencies:
- vpc
parameters:
vpc:
description: Physical ID of the VPC where the instance will be created
value:
ref:
parser: layer
parameters:
layer_name: vpc_layer
resource_name: VPC
Every reference must have a parser key that identifies the parser that should be used to parse the reference. There are also two optional keys:
• parameters: allows you to pass parameters to the reference parser. You can pass either named parameters (as a dict) or positional arguments (as a list).
• priority: the parsing priority. Parameters with a lower value in priority will be parsed before parameters with a higher value. This allows some reference parsers to refer internally to other parameters within the same layer. For example, the lambda parser, when parsing templated lambda code, it uses previously parsed layer parameters as template parameters.
More information on the reference parsers that are bundled with humilis below.
### Available reference parsers
#### layer_resource references
layer_resource references allow you to refer to the physical ID of a resource that is part of another layer.
Parameters:
• layer_name: The name of the layer you are referring to
• resource_name: The logical name of the layer resource
Example:
Consider the following environment definition:
---
my-environment:
description:
Creates a VPC with a NAT in the public subnet
layers:
- {layer: vpc}
- {layer: nat}
Obviously the nat layer that takes care of deploying the NAT in the public subnet will need to know the physical ID of that subnet. You achieve this by declaring a layer_resource reference in the meta.yaml for the nat layer:
---
meta:
description:
Creates a managed NAT in the public subnet of the NAT layer
parameters:
subnet_id:
description:
The physical ID of the subnet where the NAT will be placed
value:
ref:
parser: layer_resource
parameters:
layer_name: vpc
# The logical name of the subnet in the vpc layer
resource_name: PublicSubnet
When parsing meta.yaml humilis will replace this:
ref:
parser: layer_resource
parameters:
layer_name: vpc
# The logical name of the subnet in the vpc layer
resource_name: PublicSubnet
with the physical ID you need (something like subnet-bafa90cd). You can then use this physical ID in the resources.yaml.j2 section of the nat layer:
{# Pseudo-content of layers/nat/resources.yaml.j2 #}
resources:
{# An Elastic IP reservation that will be associated to the NAT #}
NatEip:
Type: 'AWS::EC2::EIP'
Properties: {}
{# Custom resource deploying the NAT #}
NatGateway:
Type: 'Custom::NatGateway',
Properties:
{# The ARN of the Lambda function backing the custom resource #}
ServiceToken: 'arn:aws:lambda:eu-west-1:XXXX:function:CreateNatGateway'
{# Here we use the subnet_id reference defined in meta.yaml #}
SubnetId: {{subnet_id}}
AllocationId:
Ref: NatEip
#### environment_resource references
environment_output references allow you to refer to resources that belong to other humilis environments.
Parameters:
• environment_name: The name of the environment you are referring to
• layer_name: The name of the layer you are referring to
• resource_name: The logical name of the layer resource
#### layer_output references
layer_output references allow you to refer to outputs produced by another layer.
Parameters:
• layer_name: The name of the layer you are referring to
• output_name: The logical name of the output parameter
In general you should prefer using layer_output references over layer_resource references. The output parameters produced by a layer define an informal layer interface that is more likely to remain constant than the logical names of resources within a layer.
#### boto3 references
boto3 references define arbitrary calls to boto3facade. The latter is just a simpler facade interface on top of boto3.
Parameters:
• service: The AWS service, e.g. ec2 or cloudformation. Note that only only AWS services that have a facade in boto3facade are supported.
• call: The corresponding facade method, e.g. get_ami_by_name. The value of this parameter must be a dictionary with a method key (the name of the facade method to invoke) and an optional args key (the parameters to pass to the facade method). Best to look at the example below to understand how this works.
• output_attribute: Optional. If provided the reference parser will return the value of this attribute from the object returned by the facade method.
Below an example of a layer that uses a boto3 reference:
---
meta:
description:
Creates an EC2 instance using a named AMI
# More stuff omitted for brevity
ami:
description: The AMI to use when launching the EC2 instance
value:
ref:
parser: boto3
parameters:
service: ec2
call:
method: get_ami_by_name
args:
- test-ami
output_attribute: id
humilis will parse the reference using this code:
# Import the Ec2 facade
# Make the call
# Extract the requested attribute
ref_value = ami.id
#### file references
file references allow you to refer to a local file. The file will be uploaded to S3 and the reference will evaluate to the corresponding S3 path.
Parameters:
• path: The path to the file, relative to the layer root directory.
#### lambda references
lambda references allow you to refer to some Python code in your local machine. If your code follows some simple conventions humilis will take care of building a deployment package for you, uploading it to S3, and the reference will evaluate to the S3 path of the deployment package.
Parameters:
• path: Path to either a completely self-contained .py file, or to the root directory of your lambda code. In the latter case your code needs to follow some simple conventions for this to work. More information below.
• dependencies: A list of dependencies to be included in the Lambda deployment package. Dependencies may be either pip installable packages, or paths to local Python packages or modules, or paths to local requirements files.
Example:
ref:
parser: lambda
parameters:
# Path to the root directory containing your lambda code
path: dummy_function
dependencies:
# The Lambda code requires Pypi's pyyaml
- pyyaml
# It also requires a local package in this path
- mycode/mypkgdir
# And this local module
- mycode/mymodule.py
which will evaluate to a S3 path such as:
s3://[bucket_name]/[environment_name]/[stage_name]/[func_name]-[commithash].zip
Code conventions:
Following the example above, the contents of the layer responsible of deploying the dummy_function lambda may look like this:
.
├── dummy_function
│ ├── dummy_function.py
│ └── setup.py
├── meta.yaml
├── outputs.yaml.j2
└── resources.yaml.j2
Basically all your code needs to be included under directory dummy_function. In this case there is only one file: dummy_function.py. External dependencies need to be specified in your setup.py.
#### secret references
secret references retrieve a secret using Python’s keyring module.
Parameters:
• service: The name of the service the secret is associated to.
• key: The key (e.g. the username) that identifies the secret.
Example:
ref:
parser: secret
parameters: {"service": "mysqldb", "key": "adminuser"}
### Custom Jinja2 filters
Humilis defines the following custom Jinja2 filters:
• uuid: A random UUID. Example: {{''|uuid}}.
## Project details
Uploaded source
|
2022-12-02 11:11:07
|
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|
http://umj-old.imath.kiev.ua/volumes/issues/?lang=en&year=2000&number=9
|
2019
Том 71
№ 11
# Volume 52, № 9, 2000
Anniversaries (Ukrainian)
### Anatolii Vladimirovich Skorokhod (On His 70th Birthday)
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1155-1157
Article (English)
### Regularized Brownian Motion on the Siegel Disk of Infinite Dimension
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1158-1165
We construct a process of Brownian motion on the Siegel disk of infinite dimension.
Article (Ukrainian)
### On the Asymptotic Properties of Solutions of Linear Stochastic Differential Equations in $R^d$
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1166-1175
We investigate necessary and sufficient conditions for the almost-sure boundedness of normalized solutions of linear stochastic differential equations in $R^d$ their almost-sure convergence to zero. We establish an analog of the bounded law of iterated logarithm.
Article (English)
### Stochastic Flow and Noise Associated with the Tanaka Stochastic Differential Equation
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1176-1193
We study the properties of the noise (in the Tsirelson sense) that is generated by the solutions of the well-known Tanaka equation.
Article (Russian)
### Measurable Functionals and Finitely Absolutely Continuous Measures on Banach Spaces
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1194-1204
We consider the structure of orthogonal polynomials in the space L 2(B, μ) for a probability measure μ on a Banach space B. These polynomials are described in terms of Hilbert–Schmidt kernels on the space of square-integrable linear functionals. We study the properties of functionals of this sort. Certain probability measures are regarded as generalized functionals on the space (B, μ).
Article (Ukrainian)
### A Remark on the Characterization of the Global Behavior of a Process with Independent Increments
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1205-1207
We show that the analysis of the global behavior of a process with independent increments in terms of the existence of the stationary distribution of the corresponding storage process leads to results that differ from the classical ones.
Article (Russian)
### On the Extrapolation of Entire Functions Observed in a Gaussian White Noise
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1208-1218
We solve the problem of extrapolation of an analytic function of a certain class in the case where its values are observed in a white noise whose intensity is not high.
Article (Russian)
### Estimation of the Intensity of the Flow of Nonmonotone Refusals in the Queuing System $(≤ λ)/G/m$
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1219-1225
We consider a queuing system (≤ λ)/G/m, where the symbol (≤ λ) means that, independently of prehistory, the probability of arrival of a call during the time interval dtdoes not exceed λdt. The case where the queue length first attains the level rm+ 1 during a busy period is called the refusal of the system. We determine a bound for the intensity μ1(t) of the flow of homogeneous events associated with the monotone refusals of the system, namely, μ1(t) = O r+ 1α1 m− 1α rm+ 1), where α k is the kth moment of the service-time distribution.
Article (Russian)
### Nonlinear Transformations of Smooth Measures on Infinite-Dimensional Spaces
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1226-1250
We investigate the properties of the image of a differentiable measure on an infinitely-dimensional Banach space under nonlinear transformations of the space. We prove a general result concerning the absolute continuity of this image with respect to the initial measure and obtain a formula for density similar to the Ramer–Kusuoka formula for the transformations of the Gaussian measure. We prove the absolute continuity of the image for classes of transformations that possess additional structural properties, namely, for adapted and monotone transformations, as well as for transformations generated by a differential flow. The latter are used for the realization of the method of characteristics for the solution of infinite-dimensional first-order partial differential equations and linear equations with an extended stochastic integral with respect to the given measure.
Article (Ukrainian)
### Qualitative Analysis of Systems of Itô Stochastic Differential Equations
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1251-1256
For inhomogeneous systems of Itô stochastic differential equations, we introduce the notion of local invariance of surfaces and the notion of local first integral. We obtain results that give the general possibility of finding invariant surfaces and functionally independent first integrals of stochastic differential equations.
Article (Russian)
### Properties of the Likelihood Ratio for Counting Processes in the Problem of Estimation of Unknown Parameters
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1257-1268
We obtain an asymptotic decomposition of the logarithm of the likelihood ratio for counting processes in the case of similar hypotheses. We establish the properties of the normalized likelihood ratio in the problem of estimation of an unknown parameter.
Article (Russian)
### On the Upper Limit of a Random Sequence and the Law of the Iterated Logarithm
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1269-1271
We obtain some results concerning the upper limit of a random sequence and the law of the iterated logarithm for sums of independent random variables.
Article (English)
### A Probabilistic Representation for the Solution of One Problem of Mathematical Physics
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1272-1282
We consider a multidimensional Wiener process with a semipermeable membrane located on a given hyperplane. The paths of this process are the solutions of a stochastic differential equation, which can be regarded as a generalization of the well-known Skorokhod equation for a diffusion process in a bounded domain with boundary conditions on the boundary. We randomly change the time in this process by using an additive functional of the local-time type. As a result, we obtain a probabilistic representation for solutions of one problem of mathematical physics.
Article (English)
### Multivariate Sobel–Uppuluri–Galambos-Type Bounds
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1283-1293
We improve the known upper and lower bounds for the probability of the fact that exactly k ievents should occur in a group consisting of n ievents simultaneously for all i= 1, 2, ..., d.
Article (Ukrainian)
### On Randomly Perturbed Linear Oscillating Mechanical Systems
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1294-1303
We prove that the amplitudes and the phases of eigenoscillations of a linear oscillating system perturbed by either a fast Markov process or a small Wiener process can be described asymptotically as a diffusion process whose generator is calculated.
Article (English)
### On Sums of Overlapping Products of Independent Bernoulli Random Variables
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1304-1309
We find the exact distribution of an arbitrary remainder of an infinite sum of overlapping products of a sequence of independent Bernoulli random variables.
|
2020-10-28 14:30:16
|
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|
https://www.nag.com/numeric/py/nagdoc_latest/naginterfaces.library.lapacklin.zgtsvx.html
|
# naginterfaces.library.lapacklin.zgtsvx¶
naginterfaces.library.lapacklin.zgtsvx(fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b)[source]
zgtsvx uses the factorization to compute the solution to a complex system of linear equations
where is a tridiagonal matrix of order and and are matrices. Error bounds on the solution and a condition estimate are also provided.
For full information please refer to the NAG Library document for f07cp
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/f07/f07cpf.html
Parameters
factstr, length 1
Specifies whether or not the factorized form of the matrix has been supplied.
, , , and contain the factorized form of the matrix . , , , and will not be modified.
The matrix will be copied to , and and factorized.
transstr, length 1
Specifies the form of the system of equations.
(No transpose).
(Transpose).
(Conjugate transpose).
nint
, the order of the matrix .
nrhsint
, the number of right-hand sides, i.e., the number of columns of the matrix .
dlcomplex, array-like, shape
The subdiagonal elements of .
dcomplex, array-like, shape
The diagonal elements of .
ducomplex, array-like, shape
The superdiagonal elements of .
dlfcomplex, array-like, shape
If , contains the multipliers that define the matrix from the factorization of .
dfcomplex, array-like, shape
If , contains the diagonal elements of the upper triangular matrix from the factorization of .
dufcomplex, array-like, shape
If , contains the elements of the first superdiagonal of .
du2complex, array-like, shape
If , contains the () elements of the second superdiagonal of .
ipivint, array-like, shape
If , contains the pivot indices from the factorization of .
bcomplex, array-like, shape
The right-hand side matrix .
Returns
dlfcomplex, ndarray, shape
If , contains the multipliers that define the matrix from the factorization of .
dfcomplex, ndarray, shape
If , contains the diagonal elements of the upper triangular matrix from the factorization of .
dufcomplex, ndarray, shape
If , contains the elements of the first superdiagonal of .
du2complex, ndarray, shape
If , contains the () elements of the second superdiagonal of .
ipivint, ndarray, shape
If , contains the pivot indices from the factorization of ; row of the matrix was interchanged with row . will always be either or ; indicates a row interchange was not required.
xcomplex, ndarray, shape
If the function exits successfully or = + 1, the solution matrix .
rcondfloat
The estimate of the reciprocal condition number of the matrix . If , the matrix may be exactly singular. This condition is indicated by in 1 … . Otherwise, if is less than the machine precision, the matrix is singular to working precision. This condition is indicated by = + 1.
ferrfloat, ndarray, shape
If the function exits successfully or = + 1, an estimate of the forward error bound for each computed solution vector, such that where is the th column of the computed solution returned in the array and is the corresponding column of the exact solution . The estimate is as reliable as the estimate for , and is almost always a slight overestimate of the true error.
berrfloat, ndarray, shape
If the function exits successfully or = + 1, an estimate of the component-wise relative backward error of each computed solution vector (i.e., the smallest relative change in any element of or that makes an exact solution).
Raises
NagValueError
(errno )
On entry, error in parameter .
Constraint: or .
(errno )
On entry, error in parameter .
Constraint: , or .
(errno )
On entry, error in parameter .
Constraint: .
(errno )
On entry, error in parameter .
Constraint: .
Warns
NagAlgorithmicWarning
(errno )
Element of the diagonal is exactly zero. The factorization has not been completed, but the factor is exactly singular, so the solution and error bounds could not be computed. is returned.
(errno )
Element of the diagonal is exactly zero. The factorization has been completed, but the factor is exactly singular, so the solution and error bounds could not be computed. is returned.
(errno )
is nonsingular, but is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of would suggest.
Notes
zgtsvx performs the following steps:
1. If , the decomposition is used to factor the matrix as , where is a product of permutation and unit lower bidiagonal matrices and is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.
2. If some , so that is exactly singular, then the function returns with . Otherwise, the factored form of is used to estimate the condition number of the matrix . If the reciprocal of the condition number is less than machine precision, = + 1 is returned as a warning, but the function still goes on to solve for and compute error bounds as described below.
3. The system of equations is solved for using the factored form of .
4. Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.
References
Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug
Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore
Higham, N J, 2002, Accuracy and Stability of Numerical Algorithms, (2nd Edition), SIAM, Philadelphia
|
2021-10-19 10:06:50
|
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|
https://www.chemeurope.com/en/encyclopedia/Timoshenko_beam_theory.html
|
My watch list
my.chemeurope.com
# Timoshenko beam theory
### Additional recommended knowledge
The Timoshenko beam theory was developed by Ukrainian/Russian-born scientist Stephen Timoshenko in the beginning of the 20th century. The model takes into account shear deformation and rotational inertia effects, making it suitable for describing the behaviour of short beams, sandwich composite beams or beams subject to high-frequency excitation when the wavelength approaches the thickness of the beam. The resulting equation is of 4th order, but unlike ordinary beam theory - i.e. Bernoulli-Euler theory - there is also a second order spatial derivative present. Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, why the result is a larger deflection under a static load and lower predicted eigenfrequencies for a given set of boundary conditions. The latter effect is more noticeable for higher frequencies as the wavelength becomes shorter, and thus the distance between opposing shear forces decreases.
If the shear modulus of the beam material approaches infinity - and thus the beam becomes rigid in shear - and if rotational inertia effects are neglected, Timoshenko beam theory converges towards ordinary beam theory.
This beam theory, allowing for vibrations, may be described with the coupled linear partial differential equations [1]:
$\rho A\frac{\partial^{2}u}{\partial t^{2}} = \frac{\partial}{\partial x}\left( A\kappa G \left(\frac{\partial u}{\partial x}-\theta\right)\right) + w$
$\rho I\frac{\partial^{2}\theta}{\partial t^{2}} = \frac{\partial}{\partial x}\left(EI\frac{\partial \theta}{\partial x}\right)+A\kappa G\left(\frac{\partial u}{\partial x}-\theta\right)$
where the dependent variables are u, the translational displacement of the beam, and θ, the angular displacement. Note that unlike the Euler-Bernoulli theory, the angular deflection is another variable and not approximated by the slope of the deflection. Also,
• ρ is the density of the beam material (but not the linear density).
• A is the cross section area.
• E is the elastic modulus.
• G is the shear modulus.
• I is the second moment of area.
• κ, called the Timoshenko shear coefficient, depends on the geometry. Normally, κ = 5 / 6 for a rectangular section.
• w is a distributed load (force per length).
These parameters are not necessarily constants.
Determining the shear coefficient is not straightforward (nor are the determined values widely accepted, ie there's more than one answer), generally it must satisfy:
$\int_A \tau dA = \kappa G A \theta\,$
|
2023-02-07 21:56:02
|
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|
http://www.finedictionary.com/correlativity.html
|
# correlativity
## Definitions
• WordNet 3.6
• n correlativity a reciprocal relation between two or more things
• ***
Century Dictionary and Cyclopedia
• n correlativity The character or state of being correlative; correlativeness.
• ***
## Usage
### In literature:
They are correlative and complementary expressions for the same reality.
"Evolution Of The Japanese, Social And Psychic" by Sidney L. Gulick
In the monads, the versatility, rapidity, and power of movement are always correlated with the number of these.
"Scientific American Supplement, Vol. XIX, No. 470, Jan. 3, 1885" by Various
"Mr. Britling Sees It Through" by H. G. Wells
In several ways the correlation of the musician with the non-musical world is now more intimate and conscious than ever before.
"Recent Developments in European Thought" by Various
In the mind of China, wealth is the correlative of strength.
"The Awakening of China" by W.A.P. Martin
I have a theory that any physical defect has its correlative mental and moral defect.
"Famous Modern Ghost Stories" by Various
However, the laboratory psychologists nowadays have gone far beyond such superficial claims for correlations of symptoms.
"Psychology and Industrial Efficiency" by Hugo Münsterberg
You don't give the correlative.
"Memories and Anecdotes" by Kate Sanborn
He is the correlative of Heaven, or rather Heaven itself.
"Myths and Legends of China" by E. T. C. Werner
These latter too have their properties which are also correlated to each other.
"The Mahabharata of Krishna-Dwaipayana Vyasa, Volume 1"
The relation being stated, the correlative duty arises at once.
"The Religions of Japan" by William Elliot Griffis
No attempt is made to correlate home and school work.
"Vocational Guidance for Girls" by Marguerite Stockman Dickson
Perhaps even painting may be referred to it, space being the correlative of time, and color the correlative of tone.
"The Collected Works of Ambrose Bierce, Volume 8" by Ambrose Bierce
To expiate robbery by death, and to expiate murder by the payment of a fine, are correlative ideas.
"The History of Freedom" by John Emerich Edward Dalberg-Acton
The correlation is shown in defects.
"The Voice of Science in Nineteenth-Century Literature" by Various
Here we have a direct relationship, and one which at first sight appears to be exclusive of cross-correlation.
"Territory in Bird Life" by H. Eliot Howard
The word "cause" is correlative to "effect.
"Encyclopaedia Britannica, 11th Edition, Volume 5, Slice 5" by Various
Help and hindrance, stimulation and inhibition, success and failure mean specifically different modes of correlation.
"Creative Intelligence" by John Dewey, Addison W. Moore, Harold Chapman Brown, George H. Mead, Boyd H. Bode, Henry Waldgrave, Stuart James, Hayden Tufts, Horace M. Kallen
An understanding of the latter set of data was facilitated by correlating it with the growth records of marked individuals.
"Life History and Ecology of the Five-lined Skink, Eumeces fasciatus" by Henry S. Fitch
In China there exists a peculiarly close correlation between the two.
"Government in Republican China" by Paul Myron Anthony Linebarger
***
### In poetry:
In child-way her heart's eye did see
The correlation's node:
"Yes," she said, "God takes care o' me,
An' I take care o' God."
"Reciprocity" by George MacDonald
### In news:
Age and undervotes correlated , paper finds.
UK Study Shows Correlation with Pseudoephedrine Sales and Meth Production.
The beer and voting booth correlation that might surprise you.
So it seems the fear of correlations was formalized—made into a turn of phrase, I mean—at around the time that correlations came into formal being.
Beware of spurious correlations , I know.
Debate rages over hip hop's correlation with crime.
In practice, correlations are almost always somewhere between +1.0 and -1.0.
There has been no greater correlation to the team's success this season than Robbie Keane's trademark goal celebration of a cartwheel and forward roll.
When we started using the Z-View Aberrometer, we saw that, in most cases, our refraction , autorefractor and wavefront readings all correlated.
Bush renomination of Stickler, Correll an insult to coal families, UMWA says.
Assets with high correlations tend to move in tandem, which make it harder for active managers to harvest the winners.
Differences don't necessarily correlate with skin color, but rather with geography and climate.
Bush renomination of Stickler , Correll an insult to coal families, UMWA says.
"As sunspot numbers increase, it seems to correlate to a warming of the global climate, and then as sunspot activity decreases there seems to be a general cooling," said Dr Mark Giampapa, the Deputy Director at the National Solar Observatory.
They videotaped the waves they surfed there also, so later they could correlate the video and the velocity data and check their speed at specific points on a wave.
***
### In science:
We note that correlation of local statistics arises naturally from correlation of expression among genes. A simulation study based on a real microarray data set exhibits the extreme anti-conservative behavior of Class 1 tests in the presence of realistic levels of correlation in expression.
A statistical framework for testing functional categories in microarray data
Like both Pearson’s correlation coefficients and Spearman’s rank correlation coefficients, the quotient correlation coefficients and the rank based quotient correlation coefficients measure the level of agreement between two random variables.
Quotient correlation: A sample based alternative to Pearson's correlation
Another issue that needs careful treatment is the question of correlation among different measurements, e.g. due to using the same theory for calculating acceptances. A common practice is to set the correlation coefficient to unity to indicate full correlation.
For the case of pion-pion correlations, the extracted free correlation function is almost identical with the actual correlation function of non-interaction particles.
Test of Bowler-Sinyukov Treatment of Coulomb Interaction
We calculated the correlation of gene expression patterns between each pair of individuals and used these correlations as a distance metric for clustering (distance = 1− correlation).
Bayesian testing of many hypotheses $\times$ many genes: A study of sleep apnea
This function evidences three parameters that characterize the (isotropic) stochastic field: the amplitude Φ = pE (0, 0), the correlation time τ c , which is the decay time of the Eulerian correlation and the correlation length λc , which is the characteristic decay distance.
Trapping, anomalous transport and quasi-coherent structures in magnetically confined plasmas
To implement conditional preparation requires two correlated states, however this correlation need not be entanglement—classical correlations can be suffcient.
Conditions for Factorizable Output From a Beam splitter
Packet-Peak correlation Packet-Peak correlation is a method to assess the use of the OBT as a way to correlate packets to events occurring in different time lines.
A systematic approach to the Planck LFI end-to-end test and its application to the DPC Level 1 pipeline
Later in this review, when we use this correlator to estimate the correlations between shapes, we take the ratio between the smallest and largest k to be 2/800, close to that in WMAP. A more precise correlator should be computed in the l-space in the same way that the estimator is constructed.
Primordial Non-Gaussianities from Inflation Models
Note that there obviously have to be entangled states, which do not violate any Bell inequality, since these represent an upper bound to correlations and do not distinguish between classical correlations (which result from mixing of pure product states) and quantum correlations.
Quantum Entanglement and Geometry
Although new effects from correlations are most pronounced in the regime of strong correlations Γ ≪ U , interesting quantum fluctuation effects already start to become visible in the regime of moderate correlations Γ ∼ U , where perturbative and mean-field theories are often not yet applicable.
Charge transport through single molecules, quantum dots, and quantum wires
To model the counts in correlated fields, we use a standard Cholesky decomposition technique to instill the distribution of galaxy overdensities with the correlation matrix Σ, and then Poisson-sample the correlated distribution to generate the counts in each field.
A Method for Measuring the Bias of High-Redshift Galaxies from Cosmic Variance
That is, it seems from these results that, in terms of average accuracy, it is more important to make effective use of low-order correlations from all available SNPs that are correlated with the target untyped SNP than to take account of unusual higher-order correlations that may occasionally exist.
Using linear predictors to impute allele frequencies from summary or pooled genotype data
We searched for the largest marginal probability P (a|x) numerically among all possible quantum correlations, i.e. correlations that can be achieved by measuring a quantum state, as well as among all no-signaling correlations, denoting this largest quantity by P ∗ (A|X ).
Various quantum nonlocality tests with a simple 2-photon entanglement source
The second canonical correlation λ2 is the largest correlation coefficient between linear combinations of ¯X and u that are uncorrelated with the linear combinations corresponding to the first canonical correlation.
Tests for multivariate normality based on canonical correlations
***
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2020-01-18 00:23:00
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https://proofwiki.org/wiki/Mathematician:Mathematicians/Sorted_By_Birth/1501_-_1600_CE
|
# Mathematician:Mathematicians/Sorted By Birth/1501 - 1600 CE
For more comprehensive information on the lives and works of mathematicians through the ages, see the MacTutor History of Mathematics archive, created by John J. O'Connor and Edmund F. Robertson.
The army of those who have made at least one definite contribution to mathematics as we know it soon becomes a mob as we look back over history; 6,000 or 8,000 names press forward for some word from us to preserve them from oblivion, and once the bolder leaders have been recognised it becomes largely a matter of arbitrary, illogical legislation to judge who of the clamouring multitude shall be permitted to survive and who be condemned to be forgotten.
-- Eric Temple Bell: Men of Mathematics, 1937, Victor Gollancz, London
Previous ... Next
## $\text {1501}$ – $\text {1510}$
### 1501
##### Gerolamo Cardano $($$\text {1501} – \text {1576}$$)$
Italian mathematician, physician, inventor, astrologer and gambler.
### 1510
##### Robert Recorde $($$\text {1510} – \text {1558}$$)$
Welsh physician and mathematician.
Best known for inventing the equals sign. This was just part of his contribution towards the development and systematization of mathematical notation.
show full page
## $\text {1511}$ – $\text {1520}$
### 1511
##### Erasmus Reinhold $($$\text {1511} – \text {1553}$$)$
German astronomer and mathematician, considered to be the most influential astronomical pedagogue of his generation.
show full page
### 1512
##### Gerardus Mercator $($$\text {1512} – \text {1594}$$)$
Flemish geographer, cosmographer and cartographer.
Best known for the $1569$ world map based on a new projection now referred to as Mercator's projection.
show full page
### 1514
##### Georg Joachim Rhaeticus $($$\text {1514} – \text {1574}$$)$
Austrian mathematician who was the sole pupil of Nicolaus Copernicus.
Calculated a table of sines accurate to $10$ decimals.
show full page
### 1515
##### Peter Ramus $($$\text {1515} – \text {1572}$$)$
French logician, humanist and political reformer who fell victim to religious war.
show full page
### 1520
##### Joan Francés Fulcònis $($$\text {c. 1520} – \text {?}$$)$
Mathematician from the area of southern France referred to informally nowadays as Occitania.
Notable for writing one of the earliest mathematics books printed in one of the Occitan dialects, a linguistic group which at the time was subject to political pressure.
show full page
## $\text {1521}$ – $\text {1530}$
### 1522
##### Lodovico Ferrari $($$\text {1522} – \text {1565}$$)$
Italian mathematician who was a student of Gerolamo Cardano.
First one to devise a solution to the general quartic equation, which was later published by Cardano and is now known as Ferrari's Method.
show full page
### 1526
##### Rafael Bombelli $($$\text {1526} – \text {1572}$$)$
Italian mathematician whose influence may have been greater than is currently recognised.
• Documented the rules for multiplication involving negative numbers.
## $\text {1531}$ – $\text {1540}$
##### Hudalrichus Regius $($$\text {fl. 1530s}$$)$
Dutch mathematician who showed that not all numbers of the form $2^n - 1$ for odd $n$ are prime.
He showed in $1536$ that $2^9 - 1 = 511 = 7 \times 73$, and also that $2^{11} - 1 = 2047 = 23 \times 89$.
show full page
### 1532
##### Wilhelm Xylander $($$\text {1532} – \text {1576}$$)$
German classical scholar and humanist who translated the Arithmetica of Diophantus.
show full page
### 1538
##### Christopher Clavius $($$\text {1538} – \text {1612}$$)$
German jesuit and logician.
Best known for:
• Clavius's Law (also written as Clavius' Law), otherwise known as the Consequentia Mirabilis, which states that if by assuming the negation of a proposition you can prove its truth, then that proposition is true.
• Being instrumental in the development of the Gregorian calendar.
• Writing highly-acclaimed and well-received text-books.
### 1540
##### François Viète $($$\text {1540} – \text {1603}$$)$
French amateur mathematician, trained in law, who became a privy councillor under Henry IV of France.
Contributed to many of the early developments of trigonometry and algebra.
Pioneered the use of letters in algebraic equations.
One of the first to use decimal fractions as a matter of course in his published works.
show full page
#### January
##### Ludolph van Ceulen $($$\text {1540} – \text {1610}$$)$
German-Dutch mathematician best known for his calculation of the the value of $\pi$.
The Ludolphine number is the expression of the value of $\pi$ to $35$ decimal places:
$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \ldots$
#### August
##### Joseph Justus Scaliger $($$\text {1540} – \text {1609}$$)$
Expanding the notion of classical history from Greek and ancient Roman history to include Persian, Babylonian, Jewish and ancient Egyptian history.
Also had the ambition to be a mathematician, and made a failed attempt to square the circle.
show full page
## $\text {1541}$ – $\text {1550}$
### 1546
##### Tycho Brahe $($$\text {1546} – \text {1601}$$)$
Danish nobleman famous for his contributions to the science of astronomy.
Refuted the Aristotelian view that the heavens were unchanging, by interpreting observations of supernovae and comets.
Pointed out inaccuracies in the astronomical model proposed by Nicolaus Copernicus, preferring to adhere to the geocentric model of Claudius Ptolemy.
show full page
### 1547
##### Abu al-Faiz ibn Mubarak $($$\text {1547} – \text {1595}$$)$
Indian-born Arab poet and scholar of late medieval India.
Known in the mathematical world for his translation of Lilavati into Persian.
show full page
### 1548
##### Simon Stevin $($$\text {1548} – \text {1620}$$)$
Flemish mathematician, engineer and writer most famous for inventing the decimal notation for the rendering of fractions.
Recommended the use of a decimal system be used for weights and measures, coinage and for measurement of angles.
Wrote most of his work in Dutch, believing it the best language for communication of scientific and mathematical ideas.
show full page
#### April
##### Pietro Antonio Cataldi $($$\text {1548} – \text {1626}$$)$
Italian mathematician and philanthropist who taught mathematics and astronomy.
Founded in Bologna the most ancient known academy of mathematics.
Worked on the development of perfect numbers and continued fractions.
Attempted in vain (as so many before and since) to prove Euclid's fifth postulate.
Supposed to have discovered the $6$th and $7$th Mersenne primes $M_{17}$ and $M_{19}$ in $1588$.
show full page
### 1550
##### John Napier $($$\text {1550} – \text {1617}$$)$
Scots mathematician famous for his development of natural logarithms.
show full page
## $\text {1551}$ – $\text {1560}$
### 1552
##### Jost Bürgi $($$\text {1552} – \text {1632}$$)$
Swiss clockmaker, maker of astronomical instruments and mathematician most famous for publishing a book on logarithms in 1620.
Believed to have invented his own version of logarithms as early as $1588$, but as he failed to publish, John Napier received the credit for the invention.
show full page
### 1560
##### Thomas Harriot $($$\text {c. 1560} – \text {1621}$$)$
English astronomer, mathematician, ethnographer and translator.
His name is variously reported as Harriott, Hariot, or Heriot.
Made great advances in algebra, recognising all roots of equations even whether they be negative or complex.
Had less influence than he might have done, as no mathematical writings of his were ever published in his lifetime.
Was at one point credited with the invention of $>$ and $<$ for greater than and less than, but it appears that they were in fact invented by somebody else.
show full page
## $\text {1561}$ – $\text {1570}$
### 1561
##### John Blagrave $($$\text {c. 1561} – \text {1611}$$)$
English mathematician whose main work was in the field of horology.
Designed and made instruments, including sundials and astrolabes.
show full page
##### Edward Wright $($$\text {c. 1561} – \text {1615}$$)$
English mathematician noted for his contributions to the science of cartography.
show full page
### 1564
#### February
##### Henry Briggs $($$\text {1561} – \text {1630}$$)$
English mathematician most famous for converting Napierian logarithms into Briggsian (common) logarithms.
show full page
### 1564
##### Galileo Galilei $($$\text {1564} – \text {1642}$$)$
Italian mathematician and scientist usually known as just Galileo.
At the forefront of a revolution in the understanding of physics. One of the most influential thinkers in history.
show full page
## $\text {1571}$ – $\text {1580}$
### 1571
#### December
##### Adriaan Metius $($$\text {1571} – \text {1635}$$)$
Dutch geometer and astronomer.
Best known now for his approximation $\dfrac {355} {113}$ for $\pi$ (pi), known to the Chinese and Arabic mathematical traditions centuries earlier.
show full page
##### Johannes Kepler $($$\text {1571} – \text {1630}$$)$
German mathematician and astronomer best known nowadays for Kepler's Laws of Planetary Motion.
Inherited the papers of Tycho Brahe and spent many years analysing his observations, looking for patterns.
His most significant contribution to scientific thought was his deduction that the orbits of the planets are elliptical.
Also pre-empted the methods of integral calculus to find the volume of a solid of revolution by slicing it into thin disks, calculating the volume of each, and then adding those volumes together.
show full page
### 1574
##### William Oughtred $($$\text {1574} – \text {1660}$$)$
English mathematician credited with the invention of the slide rule.
Also credited with inventing a circular version although precedence for this was disputed with his student Richard Delamain.
Experimented with notations in his famously compact writings, inventing some new symbology which stuck, notably $\times$, $\sin$ and $\cos$.
Among others, he may have been influential in the introduction of the symbol $\pi$ for pi, using an abbreviation for the Greek word for periphery (that is, περιφέρεια).
show full page
### 1580
##### Pierre Hérigone $($$\text {1580} – \text {1643}$$)$
French mathematician and astronomer of Basque origin.
Taught in Paris for most of his life.
His greatest influence was his invention of notation.
show full page
#### May
##### Johann Faulhaber $($$\text {1580} – \text {1635}$$)$
German surveyor and engineer who was also a mathematician of the cossist tradition.
A significant influence on several mathematicians, including René Descartes, Jacob Bernoulli and Carl Jacobi.
Best known for his work on series of powers.
show full page
#### June
##### Willebrord van Royen Snell $($$\text {1580} – \text {1626}$$)$
Dutch applied mathematician and astronomer who founded the modern science of geodesy, by pioneering the technique of triangulation.
Developed an improved method for determining the value of $\pi$ (pi) using polygons.
Discovered the Sine Law.
Known today for rediscovering the Snell-Descartes Law in 1621, governing the refraction of light. He did not publish himself. It first appeared in 1703 when it was published in Christiaan Huygens' Dioptrica.
show full page
## $\text {1581}$ – $\text {1590}$
### 1581
##### Edmund Gunter $($$\text {1581} – \text {1626}$$)$
British clergyman, mathematician, geometer and astronomer.
Best remembered for his contributions toward land surveying: Gunter's chain, the Gunter's quadrant and the Gunter's scale.
Credited with the first ever publication, in $1620$, of logarithms of trigonometric functions.
Invented the terms cosine and cotangent.
show full page
#### October
##### Claude Gaspard Bachet de Méziriac $($$\text {1581} – \text {1638}$$)$
First to discuss the solution of indeterminate equations by means of continued fractions.
First member to hold Seat 13 of the Académie Française.
show full page
### 1584
##### Grégoire de Saint-Vincent $($$\text {1584} – \text {1667}$$)$
Flemish Jesuit and mathematician, best remembered for his work on quadrature of the hyperbola.
Gave an early account of the summation of geometric series
Resolved Zeno's paradox by showing that the time intervals involved formed a geometric progression and thus had a finite sum.
show full page
### 1588
##### André Jumeau $($$\text {1588} – \text {1651}$$)$
French amateur mathematician and prior who found the third triperfect number $523 \, 776$.
show full page
#### April
##### Thomas Hobbes $($$\text {1588} – \text {1679}$$)$
English thinker better known for being an astute political philosopher than as a mathematician.
Best known in mathematical circles for believing that he had solved the problem of Squaring the Circle.
Generally considered a mathematical ignoramus, his influence was perhaps of greater importance than generally considered, if only because of the stimulating controversy and discussion he raised.
show full page
#### May
##### Étienne Pascal $($$\text {1588} – \text {1679}$$)$
French tax official and lawyer who also had an interest in science and mathematics.
Noted, and respected, for being unusually honest and honourable in his demanding professional position.
Investigated what is now known as the Limaçon of Pascal.
Most famous, however, for being the father of Blaise Pascal.
show full page
#### September
##### Marin Mersenne $($$\text {1588} – \text {1648}$$)$
French theologian, philosopher, mathematician and music theorist.
Most famous for his work with Mersenne primes.
Claimed in $1644$ that the only primes $p \le 257$ for which $2^p - 1$ is prime are $2, 3, 5, 7, 13, 17, 19, 31, 67, 127$ and $257$. Considering the tools he had at his disposal, he was uncannily accurate.
The first to determine the speed of sound through air.
show full page
## $\text {1591}$ – $\text {1600}$
### 1591
#### February
##### Girard Desargues $($$\text {1591} – \text {1661}$$)$
French mathematician who is considered to be one of the founders of projective geometry.
show full page
#### May
##### Jean Leurechon $($$\text {1591} – \text {1670}$$)$
French Jesuit priest, astronomer and mathematician, known for inventing the Pigeonhole Principle and naming the thermometer.
show full page
### 1595
##### Albert Girard $($$\text {1595} – \text {1632}$$)$
Professional French lutenist who also studied mathematics, working in the fields of algebra, trigonometry and arithmetic.
Gave an inductive formula for the Fibonacci numbers.
First stated in $1632$ that every prime of the form $4 k + 1$ is the sum of two squares in only one way.
show full page
### 1596
#### March
##### René Descartes $($$\text {1596} – \text {1650}$$)$
French mathematician and philosopher who is supposed to have invented the Cartesian coordinate system, and thence the field of analytic geometry.
show full page
#### November
##### Jean Appier Hanzelet $($$\text {1596} – \text {1647}$$)$
French printer and engraver who is believed by some to be the actual author of Récréations Mathématiques of $1624$, supposedly by H. van Etten.
show full page
### 1598
##### Bonaventura Francesco Cavalieri $($$\text {1598} – \text {1647}$$)$
Italian mathematician who worked on optics and motion.
His approach to geometry was a precursor to integral calculus.
Introduced the logarithm to Italy.
A disciple of Galileo.
show full page
### 1600
##### Richard Delamain $($$\text {1600} – \text {1644}$$)$
English mathematician credited with the invention of a circular slide rule although precedence for this was disputed with his tutor William Oughtred.
At one time was mathematics tutor to Charles I of England.
show full page
##### John Speidell $($$\text {fl. 1600} – \text {1634}$$)$
English mathematician known for his early work on the calculation of logarithms.
show full page
##### Pierre de Fermat $($$\text {c. 1600} – \text {1665}$$)$
French lawyer, also an amateur mathematician famous for lots of things. Especially:
Although he claimed to have found proofs of many theorems, few of these have survived.
show full page
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2021-09-21 22:40:02
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http://experiment-ufa.ru/24=120/x
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# 24=120/x
## Simple and best practice solution for 24=120/x equation. Check how easy it is, and learn it for the future. Our solution is simple, and easy to understand, so dont hesitate to use it as a solution of your homework.
If it's not what You are looking for type in the equation solver your own equation and let us solve it.
## Solution for 24=120/x equation:
D( x )x = 0x = 0x = 0x in (-oo:0) U (0:+oo)24 = 120/x / - 120/x24-(120/x) = 024-120*x^-1 = 0-120*x^-1 = -24 / : -120x^-1 = 1/5-1 < 01/(x^1) = 1/5 / * x^11 = 1/5*x^1 / : 1/55 = x^1x = 5x = 5`
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2017-11-20 07:25:15
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http://www.physicsforums.com/showthread.php?t=16868
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# Coulomb's Law-Electric Fields (help please!)
by moonlit
Tags: coulomb, fields, lawelectric
P: 1,004 1) You know that the force between two charges is: $$F_{ele} = K\frac{q_1q_2}{r^2}$$ So to compare the force between and after the change, divide them: $$\frac{F_{ele1}}{F_{ele2}} = \frac{K\frac{q_1q_2}{r_1^2}}{K\frac{q_1q_2}{r_2^2}} = \frac{r_2^2}{r_1^2}$$ And if $r_2 = 9r_1$ then: $$\frac{F_{ele1}}{F_{ele2}} = \frac{9^2r_1^2}{r_1^2} = 9^2 = 81$$ So the answer is that the magnitude of the force will be 1/81 of its original value. 2) You can answer the first part of the question with the equation above for $F_{ele}$. After they bring the spheres into contact, the charges on them move between the spheres so the after they are separated, the charge on each one is different than the charge specified in the beginning of the question. When the spheres touch each other the charges on each one equalizes, so if at first they had (-20.1c + 38.8c) = +18.7c, after they are seperated they will each have half of that i.e +9.35c. 3) Calculate the force that the charges on the circle exert on the charge in the middle, one at a time. You will get the magnitude of the force that operates along the north-south line, and the magnitude of the force that operates along the east-west line. When you have those use pythagoras to find the net force. To find the angle, use any of the trigonometric functions on the right triangle that consists of the two individual forces and the net force.
P: 1,004 Coulomb's Law-Electric Fields (help please!) You don't need to know that, the formula only requires the distance between the spheres which you have. $$F_{ele} = K\frac{q_1q_2}{r^2}$$ Just in case you are confused, r there is NOT the radius of the spheres. It is the distance between them.
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2014-09-03 07:32:11
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https://socratic.org/questions/how-do-you-condense-ln-3-2-ln-9-3
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# How do you condense ln 3 − 2 ln(9 + 3) ?
Jun 27, 2016
$\ln \left[\frac{1}{48}\right]$
#### Explanation:
Simplify the bracket:
$\ln 3 - 2 \ln \textcolor{b l u e}{\left(9 + 3\right)} = \ln 3 - 2 \ln \textcolor{b l u e}{\left(12\right)}$
If you are subtracting the logs, you must have been dividing the numbers.
Use the power rule first:
$\ln 3 - \textcolor{red}{2} \ln \left(12\right) = \ln 3 - \ln {\left(12\right)}^{\textcolor{red}{2}}$
Write as a log of a single number:
$= \ln \left[\frac{3}{12} ^ 2\right]$
$= \ln \left(\frac{3}{144}\right) = \ln \left[\frac{1}{48}\right]$
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2019-03-21 05:42:48
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http://www.j.sinap.ac.cn/hjs/EN/10.11889/j.0253-3219.2015.hjs.38.110601
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Nuclear Techniques ›› 2015, Vol. 38 ›› Issue (11): 110601-110601.
• NUCLEAR ENERGY SCIENCE AND ENGINEERING •
### Simulation study on detachment operation of snowflake divertor for CFETR
WU Haosheng1 MAO Shifeng1 CHEN Bin1 ZHANG Chuanjia1 LUO Zhengping2 GUO Yong2 PENG Xuebing2 YE Minyou1
1. 1(University of Science and Technology of China, Hefei 230026, China) 2(Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031, China)
• Received:2015-09-15 Revised:2015-10-26 Online:2015-11-10 Published:2015-11-13
Abstract:
Background: In the conceptual design of China Fusion Engineering Test Reactor (CFETR), two additional poloidal coils, with respect to International Thermonuclear Experimental Reactor (ITER), are used to generate snowflake divertor configuration proposed recently for the purpose of exploring effective way for reducing heat loads onto divertor targets. Heat flux onto divertor targets was dramatically reduced in detached regime, while the performance of impurity screening would also be reduced due to the decrease of divertor temperature. Purpose: This study aims to simulate the detachment operation of snowflake divertor for CFETR. Methods: The detachment operational status was investigated by numerical simulation based on the edge plasma simulation software SOLPS (Scrape-off Layer Plasma Simulation). A D2 gas puffing in the main chamber was used to change plasma density. Results: When the gas puffing rate was sufficiently high, snowflake divertor of CFETR was completely detached, and the ion flux and heat loads onto the targets significantly decreased. However, the plasma temperature in the divertor region was too low and the impurities could easily pass through the X-point to core plasma, which implied a risk of radiation instability. Conclusion: Therefore, a proper operational status for the snowflake divertor in CFETR should be partial detachment.
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2022-01-21 22:42:01
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https://gmatclub.com/forum/what-is-the-minimum-distance-between-the-point-3-4-and-points-o-261424.html
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# What is the minimum distance between the point (3,4) and points o
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15 Mar 2018, 02:26
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[GMAT math practice question]
What is the minimum distance between the point $$(3,4)$$and points on the circle $$x^2+y^2=1$$?
$$A. 1$$
$$B. 2$$
$$C. 3$$
$$D. 4$$
$$E. 5$$
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Intern Joined: 05 Mar 2018 Posts: 7 What is the minimum distance between the point (3,4) and points o [#permalink] ### Show Tags 15 Mar 2018, 02:31 The radius the circle is 1, the distance between (3,4) and (0,0) is 5, 5-1 = 4 Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 5839 GMAT 1: 760 Q51 V42 GPA: 3.82 What is the minimum distance between the point (3,4) and points o [#permalink] ### Show Tags 18 Mar 2018, 23:19 => Since the distance between $$(0,0)$$ the center of the circle $$x^2+y^2=1$$ and $$(3,4)$$ is $$5$$, and the radius of the circle is $$1$$, the minimum distance between a point on the circle and $$(3,4)$$ is $$5 – 1 = 4.$$ Therefore, D is the answer. Answer: D Attachments circle.png [ 10.42 KiB | Viewed 243 times ] _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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What is the minimum distance between the point (3,4) and points o [#permalink] 18 Mar 2018, 23:19
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2018-07-18 08:50:48
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https://byjus.com/questions/29-16-classify-the-decimal-form-of-the-given-rational-numbers-into-terminating-and-no-terminating-recurring-type/
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# 29/16, Classify the decimal form of the given rational numbers into terminating and non-terminating recurring type.
$$\frac{29}{16} = \frac{29}{2^{4}\times 5^{0}}$$
16 = 2 × 2 × 2 × 2
So the denominator of 29/16 is in the form of 2m × 5n
Where m, n are non-negative integers.
Hence the decimal form of 29/16 will be the terminating type.
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2021-08-02 19:03:36
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https://sites.kowsarpub.com/jcrps/articles/100923.html
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# Evaluating the Effectiveness of Self-regulation Strategy Training on Procrastination, Happiness and Academic Achievement
AUTHORS
Kheironessa Naderi 1 , Hamid Taher NeshatDoost 2 , * , Hooshang Talebi 3
1 Department of Psychology, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan, Iran
2 Department of Psychology, Faculty of Education and Psychology, University of Isfahan, Isfahan, Iran
3 Department of Statistics, University of Isfahan, Isfahan, Iran
How to Cite: Naderi K , NeshatDoost H T, Talebi H. Evaluating the Effectiveness of Self-regulation Strategy Training on Procrastination, Happiness and Academic Achievement, J Clin Res Paramed Sci. Online ahead of Print ; In Press(In Press):e100923. doi: 10.5812/jcrps.100923.
ARTICLE INFORMATION
Journal of Clinical Research in Paramedical Sciences: In Press (In Press); e100923
Published Online: March 13, 2021
Article Type: Research Article
Accepted: November 17, 2020
Uncorrected Proof scheduled for 10 (1)
Crossmark
CHECKING
### Abstract
Objectives: This study was carried with the purpose of evaluating the effectiveness of self-regulation strategy training on procrastination, academic achievement, and happiness of 1st grade high school male and female students.
Methods: It is a quasi-experimental research with a pretest-posttest design involving a control group. Sixty students were selected by means of multistage cluster sampling plan and randomly grouped in 4 groups of 15. Research instrument consisted of Solomon and Rothblum Academic Delay, Oxford Happiness, and Standard Academic Achievement Questionnaire. Test group received 8 self-regulation training sessions of 45-minute duration each.
Results: Results revealed that self-regulation training has a meaningful effect on dependent variables (P < 0.001). In the presented model, "gender", "time and group", "interactive effect of gender and group", and "interactive effect of group and time" were meaningful, as opposed to the "interactive effect of gender and time" and "interactive effect of gender, group, and time" which lacked any significance.
Conclusions: Results of LSD post hoc test showed no meaningful difference between the mean happiness, procrastination, and academic achievement scores of experimental and control groups in the pretest stage. However, in the posttest and follow-up stage, the mean scores of the two groups differed significantly.
Keywords
### 1. Background
Hence, researchers have become evermore concerned about ways of inspiring students to be self-sufficient in terms of teaching and learning and manage their learning by themselves. In otherwords, personally bearing the responsibility of learning and exert control over their responses.
Academic procrastination, as one of the several factors acting against academic achievement, manifests in the form of weak functioning and, ultimately, leads to academic failure. Procrastination involves the deliberate delay in getting assignments done. Rothblum, Solomon & Murakami (1) define procrastination as the act of delaying academic tasks to the point of experiencing subjective discomfort (2). Procrastination in academic performance creates problems such as reduced academic success or achievement, intensification of psychological complications, and loss of self-esteem and happiness in students (3). There are multiple causes for procrastination among students, the most important being stress, casualness, poor time-management, inability to control pleasing behaviors and lack of self-regulation, fear of failure, and the reluctance to do assignments. Pintrich (4) characterizes self-regulation as an active and organized process by virtue of which learners adjust their learning goals and work hard to self-monitor their cognition, motive, and behavior. Hence, self-regulated individuals initially reconsider their behavior and subsequently judge them as they proceed (4).
In recent years, procrastination has been expressed as a pattern of self-regulation failure in terms of not being able to control one's thoughts, excitements, emotions, and functioning in accordance to specific criteria. Park & Rayne (5) percieved procrastination to be positively related to weak self-regulation skills and defensive behavior, such as self-handicapping strategies. Alternative findings Troia, (6) suggest procrastination to be reversely associated to academic score, academic self-regulation, and self-esteem, and, negatively related to self-regulation (7). Meantime, Hen & Goroshit (8) conclude procrastination to be related to lower levels of self-regulated learning and academic self-efficacy.
Shortage of self-regulated behaviors, such as goal-setting, incorporation of strategies, and monitoring thought and learning process results in semi or non-accomplishment of tasks (9). Research demonstrates that happy individuals enjoy a stronger sense of self-control and, in fulfilling their tasks, are more mindful of their advantages than their disadvantages (10). Investigation on procrastrating individuals indicates that those exhibiting more procrastination in doing their assignments, make less use of cognitive strategies (11).
Self-regulation is a skill by means of which individuals can outdo procrastination and alter their academic results, as well as their physical and mental well-being. Studies (12) suggest that learners who exercise self-regulation strategies show less acts of procrastination, and that, the adoption of learning strategies reduces procrastination in students. By helping students examine their multiple failures, self-regulation strategies, ultimately, improves students' active learning (13). At the same time, research has discovered that self-regulated learners have a stronger desire to seek academic counselling, information, and positive learning environment (14).
According to fear of failure theory, stress and anxiety are one of the reasons behind procrastination. Students come to believe that their assignments are stressful and procrastinate to avoid stress. At the same time, self- handicapping theory holds that self- handicapping process begins as soon as individuals are faced with an obstacle in the way of their fine functioning. The motive behind self-handicapping is, generally, avoiding to hurt self-esteem (20). As held by temporal motivation theory, the individual fulfills the goals and assignments which are of greater value and reward to him quicker than those of lesser value and reward (21).
Multiple studies Pietrzak (22), Claessens et al. (23), Schoo (24) and Bondarenko (25), have been conducted to address self-regulation, procrastination, happiness, and academic achievement, with the results unanimously calling for the necessity to heed relentlessly to students' condition, provide them with feedback, improve their self-regulation skill, and employ cheering methods in the course of academic career to add to their happiness and content and reduce their desire for procrastination.
### 2. Methods
This study was conducted under quantitive and qualitive sections. This was done by means of semi-structured interviews held with 17 psychology and educational experts. In the subsequent stage, principle issues that represented self-regulation behaviors were classified into 8 separate categories based on which the parent, teacher, and student questionnaire consisting of 127 short questions and 5 open-ended questions was ultimately prepared. Interview data were analyzed and the result was incorporated in an educational package used for training the experimental group.
In the quantitive section, the investigation followed a quasi-experimental pretest-posttest design involving a control group follow-up. Sixty students were selected by means of multi-stage cluster sampling and asked to fill out the questionnaires. Finally, 30 female and 30 male students were randomly seeded in each of the two experimental and control groups.
#### 2.1.1. Oxford Happiness Questionnaire
This questionnaire was developed in 1989 by Argyle, Martin and Crossland. It is made up of 29 items, each item consisting of 4 phrases graded on a 0 to 3 scale. The individual ultimately obtains a score between 0 to 87.
The range of Cronbach's alpha of questionnaire narratives under separate studies was reportedly 90% with 347 subjects Argyle & Lu (26), 87% with 101 subjects Furnham & Brewin (27). The questionnaire was translated to Persian by Alipoor & Noorbala (28), and Cronbach's alpha 0.93.
#### 2.1.2. Procrastination Assessment Scale
The scale was developed in 1984 by Solomon & Rothblum (29). It is made up of 27 items. Responders have options "never", "rarely", "sometimes", "frequently", and "always" to choose from.
Solomon (30) obtained a Cronbach's reliability measure of 0.79 for the whole scale. The internal consistency value of the narratives were reportedly 0.91. In an alternative study carried out with 31 student subjects, Jokar & Rahimi (31) obtained a 0.91 and 0.84 reliability value by Cronbach's coefficient alpha and internal consistency methods respectively.
#### 2.1.3. Academic Achievement Standard Questionnaire
This 60-item dual-option questionnaire was created in 2010 by Mahmood Saatchi (32) to assess student academic achievement. Scores are added to obtain a final score of 0 to 60. The higher the score, the greater the academic achievement. Cronbach's coefficient alpha for the reliability of the questionnaire was 0.82 based on the study carried out by Saatchi to assess 40 high school students.
### 3. Results
#### 3.1. Postulation
Self regulation strategy training affects procrastination, happiness, and academic achievement.
As indicated in Table 1, the Box's M test significance level (0.128) exceeding test error (0.05) verifies the assumption of homogenity of covariance matrices.
Table 1. Box's M Test for Assumption of Homogenity of Covariance Matric
VariablesValues
Box's M test statistic86.669
F test statistic$12$
Degree of freedom
1st66
2nd30306.936
Significance Level0.128
Table 2 illustrates a meaningful difference between male and female groups in terms of their mean academic achievement, happiness, and procrastination scores. As to the effect of time variable, the mean academic achievement, happiness, and procrastination scores vary for different times (stages).
Table 2. Separate Variance Analysis for Individual Variables
VariablesSum of SquaresdfMean SquaresTest StatisticSignificance LevelEta Squared
Gender
A.a 304.2001304.20011.0790.0010.062
H2688.05012668.05028.3680.0010.144
P168.2001168.2004.7100.0310.027
Time
A.a 2649.81121324.90648.2520.0010.365
H1650.3112825.1568.7730.0010.095
P6553.81123276.90691.7620.0010.522
Group
A.a 1248.20011248.20045.4590.0010.213
H1317.60611317.60614.0090.0010.077
P9159.20019159.200256.4800.0010.604
Gender/time interaction
A.a 22.033211.0170.4010.6700.005
H4.93322.4670.0260.9740.001
P35.033217.5170.4910.6130.006
Gender/group interaction
A.a 1.80011.8000.0660.7980.001
H516.8061516.8065.4950.0200.032
P1290.68911290.68936.1430.0010.177
Time/group interaction
A.a 1290.9002645.45023.5070.0010.219
H1027.7782513.8895.4640.0050.061
P4221.30022110.65059.1030.0010.413
Gender/group/time interaction
A.a 12.23326.1170.2230.8010.003
H5.51122.7560.0290.9710.01
P14.21127.1060.1990.8200.002
Based on the contents of Table 3, mean procrastination score for the students of the control and experimental groups shows a meaningful difference. Meantime, the significance of the effect of time/group interaction on procrastination signifies that procrastination of groups differ at varying stages (times).
Table 3. Results of Between-groups Analysis of Effects for Procrastination Variable a
Source of VarianceSum of SquaresdfMean SquaredF Test StatisticSignificance LevelEta Squared
Group9159.219159.291.814*0.0010.621
Gender168.21168.21.6860.1990.029
Gender/group interaction1290.68911290.68912.938*0.0010.188
Error5586.4895699.759---
a * Significant difference at P ≤ 0.05.
Table 4 shows the significant effect of group on academic achievement. Hence, the academic achievement of the students in the two groups would be different. In Table 5, group has a meaningful effect on happiness. Hence, on the 5% mean level, happiness for the students of the two groups is not the same.
Table 4. Results of Between-groups Analysis of Effects for Academic Achievement Variable a
Source of VarianceSum of SquaresdfMean SquaredF Test StatisticSignificance LevelEta Squared
Group1248.211248.218.022*0.0010.243
Gender304.21304.24.392*0.0010.073
Gender/group interaction1.811.80.0260.8730.001
Error3878.5785669.26---
a * Significant difference at P ≤ 0.05.
Table 5. Results of Between-groups Analysis of Effects for Happiness Variable a
Source of VarianceSum of SquaresdfMean SquaredF Test StatisticSignificance LevelEta Squared
Group1317.60611317.6064.726*0.0340.078
Gender2668.05012668.0509.57*0.0030.146
Gender/Group Interaction516.8061516.8061.8540.1790.032
Error15613.256278.807---
a * Significant difference at P ≤ 0.05.
As illustrated by Table 6, based on the results of the LSD follow-up test applied on the 5% level, there is no meaningful difference between the mean procrastination, academic achievement, and happiness scores of different groups for pretest stage. However, for posttest and follow up, the mean scores significantly differ from one group to the other.
Table 6. Comparison of Mean Values of the Triple Variables in the Experimental and Control Groups Based on Time Level and by Means of LSD Follow-up Test Result a
VariablesMeanMean DifferencetdfSignificance Level
Procrastination
Pretest1.7670.972580.335
C79.1
E77.33
Posttest15.6679.00958*0.001
C77.37
E61.7
Follow-up25.36716.72658*0.001
C76.2
E50.83
Pretest2.0331.312580.195
C39.1
E37.07
Posttest-7.167-5.34858*0.001
C40.93
E48.1
Follow-up-10.667-8.72658*0.001
C41.9
E52.57
Happiness
Pretest0.70.247580.806
C44.73
E44.03
Posttest-5.967-2.26358*0.027
C45.53
E51.50
Follow-up-10.967-4.18658*0.001
C46.30
E57.27
a * Significant difference at P ≤ 0.05.
### 4. Discussion
The result of multi-variate analysis of variance for the effect of time variable reveals the difference existing in the mean scores of the triple variables, suggesting that the extent of these variables varies by time. Similary, the separate effect of gender on these triple variables was significant. At the same time, mean scores of students in the experimental group differed from those in the control group. Hence, self-regultion strategy training has a meaningful effect on at least one of the 3 variables.
Despite of the absence of a parallel study on the effects of self-regulation training on student procrastination, academic achievement, and happiness, numerous research have dealt with these variables independantly and their findings is worthy of being considered. The study conducted by Pietrzak (22) is one good example. He demonstrated procrastination to be an underlying mechanism against the unfavorable effects of failing in self-regualtion and the result of individuals failure to meet the requirements of self-regulation program. At the same time, Hen & Goroshit (8) came up with the conclusion that procrastination is associated with the lower levels of self-regulated learning and academic self-efficiency and is accompanied with high-level stress and anxiety. Alternatively, Schoo (24) and Karmen, Kinga & Edit (18) revealed how procrastination is weakly related to student's academic outlook. Hence, procrastination is generally associated with negative view for academic career. Students who fulfill their assignments with enthusiasm exhibit higher academic apprehension and lesser procrastination. In a separate effort, Claessens et al (23) suggest low level of self-regulation to act as a predictive indicator of high-level procrastination. Meantime, Randy Moore (33) shows how students with higher procrastination score are more reluctant to attend class and seek assistance prior to exams. Congruent to the findings of present study on changes in students' academic achievement, Bondarenko (25) demonstrated in his investigation, the key role of presenting students with feedback and improving their level of self-regulation in improving student academic achievement.
To conclude over the findings and assumptions discussed in this study, it is worth to elaborated, once again, on the indirect effect of self-regulation training on students by empowering them to control and adjust their academic status, which inturn, is considered as one of the main factors for renforcing self-esteem. By improving the level of self-esteem and satisfaction derived from academic achievement, the individual's level of happiness experiences similar growth. In other words, the same way academic success and achievement is expected to raise the level of satisfaction and happiness in students, happy students can be expected to perform better academically. Student happiness is of crucial importance. Joy and happiness are strongly related to the sprightful spirit of the young student generation and their proper handling would promise optimization and reward. Self-regulation can be looked upon as an academic skill, which if acquired, would ultimately raise the level of happiness in students. Self-regulation training offers the essential mental tools for enhancing the academic achievement and level of happiness, and at the same time, reducing the tendency for procrastation.
### References
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Rothblum ED, Solomon LJ, Murakami J. Affective, cognitive, and behavioral differences between high and low procrastinators. J Couns Psychol. 1986;33(4):387-94. doi: 10.1037/0022-0167.33.4.387.
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Rosario P, Costa M, Nunez JC, Gonzalez-Pienda J, Solano P, Valle A. Academic procrastination: Associations with personal, school, and family variables. Span J Psychol. 2009;12(1):118-27. doi: 10.1017/s1138741600001530. [PubMed: 19476225].
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Steel P. The nature of procrastination: A meta-analytic and theoretical review of quintessential self-regulatory failure. Psychol Bull. 2007;133(1):65-94. doi: 10.1037/0033-2909.133.1.65. [PubMed: 17201571].
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Pintrich PR. A conceptual framework for assessing motivation and self-regulated learning in college students. Educ Psychol Rev. 2004;16(4):385-407. doi: 10.1007/s10648-004-0006-x.
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Hen M, Goroshit M. Academic procrastination, emotional intelligence, academic self-efficacy, and GPA: A comparison between students with and without learning disabilities. J Learn Disabil. 2014;47(2):116-24. doi: 10.1177/0022219412439325. [PubMed: 22442254].
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Magno C. Assessing academic self-regulated learning among Filipino college students: The factor structure and item fit. Int J Educ Psychol Assess. 2010;5:61-76.
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Larson R. Is feeling "in control" related to happiness in daily life? Psychol Rep. 1989;64(3 Pt 1):775-84. doi: 10.2466/pr0.1989.64.3.775. [PubMed: 2748778].
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Meltzer L. Executive function in the classroom: Meta-cognitive strategies for fostering academic success and resilience. Learning differences conference. Cambridge, Massachusetts. Research Institute for Learning and Development; 2004.
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Clarebout G, Horz H, Schnotz W, Elen J. The relation between self-regulation and the embedding of support in learning environments. Educ Technol Res Dev. 2010;58(5):573-87. doi: 10.1007/s11423-009-9147-4.
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Pekrun R. The control-value theory of achievement emotions: Assumptions, corollaries, and implications for educational research and practice. Educ Psychol Rev. 2006;18(4):315-41. doi: 10.1007/s10648-006-9029-9.
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Lyubomirsky S. Why are some people happier than others? The role of cognitive and motivational processes in well-being. Am Psychol. 2001;56(3):239-49. [PubMed: 11315250].
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Huebner ES, Suldo SM, Smith LC, McKnight CG. Life satisfaction in children and youth: Empirical foundations and implications for school psychologists. Psychol Sch. 2004;41(1):81-93. doi: 10.1002/pits.10140.
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Kármen D, Kinga S, Edit M, Susana F, Kinga KJ, Réka J. Associations between academic performance, academic attitudes, and procrastination in a sample of undergraduate students attending different educational forms. Procedia Soc Behav Sci. 2015;187:45-9. doi: 10.1016/j.sbspro.2015.03.009.
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Kioslosky JL. Happiness; How are environment affects our well-being and performance. 2002. Available from: http://www.clearinghouseMwsc.edu.manuscripts/292-asp.
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Kandemir M, Palancı M. Academic functional procrastination: Validity and reliability study. Procedia Soc Behav Sci. 2014;152:194-8. doi: 10.1016/j.sbspro.2014.09.180.
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Zimmerman BJ. Attaining self-regulation: A social cognitive perspective. In: Boekaerts M, Pintrich PR, Zeidner M, editors. Handbook of Self-Regulation. Academic Press; 2000. p. 13-39. doi: 10.1016/B978-0-12-109890-2.X5027-6.
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Pietrzak A, Tokarz A. Procrastination as a form of misregulation in the context of affect and self-regulation. Studia Humana. 2016;5(3):70-82. doi: 10.1515/sh-2016-0016.
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Claessens BJC, van Eerde W, Rutte CG, Roe RA. A review of the time management literature. Pers Rev J. 2007;36(2):255-76. doi: 10.1108/00483480710726136.
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Schoo A. Motivational interviewing in the prevention and management of chronic disease: Improving physical activity and exercise in line with choice theory. International Journal of Reality Therapy. 2008;27(2):26-9.
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Bondarenko I. The role of positive emotions and type of feedback in self-regulation of learning goals achievement: Experimental research. Procedia Soc Behav Sci. 2017;237:405-11. doi: 10.1016/j.sbspro.2017.02.080.
• 26.
Argyle M, Lu L. Happiness and social skills. Pers Individ Dif. 1990;11(12):1255-61. doi: 10.1016/0191-8869(90)90152-h.
• 27.
Furnham A, Brewin CR. Personality and happiness. Pers Individ Dif. 1990;11(10):1093-6. doi: 10.1016/0191-8869(90)90138-h.
• 28.
Alipoor A, Noorbala AA. [Preliminary study of the reliability and narrative of oxford happiness questionnaire used for students of Tehran universities]. J Clin Psychol. 1999;5(1&2):55-65. Persian.
• 29.
Solomon LJ, Rothblum ED. Academic procrastination: Frequency and cognitive-behavioral correlates. J Couns Psychol. 1984;31(4):503-9. doi: 10.1037/0022-0167.31.4.503.
• 30.
Solomon LJ. Procrastination assessment scale-students (PASS). Measure for Clinical Practice. 1994;2:446-52.
• 31.
Jokar B, Rahimi M. [Effect of family-relation models on happiness of a group of high school students in Shiraz]. Iran J Psychiatry Clin Psychol. 2007;13(4):376-84. Persian.
• 32.
Saatchi M. [Productivity psychology]. Tehran, Iran: Virayesh Publishing; 2010. Persian.
• 33.
Moore R. Academic procrastination and course performance among developmental education students. Research and Teaching in Developmental Education. 2008;24(2):56-67.
• Copyright © 2021, Journal of Clinical Research in Paramedical Sciences. This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/) which permits copy and redistribute the material just in noncommercial usages, provided the original work is properly cited.
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2021-04-23 02:31:22
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https://www.physicsforums.com/threads/triple-integral-for-volume.395277/
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# Triple Integral for volume
#### hackmonkey7
1. Homework Statement
The region R in 3D is cut from the first octant (x,y,z >= 0) by the plane X+Z = 1, Y+2Z = 2.
Set up the volume in all 6 ways in rectangular coordinates.
Then evalute the volume in two of these ways.
Make sure to specify limits of integration in every case.
2. Homework Equations
I figure Y+2Z = 2 is the upper bound for Z and Z=0 is the lower bound.
Here's a rough sketch I drew of the base on X-Y plane, and the 3-D of what I imagined it to look like:
[PLAIN]http://img576.imageshack.us/img576/2599/calc3.png [Broken] [/PLAIN]
3. The Attempt at a Solution
So I put the triple integration in:
$$V(R) = \int_0^1{\mathrm \int_(2X)^(2-2X){\mathrm \int_0^(1-(1/2)Y){\mathrm }{\mathrm d}Z}{\mathrm d}Y}{\mathrm d}X$$
When I work this out, I keep getting a negative value!
I know the answer is suppose to be 2/3 because a rectangular pyramid's volume = 1/3 * height * base area, which the base area is 1x2 times 1 height * 1/3 = 2/3.
But I cant seem to manipulate it anyway to get it. Where am I going wrong? Am I setting the wrong plane as the Z bound?
Thanks for any help.
1. Homework Statement
2. Homework Equations
3. The Attempt at a Solution
Last edited by a moderator:
Related Calculus and Beyond Homework Help News on Phys.org
#### LCKurtz
Homework Helper
Gold Member
1. Homework Statement
The region R in 3D is cut from the first octant (x,y,z >= 0) by the plane X+Z = 1, Y+2Z = 2.
Set up the volume in all 6 ways in rectangular coordinates.
Then evalute the volume in two of these ways.
Make sure to specify limits of integration in every case.
2. Homework Equations
I figure Y+2Z = 2 is the upper bound for Z and Z=0 is the lower bound.
No, that isn't correct. Both of your slanted planes are part of the "roof". You have to break it up into two different integrals if you integrate in the z direction first taking separate xy regions under each section of the corresponding z plane.
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2019-10-17 05:20:32
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https://www.talkstats.com/threads/splitting-a-matrix.43536/
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# Splitting a matrix
#### Taibo
##### New Member
Hi,
I have a matrix x with 97 rows and used the function sample like this :
rownumbers=sample(1:97,size=70)
data1=x[rownumbers,]
I'm having trouble to get the rows which are not inside data1 into another matrix.
Does anyone have an idea?
greets,
Taibo
#### Dason
##### Ambassador to the humans
You could use setdiff on rownumbers and 1:97 to find which rows aren't in rownumbers. Alternatively you could do something like
Code:
x$id = 0 x$id[rownumbers] = 1
split(x, x$id) Note that I'm on a phone so that code might need to be slightly modified. #### trinker ##### ggplot2orBust Hello. Welcome to talkstats. A couple of logistics to make question asking and response getting smoother: When you're posting code, dataframes or computer output it's helpful to wrap this information in code tags by: 1. either clicking the pound (#) sign icon or 2. wrap with [NOPARSE] Code: some code [/NOPARSE] which produces: Code: some code For more see this (LINK) Also... If you give us a dummy data set of what x looks like it is helpful in understanding the problem. maybe something like: Code: [COLOR="#696969"]## First scale the problem down to a MWE:[/COLOR] x <- matrix(rnorm(9*3), ncol=3) rownumbers <- sample(1:nrow(x), size=5) data1 <- x[rownumbers,] c(1:nrow(x))[!1:nrow(x) %in% rownumbers] [COLOR="#696969"]## or...[/COLOR] setdiff(1:nrow(x), rownumbers) [COLOR="#696969"]## so...[/COLOR] x[setdiff(1:nrow(x), rownumbers), ] [COLOR="#696969"]## [,1] [,2] [,3] ## [1,] -0.08743614 0.7005841 0.3234109 ## [2,] -1.52568263 -0.4110520 -0.3591293 ## [3,] -1.65364148 0.2565882 0.8732189 ## [4,] 1.54629539 -1.1694642 -1.3748573 [/COLOR] #### Taibo ##### New Member seems like this is going to work.. the vector x$id seems to be correctly specified.
I still get an error after that last command, however:
Error in split[x, x$id] : object of type 'closure' is not subsettable Do you know how to fix this error? #### Taibo ##### New Member I did not know this ! will keep this in mind ! #### trinker ##### ggplot2orBust Taibo, It's difficult if not impossible to diagnose error messages if we can't reproduce them. We don't know what x is even other than it's a matrix. Please post sample data that reproduces the error. #### Taibo ##### New Member fixed the error, I made a mistake! thank you very much! #### trinker ##### ggplot2orBust Maybe try: Code: id <- rep(0, nrow(x)) id[rownumbers] <- 1 split(x, id) #### trinker ##### ggplot2orBust It depends if x is actually a dataframe or a matrix whether Dason's solution will work as is. #### Dason ##### Ambassador to the humans seems like this is going to work.. the vector x$id seems to be correctly specified.
I still get an error after that last command, however:
Error in split[x, x\$id] :
object of type 'closure' is not subsettable
Do you know how to fix this error?
split is a function so use () and not []
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2022-09-26 01:00:29
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https://undergroundmathematics.org/vector-geometry/r6491
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Review question
# Can we show the centre of mass moves in a straight line? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource
Ref: R6491
## Question
Three particles $A, B$ and $C$, each of mass $m$, are moving in a plane such that at time $t$ their position vectors with respect to the origin $O$ are \begin{align*} (2t+1)\mathbf{i}&+(2t+3)\mathbf{j} \\ (10-t)\mathbf{i}&+(12-t)\mathbf{j} \\ (3t^2-4t+1)\mathbf{i}&+ (-3t^2+2t)\mathbf{j} \end{align*}
respectively.
1. Show that the centre of mass of these three particles moves in a straight line and find the Cartesian equation of this line. Find also the value of $t$ for which the centre of mass is instantaneously at rest.
2. Verify that the particles $A$ and $B$ are both moving along the straight line with equation $y=x+2$ and that they collide when $t = 3$.
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2019-02-17 19:05:33
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https://codegolf.meta.stackexchange.com/questions/9880/necessity-of-wrapping-solution-into-method-function
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# Necessity of Wrapping Solution into Method/Function
I've noticed that sometimes people using exotic languages like Brainfuck don't ever explicitly wrap a solution into a method/function. While sometimes I see solutions in Java and such where they are wrapping their solution into a function.
Is it actually necessary (if the question didn't specify it) to wrap a solution into a reusable function?
As an example, here is my solution to The Letter A without A:
(3<2).ToString()[3-2]
Is it necessary to make it like so?
void Solution { (3<2).ToString()[3-2] }
Which also leads me to ask is it necessary to also explicitly print the result when things like PowerShell implicitly print the result all the time, and this can be done with my C# snippet using C# Interactive for instance?
## 1 Answer
The difference is, some esoteric languages (like Brainfuck) don't have a function-like construct in the language. In those languages, the submissions are full programs - you run it, pass the input in (through STDIN or whatever method it uses), and you get the output (from STDOUT or whatever method it uses).
We require that all submissions be full programs or reusable functions. For REPL (read-eval-print-loop) environments like Python, Javascript, and PowerShell consoles, having the output implicitly printed is fine - so long as you mark the answer as being a REPL answer (as opposed to a full program that doesn't require the REPL).
Here are some resources you might want to check out:
• Thanks for the quick and concise answer! – AquaGeneral Aug 21 '16 at 7:42
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2020-08-07 00:45:01
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https://www.physicsforums.com/threads/accounting-will-it-live-on.159129/
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# Accounting -Will it live on?
There is one thing my family is well known for.. It's maths.. My dad can do such complex equations in his head.. Does em faster than I can on the calculator.. Now my parents want me to become either an accountant, or an engineer..
Well I just turned 17, so My theories might be umm.. dunno how to put it.. Garbage?
Well anyway this is what I think.. I always hear everyone saying their becoming an accountant etc.. I know soooooooo many people that are becoming an accountant.. Well what I wonder is that, will accounting really survive? With computers becoming faster and more complex, I think it'll be a matter of time before computers replace accountants.. Also since there are so many people going in the accounting field will there not be more workers than work?
When I talk to people about this, they tell me that accounting is very hard and that most people that go in the field don't really graduate out as accountants..
I don't mind becoming an engineer, but I heard it's really hard (as I read the engineering thread).
Well anyway what do U guyz think? Is accounting gonna be as good of a source of bread in the house in the future as it is now? And will computers really take over?
jtbell
Mentor
With computers becoming faster and more complex, I think it'll be a matter of time before computers replace accountants..
Ever hear the expression, "Garbage in, garbage out?"
Suppose you have software that can do all the arithmetic in a flash, for a company's financial statement. Someone still has to ensure that the right numbers are going into the right inputs, which means understanding all the laws and accounting practices that deal with financial statements.
Ok I get what u mean
by the way never heard of that sayin..
I know that you're worried about competing with computers, but there's also the issue of outsourcing to worry about. As Jtbell said, you need a human being to check the math. But why pay a high salary to an American accountant when a guy in India can do the same job for half the pay? A lot of engineering jobs are being outsourced right now, and I wouldn't be surprised if the same thing happens to accounting. My dad, for example, had his engineering job outsourced somewhere, and he had a hard time finding another one (the same thing happened at his next job, btw). And he has two Master's degrees in electrical engineering! I'm not trying to scare you away, but it's something to think about as far as accounting and engineering go.
From what I can tell, in terms of job security, it looks like medicine and academics are good fields. I rarely see a doctor or a college professor get laid off.
\From what I can tell, in terms of job security, it looks like medicine and academics are good fields. I rarely see a doctor or a college professor get laid off.
I've actually a couple articles which talks about how surgeons are starting to become "outsourced" in a way. Apparently, a growing number of people are flying to places like India, where from what I understand, you can get heart surgery by U.S.-trained and licensed doctors for something like $10k. I've actually a couple articles which talks about how surgeons are starting to become "outsourced" in a way. Apparently, a growing number of people are flying to places like India, where from what I understand, you can get heart surgery by U.S.-trained and licensed doctors for something like$10k.
Really? I'd be interested to read more about this, if you have any extra information. I'm asking because I happen to have a brother who wants to go to med school (and he's motivated mostly by the job security). I think he might find this of interest...and I might be able to wrangle him into physics while I'm at it.
Really? I'd be interested to read more about this, if you have any extra information. I'm asking because I happen to have a brother who wants to go to med school (and he's motivated mostly by the job security). I think he might find this of interest...and I might be able to wrangle him into physics while I'm at it.
Sure. Here's an article that Time had on the subject. (However, if I had to guess, I'd say that the medical field as a whole isn't in danger of being outsourced. Only the really expensive surgeons have the risk: specialists and general practitioners can't really be cheaply outsourced.)
Is accounting one of those jobs where "the longer you stay, the more valuable you are to the company"? Because I know that programming/engineering AREN'T...
chroot
Staff Emeritus
Gold Member
Because I know that programming/engineering AREN'T...
This is contrary to my own experience and the experience of every engineer and programmer I know. I work in Silicon Valley, and know many...
- Warren
Let me put it this way, say you are a wiz at maths. If you do accounting you will probably excel and be fine, but if you do real maths you have a chance of taking you perceptive of the world to a whole nother level. Depends what you value more I guess, Intellectual Reward Vs Monetary Reward.
Sorry about the generalization. But does being in the accounting profession for a long time make you more valuable?
Having dealt with MANY accountants, there is no way that computers will ever replace humans when it comes to understanding tax rules. While I don't live there, I understand the USA has the most complicated tax law in the universe.
That said, accounting is about my least favorite activity in the world. Yes, you can make a lot of money shuffling other people's numbers around but, unless you have a calling for it, your life will drain away in an activity of no lasting social value. Think hard before choosing an entire life's work just based on monetary reward.
As much as many people here whine about the difficulties of finding jobs in science, there are many many many other people with well-paying jobs that they bitterly hate and can't quit because they are too heavily invested in the career.
It is not a black-and-white choice between eternal uncertainty and eternal drudgery but be cautious...
siddevt and zigzag7
Accounting is not math.
Ben Niehoff
Gold Member
Most accountants get paid peanuts, work long hours in basements, deal with horribly out-of-date software that doesn't actually work, and end up spending most of their time doing data entry and other dreadfully boring tasks. Unless you get a CPA, learn some tax law, and such; then you can do better.
Given the choice between accounting and engineering, I would definitely choose engineering. With strong math skills you could also consider being a physicist or a mathematician.
This is contrary to my own experience and the experience of every engineer and programmer I know. I work in Silicon Valley, and know many...
I just wanted to chime in "Amen." If you aren't more valuable to the company as time goes on, you just aren't doing your job properly.
So, why can't a really complicated expert system be made to process through all the tax laws? It seems to me it's only a matter of time before software automates all professions, and accounting seems to be one of the first.
turbo
Gold Member
So, why can't a really complicated expert system be made to process through all the tax laws? It seems to me it's only a matter of time before software automates all professions, and accounting seems to be one of the first.
Because the devil is in the details. Once you start going through the tax codes (if you are good at it), you will find ways to make your company more profitable simply by paying less taxes. Ever wonder why companies often lease equipment instead of buying it? There's always a reason. Ever think about who owns the company that owns the equipment that the company leases? How about paying wholly-owned offshore-based companies more than the going rate to provide services that you could have sourced domestically? There is a reason for the complexity of the tax codes. Tax laws are written by lobbyists who represent businesses.
If the tax codes were simplified dramatically, and we could keep lobbyists from having such free access (and influence) with our members of Congress, then I might be worried about whether accountants might be replaced. Not now.
siddevt
So, why can't a really complicated expert system be made to process through all the tax laws?
LOL - Go for it.
In the mean time, I think we are much closer to a solid theory of Quantum Gravity.
siddevt
Most accountants get paid peanuts, work long hours in basements, deal with horribly out-of-date software that doesn't actually work, and end up spending most of their time doing data entry and other dreadfully boring tasks. Unless you get a CPA, learn some tax law, and such; then you can do better.
Given the choice between accounting and engineering, I would definitely choose engineering. With strong math skills you could also consider being a physicist or a mathematician.
I'm actually working part-time in an accounting office at a local WalMart. It's really interesting how much of this lines up with the reality of the low-levels of accounting in an organization.
Most of what I do is what you would expect. Sort through stacks of ones and fives, all kinds of crumbly coupons and bottle return slip, and enter those into the books i.e. in WalMart's info system, which is ridiculously low-level and unhelpful on my end. The best tool I get is a linear search function to scan previous day's receipts (in a program that indexes all receipts by time and register). The room I work in is cold and windowless room, with cameras eyeing me from every direction. Anyway, after deposits I research the sales entered into the system versus what was physically sent to the office at the end of the day. Then I try to reconcile the two if I can find any cashier errors. Other than that I post a ton of random cash payments to various accounts and have head cashiers perform various randomly chosen audits on a given day.
There's not many ways you can automate that. Someone does have to check to ensure that the information systems provide accurate results. And the information systems actually help inform and speed along the process.
So there's at least two aspects of accounting that won't change. Auditing and information systems (as in building information systems and the audit process). There's also related business issues like internal controls, law (as in compliance with laws), and taxes.
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my dads a CPA. he owns his own practice in three different states. he has made quite a living on his salary alone. although he says times are getting tough now that the economy is down so i dk right now how things are going and i dont plan to ask hahas./
Because the devil is in the details. Once you start going through the tax codes (if you are good at it), you will find ways to make your company more profitable simply by paying less taxes. Ever wonder why companies often lease equipment instead of buying it? There's always a reason. Ever think about who owns the company that owns the equipment that the company leases? How about paying wholly-owned offshore-based companies more than the going rate to provide services that you could have sourced domestically? There is a reason for the complexity of the tax codes. Tax laws are written by lobbyists who represent businesses.
If the tax codes were simplified dramatically, and we could keep lobbyists from having such free access (and influence) with our members of Congress, then I might be worried about whether accountants might be replaced. Not now.
Yeah, but all of this pretty much has to do with the concern of outsourcing, doesn't it? What about automation itself?
ACCOUNTING DOES NOT REQUIRE MUCH MATH. Ok I had to put my 2 cents in because i actually work in general accounting (also have my Bach of Sci in accounting) and one day plan to become a cpa. To get to take the exam, i needed about another 36 credit hours of school , so instead of choosing to be like my friends and go for the masters in accounting then sit for the cpa(thats the usual way) I chose to go for software engineering. I hope to make tax/accounting software some day and yes it is possible, there are tons of companies that do it.
To answer the thread, yes accounting will always live on because it is much more like Law than anything else. It requires extremely little math (finance gets into math and statistics) but, accounting is essentially extreme bookeeping. All you do is put the right numbers into the correct accounts according to GAAP standards(if in US), and based off of the numbers presented, managers make their decisions. The accounting rules can and do change(albeit slowly over time - gaap will eventually be replaced by ifrs), and yes a new congress can make dramatic changes in the tax code.
Here's what I've concluded about comparing the 2 careers:
accounting pays lower STARTING salary than software engineers, but have the potential to make more (work at one of the big 4 for 7-8 years and you'll make bank). however the work requires very very little creativity and ingenuity. It can become mundane preparing the same reports/statements month to month. But the job is very secure and you know you will at least be making decent middle class living at the least.
software engineering pays higher starting salary but typically doesn't have as much potential to make the big big money. It requires alot of critical thinking and you get much more creativity. The jobs are there, but tend to be shorter lived.
After being through both colleges I can definitely say engineering school is much harder than business school and pays less if you judge it based on how much you know versus how much your paid. But it makes itself up in other ways like creativity and is kinda enjoyable.
don't look at the salary too much. yes you can make +200,000 /yr in accounting (ex:work at one of the big 4 for 8 years and move onto your own thing or a medium sized company), but there are plenty of engineers making huge money too. most accountants don't get paid nearly that amount though and are typical middle class people.
Well what I wonder is that, will accounting really survive? With computers becoming faster and more complex, I think it'll be a matter of time before computers replace accountants..
Don't think so. At the end of the day, someone has to sign a piece of paper saying, I promise that these numbers are accurate, and if they aren't, very, very bad things will happen to me. If you have a complex computer program, how do you know that the numbers it is putting out aren't total garbage?
siddevt
...accounting will always live on because it is much more like Law than anything else.
Well said. Accountants will be replaced by computers at the same time as Lawyers are.
And, yes, accounting has almost nothing to do with Mathematics.
siddevt
I really think you're right...softwares can be made which are good enough to even see if there's an error in input...at least warn you if it thinks the input is wrong.
In today's age, the only real job of human beings is to develop; not merely following the things or procedures which are already develop...cause a computer can do that. Of course excluding very complex things like application of mechanics to various bridges...but someday (very late actually), computers will takeover that job too.
And yes, engineering is NOT easy...:D...you should really think before going in there.
Ever hear the expression, "Garbage in, garbage out?"
Haha, I love this, my lecturer used to say that all the time, that and a particular conan the barbarian quote are my mantra in engineering.
Anyway, I'm from Australia, and coincidentally just lodged my Tax return tonight.
No Accountant, just plug in your numbers into the little box in an automated program commissioned by the government and bob's your uncle.
Next years tax is planned to be even simpler, basically "look and check that these automatically obtained figures are approximately right, sign here, tax done".
Of course this doesn't directly mean that accountants are gunna be out of business, big business still will probably need them to keep their accounts in order and stuff.. but the initial moves have already been made here to simplify the whole system, and at least for individuals, remove the need for an accountant.
In my opinion it's only a matter of time, though the great thing about accounting is that it's all based on silly rules to keep themselves in a job, so i'm sure they'll make up a new set that will complicate things all over again, keeping themselves 'relevant'.
:P
Oh, and as far as the initial post goes. Become an engineer. It's a total love-hate relationship at times, but the frustration is worth it when you get it right.
Like lubuntu said, it's a choice between drudgery and good pay, or average pay and the possibility of actually achieving something meaningful in your lifetime. I know what I would choose... take the money, it buys EVERYTHING.
turbo
Gold Member
Well said. Accountants will be replaced by computers at the same time as Lawyers are.
And, yes, accounting has almost nothing to do with Mathematics.
When I was close to having put in my 10 years at the paper mill (vested in the retirement program), I was looking for something else to do, and my cousin urged me to learn how to write accounting application programs for dBase. I told him I didn't know anything about accounting, and he said "You don't have to." He was right. Just learn some basic double-entry bookkeeping, and start automating peoples' businesses. I once wrote an entire inventory-control program for a manufacturer of tree-harvesting equipment that included internal, wholesale, and retail pricing for his sales accounts, and flowed the costs and profits directly to his front office's accounting program. His accountant wanted to be able to flip the program from first-in-first-out to first-in-last-out in order to explore tax advantages relating to the inventory, so I included that, too. No fancy math, there. Accounting might be deadly-dull work, but it needs to be done to keep businesses healthy and those jobs won't go away anytime soon.
I just want to be really clear here. I don't like accounting and I WISH computers could replace accountants. I don't recommend anyone (who has a soul) to go into accountancy.
...just plug in your numbers into the little box in an automated program commissioned by the government and bob's your uncle.
I can do my taxes myself as well; however, my small company has easily spent over a hundred thousand dollars on accountants in the last 10 years. This is despite me doing 5 times more of the bookwork myself than most owners do. The difference between the average Joe doing their personal tax and corporate taxes is the difference between basic Arithmatic and Differential Geometry. Sorry, bad analogy. Accounting really has no math - it is tax law.
People make corporate decisions, not computers. As a company owner you need people who know the rules better than you do (accountants) so that you can make educated decisions. I don't have the time (or desire) to learn tax law and also run a company. A computer can help push the numbers around but will never replace a human in many places.
Would you want a computer choosing your clothing for you? How about choosing where you go on holiday? Well, I wouldn't want a computer making decisions for my company - which is what people are suggesting when they want to replace my accountant with a machine. The difference between a good accountant and a bad one is staggering.
I think that most people really have NO idea what an Accountant actually does. Don't let your kids grow up to be accountants
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Here is a better analogy:
Will computers replace Mathematicians soon (not assist, replace)? Just program them with the appropriate axioms and rules of logic and send them off proving everything. We just have to come back afterward and see what new areas of Mathematics they have opened up for us!
Discuss...
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2022-05-26 17:36:33
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https://socratic.org/questions/how-do-you-find-the-integral-of-f-x-2xsin4x-using-integration-by-parts#231900
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# How do you find the integral of f(x)=2xsin4x using integration by parts?
Feb 27, 2016
int 2x sin(4x) dx=(-1)/2x cos(4x) + 1/8 sin(4x) + C, where $C$ is the constant of integration.
#### Explanation:
Let be $f \left(x\right) = 2 x \sin \left(4 x\right)$.
We want to find int 2x sin(4x) dx. We know the integral of the function $\sin$ or $\cos$ so it would be good for us to get rid of $2 x$ by differentiating it.
Let's say $g \left(x\right) = 2 x$ and $h ' \left(x\right) = \sin \left(4 x\right)$.
So $g ' \left(x\right) = 2$ and $h \left(x\right) = \frac{- 1}{4} \cos \left(4 x\right)$.
int g(x) h'(x) dx = g(x)h(x) - int g'(x) h(x) dx using integration by parts.
int 2x sin(4x) dx = 2x*(-1)/4 cos(4x) - int 2 *(-1)/4 cos(4x) dx
=(-1)/2x cos(4x) + 1/2 int cos(4x) dx
$= \frac{- 1}{2} x \cos \left(4 x\right) + \frac{1}{2} \cdot \frac{1}{4} \sin \left(4 x\right)$
$= \frac{- 1}{2} x \cos \left(4 x\right) + \frac{1}{8} \sin \left(4 x\right)$
Therefore, int 2x sin(4x) dx = (-1)/2x cos(4x) + 1/8 sin(4x) + C, where $C$ is the constant of integration.
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2021-10-15 21:32:30
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https://themeparkatitsdarkest.com/tag/lion-fight/
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## Joe Schilling: All the Glory
Posted in Glory, Joe Schilling, Journalism, Kickboxing, Martial Arts, MMA, Real Men with tags , , , , , , , , , , , , , , , , , , , , , , , , on March 29, 2016 by Jarrod Boyle
International Kickboxer Magazine, Jan/Feb 2016
Joe ‘Stitch ‘Em Up’ Schilling is a significant figure amongst the new breed of fighters. Continue reading
## Lion Fight 25: Cosmo Alexandre
Posted in 'John' Wayne Parr, Kickboxing, Muay Thai with tags , , , , , , on December 19, 2015 by Jarrod Boyle
International Kickboxer Magazine, Nov/Dec 2015
How is your body holding up to the rigors of kickboxing? Continue reading
## Lion Fight 25: ‘John’ Wayne Parr versus Cosmo Alexandre
Posted in 'John' Wayne Parr, Journalism, Muay Thai, Real Men with tags , , , , , , , , , , , , , , , , , , , , , , , , , , on December 18, 2015 by Jarrod Boyle
International Kickboxer Magazine, Nov/Dec 2015
‘John’ Wayne Parr is scheduled to take on Cosmo Alexandre at Lion Fight 25 in Las Vegas, Nevada on October 23. Continue reading
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2023-03-20 22:41:45
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http://glaucia.servers.ec-lyon.fr/spip.php?article189&lang=en
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# Fluid Mechanics and Acoustics Laboratory - UMR 5509
LMFA - UMR 5509
Laboratoire de Mécanique des Fluides et d’Acoustique
Lyon
France
## Wouter Bos
Research Team: Turbulence & Instabilities
tel: 04.72.18.62.04
email
Research in the spotlight:
### Power fluctuations in turbulence [with Rémi Zamansky (IMFT)]
How universal are the power-fluctuations measured in the electricity output of a windmill? Indeed, the power fluctuation experienced by a windmill blade are determined by the power absorbed by the incoming, turbulent airflow. The universal character of turbulent power fluctuations is, in general, of major importance in the modeling and understanding of fluid flows in industry, geophysics and astrophysics. However, the manner in which the turbulence is generated can be radically different in different types of flows, so that it seems improbable that universality is observed among different types of flow. Nevertheless, we show that different turbulence generating mechanisms can lead to similar statistics of the energy input since the input is not only determined by the forcing mechanism, but also by the turbulent flow itself.
### Football players and turbulent trajectories (with B. Kadoch and K. Schneider)
(...) Indeed, even though a football match does not seem to be ergodic, and some may argue that the trajectories of football players are not completely random, the long-time PDF of the directional change converged to a shape close to the one for fluid particles (...)
Recent experiments and simulations have shown that unsteady turbulent flows display a universal behavior at short and intermediate times, different from classical scaling relations. The origin of these observations is explained and the exact form of the observed universal scaling is derived. The derived scaling for the normalized dissipation rate $C_\epsilon$ as a function of the Reynolds number $R_\lambda$ is,
$$C_\epsilon\sim R_\lambda^{-15/14},$$
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2019-11-21 01:45:08
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https://davidlowryduda.com/2401-additional-examples-for-test-3/?replytocom=46
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# 2401: Additional Examples for Test 3
In the past, I have talked about how good a supplemental source of information the Khan Academy is. Again, it is supplementary. But it seems to have lots of fully worked and fully explained examples of the concepts of chapters 17 and chapter 18 (sections 1 through 4) — the topics for your next exam. I have placed the relevant links below.
Double Integrals ( I, II, III, IV, V, VI)
Triple Integrals( I, II, III)
Line Integrals( I, II, III, IV)
Clever Line Integrals (I, II, III, IV, V, VI, VII, VIII, IX)
As always, if you have any questions let me know. I will be hosting a review session in the Math Lab at 5 – come prepared and with questions. I suspect we’ll be focusing on the iterated integrals of Chapter 17. Good luck!
This entry was posted in Georgia Tech, Mathematics and tagged , , . Bookmark the permalink.
### 15 Responses to 2401: Additional Examples for Test 3
1. mixedmath says:
Recall that $vec{F} (x,y,z) = P hat{i} + Q hat{j} + R hat{k}$ on a curve $C$ parameterized by $vec{r} (t) = x(t) hat{i} + y(t) hat{j} + z(t) hat{k}; a leq t leq b$ has a nice relationship with our line integrals. That is:
$displaystyleint_C vec{F} cdot d vec{r} = displaystyleint^b_a (P hat{i} + Q hat{j} + R hat{k}) cdot (x’ hat{i} + y’ hat{j} + z’ hat{k})$
$= displaystyleint^b_a Px’dt + displaystyleint^b_a Qy’dt + displaystyleint^b_a Rz’dt$
$= displaystyleint_C Pdx + Qxy + Rxz$
This is so often forgotten!
• mixedmath says:
Whoops! That last line is:
$displaystyleint_C Pdx + Qdy + Rdz$
2. Anony says:
Is this only true for gradients?
• mixedmath says:
It’s true in general. At the least, it’s true for this course as we only consider well-behaved functions and relatively smooth curves.
3. Dhruv says:
Hey David…Could you please give an explained solution to question no.45 on pg 888 in the book. Thanks
• mixedmath says:
Sure. So the problem is to find the volume of the solid in the first octant bounded by the cylinders $x^2 + y^2 = a^2$ and $x^2 + z^2 = a^2$.
I will attempt to post a picture here:
http://www.math.tamu.edu/~tkiffe/calc3/newcylinder/cylinderb.jpeg
The easiest way I see to do this is with a standard double integral in rectangular coordinates. We restrict ourselves to only positive x,y,z values. So I will integrate the function $f(x,y) = sqrt{a^2 – x^2}$, gotten from the horizontal cylinder $x^2 + z^2 = a^2$.
That takes care of the height. Over what 2 dimensional domain? Well, we need only consider positive x and y values, and we are limited only be the vertical cylinder. So if I choose to integrate with respect to y first, we get the following:
$Volume = displaystyleint^a_0 displaystyleint^{sqrt{a^2 – x^2}}_0 sqrt{a^2 – x^2} dydx$
And this integrates nicely and easily to $dfrac{2a^3}{3}$.
How’s that?
4. Dhruv says:
I did not know you were the SGA rep. MATH dept. until I read your name in the following article:
http://www.nique.net/news/2011/04/15/uhr-gss-send-budget-to-conference-committee/
• mixedmath says:
Yepyep! And I’ve just spent the last 4 hours or so on the Conference Committee for the Budget. It’s about \$4.3 million or so. Exciting? Sort of. Exhausting? certainly.
5. Patrick says:
Hey David, can you perhaps post a graph of number 37 from 17.9 (page 930)? I’m having a hard time understanding why the integration is split into two parts. Thanks!
• mixedmath says:
My first embedded image! The idea is that we use spherical coordinates, and so we care about the bounding radii. Here, we can clearly see that there are parts where the boundary is on one sphere as opposed to the other.
How’s that?
As a reminder, you should do all the practice he recommends, not just the 4 problems he puts up for the test. I should also note that once I have approved a comment, your comments should appear freely (but does this work with unregistered users? who knows?).
• mixedmath says:
My first embedded image and I used the wrong one!
Attempt number 2!
6. Dhruv says:
Hey David…what r ur office hours, and where is ur office?
• mixedmath says:
I have one office hour tomorrow. It’s at 11 am and I hold it in the TA lounge, something like Skiles 233 (or maybe 230, 236, something right around there). If you have trouble finding it, ask someone and they can direct you to it.
7. Pingback: Dancing ones PhD « mixedmath
8. Pingback: My Teaching » mixedmath
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2021-09-26 00:19:37
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https://www.statistics-lab.com/category/%E9%A3%8E%E9%99%A9%E7%90%86%E8%AE%BA%E4%BB%A3%E5%86%99/
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## 金融代写|风险理论代写Risk theory代考|MATH4128
statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论Risk theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写风险理论Risk theory相关的作业也就用不着说。
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
## 金融代写|风险理论代写Risk theory代考|Bayes and Empirical Bayes
Let $\boldsymbol{X}=\left(X_1, \ldots, X_n\right)$ be a vector of r.v.s describing the outcome of a statistical experiment. For example, in the insurance context, $n$ can be the number of persons insured for losses due to accidents in the previous year, and $X_i$ the payment made to the $i$ th.
A traditional (frequentists’) model is to assume the $X_i$ to be i.i.d. with a common distribution $F_\theta$ where $\theta$ is an unknown parameter (possibly multidimensional). F.g. in the accident insurance example, one could let $b$ denote the probability that a person has an accident within one year, $b=\mathbb{P}\left(X_i>0\right)$, and one could assume that the cost of the accident has a $\operatorname{gamma}(\alpha, \lambda)$ distribution. Thus the density of $X_i$ is
$$f_{b, \alpha, \lambda}(x)=b \mathbb{1}{x=0}+(1-b) \frac{\lambda^\alpha x^{\alpha-1}}{\Gamma(\alpha)} \mathrm{e}^{-\lambda x_1} \mathbb{1}{x>0}$$
w.r.t. the measure defined as Lebesgue measure $\mathrm{d} x$ on $(0, \infty)$ with an added atom of unit size at $x=0$. Then $\theta=(b, \alpha, \lambda)$, and the conventional statistical procedure would be to compute estimates $\widehat{b}, \widehat{\alpha}, \widehat{\lambda}$ of $b, \alpha, \lambda$. These estimates could then be used as basis for computing first the expectation
$$\mathbb{E}{\widehat{\theta}} X=\mathbb{E}{\widehat{b}, \widehat{\alpha}, \widehat{\lambda}} X=(1-\widehat{b}) \widehat{\alpha} / \widehat{\lambda}$$
of $X$ under the estimated parameters, and next one could use $\mathbb{E}_{\widehat{\theta}} X$ as the net premium and add a loading corresponding to one of the premium rules discussed in Sect. I.3. For example, the expected value principle would lead to the premium
$$p=(1+\eta)(1-\widehat{b}) \widehat{\alpha} / \widehat{\lambda}$$
We now turn to the general implementation of Bayesian ideas in insurance. Here one considers an insured with risk parameter $Z^1$ and an r.v. with distribution $\pi^{(0)}(\cdot)$, with observable past claims $X_1, \ldots, X_n$ and an unobservable claim amount $X_{n+1}$ for year $n+1$. The aim is to assert which (net) premium the insured is to pay in year $n+1$
For a fixed $\zeta$, let $\mu(\zeta)=\mathbb{E}\zeta X{n+1}$, where $\mathbb{E}\zeta[\cdot]=\mathbb{E}[\cdot \mid Z=\zeta]$. The (net) collective premium $H{\mathrm{Coll}}$ is $\mathbb{E} \mu(\boldsymbol{Z})=\mathbb{E} X_{n+1}$. This is the premium we would charge without prior statistics $X_1, \ldots, X_n$ on the insured. The individual premium is $H_{\text {Ind }}=\mathbb{E}\left[X_{n+1} \mid \boldsymbol{Z}\right]=\mu(\boldsymbol{Z})$. This is the ideal net premium in the sense of supplying the complete relevant prior information on the customer. The Bayes premium $H_{\text {Bayes }}$ is defined as $\mathbb{E}\left[\mu(\boldsymbol{Z}) \mid X_1, \ldots, X_n\right]$. That is, $H_{\text {Bayes }}$ is the expected value of $X_{n+1}$ in the posterior distribution.
Note that the individual premium is unobservable because $\boldsymbol{Z}$ is so; the Bayes premium is ‘our best guess of $H_{\text {Ind }}$ based upon the observations’. To make this precise, let $H^$ be another premium rule, that is, a function of $X_1, \ldots, X_n$ and the prior parameters. We then define its loss as $$\ell_{H^}=\mathbb{E}\left[\mu(\boldsymbol{Z})-H^\right]^2=\left|\mu(\boldsymbol{Z})-H^\right|^2$$
where $|X|=\left(\mathbb{E} X^2\right)^{1 / 2}$ is the $L_2$-norm (in obvious notation, we write $\ell_{\text {Coll }}=\ell_{H_{\text {Coll }}}$ etc). In mathematical terms, the optimality property of the Bayes premium is then that it minimizes the quadratic loss:
Theorem 1.3 For any $H^, \ell_{\text {Bayes }} \leq \ell_{H^}$. That is,
$$\mathbb{E}\left(H_{\text {Bayes }}-H_{\text {Ind }}\right)^2 \leq \mathbb{E}\left(H^*-H_{\text {Ind }}\right)^2$$
## 金融代写|风险理论代写Risk theory代考|Bayes and Empirical Bayes
$$f_{b, \alpha, \lambda}(x)=b 1 x=0+(1-b) \frac{\lambda^\alpha x^{\alpha-1}}{\Gamma(\alpha)} \mathrm{e}^{-\lambda x_1} 1 x>0$$
wrt 定义为 Lebesgue 度量的度量 $\mathrm{d} x$ 上 $(0, \infty)$ 添加一个单位大小的原子 $x=0$. 然后 $\theta=(b, \alpha, \lambda)$ ，而传统的统 计程序是计算估计值 $\hat{b}, \widehat{\alpha}, \widehat{\lambda}$ 的 $b, \alpha, \lambda$. 然后可以将这些估计值用作首先计算期望值的基础
$$\mathbb{E} \hat{\theta} X=\mathbb{E} \hat{b}, \widehat{\alpha}, \hat{\lambda} X=(1-\hat{b}) \widehat{\alpha} / \widehat{\lambda}$$
$$p=(1+\eta)(1-\hat{b}) \widehat{\alpha} / \widehat{\lambda}$$
$$\mathbb{E}\left(H_{\text {Bayes }}-H_{\text {Ind }}\right)^2 \leq \mathbb{E}\left(H^*-H_{\text {Ind }}\right)^2$$
## 广义线性模型代考
statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。
## MATLAB代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。
## 金融代写|风险理论代写Risk theory代考|STAT4901
statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论Risk theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写风险理论Risk theory相关的作业也就用不着说。
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
## 金融代写|风险理论代写Risk theory代考|Reinsurance
Reinsurance means that the company (the cedent or first insurer) insures a part of the risk at another insurance company (the reinsurer). The purposes of reinsurance are to reduce risk and/or to reduce the risk volume of the company.
We start by formulating the basic concepts within the framework of a single risk $X \geq 0$. A reinsurance arrangement is then defined in terms of a function $r(x)$ with the property $0 \leq r(x) \leq x$. Here $r(x)$ is the amount of the claim $x$ to be paid by the reinsurer and $s(x)=x-r(x)$ the amount to be paid by the cedent. The function $s(x)$ is referred to as the retention function. The most common examples are the following two:
• Proportional reinsurance $r(x)=(1-\theta) x, s(x)=\theta x$ for some $\theta \in(0,1)$. Also called quota share reinsurance.
• Stop-loss reinsurance $r(x)=(x-b)^{+}$for some $b \in(0, \infty)$, referred to as the retention limit. The retention function is $x \wedge b$.
Concerning terminology, note that in the actuarial literature the stop-loss transform of $F(x)=\mathbb{P}(X \leq x)$ (or, equivalently, of $X)$ is defined as the function
$$b \mapsto \mathbb{E}(X-b)^{+}=\int_b^{\infty}(x-b) F(\mathrm{~d} x)=\int_b^{\infty} \bar{F}(x) \mathrm{d} x$$
(the last equality follows by integration by parts, see formula (A.1.1) in the Appendix). It shows up in a number of different contexts, see e.g. Sect. VIII.2.1, where some of its main properties are listed.
The risk $X$ is often the aggregate claims amount $A=\sum_1^N V_i$ in a certain line of business during one year; one then talks of global reinsurance. However, reinsurance may also be done locally, i.e. at the level of individual claims. Then, if $N$ is the number of claims during the period and $V_1, V_2, \ldots$ their sizes, then the amounts paid by reinsurer, resp. the cedent, are
$$\sum_{i=1}^N r\left(V_i\right), \text { resp. } \sum_{i=1}^N s\left(V_i\right)$$
## 金融代写|风险理论代写Risk theory代考|The Poisson Process
By a (simple) point process $\mathscr{N}$ on a set $\Omega \subseteq \mathbb{R}^d$ we understand a random collection of points in $\Omega$ [simple means that there are no multiple points]. We are almost exclusively concerned with the case $\Omega=[0, \infty)$. The point process can then be specified by the sequence $T_1, T_2, \ldots$ of interarrival times such that the points are $T_1, T_1+T_2, \ldots$ The associated counting process ${N(t)}_{t \geq 0}$ is defined by letting $N(t)$ be the number of points in $[0, t]$. Write
$$\mathscr{N}(s, t]=N(t)-N(s)=#\left{n: s<T_1+\cdots+T_n \leq t\right}$$
for the increment of ${N(t)}$ over $(s, t]$ or equivalently the number of points in $(s, t]$.
Definition 5.2 $\mathscr{N}$ is a Poisson process on $[0, \infty)$ with rate $\lambda$ if ${N(t)}$ has independent increments and $N(t)-N(s)$ has a Poisson $(\lambda(t-s))$ distribution for $s<t$.
Here independence of increments means independence of increments over disjoint intervals.
It is not difficult to extend the reasoning hehind example 1) ahnve to conclude. that for a large insurance portfolio, the number of claims in disjoint time intervals are independent Poisson r.v.s, and so the times of occurrences of claims form a Poisson process. There are, however, different ways to approach the Poisson process. In particular, the infinitesimal view in part (iii) of the following result will prove useful for many of our purposes.
## 金融代写|风险理论代写Risk theory代考|Reinsurance
• 比例再保险 $r(x)=(1-\theta) x, s(x)=\theta x$ 对于一些 $\theta \in(0,1)$. 也称为配额份额再保险。
• 止损再保险 $r(x)=(x-b)^{+}$对于一些 $b \in(0, \infty)$ ，称为保留限制。保留函数为 $x \wedge b$.
关于术语，请注意在精算文献中，止损变换 $F(x)=\mathbb{P}(X \leq x)$ (或者，等效地， $X$ )被定义为函数
$$b \mapsto \mathbb{E}(X-b)^{+}=\int_b^{\infty}(x-b) F(\mathrm{~d} x)=\int_b^{\infty} \bar{F}(x) \mathrm{d} x$$
（最后一个等式后面是分部积分，见附录中的公式 (A.1.1) )。它出现在许多不同的上下文中，例如参见 Sect。 VIII.2.1，其中列出了它的一些主要属性。
风险 $X$ 通常是总索赔额 $A=\sum_1^N V_i$ 一年内从事某项业务；然后有人谈到全球再保险。然而，再保险也可以在当 地进行，即在个人索赔层面。那么，如果 $N$ 是该期间的索赔数量，并且 $V_1, V_2, \ldots$ 他们的规模，然后是再保险公 司支付的金额，resp。分出商是
$$\sum_{i=1}^N r\left(V_i\right), \text { resp. } \sum_{i=1}^N s\left(V_i\right)$$
## 金融代写|风险理论代写Risk theory代考|The Poisson Process
Poisson rvs，因此理赔发生的次数构成一个 Poisson 过程。然而，有不同的方法来处理泊松过程。特别是，以下 结果的 (iii) 部分中的无穷小视图将证明对我们的许多目的有用。
## 广义线性模型代考
statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。
## MATLAB代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。
## 金融代写|风险理论代写Risk theory代考|STAT553
statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论Risk theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写风险理论Risk theory相关的作业也就用不着说。
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
## 金融代写|风险理论代写Risk theory代考|Actuarial Versus Financial Pricing
The last decades have seen the areas of insurance mathematics and mathematical finance coming closer together. One reason is the growing linking of pay-outs of life insurances and pension plans to the current value of financial products, another that certain financial products have been designed especially to be of interest for the insurance industry (see below). Nevertheless, some fundamental differences remain, and the present section aims at explaining some of these, with particular emphasis on the principles for pricing insurance products, resp. financial products.
In insurance, expected values play a major role. For example, let a claim $X \geq 0$ be the amount of money the insurance company has to pay out for a fire insurance on a given house next year (of course, $\mathbb{P}(X=0)$ is close to 1 !). The insurance company then ideally charges $H(X)=\mathbb{E} X$ in premium plus some loading, that is, an extra amount to cover administration costs, profit, risk etc. (different rules for the form of this loading are discussed in Sect. 3). The philosophy behind this is that charging premiums smaller than expected values in the long run results in an overall loss. This is a consequence of the law of large numbers (LLN). In its simplest form it says that if the company faces $n$ i.i.d. claims $X_1, \ldots, X_n$ all distributed as $X$, then the aggregate claim amount $A=X_1+\cdots+X_n$ is approximately $n \mathbb{E} X$ for $n$ large. Therefore, if the premium $H$ is smaller than $\mathbb{E} X$, then with high probability the total premiums $n H$ are not sufficient to cover the total aggregate claims $A$.
This argument carries far beyond this setting of i.i.d. claims, which is of course oversimplified: even in fire insurance, individual houses are different (the area varies, a house may have different types of heating, thatched roof or tiles, etc), and the company typically has many other lines of business such as car insurance, accident insurance, life insurance, etc. Let the claims be $X_1, X_2, \ldots$ Then the asymptotics
$$\frac{X_1+\cdots+X_n}{\mathbb{E} X_1+\cdots+\mathbb{E} X_n} \rightarrow 1$$
holds under weak conditions. For example, the following elementary result is sufficiently general to cover a large number of insurance settings
The standard setting for discussing premium calculation in the actuarial literature is in terms of a single risk $X \geq 0$ and does not involve portfolios, stochastic processes, etc. Here $X$ is an r.v. representing the random payment (possibly 0 ) to be made from the insurance company to the insured. A premium rule is then a $\lfloor 0, \infty)$-valued function $H$ of the distribution of $X$, often written $H(X)$, such that $H(X)$ is the premium to be paid, i.e. the amount for which the company is willing to insure the given risk. From an axiomatic point of view, the concept of premium rules is closely related to that of risk measures, to which we return in Sect. X.1.
The standard premium rules discussed in the literature (not necessarily the same as those used in practice!) are the following:
• The net premium principle $H(X)=\mathbb{E} X$ (also called the equivalence principle). As follows from a suitable version of the CLT that this principle will lead to a substantial loss if many independent risks are insured. This motivates that a loading should be added, as in the next principles:
• The expected value principle $H(X)=(1+\eta) \mathbb{E} X$, where $\eta$ is a specified safety loading. For $\eta=0$, we are back to the net premium principle. A criticism of the expected value principle is that it does not take into account the variability of $X$. This leads to:
• The variance principle $H(X)=\mathbb{E} X+\eta \operatorname{Var}(X)$. A modification (motivated by $\mathbb{E} X$ and $\operatorname{Var}(X)$ not having the same dimension) is
• The standard deviation principle $H(X)=\mathbb{E} X+\eta \sqrt{\operatorname{Var}(X)}$.
## 金融代写|风险理论代写Risk theory代考|Actuarial Versus Financial Pricing
$$\frac{X_1+\cdots+X_n}{\mathbb{E} X_1+\cdots+\mathbb{E} X_n} \rightarrow 1$$
• 净保费原则 $H(X)=\mathbb{E} X$ (也称为等价原则) 。从一个合适的 $\mathrm{CLT}$ 版本可以看出，如果许多独立风险被投 保，这一原则将导致重大损失。这促使应该添加负载，如下面的原则:
• 期望值原则 $H(X)=(1+\eta) \mathbb{E} X$ ，在哪里 $\eta$ 是指定的安全载荷。为了 $\eta=0$ ，我们又回到了净溢价原则。 对期望值原则的一个批评是它没有考虑到 $X$. 这将导致:
• 方差原理 $H(X)=\mathbb{E} X+\eta \operatorname{Var}(X)$. 修改 (动机是 $\mathbb{E} X$ 和 $\operatorname{Var}(X)$ 尺寸不同) 是
• 标准差原则 $H(X)=\mathbb{E} X+\eta \sqrt{\operatorname{Var}(X)}$.
## 广义线性模型代考
statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。
## MATLAB代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|CONSTRAINED OPTIMIZATION
statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论投资组合方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写风险理论投资组合代写方面经验极为丰富，各种代写风险理论投资组合相关的作业也就用不着说。
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|CONSTRAINED OPTIMIZATION
In constructing optimization problems solving practical issues, it is very often the case that certain constraints need to be imposed in order for the optimal solution to make practical sense. For example, long-only portfolio optimization problems require that the portfolio weights, which represent the variables in optimization, should be nonnegative and should sum up to one. According to the notation in this chapter, this corresponds to a problem of the type,
$$\begin{array}{rl} \min {x} & f(x) \ \text { subject to } & x^{\prime} e=1 \ & x \geq 0, \end{array}$$ where: $f(x)$ is the objective function. $e \in \mathbb{R}^{n}$ is a vector of ones, $e=(1, \ldots, 1)$. $x^{\prime} e$ equals the sum of all components of $x, x^{\prime} e=\sum{i}^{n} x_{i}$.
$x \geq 0$ means that all components of the vector $x \in \mathbb{R}^{n}$ are nonnegative.
In problem (2.10), we are searching for the minimum of the objective function by varying $x$ only in the set
$$\mathbf{X}=\left{x \in \mathbb{R}^{n}: \begin{array}{l} x^{\prime} e=1 \ x \geq 0 \end{array}\right}$$
which is also called the set of feasible points or the constraint set. A more compact notation, similar to the notation in the unconstrained problems, is sometimes used,
$$\min _{x \in \mathrm{X}} f(x)$$
where $\mathbf{X}$ is defined in equation (2.11).
We distinguish between different types of optimization problems depending on the assumed properties for the objective function and the constraint set. If the constraint set contains only equalities, the problem is easier to handle analytically. In this case, the method of Lagrange multipliers is applied. For more general constraint sets, when they are formed
by both equalities and inequalities, the method of Lagrange multipliers is generalized by the Karush-Kuhn-Tucker conditions (KKT conditions). Like the first-order conditions we considered in unconstrained optimization problems, none of the two approaches leads to necessary and sufficient conditions for constrained optimization problems without further assumptions. One of the most general frameworks in which the KKT conditions are necessary and sufficient is that of convex programming. We have a convex programing problem if the objective function is a convex function and the set of feasible points is a convex set. As important subcases of convex optimization, linear programming and convex quadratic programming problems are considered.
In this section, we describe first the method of Lagrange multipliers, which is often applied to special types of mean-variance optimization problems in order to obtain closed-form solutions. Then we proceed with convex programming that is the framework for reward-risk analysis. The mentioned applications of constrained optimization problems is covered in Chapters 8,9 , and 10 .
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Lagrange Multipliers
Consider the following optimization problem in which the set of feasible points is defined by a number of equality constraints,
$$\begin{array}{rl} \min {x} & f(x) \ \text { subject to } & b{1}(x)=0 \ & b_{2}(x)=0 \ \cdots \ & b_{k}(x)=0 \end{array}$$
The functions $h_{i}(x), i=1, \ldots, k$ build up the constraint set. Note that even though the right-hand side of the equality constraints is zero in the classical formulation of the problem given in equation $(2.12)$, this is not restrictive. If in a practical problem the right-hand side happens to be different than zero, it can be equivalently transformed, for example,
$$\left{x \in \mathbb{R}^{n}: v(x)=c\right} \quad \Longleftrightarrow \quad\left{x \in \mathbb{R}^{n}: h_{1}(x)=v(x)-c=0\right} .$$
In order to illustrate the necessary condition for optimality valid for (2.12), let us consider the following two-dimensional example:
\begin{aligned} \min _{x \in \mathbb{R}^{2}} & \frac{1}{2} x^{\prime} C x \ \text { subject to } & x^{\prime} e=1, \end{aligned}
where the matrix is
$$C=\left(\begin{array}{cc} 1 & 0.4 \ 0.4 & 1 \end{array}\right) .$$
The objective function is a quadratic function and the constraint set contains one linear equality. In Chapter 8, we see that the mean-variance optimization problem in which short positions are allowed is very similar to (2.13). The surface of the objective function and the constraint are shown on the top plot in Figure 2.7. The black line on the surface shows the function values of the feasible points. Geometrically, solving problem (2.13) reduces to finding the lowest point of the black curve on the surface. The contour lines shown on the bottom plot in Figure $2.7$ imply that the feasible point yielding the minimum of the objective function is where a contour line is tangential to the line defined by the equality constraint. On the plot, the tangential contour line and the feasible points are in bold. The black dot indicates the position of the point in which the objective function attains its minimum subject to the constraints.
Even though the example is not general in the sense that the constraint set contains one linear rather than a nonlinear equality, the same geometric intuition applies in the nonlinear case. The fact that the minimum is attained where a contour line is tangential to the curve defined by the nonlinear equality constraints in mathematical language is expressed in the following way: The gradient of the objective function at the point yielding the minimum is proportional to a linear combination of the gradients of the functions defining the constraint set. Formally, this is stated as
$$\nabla f\left(x^{0}\right)-\mu_{1} \nabla h_{1}\left(x^{0}\right)-\cdots-\mu_{k} \nabla h_{k}\left(x^{0}\right)=0 .$$
where $\mu_{i}, i=1, \ldots, k$ are some real numbers called Lagrange multipliers and the point $x^{0}$ is such that $f\left(x^{0}\right) \leq f(x)$ for all $x$ that are feasible. Note that if there are no constraints in the problem, then (2.14) reduces to the first-order condition we considered in unconstrained optimization. Therefore, the system of equations behind (2.14) can be viewed as a generalization of the first-order condition in the unconstrained case.
The method of Lagrange multipliers basically associates a function to the problem in $(2.12)$ such that the first-order condition for unconstrained optimization for that function coincides with (2.14). The method of Lagrange multiplier consists of the following steps.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Convex Programming
The general form of convex programming problems is the following:
$$\begin{array}{rl} \min {x} & f(x) \ \text { subject to } & g{i}(x) \leq 0, \quad i=1, \ldots, m \ & h_{j}(x)=0, \quad j=1, \ldots, k, \end{array}$$
where:
$f(x)$ is a convex objective function.
$g_{1}(x), \ldots, g_{m}(x)$ are convex functions defining the inequality constraints. $h_{1}(x), \ldots, h_{k}(x)$ are affine functions defining the equality constraints.
Generally, without the assumptions of convexity, problem $(2.17)$ is more involved than $(2.12)$ because besides the equality constraints, there are inequality constraints. The KKT condition, generalizing the method of Lagrange multipliers, is only a necessary condition for optimality in this case. However, adding the assumption of convexity makes the KKT condition necessary and sufficient.
Note that, similar to problem (2.12), the fact that the right-hand side of all constraints is zero is nonrestrictive. The limits can be arbitrary real numbers.
Consider the following two-dimensional optimization problem;
\begin{aligned} \min {\substack{x \in \mathbb{R}^{2}}} & \frac{1}{2} x^{\prime} \mathrm{C} x \ \text { subject to } &\left(x{1}+2\right)^{2}+\left(x_{2}+2\right)^{2} \leq 3 \end{aligned}
in which
$$C=\left(\begin{array}{cc} 1 & 0.4 \ 0.4 & 1 \end{array}\right) \text {. }$$
The objective function is a two-dimensional convex quadratic function and the function in the constraint set is also a convex quadratic function. In fact, the boundary of the feasible set is a circle with a radius of $\sqrt{3}$ centered at the point with coordinates $(-2,-2)$. The top plot in Figure $2.8$ shows the surface of the objective function and the set of feasible points. The shaded part on the surface indicates the function values of all feasible points. In fact, solving problem (2.18) reduces to finding the lowest point on the shaded part of the surface. The bottom plot shows the contour lines of the objective function together with the feasible set that is in gray. Geometrically, the point in the feasible set yielding the minimum of the objective function is positioned where a contour line only touches the constraint set. The position of this point is marked with a black dot and the tangential contour line is given in bold.
Note that the solution points of problems of the type $(2.18)$ can happen to be not on the boundary of the feasible set but in the interior. For example, suppose that the radius of the circle defining the boundary of the feasible set in $(2.18)$ is a larger number such that the point $(0,0)$ is inside the feasible
set. Then, the point $(0,0)$ is the solution to problem $(2.18)$ because at this point the objective function attains its global minimum.
In the two-dimensional case, when we can visualize the optimization problem, geometric reasoning guides us to finding the optimal solution point. In a higher dimensional space, plots cannot be produced and we rely on the analytic method behind the KKT conditions.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|CONSTRAINED OPTIMIZATION
X≥0表示向量的所有分量X∈Rn是非负的。
\mathbf{X}=\left{x \in \mathbb{R}^{n}: \begin{array}{l} x^{\prime} e=1 \ x \geq 0 \end{array}\对}\mathbf{X}=\left{x \in \mathbb{R}^{n}: \begin{array}{l} x^{\prime} e=1 \ x \geq 0 \end{array}\对}
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Lagrange Multipliers
\left{x \in \mathbb{R}^{n}: v(x)=c\right} \quad \Longleftrightarrow \quad\left{x \in \mathbb{R}^{n}: h_{1 }(x)=v(x)-c=0\right} 。\left{x \in \mathbb{R}^{n}: v(x)=c\right} \quad \Longleftrightarrow \quad\left{x \in \mathbb{R}^{n}: h_{1 }(x)=v(x)-c=0\right} 。
C=(10.4 0.41).
∇F(X0)−μ1∇H1(X0)−⋯−μķ∇Hķ(X0)=0.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Convex Programming
F(X)是一个凸目标函数。
G1(X),…,G米(X)是定义不等式约束的凸函数。H1(X),…,Hķ(X)是定义等式约束的仿射函数。
C=(10.4 0.41).
## 有限元方法代写
tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。
## MATLAB代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Optimization
statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论投资组合方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写风险理论投资组合代写方面经验极为丰富，各种代写风险理论投资组合相关的作业也就用不着说。
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|UNCONSTRAINED OPTIMIZATION
When there are no constraints imposed on the set of feasible solutions, we have an unconstrained optimization problem. Thus, the goal is to maximize or to minimize the objective function with respect to the function arguments without any limits on their values. We consider directly the $n$-dimensional case; that is, the domain of the objective function $f$ is the $n$-dimensional space and the function values are real numbers, $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$. Maximization is denoted by
$$\max f\left(x_{1}, \ldots, x_{n}\right)$$
and minimization by
$$\min f\left(x_{1}, \ldots, x_{n}\right)$$
A more compact form is commonly used, for example
$$\min {x \in \mathbb{R}^{n}} f(x)$$ denotes that we are searching for the minimal value of the function $f(x)$ by varying $x$ in the entire $n$-dimensional space $\mathbb{R}^{n}$. A solution to equation (2.1) is a value of $x=x^{0}$ for which the minimum of $f$ is attained, $$f{0}=f\left(x^{0}\right)=\min {x \in \mathbb{R}^{\pi}} f(x) .$$ Thus, the vector $x{0}$ is such that the function takes a larger value than $f_{0}$ for any other vector $x$,
$$f\left(x^{0}\right) \leq f(x), x \in \mathbb{R}^{n}$$
Note that there may be more than one vector $x^{0}$ satisfying the inequality in equation (2.2) and, therefore, the argument for which $f_{0}$ is achieved may not be unique. If (2.2) holds, then the function is said to attain its global minimum at $x^{0}$. If the inequality in $(2.2)$ holds for $x$ belonging only to a small neighborhood of $x^{0}$ and not to the entire space $\mathbb{R}^{n}$, then the objective function is said to have a local minimum at $x^{0}$. This is usually denoted by
$$f\left(x^{0}\right) \leq f(x)$$
for all $x$ such that $\left|x-x^{0}\right|_{2}<\epsilon$ where $\left|x-x^{0}\right|_{2}$ stands for the Euclidean distance between the vectors $x$ and $x^{0}$,
$$\left|x-x^{0}\right|_{2}=\sqrt{\sum_{i=1}^{n}\left(x_{i}-x_{i}^{0}\right)^{2}}$$
and $\epsilon$ is some positive number. A local minimum may not be global as there may be vectors outside the small neighborhood of $x_{0}$ for which the objective function attains a smaller value than $f\left(x_{0}\right)$. Figure $2.2$ shows the graph of a function with two local maxima, one of which is the global maximum.
There is a connection between minimization and maximization. Maximizing the objective function is the same as minimizing the negative of the objective function and then changing the sign of the minimal value,
$$\max {x \in \mathbb{R}^{n}} f(x)=-\min {x \in \mathbb{R}^{n}}[-f(x)] .$$
This relationship is illustrated in Figure 2.1. As a consequence, problems for maximization can be stated in terms of function minimization and vice versa.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Minima and Maxima of a Differentiable Function
If the second derivatives of the objective function exist, then its local maxima and minima, often called generically local extrema, can be characterized.
Denote by $\nabla f(x)$ the vector of the first partial derivatives of the objective function evaluated at $x$,
$$\nabla f(x)=\left(\frac{\partial f(x)}{\partial x_{1}}, \ldots, \frac{\partial f(x)}{\partial x_{n}}\right) .$$
This vector is called the function gradient. At each point $x$ of the domain of the function, it shows the direction of greatest rate of increase of the function in a small neighborhood of $x$. If for a given $x$, the gradient equals a vector of zeros,
$$\nabla f(x)=(0, \ldots, 0)$$
then the function does not change in a small neighborhood of $x \in \mathbb{R}^{n}$. It turns out that all points of local extrema of the objective function are characterized by a zero gradient. As a result, the points yielding the local extrema of the objective function are among the solutions of the system of equations,
\mid \begin{aligned} &\frac{\partial f(x)}{\partial x_{1}}=0 \ &\cdots \ &\frac{\partial f(x)}{\partial x_{n}}=0 \end{aligned}
The system of equation $(2.3)$ is often referred to as representing the first-order condition for the objective function extrema. However, it is only a necessary condition; that is, if the gradient is zero at a given point in the $n$-dimensional space, then this point may or may not be a point of a local extremum for the function. An illustration is given in Figure 2.2. The top plot shows the graph of a two-dimensional function and the bottom plot contains the contour lines of the function with the gradient calculated at a grid of points. There are three points marked with a black dot that have a zero gradient. The middle point is not a point of a local maximum even though it has a zero gradient. This point is called a saddle point since the graph resembles the shape of a saddle in a neighborhood of it. The left and the right points are where the function has two local maxima corresponding to the two peaks visible on the top plot. The right peak is a local maximum that is not the global one and the left peak represents the global maximum.
This example demonstrates that the first-order conditions are generally insufficient to characterize the points of local extrema. The additional condition that identifies which of the zero-gradient points are points
of local minimum or maximum is given through the matrix of second derivatives,
$$H=\left(\begin{array}{cccc} \frac{\partial^{2} f(x)}{\partial x_{1}^{2}} & \frac{\partial^{2} f(x)}{\partial x_{1} \partial x_{2}} & \cdots & \frac{\partial^{2} f(x)}{\partial x_{1} \partial x_{e}} \ \frac{\partial^{2} f(x)}{\partial x_{2} \partial x_{1}} & \frac{\partial^{2} f(x)}{\partial x_{2}^{2}} & \cdots & \frac{\partial^{2} f(x)}{\partial x_{2} \partial x_{n}} \ \vdots & \vdots & \ddots & \vdots \ \frac{\partial^{2} f(x)}{\partial x_{n} \partial x_{1}} & \frac{\partial^{2} f(x)}{\partial x_{n} \partial x_{2}} & \cdots & \frac{\partial^{2} f(x)}{\partial x_{n}^{2}} \end{array}\right)$$
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Convex Functions
In section 2.2.1, we demonstrated that the first-order conditions are insufficient in the general case to describe the local extrema. However, when certain assumptions are made for the objective function, the first-order conditions can become sufficient. Furthermore, for certain classes of functions, the local extrema are necessarily global. Therefore, solving the first-order conditions, we obtain the global extremum.
A general class of functions with nice optimal properties is the class of convex functions. Not only are the convex functions easy to optimize but they have also important application in risk management. In Chapter 6, we discuss general measures of risk. It turns out that the property which guarantees that diversification is possible appears to be exactly the convexity
property. As a consequence, a measure of risk is necessarily a convex functional. ${ }^{1}$
Precisely, a function $f(x)$ is called a convex function if it satisfies the property: For a given $\alpha \in[0,1]$ and all $x^{1} \in \mathbb{R}^{n}$ and $x^{2} \in \mathbb{R}^{n}$ in the function domain,
$$f\left(\alpha x^{1}+(1-\alpha) x^{2}\right) \leq \alpha f\left(x^{1}\right)+(1-\alpha) f\left(x^{2}\right)$$
The definition is illustrated in Figure 2.3. Basically, if a function is convex, then a straight line connecting any two points on the graph lies “above” the graph of the function.
There is a related term to convex functions. A function $f$ is called concave if the negative of $f$ is convex. In effect, a function is concave if it
satisfies the property: For a given $\alpha \in[0,1]$ and all $x^{1} \in \mathbb{R}^{n}$ and $x^{2} \in \mathbb{R}^{n}$ in the function domain,
$$f\left(\alpha x^{1}+(1-\alpha) x^{2}\right) \geq \alpha f\left(x^{1}\right)+(1-\alpha) f\left(x^{2}\right) .$$
We use convex and concave functions in the discussion of the efficient frontier in Chapter 8 .
If the domain $D$ of a convex function is not the entire space $\mathbb{R}^{n}$, then the set D satisfies the property,
$$\alpha x^{1}+(1-\alpha) x^{2} \in D$$
where $x^{1} \in D, x^{2} \in D$, and $0 \leq \alpha \leq 1$. The sets that satisfy equation (2.6) are called convex sets. Thus, the domains of convex (and concave) functions should be convex sets. Geometrically, a set is convex if it contains the straight line connecting any two points belonging to the set. Rockafellar (1997) provides detailed information on the implications of convexity in optimization theory.
We summarize several important properties of convex functions:
• Not all convex functions are differentiable. If a convex function is two times continuously differentiable, then the corresponding Hessian defined in equation $(2.4)$ is a positive semidefinite matrix. ${ }^{2}$
• All convex functions are continuous if considered in an open set.
• The sublevel sets
$$L_{c}={x: f(x) \leq c}$$
where $c$ is a constant, are convex sets if $f$ is a convex function. The converse is not true in general. Section $2.2 .3$ provides more information about non-convex functions with convex sublevel sets.
• The local minima of a convex function are global. If a convex function $f$ is twice continuously differentiable, then the global minimum is obtained in the points solving the first-order condition,
$$\nabla f(x)=0 .$$
• A sum of convex functions is a convex function:
$$f(x)=f_{1}(x)+f_{2}(x)+\ldots+f_{k}(x)$$
is a convex function if $f_{i}, i=1, \ldots, k$ are convex functions.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|UNCONSTRAINED OPTIMIZATION
F(X0)≤F(X),X∈Rn
F(X0)≤F(X)
|X−X0|2=∑一世=1n(X一世−X一世0)2
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Minima and Maxima of a Differentiable Function
∇F(X)=(∂F(X)∂X1,…,∂F(X)∂Xn).
∇F(X)=(0,…,0)
∣∂F(X)∂X1=0 ⋯ ∂F(X)∂Xn=0
H=(∂2F(X)∂X12∂2F(X)∂X1∂X2⋯∂2F(X)∂X1∂X和 ∂2F(X)∂X2∂X1∂2F(X)∂X22⋯∂2F(X)∂X2∂Xn ⋮⋮⋱⋮ ∂2F(X)∂Xn∂X1∂2F(X)∂Xn∂X2⋯∂2F(X)∂Xn2)
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Convex Functions
F(一种X1+(1−一种)X2)≤一种F(X1)+(1−一种)F(X2)
F(一种X1+(1−一种)X2)≥一种F(X1)+(1−一种)F(X2).
• 并非所有凸函数都是可微的。如果一个凸函数是两次连续可微的，则等式中定义的相应 Hessian(2.4)是一个半正定矩阵。2
• 如果在开集中考虑，所有凸函数都是连续的。
• 子级集
大号C=X:F(X)≤C
在哪里C是一个常数，如果是凸集F是一个凸函数。反之亦然。部分2.2.3提供有关具有凸子水平集的非凸函数的更多信息。
• 凸函数的局部最小值是全局的。如果一个凸函数F是两次连续可微的，则在求解一阶条件的点中获得全局最小值，
∇F(X)=0.
• 凸函数之和是一个凸函数：
F(X)=F1(X)+F2(X)+…+Fķ(X)
是一个凸函数，如果F一世,一世=1,…,ķ是凸函数。
## 有限元方法代写
tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。
## MATLAB代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|PROBABILISTIC INEQUALITIES
statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论投资组合方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写风险理论投资组合代写方面经验极为丰富，各种代写风险理论投资组合相关的作业也就用不着说。
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Chebyshev’s Inequality
Some of the topics discussed in the book concern a setting in which we are not aware of the particular distribution of a random variable or the particular joint probability distribution of a pair of random variables. In such cases, the analysis may require us to resort to general arguments based on certain general inequalities from the theory of probability. In this section, we give an account of such inequalities and provide illustration where possible.
Chebyshev’s inequality provides a way to estimate the approximate probability of deviation of a random variable from its mean. Its most simple form concerns positive random variables.
Suppose that $X$ is a positive random variable, $X>0$. The following inequality is known as Chebyshev’s inequality,
$$P(X \geq \epsilon) \leq \frac{E X}{\epsilon},$$
where $\epsilon>0$. In this form, equation $(1.5)$ can be used to estimate the probability of observing a large observation by means of the mathematical expectation and the level $\epsilon$. Chebyshev’s inequality is rough as demonstrated geometrically in the following way. The mathematical expectation of a positive continuous random variable admits the representation,
$$E X=\int_{0}^{\infty} P(X \geq x) d x,$$
which means that it equals the area closed between the distribution function and the upper limit of the distribution function. This area is illustrated in Figure $1.9$ as the shaded area above the distribution function. On the other hand, the quantity $\epsilon P(X \geq \epsilon)=\epsilon\left(1-F_{X}(x)\right)$ is equal to the area of the rectangle in the upper-left corner of Figure $1.9$. In effect, the inequality
$$\epsilon P(X \geq \epsilon) \leq E X$$
admits the following geometric interpretation-the area of the rectangle is smaller than the shaded area in Figure 1.9.
For an arbitrary random variable, Chebychev’s inequality takes the form
$$P\left(|X-E X| \geq \epsilon \sigma_{X}\right) \leq \frac{1}{\epsilon^{2}}$$
where $\sigma_{X}$ is the standard deviation of $X$ and $\epsilon>0$. We use Chebyshev’s inequality in Chapter 6 in the discussion of dispersion measures.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Frechet-Hoeffding Inequality
Consider an $n$-dimensional random vector $Y$ with a distribution function $F_{Y}\left(y_{1}, \ldots, y_{n}\right)$. Denote by
$$W\left(y_{1}, \ldots, y_{n}\right)=\max \left(F_{Y_{1}}\left(y_{1}\right)+\cdots+F_{Y_{n}}\left(y_{n}\right)+1-n, 0\right)$$
and by
$$M\left(y_{1}, \ldots, y_{n}\right)=\min \left(F_{Y_{1}}\left(y_{1}\right), \ldots, F_{Y_{n}}\left(y_{n}\right)\right)$$
in which $F_{Y_{i}}\left(y_{i}\right)$ stands for the distribution function of the $i$-th marginal. The following inequality is known as Fréchet-Hoeffding inequality,
$$W\left(y_{1}, \ldots, y_{n}\right) \leq F_{Y}\left(y_{1}, \ldots, y_{n}\right) \leq M\left(y_{1}, \ldots, y_{n}\right) .$$
The quantities $W\left(y_{1}, \ldots, y_{n}\right)$ and $M\left(y_{1}, \ldots, y_{n}\right)$ are also called the Fréchet lower bound and the Fréchet upper bound. We apply FréchetHoeffding inequality in the two-dimensional case in Chapter 3 when discussing minimal probability metrics.
Since copulas are essentially probability distributions defined on the unit hypercube, Fréchet-Hoeffding inequality holds for them as well. In this case, it has a simpler form because the marginal distributions are uniform. The lower and the upper Fréchet bounds equal
and
$$\begin{gathered} W\left(u_{1}, \ldots, u_{n}\right)=\max \left(u_{1}+\cdots+u_{n}+1-n, 0\right) \ M\left(u_{1}, \ldots, u_{n}\right)=\min \left(u_{1}, \ldots, u_{n}\right) \end{gathered}$$
respectively. Fréchet-Hoeffding inequality is given by
$$W\left(u_{1}, \ldots, u_{n}\right) \leq C\left(u_{1}, \ldots, u_{n}\right) \leq M\left(u_{1}, \ldots, u_{n}\right) .$$
In the two-dimensional case, the inequality reduces to
$$\max \left(u_{1}+u_{2}-1,0\right) \leq C\left(u_{1}, u_{2}\right) \leq \min \left(u_{1}, u_{2}\right) .$$
In the two-dimensional case only, the lower Fréchet bound, sometimes referred to as the minimal copula, represents perfect negative dependence between the two random variables. In a similar way, the upper Fréchet bound, sometimes referred to as the maximal copula, represents perfect positive dependence between the two random variables.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|SUMMARY
We considered a number of concepts from probability theory that will be used in later chapters in this book. We discussed the notions of a random variable and a random vector. We considered one-dimensional and multidimensional probability density and distributions functions, which completely characterize a given random variable or random vector. We discussed statistical moments and quantiles, which represent certain characteristics of a random variable, and the sample moments which provide a way of estimating the corresponding characteristics from historical data. In the multidimensional case, we considered the notion of dependence between the components of a random vector. We discussed the covariance matrix versus the more general concept of a copula function. Finally, we described two probabilistic inequalities, Chebychev’s inequality and Fréchet-Hoeffding inequality.
ε磷(X≥ε)≤和X
## 有限元方法代写
tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。
## MATLAB代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Covariance and Correlation
statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论投资组合方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写风险理论投资组合代写方面经验极为丰富，各种代写风险理论投资组合相关的作业也就用不着说。
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Covariance and Correlation
There are two strongly related measures among many that are commonly used to measure how two random variables tend to move together, the covariance and the correlation. Letting:
$\sigma_{X}$ denote the standard deviation of $X$.
$\sigma_{Y}$ denote the standard deviation of $Y$.
$\sigma_{X Y}$ denote the covariance between $X$ and $Y$.
$\rho_{X Y}$ denote the correlation between $X$ and $Y$.
The relationship between the correlation, which is also denoted by $\rho_{X Y}$ $=\operatorname{corr}(X, Y)$, and covariance is as follows:
$$\rho_{X Y}=\frac{\sigma_{X Y}}{\sigma_{X} \sigma_{Y}} .$$
Here the covariance, also denoted by $\sigma_{X Y}=\operatorname{cov}(X, Y)$, is defined as
\begin{aligned} \sigma_{X Y} &=E(X-E X)(Y-E Y) \ &=E(X Y)-E X E Y \end{aligned}
It can be shown that the correlation can only have values from $-1$ to $+1$. When the correlation is zero, the two random variables are said to be uncorrelated.
If we add two random variables, $X+Y$, the expected value (first central moment) is simply the sum of the expected value of the two random variables. That is,
$$E(X+Y)=E X+E Y .$$
The variance of the sum of two random variables, denoted by $\sigma_{X+Y}^{2}$, is
$$\sigma_{X+Y}^{2}=\sigma_{X}^{2}+\sigma_{Y}^{2}+2 \sigma_{X Y} .$$
Here the last term accounts for the fact that there might be a dependence between $X$ and $Y$ measured through the covariance. In Chapter 8, we consider the variance of the portfolio return of $n$ assets which is expressed by means of the variances of the assets’ returns and the covariances between them.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Multivariate Normal Distribution
In finance, it is common to assume that the random variables are normally distributed. The joint distribution is then referred to as a multivariate normal
distribution. ${ }^{13}$ We provide an explicit representation of the density function of a general multivariate normal distribution.
Consider first $n$ independent standard normal random variables $X_{1}, \ldots$, $X_{n}$. Their common density function can be written as the product of their individual density functions and so we obtain the following expression as the density function of the random vector $X=X_{1}, \ldots, X_{n}$ :
$$f_{\mathrm{X}}\left(x_{1}, \ldots, x_{n}\right)=\frac{1}{(\sqrt{2 \pi})^{n}} e^{-\frac{x^{\prime} x}{2}},$$
where the vector notation $x^{\prime} x$ denotes the sum of the components of the vector $x$ raised to the second power, $x^{\prime} x=\sum_{i=1}^{n} x_{i}^{2}$.
Now consider $n$ vectors with $n$ real components arranged in a matrix $A$. In this case, it is often said that the matrix $A$ has a $n \times n$ dimension. The random variable
$$Y=A X+\mu,$$
in which $A X$ denotes the $n \times n$ matrix $A$ multiplied by the random vector $X$ and $\mu$ is a vector of $n$ constants, has a general multivariate normal distribution. The density function of $Y$ can now be expressed as ${ }^{14}$
where $|\Sigma|$ denotes the determinant of the matrix $\Sigma$ and $\Sigma^{-1}$ denotes the inverse of $\Sigma$. The matrix $\Sigma$ can be calculated from the matrix $A, \Sigma=A A^{\prime}$. The elements of $\Sigma=\left{\sigma_{i j}\right}_{i, j-1}^{n}$ are the covariances between the components of the vector $Y$,
$$\sigma_{i j}=\operatorname{cov}\left(Y_{i}, Y_{j}\right) .$$
Figure $1.5$ contains a plot of the probability density function of a two-dimensional normal distribution with a covariance matrix,
$$\Sigma=\left(\begin{array}{cc} 1 & 0.8 \ 0.8 & 1 \end{array}\right)$$
and mean $\mu=(0,0)$. The matrix $A$ from the representation given in formula (1.3) equals
$$A=\left(\begin{array}{cc} 1 & 0 \ 0.8 & 0.6 \end{array}\right)$$
The correlation between the two components of the random vector $Y$ is equal to $0.8, \operatorname{corr}\left(Y_{1}, Y_{2}\right)=0.8$ because in this example the variances of the two components are equal to 1 . This is a strong positive correlation, which means that the realizations of the random vector $Y$ clusters along the diagonal splitting the first and the third quadrant. This is illustrated in Figure 1.6, which shows the contour lines of the two-dimensional density function plotted in Figure 1.5. The contour lines are ellipses centered at the mean $\mu=(0,0)$ of the random vector $Y$ with their major axes lying along the diagonal of the first quadrant. The contour lines indicate that realizations of the random vector $Y$ roughly take the form of an elongated ellipse as the ones shown in Figure 1.6, which means that large values of $Y_{1}$ will correspond to large values of $Y_{2}$ in a given pair of observations.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Copula Functions
Correlation is a widespread concept in modern finance and risk management and stands for a measure of dependence between random variables. However, this term is often incorrectly used to mean any notion of dependence. Actually, correlation is one particular measure of dependence among many. In the world of multivariate normal distribution and more generally in the world of spherical and elliptical distributions, it is the accepted measure.
A major drawback of correlation is that it is not invariant under nonlinear strictly increasing transformations. In general,
$$\operatorname{corr}(T(X), T(Y)) \neq \operatorname{corr}(X, Y)$$
where $T(x)$ is such transformation. One example which explains this technical requirement is the following: Assume that $X$ and $Y$ represent the continuous return (log-return) of two assets over the period $[0, t]$, where $t$ denotes some point of time in the future. If you know the correlation of these two random variables, this does not imply that you know the dependence structure between the asset prices itself because the asset prices $\left(P\right.$ and $Q$ for asset $X$ and $Y$, respectively) are obtained by $P_{t}=P_{0} \exp (X)$ and $Q_{t}=Q_{0} \exp (Y)$, where $P_{0}$ and $Q_{0}$ denote the corresponding asset prices at time 0 . The asset prices are strictly increasing functions of the return but the correlation structure is not maintained by this transformation. This observation implies that the return could be uncorrelated whereas the prices are strongly correlated and vice versa.
A more prevalent approach that overcomes this disadvantage is to model dependency using copulas. As noted by Patton (2004, p. 3), “The word copula comes from Latin for a ‘link’ or ‘bond,’ and was coined by Sklar (1959), who first proved the theorem that a collection of marginal distributions can be ‘coupled’ together via a copula to form a multivariate distribution.” The idea is as follows. The description of the joint distribution of a random vector is divided into two parts:
1. The specification of the marginal distributions.
2. the specification of the dependence structure by means of a special function, called copula.
The use of copulas ${ }^{19}$ offers the following advantages:
• The nature of dependency that can be modeled is more general. In comparison, only linear dependence can be explained by the correlation.
• Dependence of extreme events might be modeled.
• Copulas are indifferent to continuously increasing transformations (not only linear as it is true for correlations).
From a mathematical viewpoint, a copula function $C$ is nothing more than a probability distribution function on the $n$-dimensional hypercube $I_{n}=[0,1] \times[0,1] \times \ldots \times[0,1]:$
\begin{aligned} C: I_{n} & \rightarrow[0,1] \ \left(u_{1}, \ldots, u_{n}\right) & \rightarrow C\left(u_{1}, \ldots, u_{n}\right) \end{aligned}
It has been shown ${ }^{20}$ that any multivariate probability distribution function $F_{Y}$ of some random vector $Y=\left(Y_{1}, \ldots, Y_{n}\right)$ can be represented with the help of a copula function $C$ in the following form:
\begin{aligned} F_{Y}\left(y_{1}, \ldots, y_{n}\right) &=P\left(Y_{1} \leq y_{1}, \ldots, Y_{n} \leq y_{n}\right)=C\left(P\left(Y_{1} \leq y_{1}\right), \ldots, P\left(Y_{n} \leq y_{n}\right)\right) \ &=C\left(F_{Y_{1}}\left(y_{1}\right), \ldots, F_{Y_{n}}\left(y_{n}\right)\right) \end{aligned}
where $F_{Y_{i}}\left(y_{i}\right), i=1, \ldots, n$ denote the marginal distribution functions of the random variables $Y_{i}, i=1, \ldots, n$.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Covariance and Correlation
σX表示标准差X.
σ是表示标准差是.
σX是表示之间的协方差X和是.
ρX是表示之间的相关性X和是.
ρX是=σX是σXσ是.
σX是=和(X−和X)(是−和是) =和(X是)−和X和是
σX+是2=σX2+σ是2+2σX是.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Multivariate Normal Distribution
FX(X1,…,Xn)=1(2圆周率)n和−X′X2,
σ一世j=这(是一世,是j).
Σ=(10.8 0.81)
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Copula Functions
1. 边际分布的规范。
2. 通过称为 copula 的特殊函数指定依赖结构。
copula 的使用19提供以下优势：
• 可以建模的依赖性的性质更普遍。相比之下，相关性只能解释线性相关性。
• 极端事件的依赖性可能会被建模。
• Copulas 对不断增加的转换无动于衷（不仅是线性的，因为它对相关性也是如此）。
C:一世n→[0,1] (在1,…,在n)→C(在1,…,在n)
F是(是1,…,是n)=磷(是1≤是1,…,是n≤是n)=C(磷(是1≤是1),…,磷(是n≤是n)) =C(F是1(是1),…,F是n(是n))
## 有限元方法代写
tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。
## MATLAB代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|JOINT PROBABILITY DISTRIBUTIONS
statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论投资组合方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写风险理论投资组合代写方面经验极为丰富，各种代写风险理论投资组合相关的作业也就用不着说。
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Conditional Probability
A useful concept in understanding the relationship between multiple random variables is that of conditional probability. Consider the returns on the stocks of two companies in one and the same industry. The future return $X$ on the stocks of company 1 is not unrelated to the future return $Y$ on the stocks of company 2 because the future development of the two companies is driven to some extent by common factors since they are in one and the same industry. It is a reasonable question to ask, what is the probability that the future return $X$ is smaller than a given percentage, e.g. $X \leq-2 \%$, on condition that $Y$ realizes a huge loss, e.g. $Y \leq-10 \%$ ? Essentially, the conditional probability is calculating the probability of an event provided that another event happens. If we denote the first event by $A$ and the second event by $B$, then the conditional probability of $A$ provided that $B$ happens, denoted by $P(A \mid B)$, is given by the formula,
$$P(A \mid B)=\frac{P(A \cap B)}{P(B)}$$
which is also known as the Bayes formula. According to the formula, we divide the probability that both events $A$ and $B$ occur simultaneously, denoted by $A \cap B$, by the probability of the event $B$. In the two-stock example, the formula is applied in the following way,
$$P(X \leq-2 \% \mid Y \leq-10 \%)=\frac{P(X \leq-2 \%, Y \leq-10 \%)}{P(Y \leq-10 \%)}$$
Thus, in order to compute the conditional probability, we have to be able to calculate the quantity
$$P(X \leq-2 \%, Y \leq-10 \%)$$
which represents the joint probability of the two events.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Definition of Joint Probability Distributions
A portfolio or a trading position consists of a collection of financial assets. Thus, portfolio managers and traders are interested in the return on a portfolio or a trading position. Consequently, in real-world applications, the interest is in the joint probability distribution or joint distribution of more than one random variable. For example, suppose that a portfolio consists of a position in two assets, asset 1 and asset 2 . Then there will be a probability distribution for (1) asset 1 , (2) asset 2, and (3) asset 1 and asset 2. The first two distributions are referred to as the marginal probability distributions or marginal distributions. The distribution for asset 1 and asset 2 is called the joint probability distribution.
Like in the univariate case, there is a mathematical connection between the probability distribution $P$, the cumulative distribution function $F$, and the density function $f$ of a multivariate random variable (also called a random vector) $X=\left(X_{1}, \ldots, X_{n}\right)$. The formula looks similar to the equation we presented in the previous chapter showing the mathematical connection between a probability density function, a probability distribution, and a cumulative distribution function of some random variable $X$ :
\begin{aligned} P\left(X_{1} \leq t_{1}, \ldots, X_{n} \leq t_{n}\right) &=F_{X}\left(t_{1}, \ldots, t_{n}\right) \ &=\int_{-\infty}^{t_{1}} \ldots \int_{-\infty}^{t_{n}} f_{X}\left(x_{1}, \ldots, x_{n}\right) d x_{1} \ldots d x_{n} \end{aligned}
The formula can be interpreted as follows. The joint probability that the first random variable realizes a value less than or equal to $t_{1}$ and the second less than or equal to $t_{2}$ and so on is given by the cumulative distribution function $F$. The value can be obtained by calculating the volume under the density function $f$. Because there are $n$ random variables, we have now $n$ arguments for both functions: the density function and the cumulative distribution function.
It is also possible to express the density function in terms of the distribution function by computing sequentially the first-order partial derivatives of the distribution function with respect to all variables,
$$f_{X}\left(x_{1}, \ldots, x_{n}\right)=\frac{\partial^{n} F_{X}\left(x_{1}, \ldots, x_{n}\right)}{\partial x_{1} \ldots \partial x_{n}}$$
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Dependence of Random Variables
Typically, when considering multivariate distributions, we are faced with inference between the distributions; that is, large values of one random variable imply large values of another random variable or small values of a third random variable. If we are considering, for example, $X_{1}$, the height of a randomly chosen U.S. citizen, and $X_{2}$, the weight of this citizen, then large values of $X_{1}$ tend to result in large values of $X_{2}$. This property is denoted as the dependence of random variables and a powerful concept to measure dependence will be introduced in a later section on copulas.
The inverse case of no dependence is denoted as stochastic independence. More precisely, two random variables are independently distributed if and only if their joint distribution given in terms of the joint cumulative distribution function $F$ or the joint density function $f$ equals the product of their marginal distributions:
\begin{aligned} &F_{X}\left(x_{1}, \ldots, x_{n}\right)=F_{X_{1}}\left(x_{1}\right) \ldots F_{X_{n}}\left(x_{n}\right) \ &f_{X}\left(x_{1}, \ldots, x_{n}\right)=f_{X_{1}}\left(x_{1}\right) \ldots f_{X_{n}}\left(x_{n}\right) \end{aligned}
In the special case of $n=2$, we can say that two random variables are said to be independently distributed, if knowing the value of one random variable does not provide any information about the other random variable. For instance, if we assume in the example developed in section 1.6.1 that the two events $X \leq-2 \%$ and $Y \leq-10 \%$ are independent, then the conditional probability in equation (1.1) equals
\begin{aligned} P(X \leq-2 \% \mid Y \leq-10 \%) &=\frac{P(X \leq-2 \%) P(Y \leq-10 \%)}{P(Y \leq-10 \%)} \ &=P(X \leq-2 \%) \end{aligned}
Indeed, under the assumption of independence, the event $Y \leq-10 \%$ has no influence on the probability of the other event.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Definition of Joint Probability Distributions
FX(X1,…,Xn)=∂nFX(X1,…,Xn)∂X1…∂Xn
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Dependence of Random Variables
FX(X1,…,Xn)=FX1(X1)…FXn(Xn) FX(X1,…,Xn)=FX1(X1)…FXn(Xn)
## 有限元方法代写
tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。
## MATLAB代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|CONTINUOUS PROBABILITY DISTRIBUTIONS
statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论投资组合方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写风险理论投资组合代写方面经验极为丰富，各种代写风险理论投资组合相关的作业也就用不着说。
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Probability Distribution Function, Probability
If the random variable can take on any possible value within the range of outcomes, then the probability distribution is said to be a contimuous random variable. ${ }^{7}$ When a random variable is either the price of or the return on a financial asset or an interest rate, the random variable is assumed to be continuous. This means that it is possible to obtain, for example, a price of $95.43231$ or $109.34872$ and any value in between. In practice, we know that financial assets are not quoted in such a way. Nevertheless, there is no loss in describing the random variable as continuous and in many times treating the return as a continuous random variable means substantial gain in mathematical tractability and convenience. For a continuous random variable, the calculation of probabilities is substantially different from the discrete case. The reason is that if we want to derive the probability that the realization of the random variable lays within some range (i.e., over a subset or subinterval of the sample space), then we cannot proceed in a similar way as in the discrete case: The number of values in an interval is so large, that we cannot just add the probabilities of the single outcomes. The new concept needed is explained in the next section.
A probability distribution function $P$ assigns a probability $P(A)$ for every event $A$, that is, of realizing a value for the random value in any specified subset $A$ of the sample space. For example, a probability distribution function can assign a probability of realizing a monthly return that is negative or the probability of realizing a monthly return that is greater than $0.5 \%$ or the probability of realizing a monthly return that is between $0.4 \%$ and $1.0 \%$
To compute the probability, a mathematical function is needed to represent the probability distribution function. There are several possibilities of representing a probability distribution by means of a mathematical function. In the case of a continuous probability distribution, the most popular way is to provide the so-called probability density function or simply density function.
In general, we denote the density function for the random variable $X$ as $f_{X}(x)$. Note that the letter $x$ is used for the function argument and the index denotes that the density function corresponds to the random variable $X$. The letter $x$ is the convention adopted to denote a particular value for the random variable. The density function of a probability distribution is always nonnegative and as its name indicates: Large values for $f_{X}(x)$ of the density function at some point $x$ imply a relatively high probability of realizing a value in the neighborhood of $x$, whereas $f_{X}(x)=0$ for all $x$ in some interval $(a, b)$ implies that the probability for observing a realization in $(a, b)$ is zero.
Figure $1.1$ aids in understanding a continuous probability distribution. The shaded area is the probability of realizing a return less than $b$ and greater than $a$. As probabilities are represented by areas under the density function, it follows that the probability for every single outcome of a continuous random variable always equals zero. While the shaded area
in Figure $1.1$ represents the probability associated with realizing a return within the specified range, how does one compute the probability? This is where the tools of calculus are applied. Calculus involves differentiation and integration of a mathematical function. The latter tool is called integral calculus and involves computing the area under a curve. Thus the probability that a realization from a random variable is between two real numbers $a$ and $b$ is calculated according to the formula,
$$P(a \leq X \leq b)=\int_{a}^{b} f_{X}(x) d x$$
The mathematical function that provides the cumulative probability of a probability distribution, that is, the function that assigns to every real value $x$ the probability of getting an outcome less than or equal to $x$, is called the cumulative distribution function or cumulative probability function or simply distribution function and is denoted mathematically by $F_{X}(x)$. A cumulative distribution function is always nonnegative, nondecreasing, and as it represents probabilities it takes only values between zero and one. ${ }^{8} \mathrm{An}$ example of a distribution function is given in Figure 1.2.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|The Normal Distribution
The class of normal distributions, or Gaussian distributions, is certainly one of the most important probability distributions in statistics and due to some of its appealing properties also the class which is used in most applications in finance. Here we introduce some of its basic properties.
The random variable $X$ is said to be normally distributed with parameters $\mu$ and $\sigma$, abbreviated by $X \in N\left(\mu, \sigma^{2}\right)$, if the density of the random
$$f_{X}(x)=\frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}}, x \in \mathbb{R} \text {. }$$
The parameter $\mu$ is called a location parameter because the middle of the distribution equals $\mu$ and $\sigma$ is called a shape parameter or a scale parameter. If $\mu=0$ and $\sigma=1$, then $X$ is said to have a standard normal distribution.
An important property of the normal distribution is the location-scale invariance of the normal distribution. What does this mean? Imagine you have random variable $X$, which is normally distributed with the parameters $\mu$ and $\sigma$. Now we consider the random variable $Y$, which is obtained as $Y=$ $a X+b .$ In general, the distribution of $Y$ might substantially differ from the distribution of $X$ but in the case where $X$ is normally distributed, the random variable $Y$ is again normally distributed with parameters and $\bar{\mu}=a \mu+b$ and $\bar{\sigma}=a \sigma$. Thus we do not leave the class of normal distributions if we multiply the random variable by a factor or shift the random variable. This fact can be used if we change the scale where a random variable is measured: Imagine that $X$ measures the temperature at the top of the Empire State Building on January 1, 2008, at 6 A.M. in degrees Celsius. Then $Y=\frac{9}{5} X+32$ will give the temperature in degrees Fahrenheit, and if $X$ is normally distributed, then $Y$ will be too.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Exponential Distribution
The exponential distribution is popular, for example, in queuing theory when we want to model the time we have to wait until a certain event takes place. Examples include the time until the next client enters the store, the time until a certain company defaults or the time until some machine has a defect.
As it is used to model waiting times, the exponential distribution is concentrated on the positive real numbers and the density function $f$ and the cumulative distribution function $F$ of an exponentially distributed random variable $\tau$ possess the following form:
$$f_{\mathrm{r}}(x)=\frac{1}{\beta} e^{-\frac{x}{\beta}}, x>0$$
and
$$F_{\mathrm{r}}(x)=1-e^{-\frac{x}{\beta}}, x>0 .$$
In credit risk modeling, the parameter $\lambda=1 / \beta$ has a natural interpretation as hazard rate or default intensity. Let $\tau$ denote an exponential distributed random variable, for example, the random time (counted in days and started on January 1, 2008) we have to wait until Ford Motor Company defaults. Now, consider the following expression:
$$\lambda(\Delta t)=\frac{P(\tau \in(t, t+\Delta t] \mid \tau>t)}{\Delta t}=\frac{P(\tau \in(t, t+\Delta t])}{\Delta t P(\tau>t)} .$$
where $\Delta t$ denotes a small period of time.
What is the interpretation of this expression? $\lambda(\Delta t)$ represents a ratio of a probability and the quantity $\Delta t$. The probability in the numerator represents the probability that default occurs in the time interval $(t, t+\Delta t]$ conditional upon the fact that Ford Motor Company survives until time $t$. The notion of conditional probability is explained in section 1.6.1.
Now the ratio of this probability and the length of the considered time interval can be denoted as a default rate or default intensity. In applications different from credit risk we also use the expressions hazard or failure rate.
Now, letting $\Delta t$ tend to zero we finally obtain after some calculus the desired relation $\lambda=1 / \beta$. What we can see is that in the case of an exponentially distributed time of default, we are faced with a constant rate of default that is independent of the current point in time $t$.
Another interesting fact linked to the exponential distribution is the following connection with the Poisson distribution described earlier. Consider a sequence of independent and identical exponentially distributed random variables $\tau_{1}, \tau_{2}, \ldots$ We can think of $\tau_{1}$, for example, as the time we have to wait until a firm in a high-yield bond portfolio defaults. $\tau_{2}$ will then represent the time between the first and the second default and so on. These waiting times are sometimes called interarrival times. Now, let $N_{t}$ denote the number of defaults which have occurred until time $t \geq 0$. One important probabilistic result states that the random variable $N_{t}$ is Poisson distributed with parameter $\lambda=t / \beta$.
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Exponential Distribution
Fr(X)=1b和−Xb,X>0
Fr(X)=1−和−Xb,X>0.
λ(Δ吨)=磷(τ∈(吨,吨+Δ吨]∣τ>吨)Δ吨=磷(τ∈(吨,吨+Δ吨])Δ吨磷(τ>吨).
## 有限元方法代写
tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。
## MATLAB代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Concepts of Probability
statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论投资组合方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写风险理论投资组合代写方面经验极为丰富，各种代写风险理论投资组合相关的作业也就用不着说。
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|BASIC CONCEPTS
An outcome for a random variable is the mutually exclusive potential result that can occur. The accepted notation for an outcome is the Greek letter $\omega$. A sample space is a set of all possible outcomes. The sample space is denoted by $\Omega$. The fact that a given outcome $\omega_{i}$ belongs to the sample space is expressed by $\omega_{i} \in \Omega$. An event is a subset of the sample space and can be represented as a collection of some of the outcomes. ${ }^{3}$ For example, consider Microsoft’s stock return over the next year. The sample space contains outcomes ranging from $100 \%$ (all the funds invested in Microsoft’s stock will be lost) to an extremely high positive return. The sample space can be partitioned into two subsets: outcomes where the return is less than or equal to $10 \%$ and a subset where the return exceeds $10 \%$. Consequently, a return greater than $10 \%$ is an event since it is a subset of the sample space. Similarly, a one-month LIBOR three months from now that exceeds $4 \%$ is an event. The collection of all events is usually denoted by $\mathfrak{A}$. In the theory of probability, we consider the sample space $\Omega$ together with the set of events $\mathfrak{A}$, usually written as ( $\Omega, \mathfrak{A})$, because the notion of probability is associated with an event. ${ }^{4}$
## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Bernoulli Distribution
$$X=\left{1 如果 C 默认值 一世 0 别的。 \对。$$
(n ķ)=n!(n−ķ)!ķ!
## 有限元方法代写
tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。
## MATLAB代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。
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2022-09-25 07:31:53
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https://stats.stackexchange.com/questions/242003/anova-for-intercept-term-in-simple-linear-regression
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# ANOVA for intercept term in Simple Linear Regression
In an ANOVA framework, is it possible to test the hypothesis that $\beta_0 = 0$ given a "full" model of $\hat y_i = \beta_0 + \beta_1 x_i$? Or does the "full" model not contain the "small" model $\hat y_i = \beta_1 x_i$ (and are there bias-related issues)?
• It's not quite clear what you are asking about when you say "are there bias related issues" ... any time you have a smaller model being compared to a larger one, if the larger model is the correct one, the estimates in the smaller model are biased. If that's not what you're asking about you should clarify. – Glen_b Oct 24 '16 at 5:19
• Thanks -- I meant bias in the sense of errors not summing to zero, ans I guess my question was more along the lines of: is the soace spanned by the columns of the "small" model contained in the space of the columns spanned by the "full" model? Somehow I want to say "no" because the errors in the "small" model do not sum to zero, right? – rrrrr Oct 24 '16 at 14:35
• You mean the residuals? Sure, rather than their mean being zero you instead have a weighted mean of them being zero. But in relation to estimation, bias is about $E(T-\theta)$ for some estimator $T$ of some unknown quantity $\theta$. I'm still not entirely sure I see the issue. You still have $E(\hat{\beta}_1)=\beta_1$ for example. – Glen_b Oct 24 '16 at 21:11
• Sorry, I was referring to the sum of the residuals being nonzero, which you pointed out. Ok, so just to confirm, is it correct to think about $span(\text{small model}) \subset span(\text{full model})$ in the sense that the small model is a subspace of the columns of the larger model's design matrix? – rrrrr Oct 25 '16 at 14:56
• If you're asking "what makes models nested" you could try some of the questions on site -- e.g. maybe this one could be of some help: stats.stackexchange.com/questions/4717/… – Glen_b Oct 26 '16 at 6:15
Yes, you can test whether the intercept is 0 via ANOVA, or indeed by looking at the regression coefficient's t-value.
1. An example (in R) using ANOVA:
full <- lm(dist~speed,cars)
noint <- lm(dist~0+speed,cars)
anova(noint,full)
Analysis of Variance Table
Model 1: dist ~ 0 + speed
Model 2: dist ~ speed
Res.Df RSS Df Sum of Sq F Pr(>F)
1 49 12954
2 48 11354 1 1600.3 6.7655 0.01232 <-----
.... the p-value for the intercept term is 0.0123
1. Using the t-ratio for the intercept term:
summary(lm(dist~speed,cars))
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -17.5791 6.7584 -2.601 0.0123 <---------
speed 3.9324 0.4155 9.464 1.49e-12
--
Residual standard error: 15.38 on 48 degrees of freedom
Multiple R-squared: 0.6511, Adjusted R-squared: 0.6438
F-statistic: 89.57 on 1 and 48 DF, p-value: 1.49e-12
Again the p-value for the intercept term is 0.0123
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2020-10-19 16:03:54
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https://mathoverflow.net/questions/311178/induced-mathfrakg-k-modules
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# Induced $(\mathfrak{g},K)$-modules
Let $$G$$ be a noncompact simple Lie group, and $$G'$$ a noncompact reductive subgroup of $$G$$. Fix a maximal compact subgroup $$K$$ of $$G$$ such that the intersection $$K'=K\cap G'$$ is a maximal compact subgroup of $$G'$$. Denote by $$\mathfrak{g}$$ and $$\mathfrak{g}'$$ the complexified Lie algebras of $$G$$ and $$G'$$ respectively. Let $$\theta$$ be the Cartan involution of $$G$$ corresponding to $$K$$.
Suppose that $$\pi$$ is a nontrivial unitarizable simple $$(\mathfrak{g}',K')$$-module. Is there a common way to obtain an induced $$(\mathfrak{g},K)$$-module from $$\pi$$?
If $$\mathfrak{g}'$$ is a Levi subalgebra of a $$\theta$$-stable parabolic subalgebra of $$\mathfrak{g}$$, one may use the Zuckerman functor composed with the produced functor to obtain a $$(\mathfrak{g},K)$$-module. But what if $$\mathfrak{g}'$$ is not supposed to be a Levi subalgebra of $$\mathfrak{g}$$?
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2019-03-25 14:55:38
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https://boredofstudies.org/threads/pls-help.388062/
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pls help (1 Viewer)
mathsbrain
Member
Question: Sugar is dissolved in water in the ratio 1:3 to form sugar syrup. This sugar syrup is further diluted with the addition of water so that the ratio of the sugar syrup to the additional water is 5:7. FInd the ratio of sugar to water in the final mixture.
Accurate
member
1/4 times 5/12 = 5/48 (that's how much sugar there is )
the rest is water so 43 (48-5)
I think the answer is 5:43
uh i think i got this wrong though, but i guess this is my quick attempt
edit : its right somehow
Last edited:
Drdusk
π
$\bg_white \text{I did this in a bit of a rush but here}$
$\bg_white \text{Let the amount of syrup be 'q', and the amount of initial water be 'r'. Also let the amount of water added be 'x'}$
$\bg_white \therefore \frac{q}{r} = \frac{1}{3},\hspace{2mm} \frac{q + r}{x} = \frac{5}{7}$
$\bg_white \text{We are required to find the value of } \frac{q}{r + x} \text{as (r + x) equals the total amount of water at the end and q is the amount of sugar}$
$\bg_white \frac{q+ r}{x} = \frac{5}{7}\Rightarrow x = \frac{7(q + r)}{5}\text{. Subbing this into q/(r+x) gives}$
$\bg_white \frac{q}{r+x} = \frac{q}{r + \frac{7(q + r)}{5}} = \frac{5q}{5r + 7(q + r)}$
$\bg_white \text{Multiplying top and bottom by 1/q gives:}$
$\bg_white \frac{5q}{5r + 7(q+r)} = \frac{5q}{5r + 7(q+r)} \times \frac{1/q}{1/q} = \frac{5}{\frac{5r}{q} + 7(q+r)/q} = \frac{5}{(5r/q) + 7(1 + r/q)}$
$\bg_white \text{Now from our first equation we have:}\hspace{2mm} \frac{q}{r} = \frac{1}{3}\Rightarrow \frac{r}{q} = 3$
$\bg_white \therefore \hspace{2mm} = \frac{5}{5(3) + 7(1+3)}$
$\bg_white = \frac{5}{43}$
$\bg_white \therefore \text{The ratio is}\hspace{2mm} 5:43$
Last edited:
5:43 ?
idkkdi
New Member
If we let original value be 1000mL.
Amount of sugar = 1000 x 1/4 = 250mL, Amount of water = 1000 x 3/4 = 750mL
5:7 ratio sugar with additional water.
1000L x 7/5 = 1400mL additional water.
250: 2150
5:43
Last edited:
Accurate
member
If we let original value be 1000mL.
Amount of sugar = 1000 x 1/4 = 250mL, Amount of water = 1000 x 3/4 = 750mL
5:7 ratio sugar with additional water.
250mL x 7/5 = 350mL additional water.
250mL: 750mL+350mL
250:1100
5:22
Well using your same method I got 5:43
Let total be 1000ml
250:750 = 1:3
1000:1400 = 5:7
To calculate how much water there is we know that 750ml of that original 1000ml is water, plus add another 1400ml of water so a total of 2150 ML
There is 250 grams of sugar in the solution so 250:2150
= 5/43
5:43
Accurate
member
They added extra water to the original solution not another mixture
idkkdi
New Member
My bad, didn't notice sugar SYRUP.
Drdusk
π
Its 5:43 confirmed by my 4u state ranking friend haha.
I made a silly error when I said q/(x) = 5/7. It's actually (q+r)/x = 5/7
This should get you 5:43
Last edited:
idkkdi
New Member
Isn't 48 including sugar?
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2019-08-24 00:49:27
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https://web2.0calc.com/questions/evaluate-the-following-limit-algebraically-show-all-work
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+0
# Evaluate the following limit algebraically: Show All Work.
+2
56
1
+875
Evaluate the following limit algebraically:
Apr 21, 2020
#1
+2
do direct substitution first.
that gets you 0/0, which means that further work can be done
since theres a radical there, we can multiply the numerator and denominator by the conjugate of the part containing the square root, in this case the numerator
$$lim_{x\rightarrow 5}\frac{2-\sqrt{9-x}}{x-5}*\frac{2+\sqrt{9-x}}{2+\sqrt{9-x}}\\ =lim_{x\rightarrow 5}\frac{9-(9-x)}{(x-5)(2+\sqrt{9-x})}\\ =lim_{x\rightarrow 5}\frac{x-5}{(x-5)(2+\sqrt{9-x})}\\ =lim_{x\rightarrow 5}\frac{1}{2+\sqrt{9-x}}\\ =\frac{1}{2+\sqrt{9-5}}=\frac{1}{4}$$
.
Apr 21, 2020
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2020-05-26 21:33:34
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https://control.com/textbook/foundation-fieldbus-instrumentation/ff-function-blocks/
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# FOUNDATION Fieldbus Function Blocks
## Chapter 13 - Basics of FOUNDATION Fieldbus (FF) Instrumentation
Data-processing modules within FF systems are known as function blocks. Sometimes these blocks serve merely to catalogue data, while in other instances the blocks execute specific algorithms useful for process measurement and control. These “blocks” are not physical entities, but rather abstract software objects – they exist only as bits of data and instructions in computer memory. However, the blocks are represented on FF computer configuration displays as rectangular objects with input ports on the left-hand side and output ports on the right-hand side. The construction of a working control system comprised of FF devices consists of linking the outputs of certain function blocks with the inputs of other function blocks via configuration software and computer-based tools. This usually takes the form of using a computer to draw connecting lines between the output and input ports of different function blocks.
### Analog function blocks versus digital function blocks
Function-block programming in general strongly resembles the design philosophy of legacy analog-based computer systems, where specific functions (addition, subtraction, multiplication, ratio, time-integration, limiting, and others) were encapsulated in discrete operational amplifier circuits, and whole systems were built by connecting function blocks together in whatever patterns were desired to achieve a design goal. Here with Fieldbus programming, the function blocks are virtual (bits and data structures in digital memory) rather than real analog circuits, and the connections between blocks are merely pointer assignments in digital memory rather than actual “patch cable” connections between circuit boards.
An example contrasting analog circuit design with Fieldbus function-block design appears here, both systems selecting the greatest temperature signal to be the output. The system on the left-hand side receives analog voltage signals from three temperature sensors, using a network of operational amplifiers, diodes, and resistors to select the greatest voltage signal to be the output. The system on the right-hand side uses three Fieldbus transmitters to sense temperature, the greatest temperature signal selected by an algorithm (the ISEL function block) running in a Fieldbus device. The device running the ISEL function could be one of the three FF temperature transmitters, or another device on the segment:
Instead of analog voltage signals sent by wire to special-function circuit modules, FOUNDATION Fieldbus uses digital messages sent over an H1 network segment to special-function software “blocks” running inside ordinary Fieldbus devices. The lines connecting different function blocks together in a FOUNDATION Fieldbus system show the sources and destinations of these digital messages. If two FF function blocks reside in different FF devices, the connecting lines represent publisher/subscriber communication assignments coordinated by the Link Active Scheduler (LAS) device.
### Function block location
There is usually some freedom of choice in where various function blocks may be located in a FF segment. Take for example the following flow control loop, where a flow transmitter feeds measured flow data into a PID control function block, which then drives a control valve to whatever position necessary to regulate flow. The actual physical device layout might look something like this:
The function block connections necessary for this control scheme to work are shown in the next diagram, coupling the AI (analog input) block located in the transmitter to a PID control block to an AO (analog output) block located in the valve positioner:
All function block inputs are on the left-hand sides of the blocks, and all outputs are on the right-hand sides. In this function block program, data from the analog input (AI) block flows into the PID block. After calculating the proper output value, the PID block sends data to the analog output (AO) block where the final control element (e.g. valve, variable-speed motor) is adjusted. The AO block in turn sends a “back calculation” signal to the PID block to let it know the final control element has successfully reached the state commanded by the PID block’s output. This is important for the elimination of reset windup in the event the final control element fails to respond to the PID block’s output signal.
It should be obvious that the analog input (AI) block must reside in the transmitter, simply because only the transmitter is able to measure the process fluid flow rate. Likewise, it should be obvious that the analog output (AO) block must reside in the control valve positioner, simply because the valve is the only device capable of manipulating (exerting influence over) anything. However, given the lack of a separate controller device, the person configuring the Fieldbus loop may choose to locate the PID block in either the transmitter or the control valve positioner. So long as both the FF transmitter and the FF valve positioner possess PID function block capability, it is possible to locate the PID function block in either device.
The following illustrations show the two possible locations of the PID function block in this system:
The only factor favoring one location over another for the PID function block is the number of communication broadcasts (“Compel Data” token distributions and replies) necessary per macrocycle. Note the lines connecting function blocks between the two instruments in the previous diagrams (lines crossing from one blue bubble to another). Each of these lines represents a VCR (Virtual Communication Relationship) – an instance during each macrocycle where data must be transmitted over the network segment from one device to another. With the PID function block located in the flow transmitter, two lines connect blocks located in different physical devices. With the PID function block located in the valve positioner, only one line connects blocks in different physical devices. Thus, locating the PID function block in the valve positioner means only one CD message/reply is necessary per macrocycle, making the network communication more efficient.
To illustrate the difference this re-location of the PID block makes, we will examine the function block diagram and macrocycle timing schedule on a simple pressure control FF loop, hosted on an Emerson DeltaV distributed control system. The first composite screenshot shows the function block diagram and schedule with the PID function block located in the transmitter (PT_501):
Note the two scheduled communication events (CD tokens and responses) necessary in the macrocycle schedule to enable communication between pressure transmitter PT_501’s PID function block and valve positioner PV_501’s analog output function block. The first CD token in this macrocycle schedule compels the PID block to publish its “output” signal (subscribed to by the analog output block), while the second token compels the analog output block to publish its “back calculation” signal (subscribed to by the PID block). The amount of time required for function block execution and their publisher/subscriber communications is 330 milliseconds, with a total macrocycle time of 1 second.
Now let’s examine the same PID pressure control system with the PID function block moved to the valve. Here you see the function block diagram followed immediately by the updated macrocycle schedule:
In this macrocycle timing schedule, there is only one CD token needed: compelling the analog input block to publish its measurement signal (subscribed to by the PID block). This makes the block execution plus scheduled communication time 30 milliseconds shorter than before (300 milliseconds total as opposed to 330 milliseconds), since there is one less scheduled communications event happening. The total macrocycle time of 1 second remains unchanged, but now we have 30 milliseconds more unscheduled time during which other communication events may take place.
### Standard function blocks
The FF standard specifies many different function blocks for the construction of control algorithms. Ten of them are considered “basic” FF function blocks:
• AI – Analog Input
• AO – Analog Output
• B – Bias
• CS – Control Selector
• DI – Discrete Input
• DO – Discrete Output
• ML – Manual Loader
• PD – Proportional/Derivative control
• PID – Proportional/Integral/Derivative control
• RA – Ratio
Nineteen more “Advanced” function blocks are incorporated in the FF standard:
• Pulse Input
• Complex Analog Output
• Complex Discrete Output
• Step Output PID
• Device Control
• Setpoint Ramp
• Splitter
• Input Selector
• Signal Characterizer
• Calculate
• Arithmetic
• Integrator
• Timer
• Analog Alarm
• Discrete Alarm
• Analog Human Interface
• Discrete Human Interface
Five more function blocks are specified as well:
• Multiple Analog Output
• Multiple Digital Input
• Multiple Digital Output
• Flexible Function Block
The primary benefit of standardization is that the end-user may choose FF instruments manufactured by any standard-compliant vendor, and those function blocks should behave the same as the equivalent function blocks within any other manufacturer’s model of FF device. There are, of course, examples where manufacturers have equipped their FF devices with “extended” capability function blocks going beyond the Fieldbus Foundation standard, and the user must beware of this.
### Device-specific function blocks
In addition to the function blocks necessary to construct control schemes, all FF instruments contain one Resource block and usually one or more Transducer blocks describing details specific to that instrument. The following computer screenshot shows all function blocks within a Rosemount model 3095MV Fieldbus transmitter:
The Resource block appears first in this list, followed by three transducer blocks, then followed by the palette of general function blocks for use in constructing control algorithms. Information contained in the Resource block of an FF instrument includes the following:
• Identifier (the 32-byte code unique to every FF device)
• Type of device
• Device revision level
• Memory total and available (free) capacity
• Computation time
• Available features listing
• Current device state (Initializing, Standby, On-line, Failed, etc.)
Transducer blocks provide a means of organizing data relevant to the actual sensing inputs, outputs, calculated variables, and graphic displays of a FF device. There need not be a one-to-one correspondence between the number of transducer blocks in an FF device and the number of physical I/O channels it has. For example, in the Rosemount 3095MV multivariable transmitter, transducer block 1100 manages all physical measurement inputs (pressure and temperature sensors) while transducer block 1200 is reserved for inferred mass flow (based on calculations performed on the raw sensor measurements) and transducer block 1300 manages data for the liquid crystal display (LCD).
### FF signal status
As mentioned earlier, function block programming bears a strong resemblance to analog function-block circuit design, where specific tasks are divided up into discrete elements, those elements connected together to form a larger system with more complex functionality. One of the important distinctions between legacy analog function block circuit design and FF function block programming is the data content of the lines connecting blocks together. In the analog world, each connecting line (wire) carries exactly one piece of information: a single variable represented in analog form by a voltage signal. In the world of Fieldbus, each connecting line carries not only the variable’s numerical value, but also a status and in some cases an engineering unit (a unit of measurement). For example, a Fieldbus transmitter sensing temperature might output a digital process variable (PV) signal of “342 degrees Celsius, Good”, whereas a temperature transmitter with an analog (e.g. 4-20 mA) output is merely able to send a signal representing the temperature (no measurement unit or status information).
The inclusion of status along with data is a powerful concept, with roots in scientific practice. Scientists, as a rule, do their best to report the degree of confidence associated with the data they publish from experiments. Data is important, of course, but so is the degree of certainty with which that data was obtained. Obviously, data gathered with instruments of low quality (high uncertainty) will have different significance than data gathered with instruments of high precision and impeccable accuracy (low uncertainty). Any scientist basing research on a set of scientific data published by another scientist will have access to the data’s certainty in addition to the data itself – a very valuable detail.
By the same token, data “published” by a FF device is only as good as the health of that device. A FF transmitter exhibiting noisy or wildly fluctuating measurements might very well be nearing complete failure, and therefore its published data should be treated with skepticism. Since FF devices are “smart” (meaning, among other things, they have self-diagnostic capability), they have the ability to flag their own data as “Bad” if an internal fault is detected. The data still gets published and sent to other FF function blocks, but the status sent along with that data warns all downstream blocks of its uncertainty.
The three major status conditions associated with every FF signal passed between function blocks are Good, Bad, and Uncertain. Sub-status states also exist to further delineate the nature of the uncertainty. “Sensor Failure” is an example of a sub-status value, describing the reason for a “Bad” status value from a process transmitter.
In computer science, there is a truism that “Garbage In equals Garbage Out,” sometimes abbreviated as GIGO. No algorithm, no matter how advanced, can guarantee an output of good data from an input of bad data. This principle finds intelligent application in FF function block programming, as the blocks are programmed to switch mode when “Bad” or “Uncertain” input statuses are detected. For example, here are some of the possible actions a function block may be configured to take upon detection of a “Bad” input signal status:
• Set output signal to last “Good” value
• Fail high (set output signal to top-of-range value)
• Fail low (set output signal to bottom-of-range value)
Furthermore, status values are propagated in a FF system from the input to the output of every function block connected in series, reflecting the effect of an input signal’s uncertainty throughout the entire control loop. For example, an analog input (AI) block sending a “Bad” status signal to the process variable input of a PID control block will have its “Bad” status propagated to the output of the PID block as well. When that “Bad” PID output signal reaches the analog output (AO) function block, that final block knows the signal is not to be trusted, because its origin (the AI block) is untrustworthy. Any function blocks receiving the PID block’s output signal will likewise sense the “Bad” status and further propagate that status to their output signal(s). This “status propagation” ensures all function blocks in a Fieldbus control system are “aware” of the input data status, so that a “Bad” measurement does not result in “bad” control decisions made on that data.
### Function block modes
All FF function blocks must support multiple modes of operation, describing how the block should execute its intended function. Several different function block modes are commonly found for FF function blocks, though not all FF function blocks support all of these modes:
• OOS (Out Of Service) – All function blocks are required to support this mode, where the block freezes its output at the last calculated value and attaches a “Bad” status value
• Man (Manual) – the output of the block is fixed at a value determined by the technician, with a “Good” status value attached
• Auto (Automatic) – the function block processes information normally
• Cas (Cascade) – the function block processes information normally
• Iman (Initialization Manual) – the output of the block is fixed at its last calculated value, due to the output signal path being incomplete
• LO (Local Override) – the output of the block is fixed at its last calculated value, due to a detected fault condition within the device
• RCas (Remote Cascade) – the function block processes information normally based on a setpoint sent from a remote source to the block’s RCas_In input
• ROut (Remote Output) – the function block passes data to its output sent from a remote source to the block’s ROut_In input
Instrumentation technicians and professionals are already familiar with the concept of a controller having “Automatic,” “Manual,” and even “Cascade” operating modes, but Fieldbus function block programming extends this general concept to each and every function block. With FF, each block may be independently set into “Automatic” or “Manual” mode, which is a useful tool for testing FF algorithms and troubleshooting complex FF control schemes. The “Out of Service” mode, for instance, is commonly set when performing routine maintenance on an FF device (e.g. checking the calibration of an FF transmitter).
It is worth noting an important distinction here between Manual mode and OOS (Out Of Service) mode. In both cases, the function block’s output becomes fixed at some value, but a major difference between these two modes is their associated statuses. In Manual mode, the output value is fixed and the status is “Good,” allowing all function blocks downstream to remain operational. In OOS mode, the output value is fixed and the status is “Bad,” causing all downstream function blocks to react as they would when receiving any “Bad” signal status (usually by shedding to Manual mode themselves). Placing a function block in Manual mode is useful when performing tests on the control strategy because it allows the technician or engineer to simulate values that might come from transmitters and other “upstream” devices in the loop. All function blocks receiving a signal from a block in Manual mode will continue to operate as they are designed. However, placing a function block in OOS mode is quite different in that all function blocks receiving that signal will act as though there is a serious problem rather than acting normally.
In addition to these operating modes for FF function blocks (not all of which are supported by all FF blocks), FF function blocks also have four mode categories describing valid modes for the block to be in under various conditions:
• Target
• Actual
• Permitted
• Normal
A block’s “Target” mode is the mode it strives to be in if possible. The “Actual” mode is the mode the block is in at the present time. “Permitted” modes list all the different modes which may be used as “target” modes. “Normal” is a category describing to an operator interface what a block’s normal operation mode should be, but the block itself does not heed this setting.
Published under the terms and conditions of the Creative Commons Attribution 4.0 International Public License
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2022-01-24 17:42:37
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https://stats.stackexchange.com/questions/352012/lm-test-in-var-models
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# LM test in VAR models
One very important task, after estimating VAR(p) model, is to test the residuals for autocorrelation. One possible test is the LM test. My question is the following: How to chose the number of lags in the LM test? Since I estimated a VAR(p) model, should I also look for autocorrelation in the first p lags only?
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2019-07-23 22:51:43
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http://olivercowdery.com/texts/boud1816.htm
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OliverCowdery.com -- The Premier Web-Site for Early Mormon History
Bookshelf | Spalding Library | Mormon Classics | Newspapers | History Vault
Elias Boudinot (1740-1821) A Star in the West... (1st ed., Trenton, NJ, 1815) Title Page Contents Preface Introduction Chap. 1-4 Chap. 5-8 Chap. 9-12 Appendix transcriber's comments
Age of Revelation (1790) | Second Advent (1815) | Biographical Info.
This web-document is still under construction
A
STAR IN THE WEST;
OR,
A HUMBLE ATTEMPT TO DISCOVER
THE LONG LOST
TEN TRIBES OF ISRAEL,
PREPARATORY TO THEIR RETURN TO THEIR BELOVED CITY,
JERUSALEM.
BY ELIAS BOUDINOT, L. L. D.,
Who is wise, and he shall understand thes ethings? Prudent, and he shall know them? For all the ways of the Lord are right, and the just shall walk in them; but the transgressors shall fall therein. -- Hosea. And the Lord answered me and said, write the vision and make it plain, upon a (writing) table, that he may run who readeth it: for the vision is yet for an appointed time, but at the end it shall speak and not lie; though it tarry, wait for it, because it will surely come. It will not tarry. -- Habbak.
TRENTON, N. J.
George Sherman, Printer.
1816.
A STAR IN THE WEST. 81 CHAPTER II. An enquiry into the question, on what part of the globe is it most likely, that these descendants of Israel may now be found, arising from late discoveries and facts, that have not come to the knowledge of the civilized world, till of late years, EVERY quarter of the world has been so traversed and explored by the hardy and adventurous seamen of modern Europe and America, as well as by travellers whose curiosity and indefatigable labours, have scarcely left any considerable tract of the globe unnoticed, that we can scarcely presume on making the least discovery in any hitherto unknown part of the world. We must look yo the histories of countries already known to the geographer and traveller, and apply to the divine scriptures for the compass which is to direct our course. Hence it must answer to the following particulars -- 1. It must be a country to the north and west from Judea. Jeremiah, ii. 17-18, xxii. 7-8. Zechariah, ii. 6. 2. It must be a far country from Judea. Isaiah, xliii. 6 -- xlvi. 11.* * Remember the former things of old time, verily I am God and none elsel I am God and there is none like unto me. From the beginning, making known the end, and from ancient times the things that are not yet done, saying, my council shall stand, and whatever I have willed I will effect. Calling from the east, the eagle, and from a land far distant, the man of my council. As I have spoken, so will I bring it to pass; I have formed the design and I will execute it. Lowth's translation. 82 A STAR IN THE WEST. 3. It must answer the term, from the ends of the earth. Isaiah, xliii. 1 6. 4. It must he in the western regions, or the country of the going down of the sun. Zecharia, viii. 7. 5. It must be a land, that at the time of the tribes going to, was without inhabitants, and free from heathen neighbours. 2 Esdras, xiii. M. 6. It must be beyond the seas from Palestine, the country to which part of them are to return in ships. Isaiah, Ix. 9 xvii. 2. The scriptures are very positive in four of the above par ticulars, the fifth is founded on the text from 2cl Esdras, and although it is not pretended that the apochryphal books bear any comparison as to divine inspiration, with the bible, yet as that book was written by a Jew, somewhere about the year 100, it may, as has already been observed, be used as evidence of an historic fact, equally with any other historian, and if cor roborated by other facts, will add to the testimony. As to the sixth particular, this is not only supported by the text, but it is the opinion of that great and judicious writer, the Rev. Mr. Faber, on the whole representation of the scriptures, who certainly deserves the attention of every serious Christian. He seems very positive " that some prevailing maritime pow er of faithful worshippers, will be chiefly instrumental in con verting and restoring a part of the Jewish nation. This seems to be declared in scripture, more than once, witli suffi cient plainness." " Who are these ? like a cloud they fly, and like doves to their holes. Surely the Isles shall wait for me, and the ships of Tarshish, among the first, to bring thy sons from afar : their silver and their gold with them, unto A STAR IN THE WEST. 83 the name of the Lord thy God, unto the holy one of Israel, be cause he hath glorified me." Isaiah, Ix. 8 9. Again it is expressly said, they are to he gathered from the coasts of the earth, implying that they were to have some connection with the sea, and the address which God makes to them puts it out of doubt. " Ho ! land spreading wide the shadow of thy wings, which are beyond the rivers of Cush, or Cuthai, accus tomed to send messengers "by sea, even in Bulrush vessels upon the surface of the waters. Go swift messengers unto the nation dragged away and plucked ; unto a people wonder ful from the beginning hitherto ; a nation expecting, expect ing and trampled underfoot ; whose lands the rivers have spoiled. Isaiah, xviii. 1 2. At that season, a present shall be led to the Lord of Hosts ; a people dragged away and pluck ed ; even a people wonderful from the beginning hitherto ; a nation expecting, expecting and trampled under foot ; whose land, rivers have spoiled, unto the place of the name of the Lord of Hosts, Mount Zion." Isaiah, xviii. 7. Mr. Faber bas given a paraphrase of part of the foregoing texts, thus, (3d vol. 94) Go swift messengers, unto ti nation, long appa rently forsaken by God ; a nation dragged away from their own country and plucked; a nation wonderful from their beginning hitherto ; a nation perpetually expecting their pro- mised Messiah, and yet trampled under foot ; a nation whose land the symbolical rivers of foreign invaders have for ages spoiled. Go swift messengers ! You who by your skill in nav igation, and your extensive commerce and alliances, are so qualified to be carriers of a message to people in the remotest countries ; go with God's message unto a nation dragged away : to the dispersed Jews; a nation dragged away from its pro- 84 A STAR IN THE WEST. per seat, and plucked of its wealth and power $a people won derful from its beginning to this very time for the speeial pro vidence which has ever attended them and directed their for tunes ; a nation still lingering in expectation of the Messiah, who sd long since came and was rejected by them and now is coming again in glory ; a nation universally trampled under foot ; whose land, rivers, armies of foreign invaders, the Assy rians, Babylonians, Syromacedonians, Romans, Saracens, and Turks, have over-run and depopulated." Letter on Isa iah, 18. "My worshippers beyond the river Cush, (which must be to the northward and westward of Jerusalem) shall bring as an offering to me, the daughters of my dispersion." Zeph. iii. 10. And Zechariah treating on the same subject, says, I will hiss for them (the tribes of Ephraim and his children, mentioned in the former verses) for I have redeemed them ; and they shall increase as they have (heretofore) increased. And I will sow them among the people, and they shall remem ber me in far countries, and they shall live with their children and turn again. And I will bring them again also (that is besides those from far countries) out of the land of Egypt, and gather them out of Assyria, and I will bring them into the land of Giliad and Lebanon, and place shall not be found for them. And he (that is Ephraim) shall pass through the sea with affliction, and shall smite the wares in the sea f and all the deeps of the river shall dry up, and the pride of Assy ria shall be brought down, and the sceptre of Egypt shall pass away, and I will strengthen them in the Lord, and they shall walk up and down in his name, saith the Lord," Zechariah, x, A STAR IN THE WEST. 85 Here is an explicit difference made between the return of Judah and Ephraim, that is, between the Jews and Israel ites -the latter is to come from a far country he is to pass through a great water, or over the seas, or both. The words here made use of, may be very applicable, to people, who have no knowledge or experience of passing over the sea in ships, whose sickness is generally extremely distressing. Mr. Faber supposes that the land spreading wide the sha dow of her wings, may be some maritime nation, the sails of whose ships, and the protection given by them, are here pro phesied of. He seems to think, this may refer to Great Bri tain, in like manner, as she may be designated by Tarshish, which was formerly a great trading and maritime country. Yet he thinks it possible it may refer to some other maritime nation but it is asked, why not to a union of maritime nations, on so important and difficult an undertaking. From a serious consideration of all the foregoing circum stances, we seem naturally led to have recourse to the late discovered continent of America, which the first visitants found filled with inhabitants, and though called savages, dif fered essentially from all the savages ever known to the peo ple of the old world before. In the first place they resem bled (considerably) in appearance, the people of the oriental nations. Mr. Penn, who saw and communicated with them in a particular manner, on his first arrival in America, while in their original, uncontaminated state, before they were debased and ruined by their connection with those who called themselves civilized and Christians, was exceedingly struck with their appearance, j In one of his letters to his friends in England, he says, I found them with like countenance with 86 A STAR IN THE WEST. the Jewish race ; and their children of so lively a rcsemhlance to them, tli at a man would think himself in Duke's-place or Berry-street, in London, when he seeth them.' 5 (Penn's Works. 2d vol. 70i, year 1682.) They wore ear-rings and *imse jewels; bracelets on their arms and legs ; rings on their fingers; necklaces made of highly polished shells found in their rivers and on their coasts. Their females tied up their hair behind, worked hands round their heads, and ornamented them with shells and feathers, and are fond of strings of heads round several parts of their bodies. They use shells and tur key spurs round the tops of their mocasins, to tinkle like lit tle bells, as they walk." Isaiah proves this to have been the custom of the Jewish women, or something much like it. " In that day, says the prophet, the Lord will take away the bra very of their tinkling ornaments about their feet, and their cauls, and their round tires like the moon. The chains and the bracelets and the muflers. The bonnets and the orna ments of the legs, and the head-bands, and the tablets, and the car rings ; the rings and the nose jewels." Isaiah, iii. 18. > They religiously observed certain feasts, and feasts very similar to those enjoined on the Hebrews, by Moses, as will hereinafter more particularly be shewn.' In short, many, and indeed, it may be said, most of the learned men, who did pay any particular attention to these natives of the wilderness at their first coming among them, both English and Spaniards, were struck with their general likeness to the Jews. The Indians in New Jersey, about 1681, are described, as persons straight in their limbs, beyond the usual proportion in most nations ; very seldom crooked or deformed ; their features regular; their countenances some times fierce, in common A STAR IN THE WEST. 87 rather resembling a Jew, than a Christian. (Smith's History ( of New-Jersey, 14.) It shall now he our business to collect those facts in their history, that are well attested, with those which may he known of them from personal knowledge of men of character, or from their present manners, customs and habits ; although we are well advised, and it should be constantly borne in mind, that the corruption of both principle and practice, introduced amongst them, by their connection with Europeans, has so debased their morals and vitiated all their powers of mind, that they are quite degenerated from their ancestors. An old Cliaribbee Indian, in a very early day, thus address ed one of the white people. Our people are become almost as bad as yours. We arc so much altered since you came among us, that we hardly know ourselves, and we think it is owing to so melancholy a change, that hurricanes are more frequent than formerly. It is the evil spirit, who has done all this who has taken our best lands from us, and given us up to the dominion of christains. Edward's History West-Indies, 1 vol. 28. And yet we very gravely assert that we have bene fited the Indian nations, by teaching them the Christian religion. The Indians have so degenerated, that they cannot at this time give any tolerable account of the origin of their religious rites, ceremonies and customs, although religiously attached to them as the commands of the great spirit to their forefath ers. Suppose a strange people to be discovered, before wholly unknown to the civilized world, and an enquiry was instituted into their origin, or from what nation they had sprung, what 88 A STAR IN THE WEST. mode of examination would be most likely to succeed and lead to a rational solution of the question? In our opinion, a strict enquiry into the following particulars, would be the best means of accomplishing this valuable purpose. Their language. Their received traditions. Their known religious rites and ceremonies. And, lastly, their public worship and religious opinions and prejudices. Therefore to commence this enquiry, with some degree of method, we shall confine ourselves to these five particulars, as far as we can find well authenticated data to proceed upon. A STAR IN THE WEST. 89 CHAPTER III. An enquiry into the the language of the American Indians, WHEN we consider how soon the family of Noah, scattered throughout Asia, Africa and Europe, lost almost every trace of their original language, so far at least, as not to be easily understood by the nations into which they became divided -- established different manners and customs peculiar to each nation or people -- and finally formed for themselves respectively, such absurd and wholly differing modes of religious worship, as well as principles and doctrines, and finally became, at different times, to bear the most inveterate hatred to each other, we could no longer, at this remote period, hope for much success in looking for convincing testimony to prove the fact very satisfactorily, though we should stumble on the actual descendants of those children of Abraham, the ten lost tribes of Israel, after so long a dispersion and entire separation from the rest of the world. And if we do find any convincing testimony on this subject, we must attribute it to the over-ruling providence of that God who is wonderful in council, and true to all his promises. Hear Sir William Jones, whose authority will have great influence on all who know his character. In his discourse on the origin of the East Indians or Hindoos, Arabs, Tartar, &c. he says, "hence it follows, that the only family after the flood, established itself in the northern parts of Iran, now Persia. That as the family multiplied, they were divided into three distinct branches, each retaining 90 A STAR IN THE WEST. little, at first, and loosing the whole by degrees, of their com mon primary language ; hut agreeing severally on new expressions for new ideas." \ Father Charlevoix, a famous French writer, who came over to Canada very early, and paid particular attention to the Indian natives, says, " that the only means (which others have neglected) to come at the original of the Indian natives, are the knowledge of their languages, and comparing them with those of the other hemisphere, that are considered as primitives. Manners very soon degenerate by means of com merce with foreigners, and by mixture of several nations unit ing in one body and particularly so, amongst wandering tribes, living without principle, laws, education or civil gov ernment, especially where absolute" want of the necessaries of life takes place, and the necessity of doing without, causes their names and uses to perish together. From their dialects, we may ascend to the mother tongues themselves. These are distinguished by being more nervous than those derived from them, because they are formed from nature, and they contain a greater number of words, imitating the things where of they are the signs. Hence he concludes that if those cliar- acteristical marks which are peculiar to any oriental nation arc found in the Indian languages, we cannot reasonably doubt of their being truly original, and consequently, that the peo ple who speak them, have passed over from that hemisphere.*' This then must be an enquiry into facts, the investigation of which, from the nature of the subject, muft be wholly founded on well authenticated accounts recorded by writers of charac ter, who may be consulted on this occasion ; or from the information of such persons who have been long domesticated A STAR IN THE WEST. 91 with particular nations, suspected to have originated from the other hemisphere ; or of persons whose occupation or mode of life has led them to visit parts of the globe, the most likely to afford some light on this ahstruse subject. And even hero our assistance cannot be expected to be great ; but whatever we are able to discover, we will put together, in hopes that by pursuing this enquiry, though we should arise no farther than bare rudiments, 'the curiosity of the more learned and persevering, may produce some further and more adequate discovery, to enlighten mankind. Th^ difficulties attending this attempt must be great. The Indian languages, having never been reduced to any certainty by letters, must have been exposed to great changes and misconceptions. They are still a wandering people, having no knowledge of gram mar or of the arts and sciences. No monuments of antiquity no mechanical trades oppressed and distressed on all hands driven from their original residence into a wilder ness, and even there not suffered to remain stationary ; but still driven from place to place debased and enervated b\ the habitual use of intoxicating spirits, afforded them by tra ders for the double purpose of profit and imposition vitia ted by the awful example of white people, we are at this day confined to the few traces of their original language, their religion, rites and customs, and a few common traditions that may yet with labour be collected, to form our opinions upon. The Indian languages in general, are very copious and expres sive, considering the narrow sphere in which they move; their ideas being few in comparison with civilized nations. They have neither cases nor declensions. They have few or 92 A STAR IN THE WEST. no prepositions 'they remedy this, by affixes and suffixes, and their words are invariably the same in both numbers. All this, if the writer's information be correct, is very simi lar to the Hebrew language. He has been informed from good authority, and the same is confirmed by a writer well acquainted with the subject, that there is no language known in Europe, except the Hebrew, without prepositions ; that is, in separate and express words. The Indians have all the other parts of speech, except as above. They have no com parative or superlative degrees of comparison more than the Hebrews. They form the last, by some leading vowel of the divine name of the great spirit, added to the word. It is observed by some Jewish, as well as Christian interpreters, that the several names of God, are often given as epithets by the Hebrews to those things which are the greatest, the strongest, and the best of their kind, as ruach elohim, a migh ty wind. 1 vol. Stackhouse's History of the Bible, page 8, in a note. " Both languages are very rhetorical, nervous and em- phatical. Those public speeches of the Indians, that the wri ter of these memoirs has heard or read, have been oratorical and adorned with strong metaphors in correct language, and greatly abound in allegory. About the year 168*, the gov ernor of New- York, sent an accredited agent to the Ononda- gos, on a dispute that was likely to arise with the French. The agent (one Arnold) behaved himself very haughtily towards the Indians, at delivering his commission. One of the chiefs then answered him in a strain of Indian eloquence, in which he said among other things, " I have two arms I extend the one towards Montreal, there to support the tree of yeace j and the other towards Corlacr, (the governor of New- A STAR IN THE WEST. 93 York) who has long been my brother. Gnonthis (the governor of Canada) lias been these ten years my father. Corlaer has been long my brother, with my own good will, but neither the one or the other is my master. He 'who made tlie world, gave me this land I possess. / am free. I respect them both ; but no man has a right to command me, and none ought to take amiss, my endeavouring all I can, that this land should not be troubled. To confclude, I can no longer delay repairing to my father, who has taken the pains to come to my very gate, and who has no terms to propose, but what are reasonable/' 1 Wynne's History America, 402 3. At a meeting held with the President, General Washing ton, in 1790, to prevail upon him to relax the terms of a trea ty of peace, made with commissioners under the old confede ration, relative to an unreasonable cession of a large part of their country, which they had been rather persuaded to make to the United States, for the sake of peace, and which after wards they sincerely repented of, Cornplant who had long been a steady friend to the United States, in the most perilous part of the revolutionary war, delivered a long, persuasive and able speech, which the writer of this preserved, and has now before him, and from which are extracted the following sentences, as a proof of the above assertion. " Father, when your army entered the country of the six nations, we called you the town destroyer, and to this day, when your name is heard, our women look behind them and turn pale ; our chil dren cling close to the necks of their mothers ; but our coun cillors and warriors being men, cannot be afraid ; but their hearts are grieved by the fears of our women and children, and desire that it may be buried so deep, as to be heard of no 94 A STAR IN THE WEST. more. Father, we will not conceal from you, that the great spirit and not man, has preserved Cornplant from the hands of Jus own nation. For they ask continually, where is the land, on which our children and their children, are to lie down upon ? You told us, say they, that a line drawn from Penn sylvania to Lake Ontario, would mark it forever on the east ; and a line running from Beaver Creek to Pennsylvania, would mark it on the west. But we see that it is not so. For first one and then another comes and takes it away by order of that people, who you told us, promised to secure it to us forever. Cornplant is silent, for he has nothing to answer. When the sun goes down, Cornplant opens his heart before the great spirit ; and earlier than the sun appears again upon the hills, lie gives thanks for his protection during the night, for he feels, that among men become desperate by the injuries they sus tain, it is God only that can preserve him. Cornplant loves peace all he had in store, he has given to those, who have been robbed by your people, lest they should plunder the inno cent, to repay themselves. 66 The whole season which others have employed in providing for their families, Cornplant has spent in endeavors to preserve peace, and at this moment, his wife and children are lying on the ground, and in want of food. His heart is in pain for them ; but he perceives, that the great spirit, will try his firmness, in doing what is right. Father ! innocent men of our nation are killed one after another, though of our best families ; but none of your people, who have committed these murders, have been punished. \Ve recollect that you did promise to punish those who should kill our people; and we ask, was it intended that your people should kill the Seneca's, and not only remain un- A STAR IN THE WEST. 95 punished, but be protected from the next of kin. Father! these to us are great things. We know that you are -very strong We Itave heard that you are wise, but we sluill wait to hear your answer to this, that we may know thatijou are just." Adair records a sentence of a speech of an Indian captain to his companions, in his oration for war. Near the conclusion of his harangue, he told the warriors, he feelingly knew that their guns were burning in their hands their tomahawks were thirsty to drink the blood of their enemy, and their trusty arrows were impatient to be upon the wing ; and lest delay should burn their hearts any longer, he gave tbem the cool re freshing word, "join the holy ark" and away to cut off the de voted enemy." But a speech made by Logan, a famous Indian chief, about the year 1775, was never exceeded by Demosthenes or Cicero. In revenge for a murder committed by some unknown Indians, a party of our people fired on a canoe loaded with women and children, and one man, all of whom happened to belong to the family of Logan, who had been long the staunch friend of the Americans, and then at perfect peace with them. A war im mediately ensued, and after much blood-shed on both sides, the Indians were beat, and sued for peace. A treaty was held, but Logan disdainfully refused to be reckoned among the suppliants; but to prevent any disadvantage from his ab sence, to his nation, he scut the following talk, to be delivered to lord Dunmore at the treaty. I appeal to any white man to say, if he ever entered Logan's cabin hungry, and he gave him not meatif ever he came cold and naked, and Lo gan clothed him not. During the course of the last long and bloody war, Logan remained idle in his eabin, an advocate for 96 A STAR IN THE WEST. peace. Such was his love for the white men, that my coun trymen pointed as they passed, and said, Logan is the friend of white men. I had thought to have lived with you, but for the injuries of one man. Colonel = the last spring, in cold blood, and unprovoked, murdered all the relations of Logan. not sparing even my woman and children. There runs not a drop of his blood in the veins of any living creature. This called on me for revenge. I have sought it. I have killed many. I have fully glutted my vengeance. For my country, I rejoice at the beams of peace. But do not harbor a thought that mine is the joy of fear. Logan never felt fear. He will not turn on his heel to save his life. Who is there to mourn for Logan ? No, not one." Great allowance must be made for translations into another language, especially by illiterate and ignorant interpreters. This destroys the force as well as beauty of the original. A writer (Adair) who has had the best opportunities to know the true idiom of their language, by a residence among them for forty years, has taken great pains to shew the similarity of the Hebrew, with the Indian languages, both in their roots and general construction ; and insists that many of the Indian words, to this day, are purely Hebrew, notwithstanding their exposure to the loss of it to such a degree, as to make the preservation of it so far, little less than miraculous. Let any one compare the old original Hebrew, spoken with so much purity by the Jews before the Babylonish captivity, with that spoken by the same people on their return, after the comparatively short space of seventy years, and he will find it had become a barbarous mixture of the Hebrew and Chal- daic languages, so as not to be understood by an ancient A STAR IN THE WEST. 97 Hebrew, and in a great measure, has continued so to this day. We say such a consideration will show an almost miraculous intervention of Divine Providence, should a clear trace of the original language he discoverable among the natives of our wilderness at this day. " Their words and sentences are ex pressive, concise, emphatical, sonorous and bold." Father Charlevoix, in his history of Canada, paid more attention to the Indian languages than most travellers before him, and indeed he had greater opportunities, and was a man of learn ing, and considerable abilities. He says, " that the Algonquin and Huron languages, have, between them, that of almost all the savage nations of Canada we are acquainted with. Who ever should well understand both, might travel without an interpreter, more than fifteen hundred leagues of country, and make himself understood by an hundred different nations, who have each their peculiar tongue. The Algonquin especial ly has a vast extent. It begins at Acadia and the Gulph of St. Lawrence, and takes a compass of twelve hundred leagues, twining from the south-east by the north, to the south-west. They say also, that the Wolf Nation, or the Mohegans, and the greatest part of the Indians of New-England and Virginia, speak the Algonquin dialects. The Huron language has a copiousness, an energy, and a sublimity, perhaps not to be found in any of the finest languages we know of; and those whose native tongue it is, though now but a handful of meii, have such an elevation of soul, as agrees much better with the majesty of their language, than with the state to which they are reduced. Some have fancied they found a similarity with the Hebrew, others have thought it had the same origin with the Greek." "The Algonquin language has not so much 98 A STAR IN THE WEST. force as the Huron; but lias more sweetness and elegance. Both have a richness of expression, a variety of turns, a pro priety of terms, a regularity which astonishes -hut what is more surprising, is, that among these barbarians, who never study to speak well, and who never had the use of writing, there is never introduced a bad word, an improper term, or a vicious construction. And even their children preserve all the purity of the language in their common discourse. On the other hand, the manner in which they animate all they say, leaves no room to doubt of their comprehending all the worth of their expressions, and all the beauty of their language." Mr. Golden, who wrote the History of the Wars of the Five Nations, about the year 1750, and was a man of considerable note, speaking of the language of those nations says, " they arc very nice in the turn of their expressions, and that a few of them are so far masters of their language, as never to offend the ears of their Indian auditory by an impolite expression. They have, it seems, a certain urbanity or atticism in their language, of which the common ears are very sensible, though only their great speakers attain to it. They are so given to speech-making, that their common compliments to any person they respect, at meeting or parting, are made in harangues. They have a few radical words, but they compound them with out end. By this their language becomes sufficiently copious, and leaves room for a good deal of art to please a delicate ear. Their language abounds with gutturals and strong aspirations, which make it very sonorous and bold. Their speeches abound with metaphors, after the manner of the eastern nations." It should be noted, that Mr. Colden, though a sensible man, and of excellent character, could not speak their language, and A STAR IN THE WEST. 99 not having any considerable communication with them, took his information from others. The late Rev. Dr. Jonathan Edwards, of Connecticut, son of the late President Edwards, who was a man of great celeb rity, as a well read, pious divine, and of considerable erudi tion, was intimately associated with the Indians at Stock- bridge, of the Mohegan tribe in that state, from the age of six years. He understood their language equally with his mother tongue. He also had studied that of the Mohawks, having resided in their nation about six months for that purpose. He informs us that the name Mohegan is a corruption of Mukke- kaneaw, arising from the English pronunciation. This is a very common thing, and occasions much confusion, and great difficulties, in tracing the languages of the different tribes. For we have not only to contend with a different pronuncia tion and spelling of both English and French, but the corrup tion and ignorance of interpreters and traders, especially in an early day; and also the different modes of writing the same word by different people, arising from their different concep tions of the word as pronounced by the Indians.* As for in stance, in the same words by the English and French English. French. Owcnagunges. Abenaguies. Maques. Aniez. Odistastagheks. Mascoaties. Makihander. Mourigan. * The different sounds given by different tribes to the same letters, is also a source of difficulty. Those who write, often use the letter a, where the sound is oh, so that owoh is used in the Mohegan where a or au is used in other languages, as Moquoh for Mauquah,a bear. The sound of these two arc alike, when spoken fey an Indian. The e final, is never sorunded in any word, but a monosyllable. 100 A STAR IN THE WEST. English. Frencfi, Oneydoes. Oneyonts. Utawawas. Outawies. Todericks. Tateras. Satana's. Shaononons. The Mohcgan language was spoken by all the various tribes of New-England. Many of the tribes had a different dialect, but the language was radically the same. Mr. Elliot, called the Indian Apostle, who was among the first settlers of Massachusetts, and died in 1691, translated the bible into Indian, which is found to be in a particular dialect of the Mo- hegan language. Dr. Edwards says it appears to be much more extensive than any other language in North- Am erica. The language of the Delawares, in Pennsylvania, of the Penob- scots, bordering on Nova- Scotia, of the Indians of St. Francis, in Canada, of the Shawanese, on the Ohio, and of the CMppewas, at the westward of Lake Huron, were all radically the same with the Mohegan. The same is said of the Ottowas, Nanii- cokes, Munsees, Menomomes, Messisagas, Saukies, Ottagawnies, Killistinoes, Nipegons, Jllgonk'ms, Winmbagoes, &c. Dr. Edwards asserts, that for the pronouns common in other languages, they express the pronouns both substantive and adjective, by affixes or letters, or syllables added at the begin nings or ends, or both, of their nouns. In this particular, the structure of their language coincides with that of the Hebrew, in an instance in which the Hebrew differs from all the lan guages of Europe, ancient and modern, with this only differ ence, that the Hebrews always joined the affixes to the ends of the words, whereas the Indians, in pronouns of the singular number, prefix the letter or syllable ; but hi the plural mini- A STAR IN THE WEST. 101 ler, they add others as suffixes. Also as the word is in creased, they change and transpose the vowels, as in tmohhe- can, an hatchet ; ndumhemn, my hatchet : the o is changed into u, and transposed after the manner of the Hebrews; likewise in some instances, the t is changed into d. Besides what has heen ohserved concerning prefixes and suffixes, there is a remarkahle analogy, says Dr. Edwards, hetween some words of the Mohegan language, and the cor respondent words in the Hebrew. In the Mohegan niah is T. In Hebrew it is ani y which is the two syllables of niah trans posed. Keah, thou or thee. The Hebrews use Ka the suffix. UwoJi 9 is this man, or this thing ; very analagous to the He brew Hu 9 or Huah, ipse. Necaunuh is we : in Hebrew it is naclmu or anachnu. In Hebrew m is the suffix for me, or the first person. In the Mohegan, n, or ne, is prefixed to de note the first person, as nmeetseh, or nimeetseh, I eat. In Hebrew k or ka, is the suffix for the second person, and is indifferently either a pronoun, substantive or adjective, l^or ka, has the same use in the Mohegan language as kmeetseh or kameetseh, thou eatest. Knish, thy hand. In Hebrew the van, and the letter u and hu, are the suffixes for he or them- In the Indian the same is expressed by u, or uw, and by oo* as in uduhwhunnw, I love him. Pumissoo, he walketh. In Hebrew, the suffix to express our, or us, is nu. In Mohcgan, it is nuh, as nogknuh, our father. Nmeetschnuh, we eat, &c. To elucidate this subject still farther, a list of a few words in the different Indian dialects shall be added, with the same words in Hebrew and Chaldaick. 102 A STAR IN THE WEST. English. Charibbee. Greeks. His wife My wife Come hither Liani Yene-nori Hace-yete The heavens Chemim Jehovah Jocanna Woman Ishto Man or chief Ish I Thou or thee This man We Assembly or walled house Necklace or collar My necklace Wood My skin I am sick Good be to you To blow Roof of the house Go thy way Eat To eat The nose Give me nourishment* The great first Moihegan, and Northern Languages. Y. He. Ho. wah Ishte Niah Keah Uwoh Necaumdi Kurbet Enca Yene kali Hue Nora Nane guaete Halea tibou Phoubac Toubana ora Bayou boorkaa Baika Aika Nichiri Natoni boman Yo hewah Hebrew. Li hene Hene herranni Aca-ati (Samari tan) Shemim Jehovah Ishto Ish Ani, the 2 sylla bles transposed as ahni Ka Huah Nachnu Guir, or gra bit Ong Von gali Oa (Chaldaic) Ourni Nanceheti Ye hali ettuboa Phouhe Debona our Boua Bouak Bge Chaldaic Akl do. Neheri Natoui bamen Jehovah * Edward's West-Jndias A STAR IN THE WEST. 103 English. Charibbee. Creeks. Mohegan, and Northern Languages. Hebrew. Praise the first Halleluwah Hallelujah cause Father Abba Abba Now, the pres Na Na ent time Very hot, or Ileru, hara, or Hara haft bitter upon hala me , To pray Phale Phalac The hind parts Kesh Kish One who kills Abe, derived Abel another from Abele Gruf The war name Noabe, com who kills a pounded of rambling en Noah & Abe emy Canaan Kenaai Canaan Wife Awah Eve or eweh Winter Kora Cora Another name Ale Ale or alohim for God Do lennois* lannonf Indians of Pe- nobscot Arrarat,a high. Arrarat, a high Arrarat, a high mountain mountain. mountain As the writer of this does not understand either the Hebrew or Indian languages, so as to be a judge of their true idioms or spelling, he would not carry his comparisons of one language with the other, too far. Yet he cannot well avoid mentioning, merely as a matter of curiosity, that the Mohawks, in confed eracy with the Five Nations, as subsisting at the first arrival * Barlow. t Litterally he shall be called a son, Christian Observer for June IS13, p 349. 104 A STAR IN THE WEST. of the Europeans in America, were considered as the law givers, or the interpreters of duty, to the other tribes. Nay, this was so great, that all paid obedience to their advice. They considered themselves as supreme, or first among the rest. Mr. Colden says, that he had been told by old men in New-England, that when their Indians were at war, formerly, with the Mohawks, as soon as one appeared, their Indians raised a cry from hill to hill, a Mohawk ! a Mohawk ! Upon which all fled like sheep before a wolf, without attempting to make the least resistance. And that all the nations around them, have for many years, entirely submitted to their advice, and pay them a yearly tribute of wampum. The tributary nations dare not make war or peace, without the consent of the Mohawks. Mr. Colden has given a speech of the Mo hawks, in answer to one from the governor of Virginia, com plaining of the other confederate nations, which shows the Mohawks superiority over them, and the mode in which they corrected their misdoings. Now it seems very remarkable, that the Hebrew word Mhhokek, spelled so much like the Indian word, means a law-giver, (or leges interpres) or a superior. Blind chance could not have directed so great a number of remote and warring savage nations to fix on, and unite in so nice a religious standard of speech, and even grammatical con struction of language, where there was no knowledge of letters or syntax. For instance, A, oo, E A, is a strong religious Indian emblem, signifying, I dinib 9 ascend, or remove to another place of residence. It points to A-rio-wah, the first person singular, and E A, or Yah, He, Wah, and implies putting themselves under the divine patronage. The beginning of that most A STAR IN THE WEST. 105 sacred symbol, is by studious skill, and a thorough knowledge of the power of letters, placed twice, to prevent them from being applied to the sacred name, for vain purposes, or crea ted tilings. Though they have lost the true meaning of their religious emblems, except what a very few of the more intelligent tra ders revive in the retentive memories of the old inquisitive magi, or, beloved man; yet tradition directs them to apply them properly. They use many plain religious emblems of the divine name, as Y, 0, he, wah Yah and Ale, and these are the roots of a prodigious number of words, through their various dialects. It is worthy of remembrance, that two Indians, who belong to far distant nations, without the knowl edge of each other's language, except from the general idiom, will intelligibly converse together, and contract engagements Without any interpreter, in such a surprising manner, as is scarcely credible. In like manner we read of Abraham, Isaac and Jacob, travelling from country to country, from Chaldea into. Palestine, when inhabited by various differing nations thence into Egypt and back again, making engage ments, and treating with citizens wherever they went. But we never read of any difficulty of being understood, or their using an interpreter. The Indians generally express themselves with great vehe mence and short pauses, in their public speeches. Their periods are well turned, and very sonorous and harmonious. Their words are specially chosen, and well disposed, with great care and knowledge of their subject and language, to show the being, power and agency of the great spirit in all that concerns them. 106 A STAR IN THE WEST. To speak in general terms, their language in their rootsy idiom and particular construction, appears to have the whole genius of the Hebrew,, and what is very remarkable, and well worthy of serious observation, has most of the peculiarities of that language, especially those in which it differs from most other languages; and "often, both in letters and signification^ synonimous with the Hebrew language." They call the light ning and thunder, Eloha, and its rumbling noise Rowah, which may not, improperly, be deduced from the Hebrew word Enoch, a name of the third person in the holy Trinity, originally signifying " the air in motion, or a rushing wind." Faber. ^ The Indian compounded words are generally pretty long? but those that are radical or simple, are mostly short; very few, if any of them, exceed three or four syllables. And as their dialects are guttural, every word contains some conson ants, and these are the essential characteristics of language*. Where they deviate from this rule, it is by religious emblems, which obviously proceeds from the great regard they pay to the names of the Deity, especially to the great four lettered, divine, essential name, by using the letters it contains, and the vowels it was originally pronounced with, to convey a vir tuous idea; or by doubling or transposing them, to signify the contrary. In this all the Indian nations agree. And as this general custom must proceed from one primary cause, it seems to assure us, that this people was not in a savage state when they first separated, and varied their dialects with so much religious care and exact art* Souard, in his Melanges de Literature, or Literary Mis cellanies, speaking of the Indians of Guiana, observes, ""-on A STAR IN THE WEST. 107 the authority of a learned Jew, Isaac JVasc?, residing at Suri nam/' we are informed that the language of those Indians, which he calls the Galibe dialect, and which is common to all the tribes of Guiana, is soft and agreeable to the ear, abound ing in vowels and synonims, and possessing a syntax as regu lar as it would have been, if established by an academy. This Jew says that all the substantives are Hebrew. The word expressive of the soul in each language, means breath. They have the same word in Hebrew to denominate God, which means master, or lord." It is said there are but two mother tongues among the north ern Indians, and extending thence to % the Missisippi, the Huron and Algonquin, and there is not more difference be tween these, than between the Norman and French. Dr. Edwards asserts that the language of the Delawares, in Penn sylvania of the Penobscots, bordering on Nova-Scotia of the Indians of St. Francis, in Canada of the Shawanese, on the Ohio of the Chippewas, to the westward of Lake Huron of the Ottawas, Nanticokes, Munsees, Minoniones, Messina- gues, Saasskies, Ottagamies, Killestinoes, Mipegoes, Algon- quins, Winnebagoes, and of the several tribes in New-Eng land, are radically the same, and the variations between them are to be accounted for from their want of letters and of com munication. Much stress may be laid on Dr. Edwards' opin ion. He was a man of strict integrity, and great piety. He had a liberal education was greatly improved in the Indian languages, which he habituated himself to from early life, having lived long among the Indians. 108 A STAR IN THE WEST. [This page is blank] A STAR IN THE WEST. 109 CHAPTER IV. The Indian Traditions as received by their Nations. AS the Indian nations have not the assistance afforded by the means of writing and reading, they are obliged to have recourse to tradition, as du Pratz, 2 vol. 169, has justly observed, to preserve the remembrance of remarkable transactions or historical facts; and this tradition cannot be preserved, but by frequent repetitions; consequently many of their young men are often employed in hearkening to the old beloved men, narrating the history of their ancestors, which is thus transmitted from generation to generation. In order to preserve them pure and incorrupt, they are careful not to deliver them indifferently to all their young people, but only to those young men of whom they have the best opinion. They hold it as a certain fact, as delivered down from their ancestors, that their forefathers, in very remote ages, came from a far distant country, by the way of the west, where all the people were of one colour, and that in process of time they moved eastward to their present settlements. This tradition is corroborated by a current report among them, related by the old Chickkasah Indians to our traders, that now about 100 years ago, there came from Mexico, some of the old Chickkasah nation, or as the Spaniards call them Chichemicas, in quest of their brethren, as far north as the Aquahpah nation, above one hundred and fifty miles above the Natchez, on the south-east side of the Missisippi river; 110 A STAR IN THE WEST. but through French policy, they werre either killed or sent back, so as to prevent their opening a brtherly intercourse with them, as they had proposed. It is also said, that the Nauatalcas believe that they dwelt in another region before they settled in Mexico. -- That their forefathers wandered eighty years in search of it, through a strict obedience to the commands of the great spirit; who ordered them to go in quest of new lands, that had such particular marks as were made known to them, and they punctually obeyed the divine mandate, and by that means found out and settled that fertile country of Mexico. Our southern Indians have also a tradition among them which they firmly believe, that of old time, their ancestors lived beyond a great river. That nine parts of their nation, out of ten, passed over the river, but the remainder refused and staid behind. That they had a king when they lived far to the west, who left two sons. That one of them, with a number of his people, travelled a great way for many years, till they came to Delaware river, and settled there. That some years ago, the king of the country from which they had emigrated, sent a party in search of them. This was at the time the French were in possession of the country on the river Alleghany. That after seeking six years, they found an Indian who led them to the Delaware towns, where they staid one year. That the French sent a white man with them on their return, to bring back an account of their country, but they have never been heard of since. It is said among their principal or beloved men, that they have it handed down from their ancestors, that the book which the white people have was once theirs. That while they A STAR IN THE WEST. 111 had it they prospered exceedingly; but that the white people bought it of them, and learnt many things from it; while the Indians lost their credit, offended the great spirit, and suffered exceedingly from the neighbouring nations. That the great spirit took pity on them, and directed them to this country. That on their way they came to a great river, which they could not pass, when God dried up the waters and they passed over dry shod. They also say that their forefathers were possessed of an extraordinary divine spirit, by which they foretold future events, and controlled the common course of nature, and this they transmitted to their offspring, on condition of their obeying the sacred laws. That they did by these means bring down showers of plenty on the beloved people. But that this power, for a long time past, had entirely ceased. The reverend gentleman mentioned in the introduction, who had taken so much pains in the year 1764 or 5, to travel far westward, to find Indians who had never seen a white man, informed the writer of these memoirs, that far to the northwest of the Ohio, he attended a party of Indians to a treaty, with Indians from the west of the Mississippi. Here he found the people he was in search of -- he conversed with their beloved man who had never seen a white man before, by the assistance of three grades of interpreters. The Indian informed him, that one of their most ancient traditions was, that a great while ago, they had a common father, who lived towards the rising of the sun, and governed the whole world. That all the white people's heads were under his feet. That he had twelve sons, by whom he administered his government. That his authority was derived from the great spirit, by virtue of 112 A STAR IN THE WEST. some special gift from him. That the twelve sons behaved very bad and tyrannized over the people, abusing their power to a great degree, so as to offend the great spirit exceedingly. That he being thus angry with them, suffered the white people to introduce spirituous liquors among them, made them drunk, stole the special gift of the great spirit from them, and by this means ursurped the power over them, and ever since the Indians heads were under the white people's feet. But that they also had a tradition, that the time would come, when the Indians would regain the gift of the great spirit from the white people, and with it their ancient power, when the white people's heads would be again under the Indian's feet. Mr. M'Kenzie in his History of the Fur Trade, and his journey through North-America, by the lakes, to the South-Sea, in the year ____, says "that the Indians informed him, that they had a tradition among them, that they originally came from another country inhabited by wicked people, and had traversed a great lake, which was narrow, shallow and full of islands, where they had suffered great hardships and much misery, it being always winter, with ice and deep snows -- at a place they called Copper-mine River, where they made the first land, the ground was covered with copper, over which a body of earth had since been collected to the depth of a man's heighth. They believe also that in ancient times their ancestors had lived till their feet were worn out with walking, and their throats with eating. They described a deluge, when the waters spread over the whole earth, except the highest mountain, on the top of which they were preserved. A STAR IN THE WEST. 113 They also believe in a future judgment." M'Kenzie's history, page 113. The Indians to the eastward say, that previous to the white people coming into the country, their ancestors were in the habit of using circumcision, but latterally, not being able to assign any reason for so strange a practice, their young people insisted on its being abolished. M'Kenzie says the same of the Indians he saw on his route, even at this day. History, page 34. Speaking of the nations of the Slave and Dog-rib Indians, very far to the northwest, he says, "whether circumcision be practiced among them, I cannot pretend to say, but the appearance of it was general among those I saw." The Dog-rib Indians live about two or three hundred miles from the straits of Kamschatka. Dr. Beatty says, in his journal of a visit he paid to the Indians on the Ohio, about fifty years ago, that an old christian Indian informed him, that an old uncle of his, who died about the year 1728, related to him several customs and traditions of former times; and among others, that circumcision was practiced among the Indians long ago, but their young men making a mock of it, brought it into disrepute, and so it came to be disused. Journal, page 89. The same Indian said, that one tradition they had was, that once the waters had overflowed all the land, and drowned all the people then living, except a few, who made a great canoe and were saved in it. Page 90. And that a long time ago, the people went to build a high place. That while they were building of it, they lost their language, and could not understand one another. That while one, perhaps, called for a stick, another 114 A STAR IN THE WEST. brought him a stone, &c. &c. and from that time the Indians began to speak different languages. Father Charlevoix, the French historian, informs us that the Hurons and Iroquois, in that early day, had a tradition among them that the first woman came from heaven and had twins, and that the elder killed the younger. In an account oublished in the year 1644, by a Dutch minister of the gospel, in New-York, giving an account of the Mohawks, he says, "an old woman came to my house and told the family, that her forefathers had told her that the great spirit once went out walking with his brother, and that a dispute arose between them, and the great spirit killed his brother." This is plainly a confusion of the story of Cain and Abel. It is most likely from the ignorance of the minister in the idiom of the Indian language, misconstruing, Cain being represented as a great man, for the great spirit. Many mistakes of this kind are frequently made. Mr. Adair, who has written the History of the Indians, and who deserves great credit for his industry and improving the very great and uncommon opportunities he enjoyed, tells us, that the southern Indians have a tradition, that when they left their own native land, they brought with them a sanctified rod, by order of an oracle, which they fixed every night in the ground; and were to remove from place to place on this continent, towards the rising sun, till it buded in one night's time. That they obeyed the sacred oracle, and the miracle at last took place, after they arrived on this side of the Mississippi, on the present land they possess. This was the sole cause of their settling there -- of fighting so firmly for their reputed A STAR IN THE WEST. 115 holy land and holy things -- that they may be buried with their beloved forefathers." This seems to be taken from Aaron's rod. Col. James Smith, in his Journal of Events, that happened while he was prisoner with the Caughnewaga Indians, from 1755 to 1759, says, "they have a tradition that in the beginning of this continent, the angels or heavenly inhabitants, as they call them, frequently visited the people, and talked with their forefathers, and gave directions how to pray, and how to appease the great being, when he was offended. They told them they were to offer sacrifice, burn tobacco, buffaloe and deer's bones, &c. &c." Page 79. The Ottawas say, "that there are two great beings that rule and govern the universe, who are at war with each other; the one they call Maneto, and the other Matchemaneto. Thet say that Maneto is all kindness and love, and the other is an evil spirit that delights in doing mischief. Some say that they are equal in power; others say that Maneto is the first great cause, and therefore must be all powerful and supreme, and ought to be adored and worshipped; whereas Matchemaneto ought to be rejected and despised." "Some of the Wyandots and Caughnewaga's profess to be Roman Catholics; but even these retain many of the notions of their ancestors. Those who reject the Roman Catholic religion, hold that there is one great first cause, whom they call Owaheeyo, that rules and governs the universe, and takes care of all his creatures rational and irrational, and gives them their food in due season, and hears the prayers of all those who call upon him; therefore it is but just and reasonable to pray and offer sacrifice to this great being and to do those things that are pleasing 116 A STAR IN THE WEST. in his sight. But they widely differ in what is pleasing or displeasing to this great being. Some hold that following nature or their own propensities is the way to happiness. Others reject this opinion altogether, and say, that following their own propensities in this manner is neither the means of happiness, or the way to please the deity. My friend, Teeaughretanego said, our happiness depends on our using our reason, in order to suppress these civil dispositions; but when our propensities neither lead us to injure ourselves nor others, we may with safety indulge them, or even pursue them as the means of happiness, Page 80. Cany any man read this short account of Indian traditions, drawn from tribes of various nations, from the west to the east, and from the south to the north, wholly separated from each other, written by different authors of the best characters, both for knowledge and integrity, possessing the best means of information, at various and distant times, without any possible communication with each other, and in one instance from occular and sensible demonstration; written on the spot in several instances, with the relators before them; and yet suppose that all this is either the effect of chance, accident or design, from a love of the marvellous or a premeditated intention of deceiving, and thereby ruining their own well established reputations? Charlevoix was a clergyman of character, who was with the Indians some years, and travelled from Canada to the Mississippi, in that early day. Adair lived forty years entirely domesticated with the southern Indians and was a man of learning and great observation. Just before the revolutionary war he brought his A STAR IN THE WEST. 117 manuscript to Elizabeth-Town, in New Jersey, to William Livingston, Esq. (a neighbour of the writer) to have it examined and corrected, which was prevented by the troubles of a political nature, just breaking out. The Rev. Mr. Brainerd was a man of remarkable piety, and a missionary with the Crosweek Indians to his death. Dr. Edwards was eminent for his piety and learning, and was intimately acquainted with the Indians from his youth. Dr. Beatty was a clergyman of note and established character. Bartram was a man well known to the writer, and travelled the country of the southern Indians as a botonist, and was a man of considerable discernment, and had great means of knowledge; and M'Kenzie, in the employment of the northwest company, an old trader, and the first adventurous explorer of the country, from the lake of the woods to the southern ocean. It is now asked, can any one carefully and with deep reflection, consider and compare these traditions with the history of the ten tribes of Israel, and the late discoveries of the Russians, capt. Cook and others, in and about the peninsula of Kamschatka and the northeast coast of Asia and the opposite shore of America, of which little was before known by any civilized nation, without at least drawing strong presumptive inferences, in favour of these wandering nations being descended from some oriental nation of the old world, and most probably, all things considered, being the lost tribes of Israel. Let us look into the late discoveries, and compare them with the Indian traditions. Kamschatka is a large peninsula on the north eastern part of Asia -- It is a mountainous country, lying between fifty-one and sixty-two degrees of north latitude, and of course a very 118 A STAR IN THE WEST. cold and frozen climate. No grain can be raised there, though some vegetables are. Skins and furs are their chief exports. The natives are wild as the country itself, and live on fish and sea animals, with their rein-deer. The islands in this sea, which separate it from the northwest coast of America, are so numerous that the existence of an almost continued chain of them between the two continents is now rendered extremely probable. The principal of them are the urile Islands, those called Bherings and Copper Islands, the Aleutian Islands and Fox Islands. Copper Island which lies in fifty-four degrees north, and in full sight of Bhering's Island, has its name from the great quantities of copper with which the northeast coast of it abounds. Mr. Grieve's history. It is washed up by the sea, and covers the shores in such abundance, that many ships might be loaded with it very easily. These islands are subject to continual earthquakes, and abound in sulphur. Alaska is one of the most eastwardly islands, and probably is not far from the American coast. The snow lies on these islands till March, and the sea is filled with ice in winter. There is little or no wood growing in any part of the country, and the inhabitants live in holes dug in the earth. Their greatest delicacies are wild lily and other roots and berries, with fish and other sea animals. The distance between the most northeastwardly part of Asia and the northwest coast of America, is determined by the famous navigatior capt. Cook, not to exceed thirty-nine miles. These straits are often filled with ice, even in summer and frozen in winter, and by that means might become a safe passage for the most numerous host to pass over in safety, though these continents had never been once joined, or at a much less distance than A STAR IN THE WEST. 119 at present. The sea from the south of Bhering's Straits to the islands, between the two continents, is very shallow. From the frequent volcanoes that are continually happening, it is probable, not only that there has been a separation of the continent at Bhering's Straits, but that the whole space from the island to that small opening was once filled up by land; but that it had by the force and fury of the waters, perhaps actuated by fire, been totally sunk and destroyed, and the islands left in its room. Neither is it improbable that the first passage of the sea was much smaller than at present, and that it is widening yearly, and perhaps many small islands that existed at the first separation of the continents, have sunk or otherwise have been destroyed. These changes are manifest in almost every country. Monsieur Le Page du Pratz, in his 2d vol. of his History of Louisiana, page 120, informs us, that being exceedingly desirous to be informed of the origin of the Indian natives, made every inquiry in his power, especially of the nation of the Natchez, one of the most intelligent among them. All he could learn from them was, that they came from between the north and the sun setting -- being no way satisfied with this, he sought for one who bore the character of one of their wisest men. He was happy enough to discover one named Moneacht-ape, among the Yazous, a nation about forty leagues from the Natchez. This man was remarkable for his solid understanding and elevation of sentiments, and his name was given to him by his nation as expressive of the man -- meaning "the killer of pain and fatigue." His eager desire to see the country from whence his forefathers came, he obtained directions and set off. He went up the Missouri, where he staid a long 120 A STAR IN THE WEST. time to learn the different languages of the nations he was to pass through. After long travelling he came to the nation of the Otters, and by them was directed on his way, till he reached the southern ocean. After being some time with the nations on the shores of the great sea, he proposed to proceed on his journey, and joined himself to some people who inhabited more westwardly on the coast. The travelled a great way between the north and the sun setting, when they arrived at the village of his fellow travellers, where he found the days long and the nights short. He was here advised to give over all thoughts of continuing his journey. They told him "that the land extended still a long way in the direction aforesaid, after which it ran directly west, and at length was cut by the great water from north to south. One of them added, that when he was young he knew a very old man, who had seen the distant land before it was cut away by the great water; and when the great water was low, many rocks still appeared in those parts." Moncacht-ape took their advice and returned home after an absence of five years. This account given to Du Pratz, in the year 1720, confirms the idea of the narrow passage at Kamschatka, and the probibility that the continents once joined. It is remarkable that the people, especially the Kamschatkians, in their marches, never go but in Indian file, following one another in the same track. Some of the nations in this quarter, prick their flesh with small punctures with a needle in various shapes, then rub into them charcoal, blue liquid or some other colour, so as to make the marks to become indelible, after the manner of the more eastern nations. A STAR IN THE WEST. 121 Bishop Lowth in his notes on the 16th verse of the xlixth chapter of Isaiah, says, "this is certainly an allusion to some practice common among the Jews at that time, of making marks on their hands and arms by punctures on the skin, with some sort of sign or representation of the city or temple, to shew their affection and zeal for it. They had a method of making such punctures indelible by fire or staining -- and this art is practiced by travelling Jews all over the world at this day -- Vid. also his note on chap. xlv. 5th verse. Thus it is with our northern Indians; they always go in indian file, and mark their flesh just as above represented. The writer of this has seen an aged christian Indian Sachem, of good character, who sat for his portrait. On stripping his neck to the lower part of his breast, it appeared that the whole was marked with a deep blueish colour in various figures, very discernable. On being asked the reason of it, he answered, with a heavy sigh, that it was one of the follies of his youth, when he was a great warrior, before his conversion to christianity; and now, says he, I must bear it, as a punishment for my folly, and carry the marks of it to my grave. The people of Siberia made canoes of birch bark, distended over ribs of wood, nicely sewed together. The writer has seen this exactly imitated by the Indians on the river St. Lawrence, and it is universally the case on the lakes. Col. John Smith says, "at length we all embarked in a large birch bark canoe. This vessel was about four feet wide and three feet deep, and about thirty-five feet long; and though it could carry a heavy burthen, it was so artfully and curiously constructed, that four men could carry it several miles, from one landing place to another; or from the waters of the lake to the 122 A STAR IN THE WEST. waters of the Ohio. At night they carry it on the land, and invert it, or turn it bottom up, and convert it into a dwelling-house." It also appears from the history of Kamschatka, written by James Grieve, that in the late discoveries, the islands which extend from the south point of Kamschatka, amount to thirty-one or thirty-two. That on these islands are high mountains, and many of them smoaking volcanoes. That the passages between them, except in one or two instances, were but one or two days row, at the time of the authors writing that history. They are liable to terrible inundations and earthquakes. The following is collected from Mr. Steller's journal, as recorded in the above history. "The main land of America lies parallel with the coast of Kamschatka, insomuch that it may reasonably be concluded that these lands once joined, especially at the Techukotskoi Noss, or Cape. He offers four reasons to prove it: 1st. The appearance of both coasts, which seem to be torn asunder. 2d. Many capes project into the sea from thirty to sixty versts. 3d. Many islands are in the sea which divides Kamschatka from America. 4th. The situation of the islands, and the breadth of that sea. -- The sea is full of islands, which extend from the north-west point of America to the channel of Anianova. One follows another, as the Kuruloski islands do at Japan. The American coast at sixty degrees of north latitude, is covered with wood; but at Kamschatka, which is only fifty-one degrees, there is none for near fifty versts from the sea, and at sixty-two, not one tree is to be found. It is known also, that the fish enter rivers on the American coast, earlier than they do in the rivers of Kamschatka. There are also plenty of A STAR IN THE WEST. 123 raspberries, of a large size and fine taste, besides honey suckles, cran-berries and black-berries in great plenty. In the sea there are seals, sea-beavers, whales and dog-fish. In the country and in the rivers on the American coast, red and black foxes, swans, ducks, quails, plover, and ten kinds of birds not known in Europe. These particulars may help to answer the question, whence was America peopled; for though we should grant that the two continents never were joined, yet they lie so near to each other, that the possibility of thev inhabitants of Asia going over to America, especially considering the number of the islands, and the coldness of the climate, cannot be denied. From Bhering's Island, on its high mountains, you can see mountains covered with snow, that appear to be capes of the main land of America. From all which it appears clearly, here was a probable mean of a people passing from Asia to America, either on the main land before separation, or from island to island; or on the ice after a sseparation, by which the continent of America might have been peopled, by the tribes of Israel wandering north-east, and directed by the unseen hand of Providence, and thus they entered into a country wherein mankind never before dwelt. It is not presumed that the ten tribes of Israel alone did this. Many of the inhabitants might have gone with them from Tartery or Scythia; and particularly the old inhabitants of Damascus, who were carried away in the first place by Tiglah Pilnezer, before his conquest of the Israelites, and were their neighbours, and perhaps as much dissatisfied with their place of banishment, though for different reasons, as the Israelites, as well as from Kamschatka, on their way where 124 A STAR IN THE WEST. they were stopped some time, as the Egyptians did with the Israelites of old. And indeed it is not improbable, as has before been hinted, that some few of other nations, who traded on the seas, might, in so long a course of time, have been driven by stress of weather, and reached the Atlantic shores at different places; but the great body of people settling in North and South America, must have originated from the same source. Hence it would not be surprising to find among their descendants, a mixture of the Asiatic languages, manners, customs and peculiarities. Nay. it would appear rather extraordinary and unaccountable if this was not so. And if we should find this to be the case, it would greatly corroborate the fact of their having passed into America from the north-east point of Asia, according to the Indian tradition. We, at the present day, can hardly conceive of the facility with which these wandering northern nations removed from one part of the country to the other. The Tartars at this time, who possess that northern country, live in tents or covered carts, and wander from place to place in search of pasture, &c. A STAR IN THE WEST. 125 CHAPTER V. Their general Character and established Customs and Habits. WE will now proceed to consider the general character of the people of whom we are treating, as preliminary to the enquiring into their customs and habits. It will be necessary to the full understanding our subject, to premise a few particulars. When America was first discovered by Columbus, it was comparatively well peopled by some hundreds, if not thousands of tribes of different nations, from the coast opposite to Kamschatka to Hudson's Bay. Their numbers have not been known, neither can they be known at this day. But to form some general idea of them, by reasoning on the subject, we will give the numbers of the nations that have come to our knowledge at different times. A Abenakias Aiaouez Akamsians Algonkins Assanpinks Arrowhatoes Amelistes Aurananeaus Assinnis Assinaboils Appalachos Arathapescoas Agones Abeckas Avoyels Arkanzas Aquelou-pissasf Adaics Aughquagchs Atacapas Appomotacks Alebamons Andaslaka Accotronacks Attatramasues Attibamegues Accomacks Amdustez * Pikes Expedition. No. of Warriors. No. of Women. No. of Children, t Men who understand and see. 126 A STAR IN THE WEST. Blanes Bayoue Ogoulas C Chatkas* or flat heads Cuttatawomans Chickaliomines Cliickiaes Chesapeaks Connosidagoes Cohunnewagoes Chalas Capalmakes Coroas Christinaux Chilians Causes Caddoques Caonites Cayugas Conoies Chippewas, or An- chipawah, 345, 619, 162& Cherokees Chickasaws Catawbas Chocktaws Creeks Chouanongsf Chiahnessou Canzas Chitemachas Caonetas Chatots Chacci Cumas, or red cray fish Chaouchas or Ouachas Cadodaquioux Conestogoes Caughnewagoes Chayennes Chappunish, or pierced nose Indians Cantanyans, on the Alleghany river Ceneseans or Cenis Cahirmois Coosades Cowetas Cussutas Chukaws Colapissas Caseitas Chatkas Conchaes D Delawares Dog-rib Indians E Erics Erigas F Foxes, 400, 500, 850 G Grand Eaux Gakaos Ganawoose H Hassiniengas Hurons Houmas I Iroquois Illinois Ictans Icbewas * They reckoned formerly 85000 warriors, but it is more likely to be only men. Said to be quite peaceable. Du Pratz. f A numerous nation of 38 villages, below the Missouri, on the Missisippi. A STAR IN THE WEST. 127 loways, 300, 400, 700. K Kecoughtons Kaskkasies Killistiuoes Kickapoos Kappas Kanoatinas Kans, 465, 500, 600. L Linnilinopes Lenais Les Puans M Minatarees Messiasics Menowa Kautong, or people of the lakes, 305, 600, 1200. Mantes Machecous Mechimacks Mohiccons Munsees Manahoaes Melotaukes Monachans, now Tuscaroras, ad ded to the Five Nations in 1712. Mandans Monasiceapanoes Musquaties Monahassanoes Massin agues Mohemonsoes Mexicans Moraughtacunds Mattapomens Missinasagues Missouris Mohocs or Mohawks Mingoes Mohuccons Miamis Mynonamies 300, 350, 700. Mascoutons, or Na tion of Fire Messcothins Mencamis Mobeluns, or Mou- ville Milowacks Mertowacks Mohuccories Mahatons, or Manhattons Mohegans Muckhekanies Ministeneaux Munseys Minisinks Maherins Massawonaes Minonionees Mipegois Muskoghees Michigamias Maquas Mandans N Neshaminas Narragansetts Nepiscenicens Nassamonds Nottoways Nantieokes Natehes Nantaughtacunds Nepissens Naudowessies Natchitoches Nauatalchas Nacunes or Greens Narauwings 128 A STAR IN THE WEST. o P Q Omans Piorias Quiocohanses Onanikins Pequots Quadodaquees Ousasons Parachuctaus R Outponies Prakimines Rappahanocks Onaumanients Pimitconis Round Heads Oswagatches Piankishaws Rancokas Orundacs Patowomacks Ricoras Osages 1252, 1793, Pissassees S 974. Padoucas Sokulks Oneidas Pamunkies Skillools Onondagoes Payankatanks Seminoles Oueatonons Powhatans Schactikook,or river Ottowas Paspahegas Indians Oniscousins Panis and White Sitons, 360, 700, Ottagamies or Panis, Black 1100. Foxes Panis Susquehannas Outimacs Pouhatamies Satanas Ousasoys Penobscots Sankihani Otters Panemahas Stegerakies Oniyouths Pacha Oglouas, or Shackakonies Othouez the Nation of Secakoonies Oumas, or Red Na Bread Sivux tion Pomptons Senecas Oufe Ogulas,orthe Pawnees, 1993, Sapoonies Nation of the 2170, 2060. Shawanese Dog Pemveans Souckelas Oque-Loussas Panoses Shakies Oakfuskees Pandogas Saaskies Ouachibes Shackaxons A STAR IN THE WEST. 129 Sacs 700, 750, 1400 Tapousoas Winnebagoes 450, Shosonees or Snake Tionontates 500, 1000. Indians. Tsouonthousaas, on Washpelongorpeo- T the Ohio pie of the leaves Teganatics Tetaus 2700, 3000, ISO, 350, 530. Tauxilnanians 2500. Washpcoute 90, Tauxinentcs V 180,270. Tentilves . Vermilions Y Tuscaroras W Youghtanunds TwightwieS "Wabingies Yazous Thomez Wapings Yanetongs 900, Taensas Wighcocomicoes 1600, 2700. Tonicas \Vianoes Yatassees Theoux "Wamasqueaks Other bands gene- Titones 2000, 3600, \Vyandots rally 1704, 2565, 6000 Webings 4420. Tomaroas Whonkenties Some nations divided and settled at a distance from each ether, and after many years, their language so changed, as to form different dialects ; as was in our days, the case with the Erigas, on the Ohio, who separated from the Tuscororas, and formed si distinct dialect in the course of a few years. Here are then one hundred and ninety different nations^ each having a king or sachem over them, of whom w r e have had some knowledge, though many of them are not now known ; what then must be the number of the nations on this conti nent could they all be known ? Although we cannot with any precision know the number of the nations, on the arrival of Columbus, and much less the number of souls, yet we may as matter of curiosity give the numbers of individual nations (if 130 A STAR IN THE WEST. late years as far as the fact can be ascertained and here our labour will be greatly lessened by a late ingenious and well written pamphlet, entitled, " Discourse delivered before the New- York Historical Society, December 1811," by the hon ourable Dewitt Clinton, of the city of New- York. To the labours of this gentleman, we are greatly indebted for the substance of many of the following observations, as well as the elegant manner in which he has communicated so much infor mation to the world. Du Pratz, in his History of Louisiana, (1 vol. 107 123) gives an account of the single nation of the Padoucas, lying west by north-west of the Missouri, in 172*, which may give a faint idea of the numbers originally inhabiting this vast continent. He says "the nation of the Paduca's is very nume rous, extends almost two hundred leagues, and they have vil lages quite close to the Spaniards of New Mexico." They are not to be considered as a wandering nation, though employ ed in hunting, summer and winter page 121. Seeing they have large villages, consisting of a great number of cabins, which contain very numerous families. These are permanent abodes ; from which one hundred hunters set out at a time with their horses, their bows and a good stock of arrows." The village where we were, consisted of one hundred and forty huts, containing about eight hundred warriors, fifteen hundred women, and at least two thousand children, some Padoucas having four wives." page 124. The natives of North-America, derive their origin from the same country, since at bottom they all have the same manners and usages* as. also the same manner of speaking and thinking," A STAR IN THE WEST. 131 Mr. Jefferson, late President of the United States, in his Notes on Virginia, has also given much useful information to the world on several important subjects relating to America, and among others as to the numbers of the Indians in that then dominion. Speaking of the Indian confederacy of the war riors, or rather nations, in that state and its neighbourhood, called "the Powhatan confederacy," says, it contained in point of territory, as he supposes, of their patrimonial country " about three hundred miles in length, and one hundred in breadth. That there was about one inhabitant for every square mile, and the proportion of warriors to the whole num ber of inhabitants, was as three to ten, making the number of souls about thirty thousand." Some writers state the number of their warriors at the first coming of the Europeans to Virginia, to be fifteen thousand, and their population fifty thousand. La Houtan says that each village contained about fourteen thousand souls, that is, fifteen hundred that bore arms, two thousand superanuated men, four thousand women, two thousand maids, and four thousand five hundred children. From all which, it is but a moderate estimate to suppose that there were six hundred thousand fighting men, or warriors, on this continent at its first discovery. In 1677, col. Coursey, an agent for Virginia, had a confer ence with the Five Nations, at Albany. The number of war riors was estimated at that time in those nations at the fol lowing rate. Mohawks three hundred, Oneidas two hundred, Onondagoes three hundred and fifty, Cayugas three hundred, Senecas one thousand total two thousand one hundred and 132 A STAR IN THE WEST. fifty, which makes the population ahout seven thousand two hundred. Vide Chalmer's Political Annals, 606. Smith, in his History of New-York, says, that in 1756, the number of fighting men were abput twelve hundred. Douglass, in his History of Massachusetts, says, that they were ahout fifteen hundred in 1760. In 176, col. Boquet states the whole number of the inhab itants (he must mean fighting men) at fifteen hundred and fifty. Captain Hutching* in 1768, states them at two thousand one hundred and twenty, and Dodge, an Indian trader, in 1779, at sixteen hundred* in the third year of the American revolutionary war. Many reasons may be assigned for the above differences some may have staid at home for the de fence of their towns some might be absent treating on dis putes with their neighbours, or sickness, &c. &c. During the above war, in 1776 7, the British had in their service, according to the returns of their agent Mohawks three hundred, Oneidas one hundred and fifty, Tuscororas two hundred, Onondagoes three hundred, Cayugas two hun dred and thirty, Senecas four hundred In the whole fifteen hundred and eighty. The Americans had about two hundred and twenty, making up eighteen hundred warriors, equal to about six thousand souls. In 1783, Mr. Kirkland, missionary to the Oneidas, estima ted the number of the Seneca warriors at six hundred, and the total number of the Six Nations, at more than four thousand. In 1790, he made the whole number of Indian inhabitants then remaining, including in addition, those who reside on Grand River, in Canada, and the Stockbridge and Brother- A STAR IN THE WEST. 133 town Indians, who had then lately joined them, to be six thou sand three hundred and thirty, of which there were nineteen hundred warriors. In 1794, on a division of an annuity, by order of Congress, to be made among the Six Nations, the numbers appeared with considerable certainty, to be In the United States. In the British government. Mohawks 300 Oneidas 628 460 Cayugas 40 Onondagoes 450 760 Tuscaroras 400 Senecas 1780 Stockbridge and Brothertown In dians, about 2330 The above number of British 760 But what are these to the southern Indians, and especially those of Mexico and Peru. I will give one example. Mons, La Page Du Pratz, in his History of Louisiana, written about the year 1730, assures us, that the nation of the Natchez, from whom the town of that name on the Missisippi is called, were the most powerful nation in North America 2 vol. 146, They extended from the river Manchas or Iberville, which is about fifty leagues from the sea, to the river Wabash, which is about four hundred and sixty leagues from the sea, and that they had five hundred Sachems in the nation." He further says, that the Chatkas or Flat-heads, near the river Pacha OgiUas, had twenty-five thousand warriors, but 134 A STAR IN THE WEST. in which number, he supposes many were reckoned who had but a slight title to that name Page 140. But a short estimate of the length and breadth of different parts of America, although not pretended to be perfectly accu rate, yet having endeavoured to keep within bounds, it may serve to answer the end now proposed. Length in miles. Breadth in miles. Old Mexico , 2,000 600 New-Mexico 2,000 1,600 Louisiana 1,600 1,200 Terra Firma 1,400 700 Amazonia 1,200 960 Peru 1,800 500 Chili 1,200 500 Patagonia 700 300 La Plata 1,500 1,000 Brazil 2,500 , 700 Thirteen United States 1,250 1,040 Esquimaux 1,600 1,200 Canada 1,200 276 Nova Scotia 500 400 Floridas 600 130 Miles 20,850 11,106 Besides this immense territory, on all which there are some Indians to be found, the country from New-Mexico, west to the South seas, which is yet in a state of nature, and abounds in Indian nations, must be added to the vast amount, as more than equal to all the rest. A STAR IN THE WEST. 135 The Indians, by oppression, diseases, wars and ardent spir its, have greatly diminished in numbers, degenerated in their moral character, and lost their high standing as warriors, especially those contiguous to our settlements. "The very ancient men who have witnessed the former glory and prosperity of their country, or who have heard from the mouths of their ancestors, and particularly from their be loved men, (whose office it is to repeat their traditions and laws to the rising generations, with the heroic achievements of their forefathers) the former state of their country with the great prowess and success of their warriors of old times, they weep like infants, when they speak of the fallen condition of their nations. They derive however some consolation from a prophecy of ancient origin and universal currency among them, that the man of America, will, at some future period, regain his ancient ascendency and expel the man of Europe from this western hemisphere. This flattering and consol atory persuasion has enabled the Seneca and Shawnese prophets, to arrest, in some tribes, the use of intoxicating liquors, and has given birth, at different periods, to attempts for a general confederacy of the Indians of North America." Clinton. The writer of this was present at a dinner given by gene ral Knox, to a number of Indians in the year 1789, at New- York ; they had come to the President on a mission from their nations. The house was in Broadway. A little before dinner, two or three of the Sachems, with their chief or prin cipal man, went into the balcony at the front of the house, the drawing room being up-stairs. From this they had a view of the city, the harbour, Long-Island, &c. &e. After remain- 136 A STAR IN THE WEST. ing there a short time, they returned into the room, apparent ly dejected ; but the chief more than the rest. General Knox took notice of it, and said to him, brother ! what has happened to you ? You look sorry ! Is there any thing to distress you ? He answered I'll tell you brother. I have been looking at your beautiful city the great water your fine country and see how happy you all are. But then, I could not help think ing, that this fine countiy and this great water were once ours. Our ancestors lived here they enjoyed it as their own in peace it was the gift of the great spirit to them and their children. At last the white people came here in a great ca noe. They asked only to let them tie it to a tree, lest the waters should carry it away we consented. They then said some of their people were sick, and they asked permission to land them and put them under the shade of the trees. The ice then came, and they could not go away. They then beg ged a piece of land to build wigwams for the winter we grant ed it to them. They then asked for some corn to keep them from starving we kindly furnished it to them, they promising to go away when the ice Avas gone. When this happened, we told them they must now go away with their big canoe ; but they pointed to their big guns round their wigwams, and said they would stay there, and we could not make them go away. Afterwards, more came. They brought spirituous and intox icating liquors with them, of which the Indians became very fond. They persuaded us to sell them some land. Finally they drove us back, from time to time, into the wilderness, far from the water, and the fish and the oysters -they have destroyed the game our people have wasted away, and now >ye live miserable and wretched, while you are enjoying ouf A STAR IN THE WEST. 137 fine and beautiful country. This makes me sorry brother ! and I cannot help it." But to proceed, the colour of the Indians, generally speak ing, was red, brown, or copper coloured, differing according to climate, high and low grounds. They are universally at tached to their colour, and take every mean in their power to increase it, prefering it to the white. They give a name to the white people, which is highly contemptuous ; it is that of an heterogenous animal. Sometimes when they aim at greater severity, that of " the accursed people" The hotter or colder the country is where the Indians have long resided, the greater proportion have they of the white or red colour ; this is asserted by Adair from personal experience. He has compared the Shawanoh Indians with the Chikkasaw, and found them much fairer, though their endeavours to cultivate the copper colour were alike. He thinks the Indian colour to be the effect of climate, art and manner of living. Their tradition says, that in the country far west, from which they came, all the people were of one colour ; arid they are ignor ant which was the primitive colour. Adair has seen a white man, who, by his endeavors to change his colour, became as deeply coloured as any Indian in the camp, after he had been in the woods only four years. The Indians to the Southward are often of a deeper hue than those to the northward ; iu a high country they incline to a lighter tinge$ but then those to the northward are more ignorant, and less knowing in their traditions, rites, and religious customs. The like change is not unknown in Europe and Asia. The inhabitants of the northern countries, in many instances, are comparatively fairer than those of the southern countries. 138 A STAR IN THE WEST. In the south the Indians are tall, erect and robust their limbs are well shaped, so as generally to form a perfect human figure. They delight in painting themselves, especially with red or vermilion colour. They are remarkably vain, and suppose themselves the first people on earth. The Five Na tions called themselves < Qngue-hoTtwe, that is, men surpassing nil others? the only beloved people of the great spirit, and his peculiar people. But as to their common mode of living, they are generally all great slovens they seldom or ever wash their shirts. It is a matter of fact, proved by most historical accounts, that the Indians, at our first acquaintance with them, gener ally manifested themselves kind, hospitable and generous to the Europeans, so long as they were treated with justice and humanity; but when they were, from a thirst of gain, over reached on every occasion, their friends and relations treach erously entrapped and carried away to be sold for slaves ; themselves injuriously oppressed, deceived and driven from their Lawful and native possessions; what ought to have been expected, but inveterate enmity, hereditary animosity, and a spirit of perpetual revenge. To whom should be attributed the evil passions, cruel practices, and vicious habits to which they are now changed, but to those who first set them the ex ample ; laid the foundation, and then furnished tike continual means for propagating and supporting the evil. In a very early day, in the colony of Virginia, the first settlers, by their great imprudence, had soured the Indian temper, raised their jealousies, and provoked their free and independent spirits, so as to lead them to determine on the extirpation of the whole colony then few, weak and divided. A STAR IN THE WEST. 139 The Indians managed their intended attack with so mucli secrecy, that they surprised the colonists in every quarter, and destroyed near one fourth of them. In their turn, the survivors waged a destructive war against the Indians, and murdered men, women and children. Dr. Robertson says, *i regardless, like the Spaniards, of those principles of faith, honor and humanity, which regulate hostilities among civil ized nations, and set hounds to their rage, the English dccm- ~d every thing allowable that tended to accomplish their de signs. They hunted the Indians like wild beasts, rather than enemies; and as the pursuit of them to their places of retreat in the woods, was both difficult and dangerous, they endeav oured to allure them from their inaccessible fastnesses, by offers of peace, and promises of oblivion, made with such an artful appearance of sincerity, as deceived the crafty Indian chief, and induced the Indians to return in the year 1623, to their former settlements, and resume their usual peaceful occupations. The behaviour of the two people seemed now to be perfectly reversed. The Indians, like men acquainted with the principles of integrity and good faith, on which the intercourse between nations is founded, confided in the recon ciliation, and lived in absolute security, without suspicion of danger, while the English, with perfidious craft, were pre paring to imitate savages in their revenge and cruelty. "On the approach of harvest, when a hostile attack would be most formidable and fatal, the English fell suddenly on all the Indian plantations, murdered every person on whom they could lay hold, and drove the rest to the woods, where so many perished with hunger, that some of the tribes nearest to the English, were totally .extirpated/' History of North-America, 90. 97. 140 A STAR IN THE WEST. Robertson again, speaking of the war in New-England, between Connecticut and Providence, in their first attempt against the Pequod Indians, says, " that the Indians had se cured their town, which was on a rising ground in a swamp, with pallisades. The New-England troops, unperceived, reached the pallisades. The barking of a dog alarmed the In dians. In a moment, however, they started to their arms, and raising the war-cry, prepared to repel the assailants. The English forced their way through into the fort, or town, and setting fire to the huts, which were covered with reeds, the confusion and terror quickly became general. Many of the women and children perished in the flames, and the war riors, endeavoring to escape, were either slain by the Eng lish, or falling into the hands of the Indian allies, who sur rounded the fort at a distance, were reserved for a more cruel fate. The English resolved to pursue their victory, and hunt ing the Indians from one place of retreat to another, some subsequent encounters were hardly less fatal than the firt action. In less than three months, the tribe of the Pequods were extirpated." Ibid 18 i 5, 6. "Thus the English stained their laurels, by the use they made of victory. Instead of treating the Pequods as an inde pendent people, who made a gallant effort to defend the prop erty, the rights and freedom of their nation, they retaliated upon them all the barbarities of American war. Some they massacred in cold blood, others they gave up to be tortured by their Indian allies, a considerable number they sold as slaves iii Bermuda, the rest were reduced to servitude among them selves." A STAR IN THE WEST. 141 What I am about mentioning, may be considered as of little force while standing by itself, yet when connected with so many other circumstances, it is thought worth mentioning. This nation of Pequods were a principal nation of the cast, and very naturally reminds one of the similarity of the same name in Jeremiah 1. 21, where the inhabitants of Pekod are particularly mentioned; and also in Ezekicl xxiii. 23. The difference in spelling one with a k, and the other with a q 9 is no uncommon thing. The Indian languages being very gut tural, the k is generally used where an Englishman would use the q but many of the first names used by the English in an early day have been corrected. Sir Walter Raleigh says his " first landing in America was at Roanor, which afterwards was found to be called by the Indians, Roanoke. Another trifling observation in itself, yet will add to the pre sumption already mentioned, is the original name of a point of land on the western part of the Euxine or Black Sea, men tioned by D'Anvillc, Nagara. This is the Abydos of the Greeks, 1 D'Anville, 287, and is much the same with the point in Lake Ontario, in New -York state, well known by the Indian name Niagara. But if this character of the Indians, as originally being kind and hospitable, should be doubted, as I know it will be by many, who think themselves well acquainted with them, from being with the present race around our settlements ; let us go back and hear what idea Christopher Columbus formed of them in the very beginning of our knowledge of them. He must be the very best witness that can be produced on this subject. In his account, sent to his royal master and mis tress, of the inhabitants, on his first landing in America, h 142 A STAR IN THE WEST. says, " I swear to your majesties, that there is not a bettcV people in the world than these; more affectionate, affahle, dr mild. They love their neighbours as themselves. Their language is the sweetest, the softest and most cheerful, for they always speaking smiling." In another instance, a ven erable old man approached Columbus with great reverence, and presented him with a basket of fruit, and said, you are come into these countries, with a force against which, were, we inclined to resist, resistance would be folly. We are all therefore at your mercy. But if you are men subject to mor^ tality like ourselves, you cannot be unapprised, that after this life, there is another, wherein a very different portion is allot ted to good and bad men. If therefore, you expect to die, and believe with us, that every one is to be rewarded in a future state, according to his conduct in the present, you will do no hurt to those who do none to you/' Edwards' West-Indies, vol. 72. De las Casas, bishop of Chapia, who spent much time and labour among the Indians of New Spain, trying to serve them, says, " I was one of the first who went to America. Neither curiosity, nor interest prompted me to undertake so long and dangerous a voyage. The saving the souls of the heathen w r as my sole object- Why was I not permitted, even at the expense of my blood, to ransom so many thousands of souls, who fell unhappy victims to avarice and lust. It was said that barbarous executions were necessary to punish or check the rebellion of the Americans. But to whom was this owing? Did not this people receive the Spaniards, who first came among them, with gentleness and humanity? Did they not shew more joy in proportion, in lavishing treasure upon them, A STAR IN THE WEST. 143 than the Spaniards did greediness in receiving* it. But our avarice was not yet satisfied. Though they gave up to us their lands and their riches, we would take from them their wives, their children and their liherty. To blacken the char acters of these unhappy people, their enemies assert that they are scarce human creatures. But it is we who ought to blush for having been less men, and more barbarous than they. They are represented as a stupid people, and addicted to vice. But have they not contracted most of their vices from the examples of Christians. But it must be granted that the Indians still remain untainted with many vices usual among Europeans. Such as ambition, blasphemy, swearing, treach ery, and many such monsters, which have not yet taken place among them. They have scarce an idea of them. All na tions are equally free. One nation has no right to infringe on the freedom of another. Let us do to these people, as we would have them have done to us, on a change of circum stances. \Vhat a strange method is this of propagating the gospel; that holy law of grace, which, from being slaves to Satan, initiates us into the freedom of the children of God." The Abbe Clavigero, another Spanish writer, confirms this idea of the South- Americans. "We have had intimate con verse, says he, with the Americans j have lived some years in a seminary destined for their instruction attentively ob served their character their genius- their disposition and manner of thinking; and have besides, examined with the, utmost diligence, their ancient history their re.Iigionr their government their laws and their customs. After such long experience and study of them, w declare, that the mental 144 A STAR IN THE WEST. qualities of the Americans arc not in the least inferior to those of the Europeans." Among the many instances of provocation given to them by the white people, Neal, in his History of New-England, page 21, says, " one Hunt, an early trader with the Indians of New-England, after a prosperous trade with the natives, en ticed between twenty and thirty of them on board his vessel, and contrary to the public faith, clapped them under hatches, and took them to Malaga, and sold them to the Spaniards. This the remaining Indians resented, by revenging them selves on the next English vessel that came on their coast." In the year 1620, a sermon was preached at Plymouth by the Rev. Mr. Cushman, from which the following extract is taken, relative to the treatment they received from the na tives. The Indians are said to be the most cruel and treacherous people in all these parts, even like lions, but to us they have been like lambs, so kind, so submissive and trusty, as a man may truly say, many Christians are not so kind or sincere. Though when we came first into this coun try we were few, and many of us very sick, and many died by reason of the cold and wet, it being the depth of winter, and we having no houses or shelter, yet when there were not six able persons among us, and the Indians came daily to us by hundreds, with their sachems or kings, and might in one hour have made despatch of us; yet such fear was upon them, as that they never offered us the least injury in word or deed. And by reason of one Tisquanto, that lives among us, and can speak English, we have daily commerce with their kings, and can know what is done or intended towards us among the savages." A STAR IN THE WEST. 145 The late governor Hutchinson, in his history of New-Eng land, observes, "that the natives shewed courtesy to the English at their first arrival 5 were hospitahle, and made such as would eat their food, welcome to it, and readily instructed them in planting and cultivating the Indian corn. Some of the English who lost themselves in the woods, and must other wise have perished with famine, they relieved and conducted home." Mr. Penn, also, at his first coming amongst them, spoke and wrote of them in high terms, as a kind and benevolent people. The history of New-Jersey informs us, that " for near a century, the Indians of that state had all along maintained an intercourse of great cordiality and friendship with the inhabit ants, being interspersed among them, and frequently receiv ing meat at their houses, and other marks of their good will and esteem." Smith, page 440. Father Charlevoix, who travelled early, and for a long time among the Indians, from Quebec to New-Orleans, and had great opportunities, which he made it his business and study to improve, tells us, speaking of the real character of the In dian nations, " that with a mien and appearance altogether savage; and with manners and customs which favour the greatest barbarity, they enjoy all the advantages of society* At first view, one would imagine them without form of gov ernment, laws or subordination, and subject to the wildest ca^ price. Nevertheless, they rarely deviate from certain max ims and usages, founded on good sense alone, which holds the Jilaee of law, and supplies in some sort, the want of legal au thority. They manifest much stability in the engagements 146 A STAR IN THE WEST. they have solemnly entered upon; patience in affliction, as well as submission to what they apprehend to be the appoint ment of Providence ; in all this they manifest a nobleness of soul and constancy of mind, at which we rarely arrive, with all our philosophy and religion. They are neither slaves to ambition nor interest, the two passions that have so much weakened in us the sentiments of humanity, (which the kind author of nature has engraven on the human heart) and kind led those of covctousness, which are as yet generally unknown among them/' It is notorious, that they are generally kinder to us, though they despise us, than we are to them. There is scarce an instance occurs, but that they treat every white man who goes among them, with respect, which is not the case from us to them. The same author says, the nearer view we take of our savages, the more we discover in them some valuable qualities. The chief part of the principles by which they regulate their conduct; the general maxims by which they govern themselves; and the bottom of their characters have nothing which appears barbarous. The ideas, though now quite confused, which they have retained of a first Being ; the traces, though almost effaced, of a religious worship, which they appear formerly to have rendered to the Supreme Deity, and the faint marks which we observe, even in their most indifferent actions, of the ancient belief, and the primitive re ligion, may bring them more easily than we think of, into the way of truth, and make their conversion to Christianity more easily to be effected, than that of more civilized nations." But what surprises exceedingly, in men whose whole out ward appearance proclaims nothing but barbarity, is, to see A STAR IN THE WEST. 147 them behave to each other, with such kindness and regard, that are not to he found among the most civilized nations. Doubtless this proceeds, in some measure, from the words mine and thine, being as yet unknown to these savages. We are equally charmed with that natural and unaffected gravity, which reigns in all their behaviour, in all their actions, and in the greatest part of their diversions. Also with the civili ty and deference they shew to their equals, and the respect of young people to the aged. And lastly, never to see them quarrel among themselves, with those indecent expressions, oaths and curses, so common among us; all which are proofs of good sense and a great command of temper.* ' In short, to make a brief portrait of these people, with a savage appear ance, manners and customs, which are entirely barbarous, there is observable among them, a social kindness, free from almost all the imperfections which so often disturb the peace of society among us. They appear to be without passion ; but they do that in cold blood, and some times through prin ciple, which the most violent and unbridled passion produces in those who give no ear to reason. They seem to lead the most wretched life in the world ; and yet they were, perhaps, the only happy people on earth, before the knowledge of the objects which so work upon and seduce us, had excited in them, desires which ignorance kept in supinenessj but which have not as yet (in 1730) made any great ravages among them. We discover in them a mixture of the fiercest and most gentle manners. The imperfections of wild beasts, and * Le Page Du Pratz, says, " t have studied these Indians a considerable num ber of years, and I never could learn that there ever were any disputes or boxhig matches among either the boysor men. 2 vol. 165. 148 A STAR IN THE WEST. the virtues and qualities of the heart and mind which do the greatest honour to human nature. Du Pratz, in his history of Louisiana, says, " that upon an acquaintance with the Indians, he was convinced that it was wrong to denominate- them savages, as they are capable of making good use of their reason, and their sentiments are just. That they have a degree of prudence, faithfulness and generosity, exceeding that of nations who would he offended at being compared with them. No people, says he, are more hospitable and free than the Indians. Hence they may be esteemed a happy people, if that happiness was not impeded by their passionate fondness for spirituous liquors, and the fool ish notion they hold, in common with many professing chris- tians, of gaining reputation and esteem by their prowess in war." But to whom do they owe their uncommon attachment to both these evils ? Is it not to the white people who came to them with destruction in each hand, while we did but de ceive ourselves, with the vain notion, that we were bringing the glad tidings of salvation to them. Instead of this, we have possessed these unoffending people with so horrid an idea of our principles, that among themselves they call us the accursed people. And their great numbers, when first discov ered, shew that they had, comparatively, but few wars before we came among them. Mr. William Bartram, a gentleman well known in the state uf Pennsylvania, son to the late John Bartram, Esq. so long Botanist to Queen Caroline, of England, before the revolu tion, in the journal of his travels through the Creek country, speaking of the Siminolcs or lower Creek nation, and of their being then few in number, says, "yet this handful of people A STAR IN THE WEST. 149 possess a vast territory, all East Florida and the greatest part of West Florida, which being naturally cut and divided into thousands of islets, knolls and eminences, by the innumerable rivers, lakes, swamps, savannas and ponds, form so many se cure retreats and temporary dwelling places, that effectually guard them from any sudden invasion or attacks from their enemies. And being sucb a swampy, hammoky country, fur nishes such a plenty and variety of supplies for the nourish ment of every sort of animal, that I can venture to assert, that no part of the globe so abounds with wild game or crea tures fit for the food of man. Thus they enjoy a superabun dance of the necessaries and conveniences of life with the se curity of person and property, the two great concerns of man kind. They seem to be free from want or desires. No cruel enemy to dread ; nothing to give them disquietude but the gradual encroachments of the white people. Thus contented, and undisturbed, they appear as blithe and free as the birds of the air, and like them as volatile and active, tuneful and vociferous. The visage, action and deportment of a Siminole, being the most striking picture of happiness in this life Joy, contentment, love and friendship without guile or affectation, seem Inherent in them, or predominate in their vital princi ple, for it leaves them but with the last breath of life." To exemplify their kindness to strangers, he says, that having lost his way in travelling through their towns, he was at a stand how to proceed, when he observed an Indian man at the door of his habitation, beckoning to him, to come to him. Bartram accordingly rode up to him. He cheerfully welcom ed him to his house, took care of his horse, and with the most graceful air of respect led him into an airy, cool apartment, 150 A STAR IN THE WEST. where being seated on cabins, his women brought in a re freshing repast, with a pleasant cooling liquor to drink. Then pipes and tobacco. After an hour's conversation, and Mr. Bartram informing him of his business, and where he was bound, but having lost his way, he did not know how to go on. The Indian cheerfully replied, that he was pleased that Mr. 15. was come into their country, where he should meet with friendship and protection; and that he would himself lead him into the right path. He turned out to be the- prince or chief of \Vhatoga. How long would an Indian have rode through o:ir country, before he would have received such kindness from a common farmer, much less a chief magistrate of a country ? Mr. Bartram adds to the testimony of Father Charlevoix, in favour of their good characters among them selves. He says they are just, honest, liberal and hospitable to strangers ; considerate, loving and affectionate to their wives and relations ; fond of their children ; frugal and per severing; charitable and forbearing. He was weeks and months among them in their towns, and never observed the least sign of contention or wrangling; never saw an instance of an Indian beating his wife, or even reproving her in anger. Col. John Smith says, " when we had plenty of green corn and roasting ears, the hunters became lazy, and spent their time in singing and dancing. They appeared to be fulfilling the scriptures, beyond many of those who profess to believe them, in that of taking no thought for to-morrow, but in liv ing in love, peace and friendship, without disputes. In this last respect they are an example to those who profess Chris tianity page 29. A STAR IN THE WEST. 151 The first and most cogent article in all their late treaties with the white people is, " that there shall not he any kind of spirituous liquors brought or sold in their towns; and the traders are allowed hut ten gallons for a company, which are esteemed sufficient to serve them on their journey;, and if any of this remains on their arrival, they must spill it on the ground." { Mr B. met two young traders running about forty kegs of Jamaica spirits into the nation. They were discover ed by a party of Creeks, who immediately struck their toma hawks into every keg, and let the liquor run out, without drinking a drop of it. Here was an instance of self denial, seldom equalled by white men, for so fond are they of it, that had they indulged themselves with tasting it, nothing could have prevented them from drinking the whole of it. < Mr. B. saw a young. Indian who was present at a scene of mad in temperance and folly, acted by some white men in the town, He clapped his hand to his breast, and with a smile looking up, as if struck with astonishment, and wrapt in love and adoration of the Deity, lamented their conduct. We have thus endeavored to give some ideas of the Indian character, at the first arrival of the Europeans among them, before they were debauched and demoralized by an acquaint ance with those who pretend to be their benefactors, by com municating to them the glad tidings of salvation, through Jesus Christ. We have exhibited the testimony of the best writers, from various parts cf the continent, acquainted with very dif ferent nations, from the south to the north. It is given gen erally in the authors own words, lest we might be charged with misrepresenting their meaning, by adopting our own language, or putting a gloss on theirs; and our design has 152 A STAR IN THE WEST. been, that the reader may be made acquainted with the peo ple of whom we treat. We must confess, that we have given the fairest part of their character, while at home and among their friends, though a perfectly just one. The objects which engage their attention, and indeed their whole souls, are war and hunting. Their haughty tempers will not condescend to labour-this they leave to their women. Hence they put on rather a solemn character, except when they divert themselves with their principal amusements, dan cing and gaining. But in war, and while opposing the ene mies of their nation, they are cruel and revengeful. The* make war with unrelenting fury, on the least unatoned affront, equal to any European nation whatever. It is their custom and long continued habit. They kill and destroy their own species without regret. The warrior is the highest object of their ambition. They are bitter in their enmity, and to avenge the blood of a kinsman, they will travel hundreds of miles, and keep their anger for years, till they are satisfied.* They scalp all the slain of their enemies (as many of the Asiatics did) that they get in their power, contrary to the usage of all other savages.f They usually attack their ene mies with a most hediotis and dreadful yelling, so as to make the woods to ring. Very few of the ablest troops in the world can withstand the horror of it, who are strangers to them,, and have not before been acquainted with this kind of recep tion. They are kind to women and children whom they take * The murderer shall surely be put to death. The avenger of blood, himself, shttll slay the murderer; when lie meetelh him, he shall slay him. Numbers xxxv. 18, 19. ? David speaks of the hoary scalps of his enemies. A STAR IN THE WEST. 153 prisoners, and are remarkable for their delicacy, in their treatment of the first. To sucli prisoners as they, by certain rules, doom to death, they are insultingly cruel and ferocious beyond imagination; and their women are most ingenious and artful in the science of tormenting. All this is mutual, and it is distressing to say, with truth, that it is too much like the practice of those who call themselves a more enlightened peo ple. Had the Indians read Lucan's Pharsalia lib. iii. 400, which contains the description of the Massilian Grove of the Gallic Druids, wherein they would have found every tree reek ing with the blood of human victims or had they been ac quainted with the British Druids, " who indeed seem to have exceeded, if possible, their heathen neighbours, in savage ferocity and boundless lust of sacrificial blood, they would have, indeed, been able to settle accounts with their w r liite neighbours. [ The page of history trembles to relate the bale*- ful orgies of the Druids, which their frantic superstition cele brated, when enclosing men, women and children, in one vast wicker image, in the form of a man, and filling it with every kind of combustible, they set fire to the huge colossus. While the dreadful holocaust was offering to their sanguinary gods, the groans and shrieks of the consuming victims were drowned amidst shouts of barbarous triumph, and the air was rent with the wild dissonance of martial music." 1 vol. of Indian An tiquities. Or had the Indians read of this emperor Maxi- minian putting to death the Theban legion of six thousand, six hundred and sixty-six Christian soldiers, who had served him faithfully, because they refused to do sacrifice to the heathen gods, and persecute their brother Christians Caves primitive christ. 331 or had they beeoi acquainted with the 154 A STAR IN THE WEST. tortures oi* the martyrs for Christ, for many centuries or the European practice of burning heretics* or had they heard of the Waldenses and Albigenses of St. Bartholomews night, or the Irish massacre. They might be ignorant of the bloody torments of the Inquisition, the tortures of Amboyna, or of a French Republican Baptism or they may never have been informed of the district of La Vendee of the Convent of Cannes, or of the proceedings in France on the 12th Au gustor of the more than diabolical, cowardly murder, by the enlightened citizens of Pennsylvania, from the county of Washington, when a whole town of Christian Indians, consist ing of about ninety souls, men, women and children, were butchered in cold blood, at Muskingum, in the year 1783: and who had been our tried friends during the whole revolu tionary war. If the Indians had known these facts, and writ ten the history of the civilized white people, they might have roused the feelings of a tender conscience in their favour. t But whoever reads the history of the eulogized heroes of ancient days, will find them not much better, in this respect. Does Achille's behaviour to Hector's dead body, appear less savage or revengeful? Do the Carthagenians or Phoenicians, burning their own children alive in sacrifice, or the bloody massacres and tortures of the southern Indians, by the learned mid civilized Spaniards, claim any great preference in point of humanity and the finer feelings of the enlightened sons of science, and of the pretenders to religious knowledge. * Will any one again laugh at the strong observation of an eminent divine, ' that man in a at ate of nature, was half devil and half brute' Ciarkes'Com. l.ii. Who vill not adore the God of heaven with gratitude and thanksgiving, for the light ol the gospel, which has not only brought life and immortality to light, but MI ought so wonderful a change among the present nations of die earth. A STAR IN THE WEST. 155 But let us come nearer home. Who set them the example of cruelty and barbarity, even to those whom they invaded and plundered of their property deprived of their lands, and ren dered their whole country a scene of horror, confusion and distress. W^ynne, in his history of America, tells us, " that the New-England people, in an early day, as we have already seen, made an attack upon the Pequod Indians, and drove eight hundred of them, with about two hundred of their women and children, into a swamp a fog arising, the men escaped, except a few, who were cither killed or wounded. But the helpless women and children were obliged to surren der at discretion. The sachem's wife, who some time before, had rescued the Weathcrsfield maidens, and returned them home, was among them. She made two requests, which arose from a tenderness and virtue not common among savages. 1st. That her chastity might remain imviolatcd. 2d. That her children might not be taken from her. The amiable sweet ness of her countenance, and the modest dignity of her deport ment, were worthy of the character she supported for inno cence and justice, and were sufficient to shew the Europeans, that even barbarous nations, sometimes produce instances of heroic virtue. J It is not said by the historian, whether her requests were granted or not, but that the women and chil dren were dispersed through tire neighbouring colonies, the male infants excepted, who were sent to the Bermudas" 1 vol. 66. Indeed, had the Indians, on their part, been able to answer in writing, they might have formed a contrast be tween themselves and their mortal enemies, the civilized sub jects of Great-Britain. They might have recapitulated their conduct in the persecution of Indians, witches and quakers In 156 A STAR IN THE WEST. New-England Indians and JWgroes in New- York, and the cruelty with which the aborigines were treated in Virginia. These invaders of a country, (in the peaceahle possession of a free and happy people, entirely independent, as the deer of of the forests) made war upon them, with all the advantage of fire-arms and the military knowledge of Europe, in the most barbarous manner not observing any rules of nations, or the principles of modern warfare, much less the benign in junctions of the gospel. They soon taught the Indians by their fatal examples, to retaliate with the most inveterate malice and diabolical cruelty. The civilized Europeans, though flying from the persecution of the old world, did not hesitate to deny their professed religion of peace and good will to men, by murdering men, women and children selling captives as slaves cutting off the heads, and quartering the bodies of those who were killed, nobly fighting for their liber ty and their country, in self defence, and setting them up at various places, in ignoble triumph at their success. Philip, an independent sovereign of the Pequods, who disdained to submit, but died fighting at the head of his men, had his head cut off and carried on a pole with great rejoicings, to New- Plymouth, where, Wynne says, his skull is to be seen to this day. Vide vol. 106 to 108. This conduct produced greater violence and barbarity on the part of the other nations of Indians in the neighbourhood, often joined by French Europeans who acted, at times, worse than the native Indians, and by this means, a total disregard of promises and pledged faith on both sides, became common. Ibid. A STAR IN THE WEST. 157 I do not quote these instances of inhuman conduct to justify the Indians, but only to shew that they were not the only savages, and that the blame, as is too common, ought not to fall all on one side, because they were vanquished, but should produce some commiseration and principles of Christian be nevolence towards these highly injured and suffering sons of the wilderness./ In the beginning of the revolutionary war, the Americans were constantly styled by their invaders as rebels; and had we been conquered, I have little doubt but that we should have been treated much as the Indians have been, with the difference of having been hanged, instead of being scalped and beheaded. But as we proved successful, by the good providence of God, we are now glorious asserters of liberty and the freedom of man. The conduct of the Israelites themselves, while in a state of civilization, and under the government of a king, and with the prophets of God to direct and teacli them, did not discover a much better spirit than these supposed Israelites, wretched and forlorn, in the wilderness of America, have done. ** When Ahaz, king of Judah, had sinned against God, he delivered him into the hand of the king of Assyria; and he was also deliverod into the hand of Pekah, king of Israel, who smote him with a great slaughter, and slew in Judah one hundred and twenty thousand in one day, who were all valiant men 2 Chron. xxviii. 5. And the children of Israel carried away captive, of their brethren, two hundred thousand women, sons and daughters ; took also much spoil from them, and brought the spoil to Samaria. But a prophet of the Lord was there, whose name was Oded, and he went out before the host that tame into Samaria, and said unto them, "behold, because tne 158 A STAR IN THE WEST. Lord God of your fathers \vas wroth with Judah, and hath delivered them into your hands, and ye have slain them in a rage, that reacheth up to heaven And now ye purpose to keep under the children of Judah and Jerusalem, for bond men and hond-women unto you ; but are these not with you, even with you, sins against tire Lord your God ? Now hear me, therefore, and deliver the captives again, which yc have taken captive of your brethren ; for the fierce wratli of the Lord is upon you." Here we cannot have the same hopes of tracing the present practices of the natives of the woods to any certain source, as is in the case of their languages. When a people change from a settled, to a wandering state, especially, if thereby they be totally removed from any connection or intercourse with civilized countries, they must necessarily accommodate their actions to their then pressing wants and necessities. Their practices must change with their circumstances. Not so their language; for although it may greatly alter, and often degenerate for want of cultivation, or by separating into parties, far removed from each other ; yet the roots and principles of the language, may in remote ages, be traced in the different dialects, so as to afford tolerable proof of the original language. If a people, before their emigration, had any knowledge of the arts and sciences, although this might, and indeed would lead them, even in a wandering state, to discover more inge nuity and method in providing for their wants, yet in after ages, as they separated from each other and colonized into distant parts, they would loose this knowledge, and finally, know nothing of them but by tradition, except so far as should A STAR IN THE WEST. 159 fall within their means and absolute wants ; which in the first case must be few, and in the other many and pressing*. So that we may reasonably conclude, that the first wanderers would leave much greater evidence of their original, as well as of their knowledge of the mechanical arts, than their pos terity could possibly do. And further, that the nearer to the place of their first permanent settlement, the greater would be the remains of those arts. However, we will endeavour to search into, and enumerate those few customs that we have any account of, which pre vailed with them when the Europeans first arrived among them, and some of which they still retain. We do not mean to take up the silly and ridiculous stories published by many writers on this subject, who either had particular, and often wicked ends to answer by their publica tions, or they founded their narratives on information received on the most transient acquaintance of a few hours, with the vicious and worthless among the Indians along our frontiers; nor shall we trust to accounts related by ignorant traders, who did not comprehend either the idiom of their language, or the strong metaphorical and figurative mode of expressing themselves. This has led to the most false and absurd ac counts of both Indian manners and language. To give one instance of this, though among the best of them, the following fact is extracted from an account given of the Mohawks in 1664, by a reverend gentleman who ought to have known better, and must have bad an education, and known the prin ciples of grammar. This nation, says he, has a very heavy language, and I find great difficulty in learning it, so as to speak and preach to them fluently. There are no Christian* 160 A STAR IN THE WEST. who understand their language thoroughly. When I am among them, I ask them how things are called. One will tell me a word in the infinitive mood, another in the indica tive. One in the first, another in the second person. One in the present, another in the preterperfect tense ; so that I stand sometimes and look ; hut do not know how to put it down. And as they have their declensions and conjugations, so they have their increases, like the Greeks; and I am sometimes, as if I was distracted, and cannot tell what to do, and there is no person to set me right. I asked the commissa ry of the (Dutch West-India company) what this meant, and he answered he did not know, hut imagined they changed their language every two or three years/' He had been con nected with them twenty years. The Indians are perfect republicans, they will admit of no inequality among them hut what arises from age, or great qualifications for either council or war. Although this is the case in peace, yet in war they observe great discipline, and perfect subordination to their beloved man who carries the holy ark, and to their officers, who are appointed on account of the experience they have had of their prowess in war, and good conduct in the management and surprising of an enemy, or saving their men by a timely retreat; but this subordina tion ends with the campaign. I As the Israelites were divided into tribes, and had a chief over them, and always marched under ensigns of some ani mal peculiar to each tribe, so the Indian nations are univers ally divided into tribes, under a sachem or king, chosen by the people from the wisest and bravest among them. He has neither influence or distinction, but from his wisdom and pru- A STAR IN THE WEST. 161 dence. Hfe is assisted by a council of old, wise dnd beloved men, as they call their priests and councillors. Nothing is determined (of a public nature) but in this council, where every one has an equal voice. The chief or sachem, sits in the middle, and the council on each hand, forming a semi-cir cle, as the high priest of the Jews did in the Sanhedrim of that nation. Mr. Penn, when he first arrived in Pennsylvania, in the year 1683, and made a treaty with them, makes the following observations* in a letter he then Wrote to his friends in Eng land. e( Every king has his council, and that consists of all the old and wise men of his nation, which perhaps are two hundred people. Nothing of moment is undertaken, be it war, peace, selling of land, or traffic, without advising with them. *Tis admirable to consider how powerful the chiefs are, and yet how they move by the breath of the people. I have had occasion to be in council with them upon treaties for land* and to adjust the terms of trade. Their order is thus; the king sits in the middle of an half moon, and hath his council, the old and the wise on each hand. Behind them* at a little dis tance, sit the young fry^ in the same figure. Having con sulted and resolved their business, the king ordered one of them to speak to me. He came to me> and in the name of his king, saluted me. Then took me by the hand, and told me that he was ordered by his king to speak to me ; and that now it was not he, but the king who spoke, because what lie should say was the king's mind. During the time this person was speaking, not a man of them was observed to whisper or smile. The old were grave the young reverend in their deportment They spoke little, but fervently and with pJe* 162 A STAR IN THE WEST. gance. He will deserve the name of wise, who out- wits them in any treaty about a thing they understand. At every sen tence they shout, and say amen, in their way." Mr. Smith, in his history of New-Jersey, confirms this gen eral statement. They are grave even to sadness, upon any common, and more so upon serious occasions observant of those in company, and respectful to the aged of a temper cool and deliberate never in haste to speak, but wait, for a certainty, that the person who spake before them, had finish- ished all he had to say. They seemed to hold European vivacity in contempt, because they found such as came among them, apt to interrupt each other, and frequently speak alto gether. Their behaviour in public councils was strictly de cent and instructive. Every one in his turn, was heard, ac cording to rank of years or wisdom, or services to his country. Not a word, whisper or murmur, was heard while any one spoke : no interruption to commend or condemn ; the younger sort were totally silent. Those denominated kings, were sachems distinguished by their wisdom and good conduct. The respect paid them was voluntary, and not exacted or looked for, nor the omission regarded. The sachems direct ed in their councils, and had the chief disposition of their lands" page 142, 144. Every nation of Indians have certain customs, which they observe in their public transactions with other nations, and in their private affairs among themselves, which it is scandalous for any one among them not to observe. And these always draw after them, either public or private resentment, when ever they are broken. Although these customs may, in their detail, differ in one nation, when compared with another : yet A STAR IN THE WEST. 163 it is easy to discern that they have all had one origin. This is also apparent from every nation understanding them. Mr, Golden says their great men, both sachems and captains, are generally poorer than the common people ; for they affect to give away, and distribute all the presents or plunder they get in their treaties, or in war, so as to leave nothing to them selves. There is not a man in the ministry of the Five Na tions (of whom Mr. Golden was writing) who lias gained his office otherwise than by merit There is not the least salary, or an& sort of profit annexed to any office, to tempt the covet ous or the sordid ; but on the contrary, every unworthy action is unavoidably attended with the forfeiture of their commis sion $for their authority is only the esteem of the people, and ceases the moment that esteem is lost. An old Mohawk sachem, in a poor blanket and a dirty shirt, may be seen issu ing his orders, with as arbitrary an authority as a Roman dic tator." As every nation, as before observed, has its peculiar stand ard or symbol, as an eagle, a bear, a wolf or an otter, so has each tribe the like badge, from which it is denominated. When they encamp, on a inarch, they always cut the repre sentation of their ensign or symbol, on the trees, by which it may be known who have been there. The sachem of each tribe is a necessary party in all conveyances and treaties, to which he affixes the mark of his tribe, as a corporation does that of the public seal. If you go from nation to nation, you will not find one who doth not lineally distinguish himself by his respective family. As the family or tribe of the eagk, panther, (which is their lion) tyger, buffalo, (their ox or bull) and also the bear, deer? 164 A STAR IN THE WEST. racoon, &c. &c. So among the Jews, was the lion of the tribe of Judah Dan was known by a serpent Issachar by an ass, and Benjamin by a wolf. But the Indians, as the Jews, pay no religious respect for any of these animals, or for any other whatever. They reckon time after the manner of the Hebrews. They divide the year into spring, summer, autumn, or the falling of the leaf, and winter. Korah is their word for winter with the Cherokee Indians, as it is with the Hebrews. They number the years by any of these four periods, for they have no name for a year. And they subdivide these, and count the year by lunar months, or moons, like the Israelites, who also counted by moons. They call the sun and moon by the same word, with the addition of day and night, as the day sun, or moon- the night sun, or moon. They count the day by three sensible differences of the sun, like the Hebrews as the sun coming out mid-day, and the sun is dead, or sun set, Midnight is half way between the sun going in and coming out of the water also by mid-night and cock-crowing. They begin their ecclesiastical year at the first appearance of the first new moon of the vernal equinox, according to the ecclesiastical year of Moses. They pay great regard to the first appearance of every new moon. They name the various seasons of the year from the planting and ripening of the fruits. The green eared moon is the most beloved or sacred, when the first fruits become sanctified, by being annually offered up ; and from this period they count their beloved or holy things. The number, and regular periods of the Indian public re ligious feasfy (as will be seen hereafter) is a good historical A STAR IN THE WEST. 165 proof that they counted time, and observed a weekly Sabbath, long after their arrival on the American continent, as this is applicable to all the nations. Till the seventy years captivity commenced, according to Dr. Prideaux, the Israelites had only numeral names for the solar and lunar months, except two called Abib and Ethanaim. The former signifies a green ear of corn, and the latter robust and valiant. And by the first name the Indians term their passover, as an explicative, and which the trading people call the green corn-dance. These two months were equinoctial. Mib 9 or the present Nisan of the Jews, was the sixth month of the civil, and first of the ecclesiastical year, answering to our March or April ; and Ethanaim, which began the civil year, was the sixth of the ecclesiastical, the same as our September and October. Mr. Bartram says, while he was at Attasse, in the Creek nation, on a Sabbath day, ho observed a great solemnity in the town, and a remarkable silence and retiredness of the red inhabitants. Few of them were to be seen the doors of their dwellings were shut, and if a child chanced to stray out, it was quickly drawn in doors again. He asked the meaning of this, and was immediately answered, that it being the white people's sabbath, the Indians kept it religiously sacred to the great spirit. The writer of this being present on the Lord's day, at the worship of seven different nations, who happened (accidentally) to be at the seat of government to gether, he was pleased to see their orderly conduct. They were addressed by an old sachem, apparently with great en ergy and address. An interpreter being present, he asked him to explain what the speaker had said. The intrepreter answered that the substance of what he delivered, was a 166 A STAR IN THE WEST. warm representation to his audience, of the love the great spirit had always manifested towards the Indians, more than to any other people. That they were in a special manner, under his government and immediate direction. That it was, therefore, the least return they could make for so much good ness, gratefully to acknowledge his favour, and to be obedient to his laws to do his will, and to avoid every thing that was evil, and of course displeasing to him. Just before the service began, the writer of this observed an Indian standing at the window with the intrepreter, look ing into a small field adjoining the house, where a great many white children were playing with the Indian children, and making a considerable noise. The Indian spoke much in earnest, and seemed rather displeased. The interpreter an swered him with great apparent interest. On being asked the subject of their conversation, he said the Indian was lamenting the sad state of those white children, whom he called poor destitute orphans. The interpreter asked why he tliought them orphans ? For he believed it was not true. The Indian, with great earnestness, replied, is not this the day on which you told me the white people worshipped the great spirit ? If so, surely these children, if they had parents, or any persons to take care of them, would not be suffered to be out there, playing and making such a noise. No ! no ! they have lost their fathers and their mothers, and have no one to take care of them. When the Indians travel, they always count the time by sleeps, which is a very ancient custom, and perhaps may have been derived from the Mosaic method of counting time, mak ing the evening and the morning to be the first day, &c e A STAR IN THE WEST. 167 They have also an ancient custom of setting apart 'certain houses and towns, as places of refuge, to which a criminal, and even a captive may fly, and be safe from the avenger of blood, if he can but enter it. Mr. Bartram says, ** we arrived at the Apalachuela town, in the Creek nation. This is esteemed the mother town, sacred to peace. No captives are put to death, or human blood spilt here." The Cherokees, according to Adair, though now exceed ingly corrupt, still observe the law of refuge, so inviolably, that they allow their beloved town the privilege of protecting a wilful murderer ; but they seldom allow him to return home from it in safety. The town of refuge called Choate, is situate on a large stream of the Missisippi, five miles above where fort Loudon formerly stood. Here some years ago, a brave Englishman was protected, after killing an Indian warrior, in defence of his property. He told Adair, that after some months stay there, he intended returning to his house in the neighbour hood ; but the chiefs told him it would prove fatal to him. So he was obliged to continue there, till he satisfied the friends of the deceased, by presents to their full satisfaction. In the upper country of the Muskoge, there was an old beloved town called ICoosahy now reduced to a small ruinous village, which is still a place of safety for those who kill undesignedly. In almost every Indian nation, there are several peaceable towns, which are called old beloved, holy or white towns, --y seem to have been formerly towns of refuge, for it is not within the memory of their oldest people, that ever human 168 A STAR IN THE WEST. blood was shed in them ; although they often force persons from them, and put them to death elsewhere. It may be thought improper here, to say much of the war like abilities and military knowledge of the Indians, as it is very popular, especially with Europeans, to despise them as warriors, by which means thousands of Europeans and Americans have lost their lives. But as it may shew that they are not quite so ignorant as strangers to them have thought them, a short account of their military conduct, may illucidate our general subject. I am assisted by col. Smith, who lived long with them, and often fought against them, in what may be said on this occa sion. However despised, they are, perhaps* as well versed in the art of that kind of war, calculated for their circumstances, and are as strict disciplinarians in it, as any troops in Europe ; and whenever opposed by not more than two or three to one Indian, they have been generally victorious, or come off with small loss, while they have made their opponents repent their rashness and ignorance of war on their plan* And indeed? they were always victorious over European troops, till sad experience taught foreign officers to pay more respect to the advice of American officers, who, by adopting the Indian prin ciples of war, knew how to meet them with advantage. It is not sufficient for an army to be well disciplined on their own principles, without considering those of the enemy they are to contend with. Braddock, Boquet, and several others of great celebrity in their own country, have been defeated or sur prised, by a (comparatively) small number of these inhabit ants of the wilderness, and greatly suffered from despising A STAR IN THE WEST. 169 what they thought untutored savages ; and to save the honor and military character of those who commanded, have been led to give very false reports of the comhats. The following facts will give force to these observations " In col. Boquet's last campaign of 1764, I saw, (says col. Smith) the official return made by the British officers, of the number of Indians that were in arms against us in that year, which amounted to thirty thousand. - As I was then a lieuten ant in the British service, I told them I was of opinion, that there were not above one thousand in arms against us, as they were divided by Broadstreet's army, being then at Lake Erie. The British officers hooted at me, and said that they could not make England sensible of the difficulties they labour ed under in fighting them ; and it was expected that their troops could fight the undisciplined savages in America, five to one, as they did the East-Indians, and therefore my report would not answer their purpose, as they could not give an honorable account of the war, but by augmenting their num bers." Smith was of the opinion, that from Braddock's defeat, un r til the time of his writing, there never were more than three thousand Indians, at any time in arms against us, west of Fort Pitt, and frequently not more than half of that number According to the Indians' own account, during the wjiole of Braddock's war, or from 1755 to 1758, they killed and took fifty of our people for one that they lost. In the war of 1763, they killed, comparatively, few of our people, and lost more of theirs, as the frontier inhabitants, especially the Virginians, had learned something of their method f warj y et even in 170 A STAR IN THE WEST. this war, according to their account (which Smith believed to be true) they killed and took ten of our people for one they lost. The Indians, though few in number, put the government to immense expense of blood and treasure, in the war from 1756 to 1791. The following campaigns in the western country, will be proof of this. General Braddock's in the year 1755 col. Armstrong's against the Cattaugau town, on the Alleghany, in 1757 gen. Forbes' in 1758 gen. Stanwix's in 1759 gen. Monckton's ill 1760 col. Boquet's in 1761 and again in 1763, when he fought the battle of Brushy-Run, and lost above one hundred men ; but by taking the advice and assistance of the Virginia volunteers, finally drove the Indians col. Armstrong's up the west branch of Susquehannah in the same year gen. Broad- street's up Lake Erie in 1764 col. Boquet's at Muskingum at the same time lord Dunmore's in 1774 gen. MIntosh's in 1778, and again in 1780 col. Bowman's in 1779 gen. Clark's in 1782 and against the Wabash Indians in 1786 gen. Logan's against the Shawanese in the same year, and coL Harrner's in 1790 gen. Wilkinson's in 1791 gen. St. Clair's in 1791, and gen. Wayne's in 1794, which in all are twenty-three campaigns, besides smaller expeditions, such as the French-Creek expedition, colonels Edward's, Loughrie's, &c. All these were exclusive of the numbers of men who were internally employed as scouting parties, in erecting forts, guarding stations, &c. &c. When we take the foregoing account into consideration, may we not reasonably conclude, that the Indians are the best dis ciplined troops in the world, especially when we consider, that A STAR IN THE WEST. 171 the ammunition and arms that they are obliged to use, are of ^he worst sort, without bayonets or cartoueh boxes. No arti ficial means of carrying either baggage or provision, while their enemies have every warlike implement, and other re sources, to the utmost of their desire. Is not that the best discipline, that has the greatest tendency to ajinoy an enemy, and save their own men ? It is apprehended that the Indian discipline is better calculated to answer their purpose in the woods of America, than the British discipline in the plains of Flanders. British discipline, in the woods, is the way to have men slaughtered, with scarcely any chance to defend themselves. Privates. The Indians sum up their art of war thus " The business of the private warrior is to be under command, or punctually to obey orders to learn to march a-breast in scattered order, so as to be in readiness to surround the enemy, or to prevent being surrounded to be good marksmen, and active in the use of their musket or rifleto practice running to learn to endure hunger or hardships with patience and fortitude to tell the truth at all times to their officers, more especially when sent out to spy the enemy." Concerning Officers. They say that it would be absurd to appoint a man to an office, whose skill and courage had never been tried that all officers should be advanced only according to merit that no single man should have the absolute command of an army that a council of officers should determine when and how an attack is to be made that it is the duty of officers to lay plans, and to take every advantage of the enemy to ambush 172 A STAR IN THE WEST. and surprise them, and to prevent the like to themselves. It is the duty of officers to prepare and deliver speeches to the men, in order to animate and encourage them, and on a march to prevent the men, at any time, getting into an hud dle, because if the enemy should surround them in that posi tion, they would be greatly exposed to the enemy's fire. It is likewise their business, at all times, to endeavour to annoy the enemy, and save their own men; and therefore ought never to bring on an attack without considerable advantage, or without what appeared to them to insure victory, and that with a loss of but few men. And if at any time they should be mistaken in this, and are likely to lose many men in gain ing the victory, it is their duty to retreat, and wait for a bet ter opportunity of defeating their enemy, without the danger of losing so many men." Their conduct proves that they act on these principles. This is the statement given by those who are experimen tally acquainted with them, and as long as the British officers despised both Indians and Americans, who had studied their art of war, and formed themselves on the same plan, they were constantly beaten by those soldiers of nature, though seldom one fourth of the number of the British. But the Brit ish officers had one advantage of them. That was the art of drawing up and reporting to their superiors, plans of their bat tles, and exaggerated accounts of their great success, and the immense loss of the Indians, which were never thought of till long after the battle was over, and often while they were smarting under their severe defeat or surprise. The writer of this could give some instances, if it would an- answer any good cud, that came under his own knowledge. A STAR IN THE WEST. 173 When the Indians determine on war or hunting, they have stated preparatory, religious ceremonies, for purification, par ticularly by fasting, as the Israelites had. Father Charlevoix gives an account of this custom in his time. In case of an intention of going to war, he who is to command does not commence the raising of soldiers, till he has fasted several days, during which he is smeared with hlack has no conversation with any one invokes hy day and night, his tutelar spirit, and above all, is very careful to observe his dreams. The fast being over, he assembles his friends, and with a string of wampum in his hands, he speaks to them after this manner. Brethren ! the great spirit authorizes my senti ments, and inspires me with what I ought to do.* The blood of is not wiped away. his body is not covered, and I will acquit myself of this duty towards him," &c. Mr. M'Kenzie in some measure, confirms this account, though among different nations. " If the tribes feel them selves called upon to go to war, the elders convene the people in order to obtain the general opinion. If it be for war, the chief publishes his intention to smoke in the sacred stem (a pipe) at a certain time. To this solemnity, meditation and fasting are required as preparatory ceremonials. When the people are thus assembled, and the meeting sanctified by the * This shews the mistakes committed by writers who do not intimately under stand the idiom of the Indian languages. Above it is said, "that the warrior in voked his tutelar spirit," but by this address, it is plain that it was the great spirfo. So the translator of Charlevoix, calls a string of wampum, of which the war-belts are made, a collar of beads. Great allowance should be made for the ignorance of both travellers and writers. The secrecy of Indians, in keeping all their religious rites from the knowledge of white people, lest they should defile them by their presence, adds mm.h to their difficulty. And Charlevoix being a religious Roman easily slid into the idea of an attendant spirit. 174 A STAR IN THE WEST. custom of smoking (this may be in imitation of the smoke of the incense offered on the altar of the Jews) the chief en larges on the causes which have called them together, and the necessity of the measures proposed on the occasion. He then invites them who are willing to follow him, to smoke out of the sacred stem, which is considered as a token of enrolment." A sacred feast then takes place, and after much ceremony, usual on the occasion, " the chief turning to the east, makes a speech to explain more fully the design of their meeting, then concludes with an acknowledgment for past mercies received, and a prayer for the continuance of them, from the master of life. He then sits down, and the whole company declare their approbation and thanks by uttering the word Ho .'" (in a very hoarse, guttural sound, being the third syllable of the beloved name, " with an emphatic prolongation of the last letter. The chief then takes up the pipe, and holds it to the mouth of the officiating person," (like a priest of the Jews, with the in cense) " who after smoking three whiffs, utters a short prayer, and then goes round with it from east to west, to every per son present." The ceremony then being ended, he returns the company thanks for their attendance, and wishes them, as well as the whole tribe, health and long life." Do not these practices remind the reader of the many direc tions in the Jewish ritual, commanding the strict purification, or sanctifying individuals about to undertake great business, or to enter on important offices. Adair, who had greater opportunities of knowing the real character of the Indians to the southward, than any man that has ever written on the subject, gives the following account. "Before the Indians go to war, they have many preparatory A STAR IN THE WEST. 175 ceremonies of purification and fasting, like what is recorded of the Israelites. When the leader begins to heat up for vol unteers, he goes three times round his dark winter house, contrary to the course of the sun, sounding the warwhoop, singing the war song, and heating a drum. 5 * He addresses the croud, who come about him, and after much ceremony, he proceeds to whoop again for the warriors to come and join him, and sanctify themselves for success against the common enemy, according to their ancient religious law. A number soon join him in his winter house, where they live separate from all others, and purify themselves for the space of three days and three nights, exclusive of the first broken day. On each day they observe a strict fast till sunset, watching the young men very narrowly (who have not been initiated in war titles) lest unusual hunger should tempt them to violate it, to the supposed danger of all their lives in the war, by de stroying the power of their purifying, beloved physic, which they drink plentifully during that time. They are such strict observers of their law of purification, and think it so essential in obtaining health and success in war, as not to allow the best beloved trader that ever lived among them, knowingly, to enter the beloved ground appropriated to the duty of being sanctified for war, much less to associate with the camp in the woods, at such a time, though he is united with them in the same war design. They oblige him to walk and encamp sepa rately by himself, as an impure, dangerous animal, till the leader hath purified him, according to the usual time and meth od, with the consecrated things of the ark." With the He- * The Indians have something in imitation of a drum, made of a wet deer skm drawn over a large gourd or frame of wood. 176 A STAR IN THE WEST. brews, the ark tfBerith, (the purifier) was a small wooden chest, as has already been shewn in the first chapter, of three feet nine inches in length, and two feet three inches broad, and two feet three inches in height, and overlaid with pure gold. The Indian ark is of a very simple construction, and it is only the intention and application of it, that makes it wor thy of notice, for it is made with pieces of wood, securely fastened together in the form of a square. The middle of three of the sides extend a little out, but the fourth side is flat, for the convenience of the person's back who carries it. This ark has a cover, and the whole is made impenetrably close with hickory splinters. It is about half the dimensions of the Jewish ark, and may properly be called the Hebrew ark im itated. The leader and a beloved waiter carry it by turns. In contains several consecrated vessels, made by beloved, superanuated women, and of such various antiquated forms, as would have puzzled Adam to have given significant names to each. These two carriers are purified longer than the rest, that the first may be fit to act in the religious office of a priest of war, and the other to carry the awful, sacred ark, all the while they are engaged in tke act of fighting. And it came to pass, when the ark set forward, that Moses said, rise up Lord, and let thine enemies be scattered ; and let them that hate thee, flee before thee. And when it rested he said, return O Lord unto the many thousands of Israel" Numbers x. 35, 36. But they presumed to go up unto the hill top; nevertheless, the ark of the covenant of the Lord and Moses, departed not out of the camp. Then the Arnale- kites came down and the Canaanites who dwelt on that hill, and smote them, and discomfited them even unto Hormak" ibid xiv. *5, A STAR IN THE WEST. 177 "And David said unto them, ye are the chief of the fathers of the Levites; sanctify yourselves both ye and your brethren* that ye may bring up the ark of the Lord God of Israel unto the place that I have prepared for it" 1 Chron. xv. 12. The Hdissii) or beloved waiter, feeds each of the warriors by an exact stated rule, giving them even the water they drink, out of his own hands, lest by intemperance they should spoil the supposed communicative power of their holy things, and occasion fatal disasters to the war camp. They never place the ark on the ground, nor sit on the bare earth, while they are carrying it against the enemy. On hilly ground, where stones are plenty, they place it on them ; but on land, where stones are not to be had, they use short logs* always resting themselves in like manner. The former is a strong imitation of the pedestal on which the Jewish ark was placed, a stone rising three fingers breadth above the floor. They have as strong faith in the power and holiness of their ark, as ever the Israelites had of theirs, ascribing the superi or success of the party to their stricter adherence to the law, than the other. This ark is deemed so sacred and dangerous to be touched, either by their own sanctified warriors* or the spoiling enemy, that they will not touch it on any account* It is not to be meddled with by any but the war chieftain and his waiter, who are consecrated for the purpose * under the pen alty -of incurring great evil. Nor would the most inveterate enemy among their nations, touch it in the woods for the same reason, which is agreeable to the religious opinion and cus toms of the Hebrews, respecting the sacredness of their ark ? as in the case of Uzzah and the Philistines. 178 A STAR IN THE WEST. A gentleman who was at the Ohio in the year 1756, as- aiired the writer that he saw a stranger there, very importu nate to view the inside of the Cherokee ark, which was cov ered with a dressed deer skin, and placed on a couple of short blocks of wood. An Indian sentinel watched it, armed with a hickory how, and hrass pointed harbed arrow; and he was faithful to his trust; for finding the stranger obtruding, with apparent determination to pollute the supposed sacred vehicle, he drew his arrow to the head, and would have shot him through the body, had he not suddenly withdrawn. The leader virtually acts the part of a priest of war pro tempore, in imitation of the Israelites, fighting under the divine military banner of old. The Indians will not cohabit with women while they are out at war; they religiously abstain from every kind of inter course, even with their own wives, for the space of three days and nights, before they go out to war; and so after they re turn home, because they are to sanctify themselves. So Joshua commanded the Israelites, the night before they marched, to sanctify themselves by washing their clothes, avoiding all impurities, and abstaining from all matrimonial intercourse. When the Indians return home victorious over an enemy, they sing the triumphal song to Y. O. He. wall, ascribing the victory to him, like a religious custom of the Israelites,* who were commanded always to attribute their success in war to Jehovah, and not to their swords and arrows. The Indian method of making peace, carries the face of great antiquity. When the applicants arrive near the town, they send a messenger a head, to inform the enemy of their A STAR IN THE WEST. 179 amicable intentions. He carries a swan's wing in ids hand, painted with streaks of white clay, as an expressive emblem of his peaceful embassy. The next day, when they have made their friendly parade, by firing off their guns and whooping, they enter the beloved square. Their chief, who is a-head of the rest, is met by one of the old beloved men of the town. They approach each other in a bowing posture. The former says, Yo Ish le cher Jlnggona ? Art you come a friend, in the name of the great spirit .'" The other replies, Yah Orahre Jlnggona. " The great spirit is with me, I am come a friend in his name." The beloved man then grasps the stranger with both his hands, around the wrist of his right hand, which holds some green branches ; then again about the elbow ; then about the arm close to the shoulder, as a near approach to the heart. Then he waves an eagle's tail over the head of the stranger, which is the strongest pledge of good faith. The writer of this has been witness to this ceremony, performed by an em bassy from the Creek nation, with his excellency general Washington, president of the United States, in the year 1789. The common method of greeting each other is analogous with the above, in a great measure. The host only says, Ish la chu? Are yon a friend? The guest replies, Orahrt-0. lam come in the name of 0. E. A. or Yohetvah. " They are very loving to one another, if several came to a Christian's house, and the master of it gave to one of them victuals, and none to the rest, he would divide it into equal Chares amongst his companions. If the Christians visited them, they would give them the first cut of their victuals. They never eat the hollow of the thigh of any thing they kill; and if a Christian stranger came to one of their houses in thcjj 180 A STAR IN THE WEST. towns, he was received with the greatest hospitality, and the best of every thing was set hefore him. And this was often repeated from house to house." Smith's history of New- Jer sey, page 130. The Indians are not only religiously attached to their tribe while living; but their bodies, and especially their bones, are the objects of their solicitous care, after they are dead. Among the Mohawks, their funeral rites show they have some notion of a future state of existence. They make a large round hole, in which the body can be placed upright, or upon its haunches, which, after the body is placed in it, is covered with timber, to support the earth, which they lay over it, and thereby keep the body from being pressed, they then raise the earth in a round hill over it. They dress the corpse in all its finery, and put wampum and other things in the grave with it. The relations will not suffer grass, or any weed to grow on the grave, and frequently visit it with lamentations. Among the French Indians in Canada, as mentioned by Charlevoix, as soon as the sick person expires*, the house is filled with mournful cries; ami this lasts as long as the family is able to defray the expense, for they must keep open house all the time. In some nations the relatives fast to the end of the funeral, with tears and cries. They treat their visitors praise the dead, and pass mutual compliments. In other na tions, they hire women to weep, who perform their duty punc tually. They sing -they dance they weep without ceasing, always keeping time. He has seen the relatives in distress, walk at a great pace, and put their hands on the heads of all they met, probably to invite them to share in their grief. Those who have sought a resemblance between the Hebre\vs A STAR IN THE WEST. 181 and the Americans, have not failed to take particular notice of their manner of mourning, as several expressions in scrip ture give room to such conjectures, and to suppose them much alike to those in use with those people of God. Indeed, do not these customs and practices seem to be derived from those of the Jews burying their dead in tombs hewed out of a rock, wherein were niches, in which the dead were set in an upright posture, and often with much of their property buried with them. Josephus tells us, that from king David's sepulchre, Hyrcanus, the Maccabean, took three thousand talents^about thirteen hundred years after his death, to get rid of Antioch- us, then besieging Jerusalem. The southern Indians, when any of their people die at home, wash and anoint the corpse, and soon bring it out of doors, for fear of pollution. They place it opposite to the door in a sitting posture. They then carry it three times round the house in which he is to be interred, for sometimes they bury him in his dwelling-house, and under his bed. The re ligious man of the deceased's family, in this procession, goes before the corpse, saying each time, in a solemn tone, Yah then Ho, which is sung by all the procession. Again he strikes up He, which is also sung by the rest. Then all of them sud denly strike off the solemn chorus, by saying wok, which constitutes the divine, essential name, Fah-IIo-He-wah. In the Choktaw nation, they often sing, Hal-le-lu-yah, intermixed with their lamentations. They put the corpse in the tomb in a sitting posture, with his face towards the east, and his head anointed with bear's oil. He is dressed in the finest apparel, having his gun, pouch, and hickory bow, with a young pan ther's skin full of arrows, along side of him, and every other 182 A STAR IN THE WEST. useful thing lie had been possessed of. The tomh is made firm and clean inside. They cover it with thick logs, so as to hear several tiers of cypress hark, and then a quantity of clay over it. The graves of the dead are so sacred among the northern nations, that to profane them, is the greatest hostility that can be committed against a nation, and the greatest sign that they will come to no terms with them. The Indians imagine if a white man was to be buried in the domestic tombs of their kindred, it would be highly crim inal ; and that the spirits would haunt the eaves of the house at night, and cause misfortunes to their family. If any one dies at a distance, and they are not pursued by an enemy, they place the corpse on a scaffold, secured from wild beasts and fowls of prey. When they imagine the flesh is consumed, and the bones dried, they return to the place, bring them home, and inter them in a very solemn manner. The Hebrews, in like manner, carefully buried their dead, but on any accident, they gathered their bones, and laid them in the tombs of their fore-fathers. Thus Jacob charged his sons, and said unto them, I am to be gathered unto my people, bury me with my fathers, in the cave that is in the field of Ephron the Hittite." This was in Canaan. " There they buried Abraham and Sarah his wife ; there they buried Isaac and Rebeckab, his wife j and there I buried Leah." (e And Joseph took an oatli of the children of Israel, saying, God will surely visit you, and ye shall carry my bones from hence." " And Moses took the bones of Joseph with him."* And the bones of Joseph, which the children of Israel brought up out * Gen.xlix. 29, 31 1. 25 Exod. xiii. 19. A STAR IN THE WEST. 183 of Egypt, buried they in Shechem," as above mentioned. Joshua xxiv. 32. The Jews buried near their cities, and sometimes opposite to their houses, implying a silent lesson of friendship, and a caution to live well. They buried families together; but strangers apart by themselves. When an old Indian finds that it is probable that he must die, he sends for his friends, and with them collects his chil dren and family around him ; and then, with the greatest com posure, he addresses them in the most affectionate manner, giving them his last council, and advising them to such conduct as he thinks for their best interests. So did the patriarchs of old, and the Indians seem to follow their steps, and with as much coolness as Jacob did to his children, when he was about to die. A very worthy clergyman, with whom the writer was well acquainted, and who had long preached to the Indians, informed him, that many years ago, having preached in the morning to a considerable number of them, in the recess between the morning and afternoon services, news was suddenly brought, that the son of an Indian woman, one of the congregation then present, had fallen into a mill-dam, and was drowned. Im mediately the disconsolate mother retired to some distance in deep distress, and sat down on the ground. Her female friends soon followed her, and placed themselves in like man ner around her, in a circle at a small distance. They contin ued a considerable time, in profound and melancholy silence, except now and then uttering a deep groan. All at once the mother putting her hand on her mouth, fell with her face flat on the ground, her hand continuing on her mouth. This was followed, in like manner, by all the rest, when all cried out. 184 A STAR IN THE WEST. with the most melancholy and dismal yellings and groanings* Thus they continued, with their hands on their mouths, and their mouths in the dust a considerable time. The men also retired to a distance from them, and went through the same ceremony, making the most dismal groanings and yellings. Need any reader be reminded of the Jewish customs on occasions of deep humiliation, as in Job 21 and 5 Mark me and be astonished, and lay your hand on your mouth. 29 and 9 The princes refrained talking, and laid their hands on their mouths. 40 and 4 Behold ! I am vile, what shall 1 answer thee ? I will lay my hand on my mouth. Micah 7 and 16 .The nations shall see and be confounded ; they shall lay their hands on their mouth. Lament. 3 and 9 He putteth his mouth in the dust, if so be, there may be hope. Prov. 30 and 32 If thou hast thought evil, lay thine hand upon thy mouth. The Choktaw Indians hire mourners to magnify the merit and loss of the dead, and if their tears do not flow, their shrill voices will be heard to cry, which answers the solemn chorus much better. However, some of them have the art of shed ding tears abundantly. Jerem. ix chap. 17, 19 Thus saith the Lord of Hosts, consider ye, and call for the mourning wo men, that they may come, and send for cunning women, that they may come, for a voice of watting is heard, c. By the Mosaic law, the surviving brother was to raise up seed to a deceased brother, who should leave a widow child less. The Indian custom resembles this in a considerable de gree. A widow among the Indians is bound by a strict penal law or custom, to mourn the death of her husband, for the space of three or four years. But if it be known that the elder A STAR IN THE WEST. 185 brother of her deceased husband has lain with her, she is af terwards exempt from the law of mourning has liberty to tie up her hair, anoint and paint herself, which she could not otherwise do, under pain of being treated as an adultress. The Indians, formerly on the Juniata and Susquehannahriv- ers, placed their dead on close or covered cribs, made for the purpose, till the flesh consumed away. At the proper time they gathered the bones* scraped and washed them, and then buried them with great ceremony. There is a tribe called Nanticokes, that on their removal from an old to a new town, carry the bones of their ancestors with them. This also prevailed in particular cases among the Canada Indians. An officer of the regular troops at Oswego, upwards of sixty years ago, reported the following fact. A boy of one of the westward nations, died at Oswego the parents made a regular pile of split wood, laid the corpse upon it and burnt it. While the pile was burning, they stood gravely looking on* without any lamentation, but when it was burned down they gathered up the bones, and with many tears, put them into a box, and carried them away with them.* The Indians are universally remarkable for a spirit of independence and freedom beyond any other people, and they generally consid er death, as far preferable to slavery. They abhor covet- ousness, and to prevent it, they burn all the little property an Indian has at the time of Ins death* or bury it with him in his grave* This necessarily tempts them to frugality and abstemiousness in their manner of living. They are wholly ignorant of all kind of mechanieks, except so far as is pressed on them by necessity. They are free from hypocrisy or any * Excwl xiii. 19. Josh, xxjv, 12. 2 Sam. xxi. 12 H. 186 A STAR IN THE WEST. forced civility or politeness ; but their general conduct, show* a frank and candid, but plain and blunt hospitality and kind ness ; with a degree of faithfulness in their dealings, except with their enemies, that often astonishes white people ; who although their pretensions are so much higher, cannot, at least do not, reach them in this particular. The great author of the divine legation of Moses, in treating of the government of the Jews, both civil and religious, as ne cessarily united under one great head, the God of Abraham, Isaac and Jacob, states his subject clearly and fully, and then says, "but the poet Voltaire, indeed, has had a different rev elation. The pride of every individual among the Jews, says he, is interested in believing, that it was not their detestable policy, their ignorance in the arts or their unpoliteness, that destroyed them ; but that it is God's anger that yet pursues them for their idolatries." This detestable policy, (which I would not consider in the most obvious sense of the Mosaic institution, because that might tend to make the poet himself detestable) was a principle Of independence. This ignorance in the arts prevented the entrance of luxury ; and this unpo liteness, hindered the practice of it. And yet parsimony, frugality and a spirit of liberty, which naturally preserve oth er states, all tended in the ideas of this wonderful politician to- destroy the Jewish." How surprisingly does this observation of bishop Warburton, apply in support of these untutored In dians, and point out from whence they must have drawn their principles of conduct, A STAR IN THE WEST. 187 CHAPTER VI. The known Religious Rites and Ceremonies of the Indians TO adopt the language of Father Charlevoix, "nothing has undergone more sudden, frequent, or more surprising revolutions, than religion. When once men have abandoned the only true one, they soon lose sight of it, and find themselves entangled and bewildered in such a labyrinth of incoherent errors, inconsistencies and contradictions, that there often remains not the smallest clue to lead us back to the truth. One example. The Buccaniers of St. Domingo, who professed to be christians, but who had no commerce, except among themselves, in less than thirty years, and through the sole want of religious worship, instruction, and an authority capable of retaining them in their duty, had lost all marks of christianity, except baprism alone. Had these people continued only to the third generation, their grand children would have been as void of christianity as the inhabitants of Terra-Australis, or New Guinea. They might, possibly, have preserved some ceremonies, the meaning of which they could not account for." However, our wandering tribes of Indians have, in a most surprising manner, bordering on something rather supernatrual, preserved so many essential parts of their original plan of divine worship, and so many of their primitive doctrines, although they have at present almost wholly forgotten their meaning and their end, as to leave little doubt of their great source. 188 A STAR IN THE WEST. They are far from being idolaters, although many good men, from want of a knowledge of their language, and often having communion with the most worthless part of them, without making any allowance for their local situation and circumstances, have given terrific accounts of these children of nature. And this is not much to he wondered at. For many of our worthy, over zealous and pious Europeans, and some white Americans, deeply affected with a sense of their unhappy state, and feeling the importance of the gospel to them, have unwisely gone into the woods to them, without proper and preparatory education for so important an under taking. I mean, without understanding their language, or being well acquainted with their manners, customs and habits nay, not even making themselves acquainted with their re ligious prejudices, or by taking sufficient time and using prop er means to gain their confidence. To people so ignorant of what they ought first to have known, and wholly trusting to a heathen interpreter, unable to feel or express the nature of spiritual things, and having to deal with a most jealous and artful people, rendered so by the experience of more than a century, by the continued imposi tions and oppression of the nation to which their visitants be longed it is quite a natural thing, that they were often at first despised by the Indians, and then made a mere hutt, for the most worthless to frighten and laugh at. Hence the In dians have often in a frolic dressed themselves in the most terrific manner, and made the most frightful images, with every kind of extravagant emblem about it, to alarm and ter rify their new cornel's, of whom they thought so lightly. We speak now principally of their light, bad people, who inhabit A STAR IN THE WEST. 189 around or near our settlements. That, as a people, they are sensible of propriety, and are careful observers of characters, is well known to those who have been long conversant with them. It is a fact well attested, that a preacher went among them before the revolutionary war, and in a sudden discourse to them, began to tell them that there was a God, who crea ted all things that it was exceedingly sinful and offensive to him, to get drunk, or lie, or steal all which they must care fully avoid. They answered him " Go about your business, you fool ! Do not we know that there is a God, as well as you ! Go to your own people and preach to them ; for who gets drunk, and lies and steals more than you white people r" In short, if the Indians form their ideas of us from the com mon traders and land speculators, and common people, with whom they usually have to do, they will not rim into a greater error than we do, when we form our ideas of the character of Indians from those who generally keep about our settlements, and traffic with the frontier inhabitants. The Indians are filled with great spiritual pride we mean their chiefs and best men. They consider themselves as under a theocracy, and that they have God for their governor and head. They therefore hold all other people, comparatively, in contempt. They pay their religious worship, as Mr. Adair assures us, (and he had a great opportunity of knowing) to Loak-Ishto, Haolo-Jlbba, or the great, beneficent, supreme, holy spirit of fire, who resides above the clouds, and on earth witli unpolluted, holy people. They were never known (whatever --me Spanish writers may say to the contrary, to cover their own blood-thirsty and more than savage barbarity to the --itives they found in Mexico, at their first arrival among 190 A STAR IN THE WEST. them) to pay the least perceivable adoration to images or dead persons, or to celestial luminaries-, or evil spirits, or to any created being whatever. Their religious ceremonies are more after the Mosaic in stitution, than of pagan imitation. They do not believe the sun to be any larger than it appears to the naked eye. Not withstanding the various accounts we have had from differ ent authors, greatly exaggerating the reports of the Indian's irreligious conduct, they have taken little or no pains to be well informed (for it is attended with considerable difficulty, from their known secrecy) and have therefore grossly mis represented them, without designing to mislead. Historians ought not to be trusted, as to detailed accounts of these peo ple, with whom it seems to have been previously agreed among themselves, to charge with being red savages and bar barians, while the Indians, in return, consider as white sava ges and accursed people, those who thus traduce them. Read ers shonld carefully examine into their means of knowledge their connections with the Indians, and the length of time and opportunities they enjoyed in a social intercourse with them. Difficulties, and those very great, have arisen from the im practicability of a stranger being well informed, particularly arising from their unconquerable jealousy and great secrecy in every thing relating to their religions character. Again, historians are often fond of the marvellous, and are apt to take up with any information they can get, without examining its source, and arc too apt to make up strange stories to an swer their private purposes, or to cover base designs. This is fully exemplified in the abominable false accounts publish ed by the Spaniard*, relative to Mexico, on their first-con- A STAR IN THE WEST. 191 quering, or rather carrying destruction and blood-shed through that fine country, to gratify their covetousness and bloody dis positions, when they had not the least foundation in truth for their diabolical accounts. Adair assures us, that from the experience of forty years, he can say, that none of the various nations from Hudson's hay to the Missisippi, have ever been known by our trading peo ple, to attempt the formation of any image of the great spirit whom they devoutly worship. They never pretend to divine from any thing but their dreams, which Seems to proceed from a tradition, that their ancestors received knowledge of future events from heaven by dreams vide Job xxxiii. &c. Du Pratz had a particular intimacy with the chief of the guardians of the temple, in a nation near the Missisippi 2 vol. 173. That on his requesting to be informed of the na ture of their worship, he was told that they acknowledged a supreme being, whom they called Cayo-cop-chill, or great spirit, or the spirit infinitely great or the spirit by way of excellence. That the word chill in their language, signifies the most su perlative degree of perfection, and is added to make that ap pear, as oua is fire, and oua drill is the supreme fire, or the sun. Therefore by the word Coyo-cop-chill, tlie,y mean a spir it that surpasses other spirits, as much as the sun does com mon fire. The guardian said, that the great spirit was so great and powerful, that in comparison with him, all other things were as nothing. He had made all that we see all that we can see -and all that we cannot see. He was so good that he could not do ill to any one, even if he had a mind to do it. They believed that the great spirit had made all things by his will ; that nevertheless the little spirits who are his 192 A STAR IN THE WEST. servants, might by his orders, have made many excellent works in the universe, which we admire ; but that God him self had formed man with his own hands. They called the little spirits, free servants. That those spirits were always before the great spirit, ready to execute his pleasure with an extreme diligence. That the air was filled with other spirits, some good, some wicked, and that the latter had a chief, who was more wicked than all the rest. That the great spirit had found him so wick ed, that he had bound him forever, so that the other spiiits of the air, no longer did so much harm. He was then asked, how did God make man ? he answer ed that he kneaded some clay, and made it into a little man after examining it and finding it well formed, he blew on his work, and forthwith the little man had life grew acted walked and found himself a man, perfectly well shaped. He then was asked about the woman he said, probably she was made in the same manner as the man, but their ancient speech made no mention of any difference, only that the man was made first page 174. The Indians also, agreeably to the theocracy of Israel, think the great spirit to be the immediate head of their state, and that God chose them out of all the rest of mankind, as his peculiar and beloved people. Mr. Locke, one of the ablest men Great-Britain ever pror duced, observes, " that the commonwealth of the Jews, differ ed from all others, being an absolute theocracy. The laws established there, concerning the worship of the one invisible deity, were the civil laws of that people, and a part of their political government, in which God himself was the legislator." A STAR IN THE WEST. 193 In this, the Indians profess the same thing precisely. This is the exact form of their government, which seems unac countable, were it not derived from the same orignal source, and is the only reason that can be assigned for so extraordina ry a fact. The Indians are exceedingly intoxicated with religious pride, and hold the white people in inexplicable contempt the common name they give us in their set speeches, literally means, nothings ; but in their war speeches, ottuck ookproose, the accursed people. But they flatter themselves with the name Hothik-ore-too-pate, the beloved people. This is agree able to the Hebrew epithet Jlmmi, during the theocracy of Is rael. When their high priest (if we may be allowed the term, for their most beloved man) addresses the people, he calls them, " the beloved or holy people." These addresses are full of flourishes on the happiness of their country, calling it a land flowing with milk and honey. When any of their beloved people die, they soften the thoughts of death, by saying, he is only gone to sleep 'with tlitir beloved forefathers, and usually mention a common proverb among them, neitak intahah" the days appointed, or allow ed him, were finished. And this is their firm belief, for they affirm that there is a fixed time and place, when and where every one must die, without any possibility of averting it. They frequently say, " such a one was weighed on the path, and made to be light." They always ascribe life and death to God's unerring and particular providence. Contrary to the usage of all the ancient heathen world, they not only name God by several strong compounded appel lations, expressive of many of his divine attributes, but Kke- 194 A STAR IN THE WEST. wise say yah at the beginning of their religious dances, with a bowing posture of body then they sing y, y, y, ho, ho, ho, he, he, and repeat those sacred notes (but not the whole name) on every religious occasion. The religious attendants calling to Yah, to enable them humbly to supplicate, seems to point to the Hebrew custom of pronouncing Jah, which signifies the divine essence. It is well known, what sacred regard the Jews had to the great four lettered name, scarcely ever to mention it in the whole, but once a year, when the high priest went into the holy sanctuary on the day of expiation of sins. Might not the Indians, have copied from them this sacred in vocation, and also their religious forbearance in never men tioning the whole name, but in their sacred songs of praise. Their method of invoking the great spirit in solemn hymns, with that reverend deportment, and spending a full breath on each of the first two syllables or letters of the awful divine name, has a surprising analogy to the Jewish custom, and such as no other nation or people, even with the advantage of written records, have retained. Charlevoix, speaking of the northern Indians, observes, that the greatest part of their feasts, their songs and their dances, appeared to him, to have had their rise from religion, and yet preserve some traces of it. I have met with some persons, says he, who could not help thinking that our In dians were descended from the Jews ; and found in every thing, some affinity between them and the people of God. There is indeed a resemblance in some things, as not to use knives at certain meals, and not to break the bones of the beast that they eat at the these times, (and we may add, that they never cat the part under the lower joint of the thigh, A STAR IN THE WEST. 195 but always throw it away.) The separation of their women, at certain periods. Some persons have heard them, or thought they heard them, pronounce the word, hattalujah, in their Songs. The feast they make, at the return of their hunters, and of which they must leave nothing, lias also heen taken for a burnt offering, or for the remains of the passvwer of the Israelites : and the rather, they say, because when any one family cannot compass his portion, he may get the assistance of his neighbour, as was practised by the people of God, when a family was not sufficient to eat the whole paschal lamb. The Israelites of old were ordered by Moses to fix in the tabernacle (as Solomon did afterwards in the temple, all by command of God) Cherubim over the mercy seat. The cur tains also which lined the walls and the veil of the temple, had the like figures on them. The Cherubim are said to have represented the names, yo-Jie-wah-elohim, in redeeming lost mankind, and means the similitude of the great and mighty one, whose emblems in the congregational standards, were, * the butt, the lion, the man and the eagle." So Ezekiel in forms us the Cherubim were uniform and had these four com pounded animal emblems. Every one had four faces (ap pearances, habits or forms.) x chap. 14<, 20, 22. Each of the Cherubim, according to the prophet, had the head and face of a man$ the likeness of an eagle about the shoulders, with expanded wings ; their necks, manes and breasts re sembled those of a lion, and their feet those of a bull or calf; the soles of their feet, were like a calf's foot. Ezek. i. 4, 5, 6. 66 And I looked and behold a whirlwind came out of the north, a great cloud and a Jtre infolding itself, and a brightness was about it, and out of the midst thereof as the colour of amber, 196 A STAR IN THE WEST. out of the midst of the fire also out of the midst thereof, the likeness of four living creatures. And this was their appear ance : they had the likeness of a man, and every one had four faces, and every one had four wings," &c. &c. 10th ver. '! As for the likeness of their faces, they four had the face of a man and the face of a lion on the right side ; and they four had the face of an ox on the left side ; and they four also had the face of an eagle vide ver. 11. These are the terrestial cherubim, and the psalmist represents them as the chariot of divine majesty, and displays his transcendant and glorious title of King of Kings. Psalms xviii. 7, 11 - God sitteth between and rideth upon the cherubim" as a divine chariot ibid. xcix. 1. So the American Indians, particularly the Cherokees and Choktaws, have some very humble representation of these cherubimical figures, in their places of worship, or beloved square ; where, through a strong religious principle, they dance almost every winter's-night, always in a bowing pos ture, and frequently singing, haUduyah, yo, he, wah. They have in these places of worship, which Adair says he has seen, two white painted eagles, carved out of poplar wood, with their wings stretched out, and raised five feet from the ground, standing in the corner, close to the red and white im perial seats j and on the inner side of each of the notched pieces of wood, where the eagles stand, the Indians frequent ly paint with a white chalky clay, the figure of a man, with buffalo's horns,* and that of a panther, the nearest animal in America, to that of a lion, with the same colour. These * It was an ancient custom amongst the eastern natious, to use horns as an env Wepi of power, whih the Imliaus always do. A STAR IN THE WEST. 197 figures they paint a-fresh at the first fruit offering, or the an nual expiation of sins. Yet it has never heen known that the Indians ever substituted the eagle, panther, or the simili tude of any thing whatever, as objects of divine adoration, in the room of the great invisible divine essence. Nay, they often give large rewards for killing an eagle, and they kill the panther wherever they find him. The ideas which a people form of the supreme deity, will direct to the nature of their religious worship. Among the south ern Indians, Ish-to-hoolo is an appellation for God. It points at the greatness, purity and goodness of the creator, in form ing man. It is derived as is said from Ishto, great, which you find in all the prophetical writings, attributed to God. Also from the present tense of the infinitive mood of the active verb ahoolo, " I love," and from the preter tense of the pas sive verb hoolo, that is sanctifying, sanctified, divine or holy. Women set apart, they term hoolo, that is, sanctifying them selves to Ish-to-hook. So Netakhoolo signifies a sanctified or holy day. So Okka hodo, water sanctified. Thus Ish-to-hoo!o, when applied to God, in its true radical meaning, imports the great Moved holy cause, which is exceedingly comprehensive and more expressive of the true nature of God, than the He brew name Jldonai, which may be applicable to a human be ing. When they apply the epithet, compounded, to any of their own religious men, it signifies, tJie great holy, beloved^ sanctified man of the holy one. They make the divine name point yet more strongly to th^ supreme author of nature. For as abba, signifies father, so, to distinguish God, as the king of kings, by his attributes, from their own Minggo Ishto, or great chief, they frequently* 198 A STAR IN THE WEST. name God Minggo Ishto J&ba, Ishto Allba, Mnggo Alba, &e. and when they strive to move the passions, Ishto Hoolo Mba. They have another more sacred appellative, which with them is the mysterious essential name of God. The tetragrammana- ton of the Hebrews, or the great four lettered name already mentioned, F. 0. He. ivah. This they, like the Hebrews, never mention altogether in common speech. Of the time and place, when and where they mention it, they are very particular, and always with a solemn air. The Indians have among them orders of men answering to our prophets and priests. In the Muskohge language, Hitch Lalage, signifies cunning men, or persons prescient of futurity, much the same with the Hebrew seer. But the Indians in general call their pretended prophets, Loa-che, men resem bling the holy fire, or elohim. Their tradition says, that their forefathers were possessed of an extraordinary divine spirit, by which they foretold things future, and controled the com mon course of nature ; and this they transmitted to their off spring, provided they obeyed the sacred laws annexed to it. They believe that by the communication of the same divine fire, working in their Zoo-c/te, they can yet effect the like. But they say it is out of the reach of Nana Ookproo, or bad people, either to comprehend or perform such things, because the holy spirit of fire will not co-operate with or actuate Hot- luch OokproOf the accursed people. " A sachem of the Mingo tribe, being observed to look at the great comet which appear ed the first day of October, one thousand six hundred and eighty, was asked, what he thought was the meaning of that prodigious appearance ? answered gravely, " It signifies that we Indians shall melt away, and tbk country be inhabited by A STAR IN THE WEST. 199 another people." Smith's New- Jersey, 136, in a note. How this Indian came hy his knowledge, without the learned Whis, ton's astronomical tables, or whether he had any knowledge, is not so material. He will, however, be allowed as good a right to pretend to it, when the event is considered, as the other had in his conjectures concerning the cause of Noah's flood. At all events, this Indian must have reasoned well, and had pretty clear conceptions of the effects that would nat urally follow such causes. Mr. Beatty gives much the same account of their prophets among the Delaware nations or tribes, above forty-five years ago. They consult the prophets upon any extraordinary occa sion as in great or uncommon sickness, or mortality, &c. This, he says, seems to be in imitation of the Jews of old, en quiring of their prophets. Ishto Hoolo is the name of all their great beloved men, and the pontifical office descends by inheri tance to the eldest. It cannot be expected but that the dress of the old Indian high-priest, or rather, their great Moved man, or the first and oldest among the lelorced men, should be different from that of the high-priest of the Jews. The poverty and distressed con dition of the Indians, renders such a conformity impossible ; but notwithstanding the traces of agreement are really aston ishing, considering their circumstances, and their having no means of knowing what it was, but by tradition, being deprived of all records relative to it. Before the Indian Archi-magus, or high-priest, officiates in making the supposed holy fire, for the yearly atonement for sin, as will soon be shewn, he clothes himself with a white garment, resembling the ephod of the Jews, being made of 200 A STAR IN THE WEST. a finely dressed deer or doe skin, and is a waistcoat without sleeves. When he enters on that solemn duty, a beloved at tendant spreads a white dressed buckskin* on the white seat, which stands close to the supposed holiest division of their place of worship, and then puts some white beads on it, that are offered by the people. Then the Archi-magus wraps round his shoulders a consecrated skin of the same sort, which reaching across under his arms, he ties behind his back, witli two knots on his legs, in the form of a figure of eight. Instead of going barefoot, he wears a new pair of white buckskin moc- asins, made by himself, and stiched with the sinews of the animal. He paints the upper part of them across the toes, with a few streaks of red, made of the red root, which is their symbol of holy things, as the vermilion is of war. These shoes he never wears at any other time, and leaves them with the other parts of his pontifical dress, when the service is over, in the beloved place. In resemblance of the sacred breast-plate, the American priest wears a breast -plate, made of a white conck-shell, with two holes bored in the middle of it, through which he puts the ends of an otter skin strap, and fastens a buckhorn white button to the outside of each, as if in imitation of the precious stones of urim and thumim, which miraculously blazoned on the high- priest's breast, the unerring words of the divine oracle. In stead of the plate of gold which he wore on his forehead, witli the words holy, or separated to God, the Indian wears around his temples either a wreath of swan's feathers, or a long piece of swanskin doubled, so as only the fine snowy down appears * When the high-priest of the Jews went into the holy of holies, on the day of xpiatioM, he clothed himself in white ; and \vheu the service was over, he left those clothes in the tabernacle. JLevit. xvi. 4-23. A STAR IN THE WEST. 201 eh each side. And in likeness of the tiara of the former, the latter wears on the crown of his head a tuft of white feathers, which they call yaterah, but the meaning of the word is not known. He also fastens a number of blunted wild turkey cocks' spurs towards the toes of his mocasins, as if in resem blance of the bells which the Jewish liigh-priest wore on his coat of blue. Bartram assures us, " that there is in every town or tribe, an high-priest, usually nick-named by the white people, the juggler or conjurer, besides several of inferior rank. But that the oldest high priest or seer presides always in spiritual things, and is a person of great consequence. He maintains and exercises great influence in the state, particularly in mil itary affairs ; their senate or great council never determining on an expedition without his council and assistance* These people believe most firmly, that their seer or high priest has communion with powerful invisible spirits, who they suppose have some share in the rule and government of human affairs* as well as irt that of the elements. He further adds, that these Indians are by no means idolaters* unless their puffing the tobacco smoke towards the sun, and rejoicing at the ap* pearance of the new moon, may be termed so.* So far from idolatry are they that they have no images among them, nor any religious rite or ceremony relating to them, that I could ever perceive* " They adore the great spirit, the giver and taker away of the breath of life, with the most profound and respectful horn* * It is rather supposed that they use the smoke of the sacred stem or pipe, as the Jews did their incense and as to the hew moon, as they reckon thfeit time by it, they are as careful observer^ of it, as the Jew* were. 202 A STAR IN THE WEST. age. They believe in a future state, where the spirit exists?. which they call the world of spirits, where they enjoy differ ent degrees of tranquility and comforts agreeably to their life spent here. They hold their beloved man or priest in great respect, and pay strict obedience to what he directs." These religious beloved men, are also supposed to be in great favour with the deity y and able to procure rain when- they please. In this- respect also, we may observe a great conformity to the practice of the Jews, Their records inform? us, that in the moon Abib or Nisan, they prayed for the spring or latter rain, to be so seasonable and sufficient as to give them a good harvest; and the Indians have a tradition, that their fore-fathers sought for, and obtained such seasonable rains, as gave them plentiful crops continually jvand they now seek them, in a manner agreeable to a shadow of this tradition. In the year 1747, a Natchez warrior told Adair, that while one of their prophets was using his divine invocations for rain,, he was killed by thunder on the spot ; upon which account the- spirit of prophecy ever after subsided among them, and he became the last of their reputed prophets. They believed that the holy spirit ofjlre had killed him with some of his an gry darting fire, for wilful impurity ; and by his threatening voice, forbad them to renew the like attempt; and justly con cluded, that if they all lived well, they should fare well and have proper seasons. This opinion coincides with that of the- Hebrews, who esteemed thunder-struck individuals as under the displeasure of heaven, and they also observed and enforc ed such rules of external purity as none of the nations observ ed, except the Hebrews* A STAR IN THE WEST. 203 As the Jewish prophets had oracular answers to their pray ers, so the Indian prophets, who invoke yo-he-ivah and medi ate with the supreme holy fire, that he may give seasonable rains, have a transparent stone of supposed great power m assisting to hring down the rain, when it is put in a basin of water agreeably to a reputed divine virtue impressed on one of the like sort, in times of old, which communicates it circularly. This stone would suffer great injury, as they assert, were it even seen by their own laity ; but if by foreigners, it would be utterly despoiled of its divine communicative power. This looks something like a tradition of the blazing stones of Urira and Thumim. As the Jews had a sanctum sanctorum, or most holy place in their tabernacle and temple, so have all the Indian nations, particularly the Muskohge nation. It is par titioned off by a mud wall, about breast high,4>etween the white seat, winch always stands to the left hand of the red painted war seat, There they deposit their consecrated ves sels and supposed holy utensils, none of the laity daring to approach that sacred place for fear of particular damage to themselves, and a general hurt to the people, from the sup posed divinity of the place. According to Mr. Bartram, the great or public square of the southern towns, generally stands alone, in the centre and highest part of the town. It consists of four square or cubical buildings of one story high- uniform and of the same dimen sions, so situated as to form an exact tetragon, encompassing an area of half an acre of ground, more or less, according to the strength and size of the town, or will of the inhabitants. One of these buildings is the council-house, where all public 204 A STAR IN THE WEST. business is done. Another of these huildings differs from the rest It is closely shut up on three sides, and has a partition wall run through it, longitudinally from end to end, dividing it into two apartments, the hack part is dark, having only three small arched apertures or holes opening into it from the front apartment, and are but just sufficient for a man to go in at. This secluded place, appears to be designed as a sanctuary or sacred part of the temple, as it is said among them, to be death for any person, but the Mico, or high priest, to enter into it, and none are ever admitted, unless by permission of the priests, who guard it night and day. Here are deposited all the sacred things, as the physic-pot, rattles, chaplets, ea gle's tail, calumet or sacred stem, the pipe of peace, &c. But children and females are never admitted. At this time the people of the town were fasting, taking medicine, and praying to avert a grievous calamity of sick ness which then afflicted them. They fasted seven or eight days, during which they neither eat or drank any thing, but a meagre gruel made of corn flour and water, at the same time drinking their black drink or physic, which acts as a severe emetic.
A STAR IN THE WEST. 205 CHAPTER VII. Their Public Worship and Religious Opinions. THE Indians, in general, keep the following religious fasts and festivals -- 1. Their Feast of First Fruits, and after it, on the evening of the same day, one something like the Passover. 2. The Hunter's Feast, like that of Pentecost. 3. The Feast of Harvest and day of expiation of sin. 4. A daily Sacrifice. A Feast of Love. 1st. Their Fiest of First Fruits and Passover. Mr. Penn, who found them perfectly in a state of nature, and wholly a stranger to their manners and characters, and who could not have had any knowledge of them but from what he saw and heard for some months he remained with them, on his first visit to their country, informs his friends in England, in one of his letters, in 1683. "that he considered these poor people as under a dark night in things relating to religion; yet that they believed in God, and immortality, without the help of metaphysics, for they informed him that there was a great king who made them, who dwelled in a glorious country to the southward of them; and that the souls of the good will go thither, where they shall live again. Their worship consists of two parts -- sacrifice and cantico. The first is with 206 A STAR IN THE WEST. their first fruits. The first and fattest huek they kill goeth to the fire, where he is all burnt with a doleful ditty of him who performs the ceremony, but with such marvellous ferven cy and labour of body, that he will even sweat to a foam. The other part is their cantico, performed by round dances -sometimes words sometimes songs then shouts two are in the middle, wh begin, and by singing and druming on a board, direct the chorus. This is done with equal earnest* ness and labour, but with great appearance of joy. In the fall when the corn cometh in, they begin to feast one another. There have been two great festivals already, to which all come, who wifl. Mr. Penn was at one himself. Their en tertainment was at a great seat by a spring, under some sha dy trees. It consisted of twenty bucks, with hot cakes made ef new corn, with both wheat and beans, which they make up in a square form, in the leaves of the corn, and then bake them in the ashes they then fall to dancing : But all who go to this feast must take a small present in their money, it might be but six pence, which is made of the bone of a fish. The black is with them as gold, and the white as silver they call it wampum." Afterwards speaking of their agree ment in rites with the Hebrews, lie says that "they reckon by moons they offer their first fruits they have a kind of Feast of Tabernacles they are said to lay their altars upon twelve stones they mourn n year they have a separation of women; with many other things that do not now occur. From Mr. Adair, the follow ing account, or rather abstract, of his account of the feast and fast of what may be called tlicir Passover, and Feast of First Fruits, is made. A STAR IN THE WEST. 207 On the day appointed (which was among the Jews, generally in the spring, answering to our March and April, when their barley was ripe, being the first month of their ecclesiastical, and the seventh of their civil year, and among the Indians, as soon as their first spring produce comes in) while the sanc tified new fruits are dressing, six old beloved women come to their temple, or sacred wigwam of worship, and dance the beloved dance with joyful hearts. They observe a solemn procession as they enter the holy ground, or beloved square, carrying in one hand a bundle of small branches of various green trees 5 when they are joined by the same number of beloved old men, who carry a cane in one hand, adorned with white feathers, having green boughs in the other hand. Their heads are dressed with white plumes, and the women in their finest clothes and anointed with bear's grease or oil, having also small tortoise shells and white pebbles fastened to a piece of white dressed deer skin, which is tied to each of their legs. The eldest of the beloved men, leads the sacred dance at the head of the innermost row, which of course is next the holy fire. He begins the dance, after once going round the holy fire, in solemn and religious silence. He then in the next circle, invokes yah, after their usual manner, on a bass key and with a short accent. In another circle, he sings ho, lio, which is repeated by all the religious procession, tili they finish that circle. Then in another round, they repeat he, he, in like manner, in regular notes, and keeping time in the dance. Another circle is continued in like manner, with repeating the word wah, umh (making in the whole, the di vine and holy name of yah, ho, he, wa/j.) A little after this is finished, which takes considerable timo, they begin again, 208 A STAR IN THE WEST. going fresh rounds, singing hal-hal-le-k'lu'lu-ijah-yah, in like manner; and frequently the whole train strike up hattelu, hot- klu 9 hatteluyah, halleluyah, with great earnestness, fervour and joy, while each strikes the ground with right and left feet al ternately, very quick, but well timed. Then a kind of hol low sounding drum, joins the sacred choir, which excites the old female singers to chant forth their grateful hymns and praises to the divine spirit, and to redouble their quick, joyful steps, in imitation of the leader of the beloved men, at their head. This appears very similar to the dances of the Hebrews, and may we not reasonably suppose, that they formerly under stood the psalms and divine hymns, at least those which be gin or end with hallelujah ; otherwise how comes it to pass, that all the inhabitants of the extensive regions of North and South America, have and retain these very expressive He brew words, and repeat them so distinctly, applying them after the manner of the Hebrews, in their religious acclamations. On other religious occasions, and at their Feast of Love, they sing ale-yo, ak-yo, which is the divine name by the attribute of omnipotence. They likewise sing he-wah, he-wah, which is the immortal soul, drawn from the divine essential name, as deriving its faculties from yo-he-wah. These words of their religious dances, they never repeat at any other time, which has greatly contributed to the loss of their meaning ; for it is believed they have grown so corrupt, as not now to understand either the spiritual or literal meaning of what they sing, any farther than by allusion to the name of the great spirit. A STAR IN THE WEST. 209 In these circuitous dances, they frequently also sing on a bass key, aluhe, aluhe, aluwah, 'aluivah. Also shilu-yo, shilu- fyo, shilu-he, shilu-he, sh&u^tvak', shilu+wahi and shilu-hah, shilu-hah.* Tliey transpose them also several ways, but with the very same notes. The three terminations make up the four lettered divine name. Hah is a note of glad ness and joy. The word preceding itj shihi, seems to ex press the predicted human and divine Shiloh, who was to be the purifier and peace maker. They continue their grate ful divine hymns for the space of about fifteen minutes, and then break up. As they degenerate, they lengthen their dances, and shorten the time of their fasts and purifications 5 insomuch, that they have so exceedingly corrupted their prim- itive rites and customs, withia the space of the last thirty years, (now about eighty years) that, at the same rate of de clension, there will not long be a possibility of tracing their origin, but by their dialects and war customs. At the end of this notable religious dance, the old beloved women return home to hasten the feast of the new sanctified fruits. In the mean time, every one at the temple drinks plentifully of the cussena and other bitter liquids, to cleanse their sinful bodies, as they suppose. After which, they go to some convenient deep water, and there, according to the ceremonial law of the Hebrews, they wash away their sins with water. They then return with great joy, in solemn procession, singing their notes of praise, till they again enter their holy ground, to eat of the new delicious fruits, which are brought to the outside of the * Cruden, in his Concordance, says " All Christian commentators agree/ that the word Shiloh ought to be understood of theMessiah, of Jesus Christ. Jerome translates it, by qui metbendus est He who is to be sent ; and manifestly reads Shi loach, sent, instead of Shiloh." 210 A STAR IN THE WEST. square by the old beloved women. They all behave so mod estly, and are possessed of such an extraordinary constancy and equanimity in pursuit of their religious mysteries, that they do not shew the least outward emotion of pleasure at the first sight of the sanctified new fruits. If one of them should act in a contrary manner, they would say to him che-hakset- Kanaha You resemble such as were beat in Kanaha. Formerly, on the north side of the Susquehannah river, in Pennsylvania, were some old Indian towns, called Kanaa, and now about eighty years ago, there was a remnant of a nation, or a subdivided tribe of Indians, called Kanaai, which greatly resembles the Hebrew name Canaan. Mr. Smith, in his History of New-Jersey, speaking of the Indians in the year 1681, says " Very little can be said as to their religion. They are thought to believe in a god and immortality, and seemed to aim at public worship. When they did this, they sometimes sat in several circles, one within another. The action consisted of singing, jumping, shouting and dancing; but mostly performed rather as something handed down from their ancestors, than from any knowledge or enquiry into the serious parts of its origin. They said that the great king who made them, dwelt in a glorious country to the southward, and that the spirits of the best should go there and live again. Their most solemn worship was the sacrifice of the first fruits, in which they burnt the first and fattest buck, and feasted together on what else they had col lected. But in this sacrifice broke no bones of any creature they eat. When done, they gathered the bones and buried them very carefully : these have since been frequently ploughed up." page 140. A STAR IN THE WEST. 211 Among the Indians on the northwest side of the Ohio, the Feast of the First Fruits is thus described by the Rev. Dr. Charles Beatty, who was an eye witness of the ceremony : Before they make use of any of the first or spring fruits of the ground, twelve of their old men meet, when a deer and some of the first fruits are provided. The deer is divided into twelve parts, according to the number of the men, and the tiorn beaten in a mortar and prepared for use by boiling or baking into cakes under the ashes, and of course unleavened. This also is divided into twelve parts. Then these men hold up the venison and first fruits, and pray with their faces to the east, acknowledging, as he supposed, the goodness and bounty of heaven towards them. It is then eaten ; after which they freely enjoy the fruits of the earth. On the evening of the same day, they have another public feast, besides that of the First Fruits, which looks somewhat like the Passover ; when a great quantity of venison is pro vided, with other things, dressed in the usual way, and dis tributed to all the guests ; of which they eat freely that even ing ; but that which is left, is thrown into the fire and burned, as none of it must remain till sun -rise on the next day, nor must a bone of the venison be broken. The writer of these sheets has made great use of Mr. Adair's history of the Indians, which renders it necessary that something should be further said of him. Sometime ahout the year 177*, or 1775, Mr. Adair came to Elizabeth- Town, where the writer then lived, with his manuscript, and applied to Mr. Livingston, afterwards governor of the state of New-Jersey, a correct scholar, well known for his literary abilities and knowledge of the belle-lettres, requesting him 212 A STAR IN THE WEST. to correct his manuscript for him. He brought ample recom mendations, and gave a good account of himself. Our political troubles then increasing, Mr. Adair, who was on his way to Great-Britain, was advised not to risk being detained from his voyage, till the work could be critically ex amined, but to get off as soon as possible. He accordingly took passage in the first vessel that was bound to England. As soon as the war was over, the writer sent to London and obtained a copy of the work. After reading it with care, he strictly examined a gentleman, then a member with him in Congress, of excellent character, who had acted as our In? dian agent to the southward, during the war, (without letting him know the design) and from him found all the leading facts mentioned herein, fully confirmed, by his own personal knowledge. The Feast of Weeks, or the Hunter's Feast, or Pentecost. An ancient missionary, who lived a long time with the Qutaowaies, has written, that among these savages, an old man performs the office of a priest at the feasts. That they begin by giving thanks to the great spirit for the success of the chase, or hunting time. Then another takes a cake, breaks it in two, and casts it in the fire. This was upwards of eighty years ago. Dr. Beatty says, that once in the year, some of the tribes of Indians beyond the Ohio, choose from among themselves twelve men, who go out and provide twelve deer ; and each of them cuts a small saplin, from which they strip the bark, to make a tent, by sticking one end into the ground, bending A STAR IN THE WEST. 213 the tops over one another, and covering the poles with blankets. Then the twelve men choose, each of them, a stone, which they make hot in the fire, and place them to gether, after the manner of an altar, within the tent, and then hum the fat of the insides of the deer thereon.* At the time they are making this offering, the men within cry to the Indians without, who attend as worshippers, " we pray or praise." They, without, answer, " we hear." Then those in the tent cry ho-hah, very loud and long, which appeared to he something in sound like halle-lujah. After the fat was thus offered, some tribes burned tobacco, cut fine, upon the same stones, supposed in imitation of incense. Other tribes choose only ten men, who provide but ten deer, ten saplins, or poles, and ten stones. The southern Indians observe another religious custom of the Hebrews, as Adair asserts, by offering a sacrifice of grat itude, if they have been successful, and have all returned safe home. But if they have lost any in war, they generally de cline it, because, they imagine, by some neglect of duty, they are impure ; then they only mourn their vicious conduct, which defiled the ark, and thereby occasioned the loss. Like the Israelites, they believe their sins are the procur ing cause of all their evils, and that the divinity in the ark will always bless the more religious party with the best suc cess. This is their invariable sentiment, and is the sole reason for mortifying themselves in so severe a manner while they are out at war ; living very scantily, even in a buffalo range, * Thou shalt sprinkle the blood upon the altar, and shalt burn their fat for an Bering made by fire, for a sweet savour unto the Lord. Xumb. xviii. 17. 214 A STAR IN THE WEST. under a strict rule, lest by luxury, their hearts should grow evil, and give them occasion to mourn. The Rev. Dr. Beatty, who went into the Delaware nation so long ago, informed the writer of this, that he was present when there was a great meeting of the nation, consulting on a proposition for going to War with a neighbouring nation. At this time they killed a buck and roasted it, as a kind of sacrifice, on twelve stones, on which they would not suffer any tool or instrument to be used. That they did not eat the mid dle joint of the thigh. In short, he assured the writer, that he was astonished to find so many of the Jewish customs pre vailing among them, and began to conclude that there was some affinity between them and the Jews. The Muskohgee Indians sacrifice a piece of every deer they kill at their hunting camps, or near home. If the latter, they dip their middle finger in the broth, and sprinkle it over the domestic tombs of their dead, to keep them out of the power of evil spirits, according to their mythology. This seems to proceed from a traditional knowledge, though cor rupt, of the Hebrew law of spririgling with blood. Charlevoix informs us, that to be esteemed a good hunter among the northern Indians, a man must fast three days to gether, without taking the least nourishment, having his face smeared with black all the time. When the fast is over, the candidate sacrifices to the great spirit a piece of each of the beasts he intends to hunt. This is commonly the tongue and muzzle, which at other times are the hunter's peculiar share, to feast his friends and strangers with. His family and rela tions do not touch them ; and they would as soon die with hunger as eat any of them. A STAR IN THE WEST. 215 Though the Indians in general believe the tipper heavens are inhabited by Ishto-hoolo Aba, and a great multitude of inferior good spirits, yet they are firmly persuaded that the divine omnipresent spirit of fire and light, resides also on earth, in their annual sacred fire, while it is unpolluted, and that he kindly accepts their lawful offerings, if their own con duct is agreeable to the old divine law, which was delivered to their forefathers. The former notion of the deity, is agree able to those natural images with w r hich the divine penmen, through all the prophetic writings, have drawn of Yo. lie. Wah, Elohim. When God was pleased with Aaron's priesthood and offerings, the holy fire descended and consumed the burnt offering on the altar, &c. Throughout the Old Testament, this was an emblematic token of the divine presence, and the smoke of the victims ascending towards heaven, is repre sented as a sweet savour to God and the incense from the altar is emblematic of the prayers of the saints. And God is said in scripture to be a consuming fire-r-Deut. iv. 24. He shewed himself to the prophets David, Ezekiel, and his apos tle John, in the midst of fire Psalms civ. 4, Ezekiel i. *, Daniel vii. 9 and 10, Acts ii. 3. God also appeared sur rounded by a flame of fire at the burning bush. And when descending on Mount Sinai, the mountain appeared enveloped in flaming fire Exodus iii. 2 .xix. 18. The people who have lived so long apart from the rest of mankind, are not to be wondered at, if they have forgotten the meaning and end of the sacrifices. They are rather to be pitied for seeming to believe, like the ignorant part of the Israelites of old, that the virtue is either in the form of offering the sacrifice, or iu the divinity, who they imagine resides on earth, in the sacred 216 A STAR IN THE WEST. annual fire : likewise, for having forgotten that the blessing was not in the outward sign, but in the thing signified or typified by that sign. The Feast of Harvest and Day of Expiation of Sin. We shall now proceed to their most solemn and important feast and fast, answerable to the Jewish Feast of Harvest and Day of Expiation of Sin. The Indians formerly observed this grand festival of the annual expiation of sin, and the offering of the first fruits of the harvest, at the beginning of the first new moon in which their corn became full eared, as we learn from Adair. But for many years past, they are regulated by the season of their harvest. Yet they are as skilful in observing the revolutions of the moon, as ever the Israelites were, at least till the end of the first temple. For during that period, instead of mea suring time by astronomical calculations, they knew it only by the phases of the moon. In like manner the Indians annually observed their festi vals and Nectak'Ya-ah, or days of afflicting themselves before the great spirit, at a prefixed time of a certain moon. According to Charlevoix, the harvest among the Jfatchex, on the Missisippi, is in common. The great chief fixes the day for the beginning of the festival of the harvest, which lasts three days, spent in sports and feasting. Each private person contributes something of his hunting, his fishing, and liis other provisions, as maize, beans and melons. The great chief presides at the feast all the sachems are round him, in a respectful posture. The last day, the chief makes a A STAR IN THE WEST. 217 apeech to the assembly. He exhorts every one to be exact in the performance of his duties, especially to have a great Veneration for the spirit which resides in the temple, and to be careful in instructing their children. The fathers of families never fail to bring to the temple the first produce of their harvest, and of every thing that they gather, and they do the same by all the presents that are made to their natiom They expose them at the door of the temple, the keeper of which, after presenting them to the spirit, carries them to the king, who distributes them to whom he pleases. The seeds are in like manner offered be* fore the temple, with great ceremony. But the offerings which are made of bread and flour every new moon, are for the use of the keepers of the temple. As the offerings of the fruits of the harvest precede a long strict fast of two nights and a day, they gormandize such a prodigious quantity of strong food, as to enable them to keep inviolate the succeeding fast. The feast lasts only from morn ing to sunset* As we have already seen, this Feast with the Hebrews be gan in the month Tizri, which was the first month of the civil year, answerable to our September and October. The feast took place previous to the great day of expiation, which was the tenth day of the month. So the Indian corn being gen erally full eared and fit to eat about this time, they are not far from the very time directed in the Mosaic appointment for keeping it. The feast being over, some of their people are carefully employed in putting their temple in proper order for the annual expiation, while others are painting the white cabin 218 A STAR IN THE WEST. and the supposed holiest with white clay ; for it is a sacred and peaceahlc place, and white is its emblem. Others of an inferior order are covering all the seats of the beloved square with new matrasses, made out of fine splinters of long canes, tied together with flags. Several are busy in sweeping the temple, clearing it of every supposed polluted thing, and car rying out the ashes from the hearth, which, perhaps, had not been cleaned but a few times since the last year's annual offering. Every thing being thus prepared, the chief beloved man, or high-priest, orders some of his religious attendants to dig up the old hearth or altar, and to sweep out the remains, that by chance might either be left or dropped down. He then puts a few roots of the button-snake root, with some green leaves of an uncommon small sort of tobacco, and a lit tle of the new fruits, at the hottom of the fire-place, which he orders to be covered up with white marley clay, and wetted over with clean water. Immediately the magi or priests, order a thick arbor to be made over the altar with green branches of the various young trees, which the w r arriors had designedly chosen and laid down on the outside of the supposed holy ground. The women in the interim are busy at home, clear ing out their houses, putting out all the old fire, renewing the old hearths, and cleansing all their culinary vessels, that they may be fit to receive the pretended holy fire, and the sancti fied new fruits, according to the purity of the law, lest by an improper conduct, they should incur damage in life, health, or future crops, &c. ^p It is fresh in the memory of the old traders, as we are as sured by those who have lived long with them, that formerly none of those numerous nations of Indians would eat, or even A STAR IN THE WEST. 219 handle, any part of the new harvest, till some of it had been offered up at the yearly festival hy the hcloved man or high- priest, or those of his appointment at their plantations,^ al though the light harvest of the past year should almost have forced them to give their women and children of the ripening fruits to sustain life. But they are visibly degenerating more and more, both in this and every other religious observance, except what con cerns war ; yet their magi and old warriors live contentedly on such harsh food as nature affords them in the woods, rather than transgress the divine precept given to their forefathers. Having every thing in order for the sacred solemnity, the religious waiters carry off the remains of the feast, and lay them on the outside of the square. Others, of an inferior order, carefully sweep out the smallest crumbs, for fear of polluting the first fruit offering ; and before sunset, the tem ple must be cleared, even of every kind of vessel or utensil that had contained any thing, or had been used for any kind of provision during the past year. Now one of the waiters proclaims with a loud voice, for all the warriors and beloved men, whom the purity of their law admits, to come and enter the beloved square and observe the fast. He also exhorts the women and children, with those wh& have not been initiated in war, \to keep apart, according to the law. Four centin,els are now placed, one at each corner of the holy square, to keep out every living creature as impure, * Vide Luke, vi. 1, relating to the second sabbath, but not the seventh-day sab bath, it was the day of offering up the first fruits, before which it was cot lawful to eat of the harvest. 220 A STAR IN THE WEST. except the religious order, and the warriors who are not known to have violated the law of the first fruit offering, and that of marriage, since the last year's expiation. They ob serve the fast till the rising of the second sun ; and he they ever so hungry in that sacred interval, the healthy warriors deem the duty so awful, and disobedience so inexpressibly vi cious, that no temptation would induce them to violate it. They at the same time drink plentifully of a decoction of the button-snake root, in order to vomit and cleanse their sinful bodies. When we consider their earnest invocations of the divine essence in this solemnity -that they never apply this root only oji religious occasions that they frequently drink it to such excess as to impair their health ; and take into consideration its well known property of curing the bite of the rattle snake, must not it be concluded, that this has some reference to the cure of the bite of the old serpent in Eden, or to the serpent lifted up in the wilderness. In the general fast, the children, and men of weak consti tutions, are allowed to eat, as soon as they are certain that the sun has begun to decline from his meridian altitude. This seems to be founded on the principle of mercy before sacrifice and the snake root used by those in the temple, and the bitter green tobacco, which is eaten by the women and those too wicked to be admitted to the fast held therein, seem to point to eating of the paschal lamb with bitter herbs. Being great loyers of ripe fruit, and as yet only tantalized with the sight of them, thiajlfray, with justice, be said to be a fast to afflict their souls, and to be a sufficient trial of their religious principles. At the end of this solemn fast, the wo- A STAR IN THE WEST. 221 men, by the vofee of a crier, bring to the outside of the holy square, a plentiful variety of the old year's food newly dressed, which they lay down and immediately return home. The waiters then go, and reaching their hands over the holy ground, they bring in the provisions and set them down be fore the famished multitude. They think it wholly out of order to show any joy or gladness for the end of their reli gious duties. They are as strict observers of their set forms, as the Israelites were of those they received from divine appointment. As soon as the sun is visibly declining from the meridian, the third day of the fast, the chief beloved man orders a religious attendant to cry aloud to the crowded town, that the holy fire is to be brought out for the sacred altar commanding every person to stay within his house, as becomes the beloved people, without doing the least bad thing ; and to be sure to extinguish every spark of the old fire, otherwise the divine fire will bite them severely. Now every thing is hushed. Nothing but silence all around. The great beloved man, and his beloved waiter, rising up with a reverend carriage, steady countenance, and composed be haviour, go into the beloved place, or holiest, to bring them out the beloved fire. The former takes a piece of dry poplar, willow, or white-oak, and having cut a hole, but not so deep as to reach through it ; he then sharpens another piece, and placing that in the hole, and both between his knees, he drills it briskly for several minutes, till it begins to smoke or, by rubbing two pieces together for a quarter of an hour, lie col lects, by friction, the hidden fire, which they all consider as proceeding from the holy spirit of fire. 222 A STAR IN THE WEST. They then cherish it with fine chips, till it glows into a flame, by using a fan of the unsullied wing of a swan. On this the beloved man hrings out the fire, in an old earthen vessel, and lays it on the altar, which is under the arhor, thick weaved on the top with green boughs.* They rejoice exceedingly at this appearance of the reputed holy fire, as it is supposed to atone for all their past crimes, except murder. Although the people without, may well know what is doing within, yet by order, a crier informs them of the glad tidings, and orders a beloved old woman to pull a basket full of the new ripened fruits, and bring them to the beloved square. As she is prepared for the occasion, she readily obeys, and soon lays it down at the corner thereof. Then the fire-maker rises from his white seat, and walks northward three times round the holy fire with a slow pace, and in a sedate and grave manner, stopping now and then, and saying some old ceremonial words with a low voice and a rapidity of expression, which none un derstand but a few of the beloved old men, who equally secrete their religious mysteries, that they may not be profaned. He then takes a little of each sort of the new fruits, rubs some bear's oil over them, and offers them up, together with some flesh, to the bountiful spirit of fire, as a fruit offering and an annual oblation for sin. He likewise pours a little of a strong decoction of the button-snake root, and of the cusseena, into the pretended holy fire. He then purifies the red and white seats with those bitter liquids, and sits down. All culprits may now come forth from their hiding places, dressed in their finest; clothes, to pay their thanks, at an awful distance, to the forgiv- * Even among the Romans, if the sacred fire at any time happened to be extinguished; it could only he lighted again at the rays of the sun A STAR IN THE WEST. 223 ing divine Jire. Orders are now given to call the worn on to come for the sacred fire They gladly obey. The great be- foved man, or high-priest, addresses the warriors and women ; giving all the particular positive injunctions and negative pre cepts they yet retain of the ancient law. He uses very sharp language to the women. He then addresses the whole multi tude. He enumerates the crimes they have committed, great and small, and bids them look at the holy Jire which has for given them. He presses on his audience, by the great motives of temporal good and the fear of temporal evil, the necessity of a careful observance of the ancient law, assuring them that the holy Jire will enable their prophets, the rain-makers, to pro cure them plentiful harvests, and give their war leaders vic tory over their enemies. He then orders some of the fire to be laid down outside of the holy ground, for all the houses of the various associated towns, which sometimes lay several miles apart.* If any are sick at home, or unable to come out, they are allowed one of the old consecrated conch shells full of their sanctifying bitter cusseena, carried to them by a beloved old - * Dr. Hyde says, that the third state of the Persian religion commenced, when, in imitation of the fire preserved upon the altar in the temple at Jerusalem, they kept also a perpetual fire upon an altar. This gave occasion to the common opin ion, that the ancient Persians worshipped fire ; but Dr. Hyde justifies them from that imputation. He owns that they regarded this fire as a thing sacred, and paM it a kind of service ; but he denies that they ever paid to it a proper adoration. One of their priests said, that they did not pay any divine worship to mithra, which is the sun ; or to the moon, or the stars, but only turned towards the sun when they prayed, because the nature of it nearly resembled that of fire. They regarded it as an image of God, and some said God resided in it, and others, that it will be the seat of the blessed. Oa the twenty-fourth March all the inhabitants of a parish in Persia extinguish the fire in their houses, and go to light it again by the fire of the priest, each paying him about six shillings and three pence, which serves for his support. They must have taken this custom from the Jews. ^S r 224 A STAR IN THE WEST. man. This is something like the second Passover of the Jews. At the conclusion, the beloved man orders one of his religious waiters to proclaim to all the people that the sacred annual solemnity is now ended, and every kind of evil avert ed from the beloved people, according to the old straight be loved speech. They are then commanded to paint themselves, and go along with him, according to ancient custom. They immediately fly about to grapple up a kind of chalkey clay to paint themselves white. They soon appear, all over, as white as the clay can make them. Then they follow on, in an orderly slow procession, to purify themselves in running water. The beloved man, or high-priest, heads the holy train> -his waiter next the beloved men according to their seniority and the warriors according to their reputed merit. The women follow in the same orderly manner, with all the children who can walk, ranged according to their height. The very little ones, are carried in the mothers arms. In this manner they move along, singing halleluyah to Y. 0. He-ivah, till they get to the water, when the high-priest jumps into it, and all the train follow him.* Having thus purified them selves, and washed away their sins, as they suppose and verily believe, they consider themselves as out of the reach of tem poral evil, for their past vicious conduct. They now return * The Indian women never perform their religious ablutions in presence of the men, but purify themselves, not at appointed times, with the men, but at their dis cretion. Tlvey are also e ntirely excluded from their temples by ancient custom^ except the six old beloved women, who are permitted to sing, dance, and rejoice at their annual expiation for sin ; but they must retire before the other solemnities begin. So the Hebrew women performed their ablutions, separated from the men, by themselves. They also worshipped apart from the men, lest they should attract each others attention in divine worship. A STAR IN THE WEST. 225 to the centre of the holy ground, where having made a few circles, dancing round the altar, they finish their annual great festival, and depart in joy and peace. Mr. Bartram, who visited the southern Indians in 1778, gives an account of the same feast, hut in another nation. Hs says that the Feast of First Fruits is their principal festival. This seems to end the old and heigin the new ecclesiastical year. It commences when their new crops are arrived to ma turity This is their most solemn celebration.* "When a town celebrates the husk, or first fall fruits, having previously provided themselves with new clothes, new pots<> pans, and other household utensils and furniture, they collect all their worn out clothes and other despicable things, sweep and clean their houses, squares, and the whole town, of their filth, which, with all the remaining grain and other old provi sions, they cast together in one common heap, and consume it with fire. After taking medicine, and fasting for three days, all the fire in the town is extinguished. During this fast, they abstain from the gratification of every appetite and passion whatever. A general amnesty is proclaimed. All malefac tors may return to their town, and they are absolved from their crimes, which are now forgotten, and they are restored to favour. On the fourth morning, the high-priest, or chief beloved man, by rubbing dry wood together, produces new fire in the public square, from Whence every habitation in the town is supplied with the new and pure flame. Then the wo men go forth to the harvest fields and bring from thence new * This is plainly the great feast on the day of expiation, and that of harvest, when they oiFer up their fall fruits, and not the spring first fruit feast, have been caHed the new civil year. 226 A STAR IN THE WEST. corn and fruits, which being prepared in the best manner, in various dishes, and drink withal, is brought with solemnity to the square, where the people are assembled, appareled in their new clothes and decorations. The men having regaled them selves, the remainder is carried off and distributed among the families of the town. The women and children solace them selves in their separate families, and in the evening repair to the public square, where they dance, sing and rejoice, during the whole night, observing a proper and exemplary decorum. This continues three days, and the four following days they receive visits and rejoice with their friends from neighbouring towns, who have also purified and prepared themselves. The Rev. Mr. Brainerd, in his journal says, he visited the Indians on the 20th of September, 1745, at the Juniata, near the Susquehannah, in Pennsylvania. This is the first month of their civil year, and the usual time of the feast of fruits, or harvest. It ought to be noted, that Mr. Brainerd, though an excellent man, was at this time wholly unacquainted with the Indian language, and indeed with their customs and manners. These Indians in particular, were a set of the lowest grade ; the most worthless, of the nations wholly ruined by the exam ple and temptations of the white people. Mr. Brainerd's interpreter was a common Indian, greatly attached to the habits of his countrymen, and much in their interest. He says lie found the Indians almost universally busy in making prep arations for a great sacrifice and dance. In the evening they met together, to the number of about one hundred, and danced round a large fire, having prepared ten fat deer for the sacri fice. They burned the fat of the inwards in the fire, while they were dancing, and sometimes raised the flame to a pro- A STAR IN THE WEST. 227 digious height, at the same time yelling and shouting in such a manner that they might easily be heard two miles off. They continued their sacred dance nearly all night ; after which, they eat the flesh of the sacrifice, and then retired each to his lodging. As Mr. Brainerd acknowledges, that he dared not go among them, he could give a very imperfect account of their proceedings, as he must have received it from the inter preter. The Feast of the Daily Sacrifice. The next remarkable feasts they religiously ohserve, are those of the Daily Sacrifice and some occasional ones. The Hebrews, it is well known, offered daily sacrifices of a lamb every morning and evening, and except the skin and entrails, it was burnt to ashes. The Indians have a very humble imitation of this rite. The women always throw a small piece of the fattest of the meat into the fire, before they begin to eat. At times they view it with pleasing attention, and pretend to draw omens from it. This they will do, though they are quite alone, and not seen by any one. Those who have been adopted by them, and fully considered as belonging to their nation, say, that the Indian men observe the Daily Sacrifice both at home and in the woods, with new killed venison. They also draw their new killed venison, before they dress it, several times through the smoke and flame of fire, both by way of an offering as a sacrifice, and to consume the blood, which, with them, as with the Hebrews, would be a 228 A STAR IN THE WEST. most horrid abomination to eat. They also sacrifice, while ui the woods, the melt, or a large fat piece of the first huck they kill. They imagine that their temples have such a typical holi ness, beyond any other place, that if they offered up the an nual sacrifice elsewhere, it would not atone for the people, but rather bring down the anger of Ish-to-hoolo Aba, and utterly spoil the power of their holy place and holy things. They who sacrifice in the woods, do it only on particular occasions, allowed by their laws and customs. Their Feast of Lave, $c. Every spring season, one town or more, of the Missisippi Floridians, keep a solemn Feast of Love, to renew their old friendships. They call this annual feast Hottuck Jfiiiipa, Heet- tla Tanaa, that is, " the people eat, dance and walk, as twined together." The short name of the feast is, "Hottuk Impanaa," that is, "eating by a strong religious and social principle." Impanaa signifies, as I am informed, several threads or strands twisted together. They assemble three nights before the feast. On the fourth night they eat together. During the interme diate space, the young men and women dance in circles, from the evening till the morning. When they meet at night, it is professed to be to gladden and unite their hearts before Y. O. He. wah. They sing Y. O. He. wah. shoo -Y. O. He. wjih. shoo Y. O. He. wah. shee Y. O k He. wah. shec Y. O. He. wah. shai Y. O. He. wah. shai with great energy. The first word is nearly in the Hebrew characters, the name of Joshua or Saviour. A STAR IN THE WEST. 229 CHAPTER VIII. Of Miscellaneous Facts omitted. THE writer of these sheets was himself present at a religious dance of six or seven nations, accidently meeting together, and having been hospitably entertained by the governor and inhabitants, they gave this dance to the governor and such as he should invite, by way of showing their gratitude. The writer was invited, with a very large company of gentlemen and ladies. The following is an exact account of what passed; to every circumstance of which he was critically attentive. After the company had assembled in a very large room, the oldest sachem of the Senecas, and a beloved man, entered, and took their place in the midle of the room, having something in imitation of a small drum, on which the old sachem beat time at the dance. Soon after, between twenty and thirty Indians came in, wrapped in their blankets. These made a very solemn and slow procession around the room, keeping the most profound silence, the sachem sounding his drum to direct their motion. The second round, they began to sing on a bass key y. y. y. till they completed the circle, dancing the whole time, to the sound of the drum, in a very solemn and serious manner. The third round, their ardor increased to such a degree, while they danced with a quicker step, and sang he-he-he, so as to make them very warm, and they began 230 A STAR IN THE WEST. to perspire freely, and to loosen their blankets. The fourth round they sang ho 9 ho, ho 9 with great earnestness, and by dancing with greater violence, their perspiration increased, and they cast off their blankets entirely, which caused some confusion. The next and last round put them in a mere frenzy, twisting their bodies, and wreathing like so many snakes, and making as many antic gestures as a parcel of monkies, singing the whole time, in the most energetic man ner, vvah-wah'Wah. They kept time in their dancing, as well as any person could do, who had been taught by a master. Each round took them between ten and fifteen minutes. They then withdrew in indian file, with great silence, except the two with the drum. The company had supposed that they were invited to a war-dance. The writer, desirous of ascer taining the nature of the dance, went to the interpreter, and asked him if what they had seen was intended as a war- dance ; he seemed much displeased, and in a pettish manner, answered, a war-dance, no! Indians never entertain civil people with a war-dance. It was a religious dance. In a short time, a considerable bustle being heard at the door, the company came to order, when the Indians re-entered in indian file, and danced one round then a second, singing, in a more lively manner, hal-hal-hol till they finished the round. They then gave us a third round, striking up the word, le-le-le. On the next round, it was the word lu-lu-lu, dancing naked, with all their might, having again thrown off their blankets. Dur ing the fifth round, was sung the syllable yah-yah-yah. Then all joining, as it were, in a general, but very lively and joy ous chorus, they sang hal-le-lu-yah, dwelling on each syllable with a very long breath, in the most pleasing manner. A STAR IN THE WEST. 231 There could be no deception in all this the writer was near them paid great attention and every thing was obvi ous to the senses, and discovered great fervor and zeal in the performers. Their pronunciation was very guttural and son orous, but distinct and clear. ' "The compiler of these facts, rode in the stage to Elizabeth- Town, sometime about the year 1789, with an Indian sachem from the Creek or Chikkesah nation, and his retinue, who was going, under the care of col. Butler, to New- York, to establish or renew a peace with the United States. He was a strong, tall, well proportioned man, of great gravity in his appearance, and all kis behaviour. He was well dressed* and a much better demeanor in his whole conduct, than any Indian the writer had ever seen. Neither he nor one of his attendants could speak English. From the extraordinary respect paid him by his attendants, he was certainly a sachem of high reputation. At dinner, though hard pressed by some of the gentlemen at table* he could not be persuaded to drink more than three glasses of wine, and he would not taste bran dy. "When in Philadelphia, he drank tea in company with a number of ladies, among whom was a Miss P e, who painted minature pictures very well. She being prepared for it, took his face with a strong likeness, without his perceiving it. When it was finished, she gave it to the interpreter, who put it into the hands of the chief. He appeared in perfect aston ishment; he looked wildly about him, and spoke to the inter preter in Indian, in a very emphatical manner, asking him (as he said) where that had come from, and what was the meaning of it. The interpreter introduced the young lady to him. and told him that she had done it while sitting in the 232 A STAR IN THE WEST. room. He expressed himself very much gratified with it, offered to return it to her, but she desired the interpreter to inform him that she wished to present it to him. He made great acknowledgments for the favour, saying, that lie was a poor Indian, and had nothing to give her in return ; hut that he often spoke to the great spirit, and the next time he did, he would remember her. When the stage drove up to the tavern at Frankfort, the stage-driver got out to get a dram, the horses took fright and ranaway with the stage and overset it, hy which the chief, received a large and very severe cut on his forehead ; and col. Butler, was also wounded, but all the rest got off unhurt. The chief jealous that it was done to injure him, seemed terri fied and alarmed. But when he observed that col. Butler was also hurt, and that it was an accident, he seemed imme diately to become calm and easy A surgeon soon came in, and sewed up the wound, in a manner that must have given the chief great pain ; but he would not acknowledge it, neith er did he discover the least symptom of it. As soon as he was dressed, he arose up and addressed col. Butler, which the in terpreter explained, saying, "never mind this brother it will soon be all well. This is the work of the evil spirit he knows we are going to effect a work of peace he hates peace and loves war never mind it let us go on and accomplish our business we will disappoint him." The writer of these sheets, many years ago, was one of the. corresponding members of a society in Scotland, for promoting the gospel among the Indians. To further this great work, they educated two young men of very serious and religious dispositions, and who were desirous of undertaking the mis A STAR IN THE WEST. 233 sion, for this special purpose when they were ordained and ready to depart, we wrote a letter in the Indian style, to the Delaware nation, then residing on the north-west of the Ohio, informing that we had, hy the goodness of the great spirit, been favoured with a knowledge of his will, as to the worship lie required of his creatures, and the means he would hless to promote the happiness of man, hoth in this life and that which was to come. That thus enjoying so much happiness our selves, we could not but think of our red brethren in the wil derness, and wished to communicate the glad tidings to them, that they might be partakers with us. We had therefore sent them two ministers of the gospel, who would teach them these great things, and earnestly recommended them to their care ful attention. With proper passports the missionaries set off and arrived in safety at one of their principal towns. The chiefs of the nation were called together, who answer ed them that they would take it into consideration, and in the mean time they might instruct their women, but they should not speak to the men. They spent fourteen days in council, and then dismissed them very courteously, with an answer to us. This answer made great acknowledgments for the favour we had done them. They rejoiced exceedingly at our happi ness in thus being favoured by the great spirit, and felt very grateful that we had condescended to remember our breth ren in the wilderness. But they could not help recollecting that we had a people among us, who, because they differed from us in colour, we had made slaves of, and made them suf fer great hardships and lead miserable lives. Now, they could not see any reason, if a people being black, entitled us thus to deal with them, why a red colour would not equally justify the 234 A STAR IN THE WEST. same treatment. They therefore had determined to wait, to see whether all the black people amongst us were made thus happy and joyful, before they could put confidence in our promises 5 for they thought a people who had'suffered so much and so long hy our means, should be entitled to our first at tention ; that therefore they had sent hack the two missiona ries, with many thanks, promising that when they saw the black people among us restored to freedom and happiness, they would gladly receive our missionaries. This is what in any other case, would be called close reasoning, and is too mortifying a fact to make further observations upon. The Indians to the northward, are said, by Mr. Golden, a laborious, sensible writer, in the times of their rejoicings, to repeat yo-ha-han, which, if true, evinces that their corrup tion advances in proportion as they arc distant from South- America. But Mr. Golden, was an utter stranger to their language and manners, and might have mistaken their pro nunciationor if he wrote from information of others, he has not been accurate,$*c. It was a material, or rather an essential mistake to write yo-ha-han, as it is confounding their two religions words to gether. Mr. Adair was assured by Sir William Johnson, who had the management of Indian affairs for many years under the British government, as well as by the Rev. Mr. Ogilvie, a missionary" with the Mohawks* that the northern Indians, always pronounced the words of their songs, y-ho- lie, a or ah, and so Mr. Golden altered them in the second edition of his history. He also says, when the northern In dians, at a treaty or conference would give their assent, they answered y. o. hah -The speaker called out, y. o. hah, the A STAR IN THE WEST. 235 rest answered in a sound, which could not be expressed in English letters, but seemed to consist of two words, remark ably distinguished in their cadence* The sachem of each na tion, at the close of their chief's speech, called out severally, ij. o. hah. Charlevoix, in his history of Canada, says, that Father Grillon often told him, that after having laboured some time in the missions in Canada, he returned to France and went to China. One day as he was travelling through Tartary, he met a Huron woman, whom he had formerly known in Can ada. She told him, that having been taken in war, she had been conducted from nation to nation, till she arrived at the place where she then was. There was another missionary, passing by the way of Nantz, on his return from China, who related the like story of a wo-^ man he had seen from Florida, in America. She informed him, that she had been taken by certain Indians, and given to those of a distant country ; and by these again to another na tion, till she had been thus successively passed from country to country ; had travelled regions exceedingly cold, and at last found herself in Tartary, and had there married a Tartar, who had passed with the conquerors into China, and there settled. The Cherokees had an honourable title among them, called the deer-killer of the great spirit, for his people." Every town had one solemnly appointed, who killed deer for the holy feasts. Thus Nimrod is said to have been " a mighty hunter before the Lord." Gen. x. 9. The Indian nations, in the coldest weather, and when the ground is covered with snow, practice their religious ablutions. 236 A STAR IN THE WEST. Men and children turn out of their warm houses, singing their usual sacred notes, at the dawn of day, F. 0. Hc-wah, and thus they skip along, singing till they get to the river, when they instantaneously plunge into it. The Hebrews also had various washings and anointings. They generally, after bathing, anointed themselves with oil. Their kings, prophets and priests, were anointed with oil, and the Saviour himself is described as " the Anointed." The In dian priests and prophets, or beloved men, are always initiated by unction. The Chickesaws some time ago set apart some of their old men. They first obliged them to sweat themselves for the space of three days and nights, in a small hut made for the purpose, at a distance from the town, for fear of pollu tion, and from a strong desire they all have of secreting their religions mysteries. They eat nothing but green tobacco leaves, and drink only of button-snake wood tea, to cleanse their bodies, and prepare them to serve in the beloved, holy office. After which, their priestly garments are put on, with the ornaments before described, and then bear's oil is poured upon their heads. Like the Jews, both men and women fre quently anoint themselves with bear's oil. It may not be amiss to mention, that Indians never prostrate themselves, nor bow their bodies to each other, by way of sa lute or homage, except when they are making or renewing peace with strangers, who come in the name of Yah ; then they bow their bodies in that religious solemnity. Also in their religious dances, for then they sing their hymns addressed to F. 0. He-wah. The Indians would not eat either the Mexican hog, or of the sea-cow, or the turtle, as Gumilla and Edwards inform us; A STAR IN THE WEST. 237 but they held them in the greatest abhorrence. Neither would they eat the eel ; nor of many animals and birds they deemed impure. It was foretold by Moses, that the Israelites should wcdk in the stubbornness of their own liearts, to add drunkenness to thirst." God, by his prophet, threatens them in the severest manner for this abominable crime: 66 Wo to the. proud crown of the drunkards of Ephraim, And to the fading flower of their glorious beauty ! To those that are at the head of the rich valley, that are stupified with wine ! Behold the mighty One ! the exceedingly strong One * Like a storm of hail, like a destructive tempest ; Like a rapid flood of mighty waters pouring down ; He shall dash them to the ground with his hand. They shall be trodden under foot, The proud crown of the drunkards of Ephraim. In that day shall Jehovah, God of Hosts, become a beauteous crown, And a glorious diadem to the remnant of his people: But even these have erred through wine, and through strong drink they have reeled; The priest and the prophet have erred through strong drink; They are overwhelmed witli wine, they have reeled through strong drink; They have erred in vision, they have stumbled in judgment, For all their tables are full of vomit; Of filthiness, so that no place is free." (Isaiah xxviii. 1-8. LoivtlCs translation) 238 A STAR IN THE WEST. This is one of the most terrible predictions denounced against them, and has hecn most awfully verified, should it turn out that the Indians in truth are of the lost ten tribes of Israel. Among all their vices, this seems the most predominant, and destroys every power of soul and body. It is not of this na tion or that of one tribe or another or of one rank or the other; but it is universal, among men, women and children. In sltort, it is one, among a great number, of the unnatural returns made them by the Europeans of every nation, for the Indian's kindness at first, and their giving up their lands after wards, the bringing in ardent spirits among them for lucre of gain, and by this means have reduced their numbers, and driven them into the wilderness. They have themselves long seen their misery in this respect, and have long been struggling to get rid of it ; but all in vain, till of late years, many men of virtue and of real religion, have united with them, to aid them, without which it seems impossible that they can withstand this all-conquering enemy. They will make laws against it they will determine to expel all spiritous liquors from their towns, and they will with philosophical firmness, destroy large quantities of it, brought in by the traders by stealth. But if they once taste it, all the reasoning of the most beloved man will not prevent them drinking as long as a drop lasts, and generally they transform themselves into the likeness of mad foaming bears. Mr. Golden says, "there is one vice which the Indians have fallen into since their acquaintance with the Christians, and of which they could not have been guilty before that time, that is drunkenness. It is strange how all the Indian nations, and almost every person among them, male and female, are infatu- A STAR IN THE WEST. 239 ated with the love of strong drink. They know no bounds to their desires, while they can swallow it down, and then, in deed, the greatest men amongst them scarcely deserve the name of a brute." They complained heavily to the Rev. Mr. Brainerd, that before the coming of the English they knew of no such thing as strong drink. That the English had, by these means, made them quarrel with, and kill one another, and in a word, brought them to the practice of all those vices, that then pre vailed among them. In an address, or rather an answer, made by the Delaware Indians in 1768, they say, brothers! you have spoken to us against getting drunk. What you have said is very agreeable to our minds. We see it is a thing that is very bad, and it is a great grief to us that rum or any kind of strong liquor should be brought among us, as we wish the chain of friendship, which now unites us and our brethren the English together, may remain strong. Brothers ! the fault is not all in us. It begins with our brothers, the white people. For if they will bring us rum, some of our people will buy it; it is for that purpose it is brought. But if none was brought, then we could not buy it. Brothers! we beseech you, be faithful and desire our brothers, the white people, to bring no more of it to us. Shew this belt to them for this purpose. Shew it to the great man of the fort (meaning the commandant at Fort Pitt) and to our brothers on the way as you return, and to the great men in Philadelphia, and in other places, from which rum may be brought, and intreat them not to bring any more." There is a very early record in the history of New-Jersey, to the credit of both Indians and white inhabitants of that 240 A STAR IN THE WEST. day. At a conference held with them, when eight kings or sachems were present, the Indian speaker said, " strong liquors were sold to us by the Swedes and by the Dutch. These people had no eyes. They did not see that it was hurtful to us. Nevertheless, if people will sell it to us, we are so in love with it we cannot forbear. But now, there is a people come to live among us that have eyes. They see it to be for our hurt. They are willing to deny themselves the profit for our good. This people have eyes. We are glad such people have come. We must put it down by mutual con sent. We give these four belts of wampum to be witnesses of this agreement we make with you, and would have you to tell it to your children." Several nominal prophets have lately risen among them, and have become very popular, by taking advantage of their su perstition, and declaring themselves messengers from heaven. Whatever they may be in reality, they have done some good. The Onondagoes, greatly addicted to drunkenness, have, by the influence of the brother of Corn-Planter, a Seneca chief, been prevailed on to give up the use of spiritous liquors, and to become comparatively moral. Another of these prophets among the Shawanese and north-western Indians, has been equally successful. All the promises of a God of truth, to his faithful servants, Abraham, Isaac and Jacob, must be strictly fulfilled, as well as the threatnings of his abused justice. God did make a solemn and special promise to Abraham, which was after wards repeated to Isaac and Jacob, in very strong and ex pressive terms. And God said, by myself have I sworn, saith the Lord, for because thou hast done this thing, and A STAR IN THE WEST. 241 hast not withheld thy son, thine only son, that in Messing, I will bless thee, and in multiplying, I will multiply thy seed as the stars of heaven, and as the sand upon the sea shore, and thy seed shall possess the gates of his enemies" Gen. xxii. 16, 17. Yet this was on condition of their observing the com mandments that he had given them, for in case of disobedience* the threatnings were as explicit as the blessings. "Jehovah hath sent a word against Jacob, and it hath lighted upon Israel because the people all of them, carry themselves haughtily ; Ephraim and the inhabitants of Sa maria, and Jehovah, God of Hosts, they have not sought;" Yet his mercy will not finally forsake them. For " it shall come to pass in that day, no more shall the remnant of Israel^ and the escaped of the house of Jacob, lean upon him who smote them, but shall lean upon Jehovah, the holy one of Israel, in truth. A remnant shall return, even a rcmriant of Jacob unto the mighty God, for though thy people Israel be as the sand of the sea, yet a remnant of them only shall return : the consummation decided shall overflow with strict justice"- Lowth's Isaiah, x. 23. The learned Dr. Bagot* Dean of Christ's Church* Oxford, translates the last clause of the verse thus, the accomplishment determined, overflows with justice ; for it is accomplished, and that which is deter mined* the Lord of Hosts dotli in the midst of the land" vide Lowth's notes on Isaiah, page 81. Hosea also repeats the affecting fate of Israel. " And the Lord said unto him, I will cause to cease, the kingdom of the house of Israel, for I will no more have mercy on the house of Israel; but I will utterly take them away. Yet the number of the children of Israel shall be as the sand of the sea, 242 A STAR IN THE WEST. which caimot be measured or numbered ; and it shall come to pass, that in the place where it was said unto them, ye are not my people,, there it shall be said unto them, ye are the sons of the living God. Then shall the children of Judah, and the chil dren of Israel be gathered together, and shall appoint them selves one head, and they shall come up out of the land, for great shall be the day of Jezreel." And St. John says, " and the sixth angel poured out his vial on the great river Euphrates, and the waters thereof were dried up, that the way of the Jdngs of the east might be pre pared."^ The Indian nations will answer, in a great measure, the description here given. That they have long been confined to wander in the wilderness of America, and that the con sumption decreed has been awfully executed on them, cannot be denied* That they have been despised, and considered as barbarians, and children of the devil, is too true. We have already enumerated one hundred and ninety na tions within our scanty means of knowledge, and though many of them are destroyed and done away, for the consump tion was decreed, yet if we look at the maps of travellers, and attend to the account given of the nations from Green land to Mexico, and from thence to the nation of the Dog- ribbed Indians ; thence to the Southern ocean, and along its coast northward to the Lake of the Woods, and thence to Hudson's Bay and Greenland, and estimate in addition, the nations of the interior, what nation or people in the world, can so literally answer to the strong figures, of the stars of heaven, and the sands of the sea. A STAR IN THE WEST. 243 Again, the tribes of Judah and Benjamin, attended by a few of the Israelites among them, scattered throughout Asia, Africa and Europe, have no pretensions to any king among them. But the Indians have a king to every tribe, and as we have seen, the Natchez had once five hundred kings in that one nation. Now if part of the nations to the north -west, should again return over the straits of Kamschatka, and pass on from the north-eastern extremity of Asia, by the way between the Etixine and the Caspian sea, through ancient Media, which formerly extended west to the river Halys, on the Black or Euxine sea 5 * and Asia Minor, into Palestine, then they must pass through the territory of the Grand Porte. Therefore that government must necessarily be destroyed, to make way for these kings from the east, as it is not likely that despotic power would consent to their passing through iu peace, to deprive her of the region of Palestine. Another remarkable circumstance attending the foregoing account is, that before the Babylonish captivity, the Jews had hut one temple for public worship, whither the males assem bled three times in the year. The Samaritans, after the cap tivity, observed the same at Samaria, the capital of their kingdom. The ten tribes were carried captives into the north-west parts of Assyria, before the Babylonish captivity, and therefore had no idea of hut one place of worship for a nation. * The different empires of the Lydians and the Medes, were divided by the river Haly's (which has two branches,) which rising in a mountain of Armenia, passing through Celicia, leaving in its progress the Matenians on the right, and 1'hrygia on the left ; then stretching towards the north, it separates the Cappado- cian Syrians from Paphlagouia, which is on the left of the stream. Thus the river Halys separates all the lower parts of Asia from the sea, which flows opposite to Cyprus, as far as the Euxine, a space over which an active man could not travel m less than five daysI Heredotus 112, 113. 244 A STAR IN THE WEST. The Indians have also but one temple, or beloved square for a nation, whither their males also assemble three times in the year, to wit : at the Feast of First Fruits, generally the latter end of March and April, it being the beginning of their ecclesiastical year : at the end of wliich they have another, in imitation of the Passover. The feast for success in hunting, about the time of Pentecost, called the Hunter's Feast ; and their great feast for the Expiation of Sin, which is about the time of the ripening of their indian corn and other fall fruit. These form a coincidence of circumstances in important and peculiar establishments, that could not, without a miracle, be occasioned by chance or accident. And though if considered individually, or each by itself, might be said, not to be con clusive evidence, yet taken altogether and compared with many other peculiarities of the Jewish people, they carry strong conviction to the understanding, that these wandering nations have some how or other had intimate connection with those once people of God.
A STAR IN THE WEST. 245 CHAPTER IX. The testimony of those who had an opportunity of judging, from the appearance and conduct of the Indians at the first discovery of America, as well as of some who have seen them since, in a state of nature. AND first, that of Spanish authors. And here proper allowance must be made for the prevailing intentions of the first Spanish visitors, in their coming to America, which (with some few exceptions) were principally from the most covetous desires of amassing wealth, and obtaining immense riches at all risques, and by every means. Also it must be remembered, how few concerned themselves about the religious state of the natives, if they could but get their property; neither did they give themselves any trouble to know their history, their origin, customs, or future expectations; but their gold, their silver, their lands, and their furs, were the whole objects of their attention. We thank God, there were some favourable exceptions. The learned world are by this time pretty well acquainted with the degree of confidence that ought to be put in the Spanish historians in general, further than their accounts are confirmed and supported by after labours of historians of character among other nations. Few of them conversed with the natives, in such a manner as to gain their confidence, or obtain any intimate knowledge 246 A STAR IN THE WEST. of their customs and manners, with any tolerable degree of certainty. They did not treat them as friends, but as the most inveterate enemies, and despised, hated and murdered them, without remorse or compunction, in return for their kindness and respect. And to excuse their own ignorance, and to cast a mantle over their most shocking, barbarous, cool and premeditated murders, they artfully described tfiem. as an abominable swarm of idolatrous cannibals, offering human sacrifices to their false deities, and eating the un natural victims. Notwithstanding, from even many of these partial accounts, we can trace a near agreement between the civil and martial customs, the religious worship, traditions, dress, ornaments, and other particulars of the ancient Peru vians and Mexicans, and those of the Indians of North- America. Acosta tells us, that the Mexicans had no proper name for God, yet that they allowed a supreme omnipotence and provi dence. His capacity was not sufficient to discover the former, however, the latter means that very being, and agrees with the religious opinion of their North- American brethren. Lopez de Gamara, tells us that the Americans were so de vout as to offer to the sun and earth, a small quantity of every kind of meat and drink, before any of themselves tasted of it, and that they sacrificed a part of their corn, fruits, &c. in like manner. Is not this a confused Spanish account of the imitation of the Jewish daily sacrifice, which we have before seen our more northern Indians, in the constant habit of offering to the holy spirit of fire, whom they invoke in their sacred A STAR IN THE WEST. 247 song of F. Ho. He-wali, and loudly ascribe to him, hal-k-lu- ivah, for his continued goodness to them. The Spanish writers say, that when Cortes approached Mexico, Montezuma shut himself up, and continued for the space of eight days, in prayer and fasting; but to blacken him, and excuse their own diabolical conduct, they assert, that he offered human sacrifices at the same time, to abominable and frightful idols. These prayers and fastings, were doubt less the same with those of the northern Indians, who on par ticular occasions, seek to sanctify themselves, and regain the favour of the deity. Yet these same authors tell us, that they found there, a temple called Teucatti, or the house of the great spirit, and a person belonging to it, called Cfiacalmua, that is, a minister of holy things. They likewise speak of the hearth of the great spirit the continual fire of the great spirit the holy ark, &c. Acosta says, that the Peruvians held a very extraordinary feast, called Ytu, which they prepared themselves for by fast ing two days, not accompanying with their wives, or eating salt meat or garlic, or drinking chicca during that period. That they assembled altogether in one place, and did not allow any stranger or beast to approach them. That they had clothes and ornaments which they wore only at that great festival. That they went silently and sedately in procession, with their heads veiled and drums beating ; and that this con tinued one day and night. But the next day they danced and feasted, and for twa days successively, their prayers and praises were heard. This appears no other than our northern Indians' great festival to atone for sin, according to the Mosaic system. 248 A STAR IN THE WEST. Lericus tells you, that he was present at the triennial feast of the Charibbeans, where a multitude of men, women and children, were assembled. That they soon divided themselves into three orders, a part from each other, the women and children being strictly commanded to stay within, and attend diligently to the singing. That the men sang in one house, he-he-he, while the others, in their separate houses, answered by a repetition of the like notes. Thus they continued a quar ter of an hour, dancing in three rings, with rattles. They also tell us, that the high-priest) or beloved man, was anointed with holy oil, and dressed with pontifical ornaments peculiar to himself, when he officiated in his sacred function. Ribault Landon describing the annual festival of the Flori- dians, says, that the day before it began, the women swept out a great circuit of ground, w T here it was observed with solemnity. That when the main body of the people entered the holy ground, they all placed themselves in good order, decked in their best apparel, when three beloved men, or priests, with different painting and gestures, followed them, playing on musical instruments, and singing with solemn voices, the others answering them. And when they made three circles, the men ran off to the woods, and the women staid weeping behind, cutting their arms with muscle shells, and throwing the blood towards the sun. And when the men returned, the three days were finished. This is no other than the northern Indians' Passover, or the Feast of Love, badly told, attended with their universal custom of bleeding themselves after great exercise, which the Spaniards foolishly supposed they offered up to the sun,. A STAR IN THE WEST. 249 These Spanish writers also assure us, that the Mexicans had a feast and month, which they called Hueitozolti, when the Indian corn was ripe. Every man at that time bringing an handful to he offered at the temple, with a kind of drink made out of the same grains. This is no other than the first fruit offering of the northern Indians. Don Jlntonio de Ulloa informs us, that some of the South- American natives cut the lobes of their ears, and fasten small weights to them, in order to lengthen them ; and others cut holes in their upper and under lips, in which they hang pieces of shells, rings, &c.^ This also agrees with the practice of every nation of the northern Indians. Mr. Bartram says, " their ears are lacerated, separating the border or cartilagenous limb, which is first bound round, very close and tight, with leather strings or thongs* and anoint ed with fresh bear's oil, until healed. The weight of the lead which they hang to it, extends the cartilage, which after being craped or bound round with brass or silver wire, extends it semi circularly, like a bow or crescent, and it is then very elastic. It is then decorated with a plume of white herons feathers. Acosta says, that the clothes of the South- Americans are shaped like those of the ancient Jews, being a square little cloak, over a little coat. Lact, in his description of South-America, as well as Escar- botus, assures us, that he often heard the South-Americans repeat the word hallelujah. And Mal-venda says that the na- * Mr. Bruce in his travels, speaking of a sect of Christians called Remmout, says, " their women pierce their ears, and apply weights to make them hang down and enlarge the holes, into which they put ear-rings almost as big as shackles, in th same manner as do the Bedowise, in Syria and Palestine" 4vol. p. 275. 250 A STAR IN THE WEST. tives of St. Michael had tomb-stones with several ancient Hebrew characters upon them, as " why is God gone away ?" and " he is dead, God knows." The Michuans, one of the original nations of Mexico, held, according to the Me Clavigero's declaration, this tradition, that " there was once a great deluge, and Tepzi, as they call Noah, in order to save himself from being drowned, embarked in a ship formed like an ark, with his wife, his children, and many different animals, and several seeds and fruits. As the waters abated, he sent out the bird, which bears the name of aura, which remained eating dead bodies. He then sent out other birds, which did not return, except the little bird called the Jlower sucker, which brought a small branch with it" Panoplist for June 1813, page 9. From this family of Tepzi, the Michuccans all believed they derived their origin. Both Malvenda and Acosta affirm that the natives observed a year of jubilee, according to the usage of the Israelites. Emanual de Moraez, a Portuguese historian, in his history of Brazil, says, " America has been wholly peopled by the Carthagenians and Israelites. As to the last, he says nothing but circumcision is wanting to constitute a perfect resemblance between them and the Brazilians." And we have seen, that some of the nations practice it to this day. Monsieur Poutrincourt says, that at an early day, when the Canada Indians saluted him, they said ho-ho-ho. Mr. Edwards, in his history of the West-Indies, says, that the striking conformity of the prejudices and customs of the Charibbee Indians, to the practices of the Jews, has not escaped the notice of historians, as OuineUa, Du Tertre, and others." A STAR IN THE WEST. 251 Adair, who was the most careful observer of the Indians* whole economy, both public and private, and had the best op portunity of knowing it, without much danger of deception, beyond any other writer, gives his opinion in these words, " It is a very difficult thing to divest ourselves of prejudices and favourite opinions, and I expect to be censured for op posing commonly received sentiments* But truth is my ob ject, and from the most exact observations I could make in the long time I traded among the Indian Americans, I was forced to believe them to be lineally descended from the Israelites." The Rev. Mr. Beatty says, " I have before hinted that I have taken great pains to search into the usages and customs of the Indians, in order to see what ground there was for sup posing them to be part of the ten tribes of the Jews, and I must own, to my no small surprise, that a number of their customs appear so much to resemble those of the Jews, that it is a great question with me, whether we can expect to find among the ten tribes (wherever they are) at this day, all things considered, more of the footsteps of their ancestors than among the different Indian tribes. It is not forgotten that the Indians are charged, as a barbarous, revengeful, cruel and blood thirsty race deceitful, ungrateful, and ever ready for murder and rapine. Most of this will not be disputed. They are educated from their infancy to make war in this cruel manner. They scalp their fallen enemy, and most cruelly torment and burn some of those whom they take prisoners. This they think lawful, and often plead the will of the great spirit for it. It is their habitual custom, and they make war on these principles. But they have their virtues too. They pay the greatest respect to female prisoners, and are never 252 A STAR IN THE WEST. known to offer them the least indecency. "Whenever they de termine to spare their enemies, which is often done, they not only make them free, hut they adopt them into their families, and make them a part of their nation, with all the privileges of a native Indian. This is an instance of mildness and gen erosity known to very few savages in the world, but rather resembles the Romans. They are generous, hospitable, kind and faithful to their friends or strangers, in as great a degree as they are vindic tive and barbarous to their enemies in war. Col. Smith, in his journal mentions, that he went a great distance hunting with his patron Tontileaugo, along the shore of Lake Erie. Here we staid several days on account of the high winds, which raised the lake in great billows. Tonti leaugo went out to hunt. When he was gone a Wiandot came to the camp I gave him a shoulder of venison well roasted. He received it gladly told me he was hungry, and thanked me for my kindness. When my patron came home, I told him what I had done he answered, it was very well, and supposed I had given him also sugar and bears oil to eat with his venison I told him I did not, as both were down in the canoe, and I did not go for them. He replied, you have behaved just like a Dutchman. Do you not know, that when strangers come to our camp, we ought always to give them the best that v^.have. I acknowledged my fault. He said that he would excuse this as I was but young $but I must learn to behave like a warrior, and do great things, and never be found in such little actions." Page 25, 26. Smith 9 in his history of New-Jersey, informs us, that the Indians long remembered kindnesses families or individuals A STAR IN THE WEST. 253 had shewn them. This also must undoubtedly he allowed, that the original and more incorrupt among them, very seldom for- got'to he grateful, where real benefits had been received. And notwithstanding the stains of perfidy and cruelty, which lat terly, in 1754, and since, have disgraced the Indians on the frontiers of these provinces, (but which the writer well knows had been produced by the wicked and unjust oppression of these sons of nature, by the white people) even these, by the uninterrupted intercourse of seventy years, had, on many oc casions, given irrefragable proofs of liberality of sentiment, hospitality of action and impressions, that seemed to promise a continuation of better things. Witness their first reception of the Englishtheir selling their lands to them afterwards- their former undeviating candor at treaties in Pennsylvania, and other incidents." Page 144. But however guilty these unhappy wandering nations may have been, neither Europeans or Americans ought to com plain so heavily of Indian cruelties, particularly in scalping their enemies, which is one of their most habitual cruelties, and in which they glory. They are too fully justified in this horrible practice, by the encouragement and example of those who call themselves civilized, and even Christians. Herodotus informs us that the Scythians scalped their enemies, and used them as trophies of victory. Polybius says, in the war with the Mercenaries, Gisco, the Carthagenian general, and seven hundred prisoners were scalped alive. Varrus, the Roman general, caused two thousand Jews, whom lie had taken pris oners, to be crucified at one time Josephus, 4 vol. chap. iii. page 12. 254 A STAR IN THE WEST. Under the mild government of Great-Britain, and that of France, premiums have been promised and given to the In dians, by their governors and generals, for the scalps of their enemies. Nay, even in America, acts of assembly have been passed, giving rewards to the civilized inhahitants, for scalps and prisoners, even so high as one hundred pounds for an Indian scalp 2d Golden, 120. If it should be said the gov ernment of Great-Britain ought not to be charged with this, It is answered that government not only knew of all this, but during our revolutionary war, the British secretary of state, in the House of Lords, supported its policy and necessity, as they ought to use every means that God and nature had put into their hands. JJelsharn. They had in their service at that time, at least fifteen hundred Indian warriors. Mr. Belsham says, that in the revolutionary war with America, the son of Sir William Johnson, " held a great war feast with the Indians, chiefly Iroquois, when he invited them to banquet upon a Bostonian and to drink his blood." And though I doubt not but this was mere hyperbolical language, yet did it not countenance and encourage the Indians in their customary cruelty and vindictive rage ?* * But are the United States, with all their boasted freedom and philanthropy, free from blame on this subject ? The following is an extract from a report from Brigadier General Clayborne, to the Secretary of War, since these sheets have been prepared for the press, even so late as 1st January, 1814. " Sir, on the 13th ultimo, I marched a detachment from this post, with a view of destroying the towns of the inimical Creek Indians, on the Alabama, ;ibove the mouth of the Cahaba, After having marched about eighty miles, I was within thirty miles of a town new ly erected on ground called holy, occupied by a large body of the enemy.** " About noon of the 23d, the right column, commanded by col. Joseph Carson, came in view of the town called Eckanachacu, (or holy ground) and was vigo rously attacked" "Thirty of the enemy were killed, and judging from every ap pearance, many were wounded." "In the town we found a large quantity of pro Tisions, and immense property of various kinds, which Ihe enemy, flying prtcijO A STAR IN THE WEST. 255 111 1T94, the six nations, including a late addition of those of Grand River in Canada, the Stockbridge and Brotherton Indians, consisted of about six thousand souls. They now do tately, were obliged to leave behind, and which together with two hundred houses were destroyed. They had barely time to remove their women and children across the Alabama, which runs near where the town stood. The next day was occupied in destroying a town consisting of sixty houses, eight miles higher up the river. 5 ' "The town first destroyed, was built since the commencement of hostili ties, and was established as a place of security for the inhabitants of several villa ges." Three principal prophets resided there United States Gazette, 15th Feb ruary, 1814. In Nile's Register, of September 20, 1812, we find this pleasing flight of the imagination of the friends of the war. " Imagination (says the Reg ister) looks forward to the moment, when all the southern Indians [[meaning as well in Florida as in Georgia] shall be pushed across the Missisippi." And again in the same paper "fortunately this nation [meaning the Creeks in Georgia] have supplied us M ith a pretext for dismembering their country." Now the southern Indians had not at that time taken up the hatchet against the United States. In proof of this, we have the assertion of Governor Mitchel, who in his speech to the legislature of Georgia, October 1812, (the next month after the above publicatioa iu the Register) said, "as yet those [Indians] within the United States lines, pro fess peace and friendship." Shortly after this speech the war with the southern Indians was commenced. The radical cause of it is more than broadly hinted at in the letter of the Governor of St. Augustine, to Governor Mitchel, dated Decem ber 12, 1812 He, along with other warm expostulations, regarding the conspira cy of the people of Georgia, to expel or destroy the Indians, has the following ; "The Indians are to be insulted, threatened and driven from their lands ; if they resist, nothing less than extermination is to be their fate ; but you deceive your self sir, if you think the world is blind to your motives ; it is not long since the state of Georgia had a slice of Indian lands, and the fever is again at its height." Accordingly, in 1813, Nile's Register sounded .the tocsin for their extermination. " AH these pleasing prospects, says he, are clouded by blood, and forever blasted by that treacherous people [meaning the Creeks] for whom we have done so much, so that mercy itself seems to demand their extermination. And afterwards, " the fighting continued, with some severity, about five hours, but we continued to destroy many of them, "that is after the fighting was over,*' who had concealed, themselves under the bank of the river, until we were prevented by night This, morning we killed sixteen, who had been concealed."Poulson's Daily Advertiser, June 24, 1814. Yet we are the people who remonstrate with zealous warmth and loud recrimi nation against the barbarism of the British army, in wantonly burning our towns and injuring the defenceless inhabitants, contrary to the rules of civilized warfare a strange warfare it must be Civilized warfare, what a contradiction in express terms. Alas ! what has not our nation to answer for at the bar of retribiuiv tic(?. The capitol of Washington, in flarnes> instructs on this occasion. 256 A STAR IN THE WEST. not exceed half that number. They have not reserved to them now, above two hundred thousand acres of land out of their immense territory of at least one thousand miles long, and five hundred miles broad. Clinton 48, 53. The famous eapt. Cook, in his visit to the coast of America, in the south seas, without any reference to this great question, barely gives you the facts that appeared to him during the ve ry short intercourse he had with them 2 vol. 266, 283. He says that " the inhabitants met them, singing in slow and then quicker time, accompanying their notes with beat ing time in concert, with their paddles, and regular motions of their hands, and other expressive gestures. At the end of each song, they remained silent, and then began again pro nouncing ho-ho-ah, forcibly as a chorus. The ship's crew list ened with great admiration the natives behaved well. "The people of Nootka Sound, keep the exactest concert in their songs, by great numbers together they are slow and solemn their variations are numerous and expressive, and the cadence or melody powerfully soothing >their music was sometimes varied from its predominant solemnity of air, and sung in a more gay and lively strain they have a weapon made of stone, not unlike the American tomahawk, they call it Taaweesh and Tsusknah. Page 310. Their manufactures and mechanic arts are far more exten sive and ingenious than the savages of the South Sea Islands, whether we regard the design or the execution. Their flan nel and woollen garments, made of the bark of a pine tree beaten into an hempen state, with various figures artificially inserted into them, with great taste, and of different colours of exquisite brightness. They are also famous for painting A STAR IN THE WEST. 257 and carving -ibid 30&. Their common dress is a flannel garment or mantle, ornamented on the upper edge by a narrow strip of fur, and at the latter edge by fringes or tassels. Over this, which reaches below the knees, is worn a small cloak of the same substance, likewise fringed at the lower part. Every reader must be reminded by this of the fringes and tassels of the Jews on their garments. In Prince William's Sound, the common dress is a kind of frock or robe, reaching to the knees, and sometimes to the ankles, made of the skins of animals; and in one or two in stances they had woollen garments. All are ornamented with tassels or fringes. A few had a cape or collar, and some a hood. This bears a great resemblance to the dress of the Greenlanders, as described by Crantz ibid 3678. The reader will find in Cranfa, many striking instances, in which the Greenlanders and Americans of this part of America re semble each other, besides those mentioned by capt. Coook" vol. 1, 136, 138. Father Joseph Gumella, in his account of the nations bor dering on the Oronoko, relates that the Charibbee Indians of the continent, punished their women caught in adultery, like the ancient Jews, by stoning them to death before the assem bly of the people Edward's West-Indies, 1 vol. 39, in a note. 258 A STAR IN THE WEST. This page has yet to be transcribed. A STAR IN THE WEST. 259 CHAPTER X. The Indians have a system of morality among them that is very striking. -- They have teachers to instruct them in it -- of which they have thought very highly, till of late years, they begin to doubt its efficacy, WE are indebted to Dobson's Encyclopedia for the following testimony in favour of Indian morality -- vol. 1, page 557. It is the advice given from a father to a son, it is believed, taken from a Spanish author. "My son, who art come unto the light from the womb, we know not how long heaven will grant to us the enjoyment of that precious gem, which we possess in thee. But however short the period, endeavour to live exactly -- praying to the great spirit continually to assist thee. He created thee -- thou art his property. He is thy father, and loves thee still more than I do. Repose in him thy thoughts, and day and night direct thy sighs to him. Reverence and salute thy elders, and hold no one in contempt. To the poor and distressed be not dumb, but rather use words of comfort." "Mock not, my son, the aged or the imperfect. Scorn not him who you you see fall into some folly or transgression, nor make him reproaches; and beware lest thou fall into the same error, which offends thee in another. Go not where thou art not called, nor interfer in that which does not concern thee." "No more, my son. Enough has been said in discharge of the duties of a father. With these councils I wish to fortify 260 A STAR IN THE WEST. thy mind. Refuse them not, nor act in contradiction to them ; for on them, thy life and all thy happiness depend." Mr. Beatty, when among the Indians on the Ohio, address ed them. In answer, the speaker said, that they believed that there was a great spirit above, and desired to serve him in the best manner they could. That they thought of him at their rising up, and lying down ; and hoped he would look upon them, and be kind to them, and do them good." In the even ing several came to their lodging. Among these was one called Neolin, a young man, who used for some time past to speak to his brethren, the Indians, about their wicked ways. He had taken great pains with them, and so far as Mr. Beatty could learn, he had been the means of reforming a number of them. He was informed by a captive, who had been adopted into Neolin's family, that he frequently used to boil a quantity of bitter roots, till the water became very strong that he drank plentifully of this liquor, and made his family and relatives drink of it. That it proved a severe emetic. The end of which, as Neolin said, was to cleanse them from their inward sins." The following is an account of their evening entertainment at Altasse, one of the Creek towns, in the year 1778. The writer, after describing the council house, where the Indians met, says, the assembly being now seated in order, and the house illuminated by their mystical cane fire in the middle ; two middle aged men came in together, each having a very large conck shell, full of black drink, advancing with slow, uniform and steady steps, their eyes and countenances lifted up, and singing very low, but sweetly, till they came witbin six or eight steps of the king's and white people's seats, when A STAR IN THE WEST. 261 they stopped, and each rested his shell on a little table ; but soon taking it up again, advanced, and each presented his shell, one to the king, and the other to the chief of the white people ; and as soon as he raised it to his mouth, they uttered or sang two notes, each of which continued as long as he had breath, and as long as these notes continued, so long must the person drink, or at least keep the shell to his mouth. These long notes are very solemn, and at once strike the imagination with a religious awe and homage to the Supreme Being, sound ing somewhat like a-hoo-o-jah and a-lu-yah. After this manner the whole assembly were treated, as long as the drink and light continued to hold out. As soon as the drink began, to bacco and pipes were brought in. The king or chief smoked, first in the great pipe, a few whiffs, blowing it off ceremonious ly, first towards the sun, or as it is generally supposed, to the great spirit, for it is puffed upwards ; next towards the four cardinal points ; then towards the white people in the house. Then the great pipe is taken from the hand of the king 9 and presented to the chief white man, and then to the great war chief, from whence it is circulated through the ranks of head men and warriors ; and then returned to the chief. After this, eacli one filled his pipe from his own, or his neighbour's ponch. Here all classes of citizens resort every night in the summer or moderate season. The women and children arc not allowed, or very seldom, to enter the public square." In this same year, the son of the Spanisli governor of St. Augustine, in East Florida, with two of his companions, were brought in prisoners, they being then at war with that prov ince. They were all condemned to be burned. The English traders in the town petitioned the Indians in their behalf, expressing 262 A STAR IN THE WEST. their wishes to obtain their pardon, offering a great ransom, acquainting them at the same time with their rank. Upon this, the head men, or chiefs, of the whole nation, were convened ; and after solemn and mature deliberation, returned the traders their final answer, in the following address : Brothers and friends we have been considering upon this business concerning the captives, and that under the eye and fear of the great spirit. You know that these people are our cruel enemies they save no lives of us red men, who fall in their power. You say that the youth is the son of the Spanish governor we believe it. We are sorry that he has fallen into our hands, but he is our enemy. The two young men, his friends, are equally our enemies. \Ve are sorry to see them here. But we know no difference in their flesh and blood. They are equally our enemies. If we save one, we must save all three. But we cannot do this. The red men require their blood to appease the spirits of their slain relatives. They have entrusted us with the. guardianship of our laws and rights we cannot betray them. However, we have a sa cred prescription relative to this affair, which allows us to ex tend mercy to a certain degree. A third is to be saved by lot. The great spirit allows us to put it to that decision. He is no respecter of persons." The lots were cast. The governor's son and one of his friends were taken and burnt. This must certainly appear to some as the act of barba rians, but how far is it removed from the practice of the Jews, when they so vociferously called out, crucify him, crucify him ? And Pilate said ye have a custom that I should release a prisoner to you at the feast, but they cried more bitterly, not this man, but Barabbas. A STAR IN THE WEST. 263 A minister preaching to a congregation of Christian Indians* west of the Delaware, observed a stranger Indian, listening with great attention. After the service, the minister enquir^ ed who he was ? It appeared on enquiry, that he lived three hundred miles to the westward that he had just arrived and gave this account of himself. " That his elder brother living in his house, had been many days and nights in great per plexity, wishing to learn to know the great spirit, till at length he resolved to retire into the woods, supposing that he should succeed better in a state of separation from all mankind. Having spent many weeks alone in great affliction, he thought he saw a man of majestic appearance, who informed him that there were Indians living to the south-east, who were acquaint ed with the great spirit and the way to everlasting life; add ing that he should go home and tell his people, what he had seen and heard. For this reason, as soon as he heard his brother speak, he determined to travel in search of the peo ple he had described, till he found them ; and since he had heard what had been said that day, the words had been wel come to his heart." A missionary made a journey to the Shawanese country, the most savage of the Indian nations. He stopped at the first village he came to, and lodged with one of the chief men. He informed the chief of his business, and opened some truths of the gospel to him by means of an interpreter who ac companied him. The chief paid great attention, and after sometime told him, that he was convinced that the mis sionary's doctrines were true, pointing out the right road. That the Shawanese had been long striving to find out the way of life; but that he must own, with regret, that all their 264 A STAR IN THE WEST. labour arid researches had been in vain. That they, there fore, had lost all courage, not knowing what they should do further, to obtain happiness. The chief accompanied the missionary to the next village and persuaded him to lodge with a heathen teacher. The missionary then preached to him, and told him that he had brought him the words of eternal life. This the In*- dian said was what they wanted, and they would hear him with pleasure. After some days, the heathen teacher said, I have not been able to sleep all night, for I am continually meditating upon your words, and will now open to you my whole heart. I believe what you say is the truth. A year ago I became convinced, that we are altogether sinful crea tures, and that none of our good works can save us ; but I did not know what to do to get relief. I have therefore always comforted my people, that some body would come and shew us the true way to happiness, for we are not in the right way. And even but the day before you came, I desired my people to have a little patience, and that some teacher would certain ly come. Now you are come, and I verily believe that the great spirit has sent you to make known his word to us." Monsieur De Lapoterie, a French author, speaking of the Cherokces and other southern Indians, gives this account of them : " These Indians look upon the end of life, to be living happily ; and for this purpose their whole customs are calcu lated to prevent avarice, which they think embitters life. Nothing is a more severe reflection among them than to say, tJiat a man loves his own. To prevent the use and propaga tion of such a vice, upon the death of an Indian, they burn all that belongs to the deceased* that there may be no tempta- A STAR IN THE WEST. 265 tion for the parent to hoard up a superfluity of arms or domes tic conveniences for his children. They cultivate no more land than is necessary for their plentiful subsistence and hos pitality to strangers. At the feast of expiation, they also burn all the fruits of the earth and grain left of the past year's crops* Mr. Brainerd informs us, that at about one hundred and thirty miles from our settlements, he met with an Indian, who was said to be a devout and zealous reformer. He was dres sed in a hideous and terrifick manner. He had a house con* secrated to religious purposes. Mr. Brainerd discoursed with him about Christianity, and some of the discourse he seemed to like, but some of it he wholly rejected* He said that God had taught him his religion, and that he would never turn from it ; but wanted to find some who would heartily join him in it, for the Indians had grown very degenerate and corrupt. He said he had thoughts of leaving all his friends and travel ling abroad in order to find some who would join with him, for he believed that the great spirit had good people some where, who felt as he did. He said that he had not always felt as he then did, but had formerly been like the rest of the Indians, until about four or five years before that time. Then he said, that his heart was very much distressed, so that he could not live among the Indians, but got away into the woods and liv ed for some months. At length he said the great spirit had comforted his heart and shewed him what he should do ; and since that time he had known the great spirit and tried to serve him, and loved all men, be they who they may, so as he never did before. He treated Mr. Brainerd with uncommon courtesy, and seemed to be hearty in it. 266 A STAR IN THE WEST. The other Indians said, that he had opposed their drink ing strong liquor with all his power ; and if at any time he could not dissuade them from it, he would leave them and go crying into the woods. It was manifest that he had a set of religious notions of his own, that he had looked into for him self, and had not taken for granted upon bare tradition ; and he relished or disrelished* whatever was spoken of a religious nature, according as it agreed or disagreed with his standard. He would sometimes say, now, that I like, so the great spirit has taught me, &c. Some of his sentiments seemed very just; yet he utterly denied the existence of an evil spirit, and declared there was no such a being known among the Indians of old times, whose religion he supposed he was attempting to revive. He also said that departed souls went southward, and that the difference between the good and bad was, that the former were admitted into a beautiful town with spiritual walls, or walls agreeably to the nature of souls. The latter would forever hover round those walls, and in vain attempt to get in. He seemed to be sincere, honest and conscientious in his own way, and according to his own religious notions, which was more than could be said of most other pagans Mr. Brain- erd had seen. He was considered and derided by the other Indians as a precise zealot, who made an unnecessary noise about religious matters, but in Mr. Brainerd's opinion, there was something in his temper and disposition that looked more like true religion, than any thing he had observed among oth er heathen Indians. Smith, in his history of New-Jersey, gives the following extract from a letter on this subject, from an Indian interpreter, the well known Conrad Wiser 145. A STAR IN THE WEST. 267 I write this to give an account of what I have observed amongst the Indians, in relation to their belief and confidence in a divine being, according to the observations I have made from the year 1714, in the time of my youth to this day. If by the word religion, is meant an assent to certain creeds, or the observation of a set of religious duties, as appointed pray ers, singing, preaching, baptism, &c. or even heathenish wor ship, then it may be said, the Five Nations have no religion ; but if by religion we mean, an attraction of the soul to God, whence proceeds a confidence in and an hunger after the knowledge of him, then this people must be allowed to have some religion among them, notwithstanding their some times savage deportment ; for we find among them some traits of a confidence in God alone, and even some times, though but seldom, a vocal calling upon him. In the year 1737, I was sent for the first time to Ononda- go, at the desire of the governor of Virginia. I sat out the lat ter end of February, for a journey of five hundred English miles, through a wilderness where there was neither road nor path ; there were with me a Dutchman and three Indians." He then gives a most fearful account of the distresses to which they were drivenparticularly on the side of a mountain where the snow was so hard, that they were obliged to make holes in it with their hatchets to put their feet in, to keep them from sliding down the mountain. At length one of the In dians slipped and went down the mountain, but on his way was stopped by the string of his pack hitching fast to a stump of a small tree. They were obliged then to go down into the yalley, when they looked up and saw that if the Indian had slipped four or five paces further he would have fallen over a 268 A STAR IN THE WEST. rock, one hundred feet perpendicular, upon craggy pieces of rock below. The Indian was astonished and turned quite pale then with out-stretched arms, and great earnestness, spoke these words, / thank tlie great Lord and Governor of this 'world that lie has had mercy upon me, and has been witting that I should live longer ; which words I at that time sat down in my journal. This happened on the 25th March, 1737." On the 9th April following, he was reduced so low that he gave up all hopes of ever getting to his journey's end. He stepped aside and sat down under a tree, expecting there to die. His companions soon missed him they came back and found him sitting there. " I told them that I would go no further, but would die there." They remained silent awhile, at last the old Indian said, my dear companion, thou hast hith erto encouraged us, wilt thou now quite ,give up ? Remember tJiat evil days are better tJiangood days, for when we siiffermuch, we do not sin ; and sin will be drove out of us by suffering ; but good days cause men to sin, and God cannot extend his mercy to tJiem, but contrarywise, when it goeth evil with us, God hath compas sion on us. These words made me ashamed ; I rose up and travelled on as well as I could." Two years ago I was sent by the governor to Shamoken, on account of the unhappy death of John Armstrong," after he had performed his er rand, which was to make peace by the punishment of the mur derer. The Indians made a great feast for him ; and after they had done, the chief addressed his people, and exhorted them to thankfulness to God then began to sing with an aw ful solemnity, but without expressing words, the others ac companied him with their voices. After they had done, the same Indian, with great earnestness said, thanks ! thanks ! be A STAR IN THE WEST. 269 to thee, thou great Lord of the world, in that thouJiast again caus ed the sun to shine and hast dispersed the dark cloud. The Indians are thine." The old king Ockanickon, who died in 1681, in Burlington, New- Jersey, just before his death, sent for his brother's son, whom he had appointed to be king after him ; he addressed him thus, " My brother's son, this day I deliver my heart into your bosom mind me. I would have you love what is good, and keep good company ; refuse what is evil, and by all means avoid bad company." "Brother's son! I would have you cleanse your ears, that you may hear both good arid evil ; and then join with the good and refuse the evil ; and where you see evil, do not join with it, but join to that which is good." " Brother's son! I advise you to be plain and fair, with all, both Indians and Christians, as I have been. I am very weak, otherwise I would have spoken more." After lie stopped, Mr. Budd, one of the proprietors of West- Jersey, said to him, " there is a great God, who created all things ; that he had given man an understanding of what was good and bad ; and after this life rewarded the good with blessings, and the bad according to their doings." The king answered, " it is very true. It is so. There are two ways, a broad and a straight way ; tJiere are two patJis, a broad and a straight path ; the worst and the greatest number go in the broad, the best and fewest, in the straight path." Smith's history New-Jersey, 149. The Indians originally shewed great integrity in their dealings, especially with one another. Col. Smith informs us that going a hunting to a very great distance, and having got many skins and furs by the way, 270 A STAR IN THE WEST. very inconvenient to carry, they stretched them on scaffolds and left them till their return. When they returned some considerable time after, they found their skins and furs all safe. Though this was a pub lic place and Indians often passing and our skins hanging up to view, yet there were none stolen, and it is seldom that In dians do steal any thing from one another ; and they say they never did, until the white people came among them, and learned some of them to lie, cheat and steal." Page 42. He further informs us that being in the woods in the month of February, there fell a snow and then came a severe frost that when they walked caused them to make a noise by break ing through the crust, and so frightened the deer that they could get nothing to eat. He hunted two days without food, and then returned fatigued, faint and weary. He related his want of success. Tontileaugo asked him if he was not hun gry he said he was he ordered his little son to bring him something to eat. He brought him a kettle with some bones and broth, made from those of a fox and wild cat that the ra vens and turkey buzzards had picked, and which lay about the camp. He speedily finished his repast and was greatly refreshed. Tontileaugo gave him a pipe and tobacco and when he had done smoking, he said that he had something of importance to tell him Smith said he was ready to hear, He said he had deferred his speech, because few men were in a right humor to hear good talk when they are extremely hungry, as they are then generally fretful and discomposed ; but as you appear now to enjoy calmness and serenity of mind, I will now communicate the thoughts of my heart, and those things which I know to be true. Brother! As you have A STAR IN THE WEST. 271 lived with the white people, you have not had the same ad vantage of knowing that the great being above, feeds his peo ple and gives them their meat in due season, as we Indians have, who are frequently out of provisions, and yet are won derfully supplied, and that so frequently, that it is evidently the hand of the ^reat Owaneeyo, (this in their language sig nifies the owner and ruler of all things) that doeth this. Whereas the white people have large stocks of tame cattle that they can kill when they please, and also their barns and crihs filled with grain, and therefore have not the same oppor tunity of seeing and knowing that they are supported by the ruler of heaven and earth. Brother ! I know that you are now afraid that we will all perish with hunger ; but you have no just reason to fear this. Brother ! I have been young but am now old ! I have frequently been under the like circum stances that we now are, and that, sometime or other, in al most every year of my life ; yet I have hitherto been support ed and my wants supplied in times of need. Brother ! Owan eeyo ! sometimes suffers us to be in want, in order to teach us our dependance upon him, and to let us know that we are to love and serve him$ and likewise to know the worth of the favours that we receive and to make us more thankful." Was not this one of the great ends designed by a gracious God, in leading the Israelites through the wilderness for for ty years vide Lowth's Isaiah, xli. 17, &e. vide 2 Du Pratz, 172, for account of great spirit. " Brother ! be assured that you will be supplied with food and that just in the right time ; but you must continue diligent in the use of means go to sleep and rise early in the morning and go a hunting be strong and exert yourself like a man, and the great spirit will direct 272 A STAR IN THE WEST. your way." The next morning, Smith rose early and set off. He travelled near twelve miles and was just despairing* when he came across a herd of buffaloes and killed a large cow* He loaded himself with the beef, and returned to his camp and found his patron, late in the evening in good spirits and humor. The old Indian thanked him for his exertion and commanded his son to cook it which he did, but eating some himself almost raw. They put some on to boil^ and when Smith was hurrying to take it off his patron calmly said, let it be done enough, as if he had not wanted a meal. He pre vented his son from eating but a little at a time, saying it would hurt him, but that he might sup a few spoonsful of the broth. When they were all refreshed, Tontileaugo delivered a speech upon the necessity and pleasure of receiving the ne cessary supports of life with thankfulness, knowing that Owa- neeyo is the great giver. Sometime after they set off for home, Tontileaugo on the w r ay, made himself a sweat-house and went into it, and put himself in a most violent perspira tion for about fifteen minutes, singing aloud. This he did in order to purify himself before he would address the Supreme Being. He then began to burn tobacco and to pray He be gan each petition with Oh! Oh! Oh! Oh! He began his address in the following manner. O great being ! I thank thee that I have obtained the use of my legs again (he had been ill with the rheumatism) that I am now able to walk about and kill turkeys, &c. without feeling exquisite pain and misery. I know that thou art a hearer and a helper, and therefore I will call upon thee. Oh, Oh, Oh, Oh! grant that my knees and ankles may be right well, and that I may be able not only to walk, but to run and A STAR IN THE WEST. 273 to jump logs, as I did last fall. Oh! Oh! Oh! Oh! grant that on this voyage we may frequently kill hears, as they maybe crossing the Seiota and Sandusky. Oh! Oh! Oh! Oh ! grant that rain may come to raise the Ollentangy about two or three feet, that we may cross in safety down to Sciota> without danger of our canoe being wrecked on the rocks. And now, O great being ! tliou knowest how matters stand thou knowest that I am a great lover of tobacco, though I know not when I may get any more, I now make a present of the last I have unto thee, as a free burnt offering ; therefore I expect thou wilt hear and grant these requests, and I thy servant will return thee thanks and love thee for thy gifts." During this time Smith was greatly affected with his pray ers, until he came to the burning of the tobacco, arid as he knew that his patron was a great lover of it, when he saw him cast the last of it into the fire, it excited in him a kind of meriinent, and he insensibly smiled. Tiie Indian observed him laughing, which displeased him and occasioned the follow ing address "Brother! I have somewhat to say to you and I hope you will not be offended, when I tell you of your faults. You know that when you were reading your books in town, I would not let the boys or any one disturb you ; but now When I was praying, I saw you laughing. I do not think that you look upon praying as a foolish thing. I believe you pray yourself. But perhaps you may think my mode or manner of praying, foolish. If so you ought in a friendly manner to instruct me, and not make sport of sacred things." Smith acknowledged his error. On this the Indian handed him his pipe to smoke in token of friendship* though he had nothing to smoke but red willow bark. Smith then told him 274 A STAR IN THE WEST. something of the method of reconciliation with an offended God, as revealed in his bible, that he had with him. The In dian said, " that lie liked that story better than that of the French priest's ; but that lie thought he was now too old to begin to learn a new religion ; lie should therefore continue to worship God in the way that he had been taught, and that if future happiness was to be had in his way of worship, he ex pected he would obtain it ; and if it was inconsistent with the honor of the great spirit to accept of him in his own way of worship, he hoped that Owaneeyo would accept of him in the way Smith had mentioned, or in some other way, though he might now be ignorant of the channel through which favour OF mercy might be conveyed. Page 54, 55. He added, that he believed that Owaneeyo would hear and help every one who sincerely waited upon him. Here we see, notwithstanding the just views this Indian entertained of Providence, yet though he acknowledged his guilt, he expected to appease the deity and procure his favour by burning a little tobacco. Thus the Indian agreed with revelation in this, that sacrifice is necessary, or that some kind of atonement is to be made in order to remove guilt and reconcile the sinner to God. This, accompanied with numberless other witnesses, is sufficient evidence of the truth of the scriptures. At another time Tontileaugo informed him that there were a great many of the Caughnawagas and Wiandots, a kind of h.ilf Roman Catholics ; but as for himself, he said, that the priest and he could not agree; as the priest held notions that contradicted both sense and reason and had the assur ance to tell him, tluit the book of God taught them those foolish A STAR IN THE WEST. 275 absurdities ; but he could not believe the great and good spirit ever taught them any such nonsense. And therefore he concluded that the Indian's old religion was better than this new way of worshipping God. 276 A STAR IN THE WEST. [This page is blank] A STAR IN THE WEST. 277 CHAPTER XI. Separation of the Indian Woman, THE last remarkable fact to be mentioned is, the constant practice of the Indian nations, in the separation of their women, on certain occasions. The southern Indians oblige their women, in their lunar retreats, to build small huts, at a considerable distance from their dwelling houses, as they imagine to be sufficient, where they arc obliged to stay, at the risque of their lives. Should they be known to violate this ancient law, they must answer for every misfortune that the people should meet with. Among the Indians on the north-west of the Ohio, the conduct of the women seems perfectly agreeable (as far as circumstances will permit) to the law of Moses. A young woman, at the first change in her circumstances, immediately separates herself from the rest, in a hut made at some distance from the dwelling-houses, and remains there, during the whole time of her malady, or seven days. The person who brings her victuals, is very careful not to touch her, and so cautious is she herself of touching her own food with her hands, that she makes use of a sharpened stick, instead of a fork, with which to take up her venison, and a small ladle or spoon for her other food. When the seven days are ended, she bathes herself in water, washes all her clothes and cleanses the vessels she has made use of. Such as are 278 A STAR IN THE WEST. made of wood, she scalds and cleans with lye made of wood ashes, and such as are made of earth or iron, she purifies by putting into the fire. She then returns to her father's house and is after this looked upon fit for marriage; but not before. A Muskoghe woman, delivered of a child, is separated in like manner for three moons, or eighty-four days. Crossweeksung (the once Indian town in New-Jersey,) signifies, the house of separation. By the Levitical Law, a woman was to be separated and unclean forty days for a man child, and eighty days for a female child; from which law alone it appears that the Indians could have adopted this extraordinary custom, as they must have done all their numerous laws of purity and more especially as some of the nations observe the like distinction between male and female children. The young women, at our people's first coming among them were very modest and shame-faced -- both young and old women would be highly offended at indecent expressions, unless corrupted by drink: They were very neat and cleanly except in some instances when they neglected themselves. Smith 138. A STAR IN THE WEST. 279 CHAPTER XII. The Conclusion HAVING thus gone through with a collection of facts, that has taken much time, great attention and strict enquiry, in order to prevent the writer from being deceived himself; or his being the innocent cause of deceiving others; he is now brought to draw some conclusions from the whole taken together. On a subject like this, where there is so much to hope, and so much to fear, he would use great modesty and diffidence. He would avoid all dogmatical assertions, or unreasonable confidence in any thing that he has collected, or any observations he has made, as he considers this a subject for the exercise of wisdom, research, enquiry and mature reflection. But nevertheless, while he uses every necessary precaution, and wishes perfect freedom of inquiry on the best evidence, yet he earnestly solicits the reader to keep in mind that his principal design, in these his labours, has been to invite and tempt the learned and the industrious, as far as they can obtain opportunities, to enquire further into this important and useful subject. What could possibly bring greater declarative glory to God, or tend more essentially to affect and rouse the nations of the earth, with a deeper sense of the certainty of the prophetic declarations of the holy scriptures, and thus call their attention to the truth of divine revelation, 280 A STAR IN THE WEST. than a full discovery, that these wandering nations of Indians are the long lost tribes of Israel; but kept under the special protection of Almighty God, though despised by all mankind, for more than two thousand years, separated from and unknown to the civilized world? Thus wonderfully brought to the knowledge of their fellow men, they may be miraculously prepared for instruction, and stand ready, at the appointed time, when God shall raise the signal to the nations of Europe, to be restored to the land and country of their fathers, and to Mount Zion the city of David, their great king and head, and this in direct, positive and literal fulfillment of the numerous promises of the God of Abraham, Isaac and Jacob, their pious progenitors and founders, near four thousand years ago. Would not such an event be the most ample mean of publishing the all important facts of both the Old and New Testament to all the nations of the earth, and thereby lead all men to the acknowledgment, that the God of Israel, is a God of truth and righteousness, and that whom he loves, he loves unto the end? They would be convinced that his all seeing eye had been open upon them in all their wanderings; under all their suffering, and that he had never forsaken them, but had shown his watchful providence over them, and that in the latter day, "it shall come to pass, that the mountain of the Lord's house shall be established in the top of the mountains, and shall be exalted above the hills; and all nations shall flow unto it. And many people shall go and say, come ye, let us go up to the mountain of the Lord; to the house of the God of Jacob; and he will teach us of his ways, and we will walk in his paths: for out of Zion shall go forth the law, and the word of the Lord from Jerusalem." -- Isaiah ii. chap. 1, 3. A STAR IN THE WEST. 281 St. Paul certainly entertained some such views of this extraordinary event, when he so pathetically sets forth this glorious issue of the providence of God. --Speaking of Israel, "I say then, have they stumbled, that they should fall? God forbid, but rather, through their fall, salvation is come unto the gentiles to provoke them to jealousy. Now if the fall of them be the riches of the world, and the diminishing of them, the riches of the gentiles, how much more their fulness. For if the casting away of them, be the reconciling of the world, what shall the receiving of them be, but life from the dead.'' * The writer will not determine with any degree of positiveness on the fact, that these aborigines of our country are, past all doubt, the descendants of Jacob, as he wishes to leave every man to draw the conclusion from the facts themselves. But he thinks he may without impeachment of his integrity or prudence, or any charge of over credulity, say, that were a people to be found, with demonstrative evidence that their descent was from Jacob, it could hardly be expected, at this time, that their languages, manners, customs and habits, with their religious rites, should discover greater similarity to those of the ancient Jews and of their divine law, without supernatural revelation, or some miraculous interposition, than the present nations of American Indians have done, and still do, to every industrious and intelligent enquirer. This not the first time, that the idea has been advanced, of the possibility of these tribes emigrating to America, over the straits of Kamschatka, and preserving the indelible marks of the children of Abraham, as has been already shewn in the __________ * Rom. chap. xi. 11, 15. 282 A STAR IN THE WEST. foregoing pages. In addition to which, many of the first European visitants, in a very early day, drew this conclusion from personal observation, of the then appearance of things and persons. Mons. De Guignes, who wrote so long ago, in one of his memoirs, speaking of the discoveries made of America, before the time of Columbus, says, "these researches, which of themselves give us great insight into the origin of the Americans, leads to the determination of the route of the colonies sent to the continent. He thinks the greater part of them passed thither by the most eastern extremities of Asia, where the two continents are only separated by a narrow strait, easy to cross. He reports instances of women, who from Canada and Florida; have travelled to Tartary without seeing the ocean." In this case they must have passed the straits on the ice. Let the foregoing facts, collected in these pages, however imperfectly and immethodically put together by one whose means of knowledge have been very scanty, be impartially examined without prejudice, and weighed in the scale of testimony, compared with the language, customs, manners, habits, religious prejudices and special traditions of the Hebrews, especially under the impression of their being related and confirmed by so many authors, separated by birth, national manners, distance of time, strong prejudices, religious jealousies, various means of knowledge and different modes of communication, the facts, from Christopher Columbus, of glorious memory, and first discoverer of America, down to Mr. Adair, who lived with them in social intercourse and great intimacy for more than forty years, and Mr. M'Kenzie, traveller of a late day, but the first who crossed from the Atlantic to the A STAR IN THE WEST. 283 southern ocean -- Portuguese, Spaniards, English, French, Jew and Christian, men of learning -- plain, illiterate travellers and sea-faring men, all -- all combining without acquaintance or knowledge of each other, to establish the material facts, such as they are. Is it possible that the languages of so many hundred nations of apparent savages, scattered over a territory of some thousands of miles in extent, living excluded from all civilized society, without grammar, letters, arts or sciences, for two thousand years, should, by mere accident, be so remarkable for peculiarities, known in no other language, but the Hebrew -- using the same words to signify the same things -- having towns and places of the same name? A gentleman of the first character of the city of New-York, well acquainted with the Indians in that state from his childhood, assured the writer of this, that when with them at a place called Cohock or Owl-flat, now degenerated to Cook-house,, yet well known, they shewed him a mountain to the west, very high, and that appeared from Cohock much as the Neversinks do from the sea, at first approaching the American coast, and told him the Indians called it Ararat. Is there no weight of evidence, in finding peculiar customs among the Indians, of the same import as those enjoined on the ancient people of God, and held sacred by both? Or in each people having three sacred feasts, religiously attended every year, with peculiar and similar rites and dress, to which the males only should be admitted, and these held at certain periods and at one special place of worship in a nation, and conforming, with astonishing precision, to each other; while the women were wholly excluded by both people, and particularly that connected with one of them, each people should 284 A STAR IN THE WEST. have another of a very singular and extraordinary nature in the evening, being in part a sacrifice, in which not a bone of the animal, provided for the occasion, should be broken, nor a certain part of the thigh eaten -- that if a family were not sufficient to eat the whole, a neighbor might be called in to partake with them; and if any should be still left it must religiously be burned in the fire before the rising of the next sun. That their houses and temple, at one of these feasts, we to be swept with the greatest care, and searched in every part, with religious scrupulosity, that no unhallowed thing should remain unconsumed by fire. And that the altars for the sacrifices were to be built of unhewen stone, or on stones on which a tool had not been suffered to come. That the entrails and fat of the sacrifice, were to be burned on the altar, and the body of the animal only to be eaten? When all these are compared with the Hebrew divine law, given by God himself from heaven, we find every article rigidly commanded and enforced by sovereign authority. Then examine their other religious feasts of different kinds, and reflect on their conformity, in a surprising manner, in times, causes and effects, to the Hebrew rites and ceremonies, and what rational man, of sound judgment, but must, at least acknowledge, that there is great encouragement to the inquisitive mind, to proceed farther, and make these people the subject of attentive and unwearied inquiry. Add to all this, their general appearance -- their customs and manners in private life -- their communion with each other -- their ceremonies and practices in society -- their common religious and moral observations -- their belief in a future state -- their religious observation of the most sacred respect to an ark in A STAR IN THE WEST. 285 going to war, and even their cruelties and barbarous customs in the treatment of their enemies, and ought they not to be included in the enumeration. The strong bearings that many of the foregoing traditions have on their origin and descent -- their manner of coming into this country and their future expectations, being so very similar to the experience of the Jews in their exodus from Egypt, should not be left out of the scale of testimony. Can it be probable -- nay, if we judge from past experience, may we not ask with propriety, can it be possible, unless a miracle is acknowledged, that so many Indian words should be purely Hebrew, and the construction of what little we know of their languages, founded on the same principles, if there never had been any intercommunion between the two people? There can be but little doubt, were their language well known to the learned in Europe and America, but that many more important discoveries might be made, convincing to every judicious mind, that now lie in utter oblivion. Let it now be asked -- What, then, is the use that should be made of the facts that are thus brought to light, partuial as they are? It is answered. Ought not the nations of Europe and America to make a solemn pause, and consider the Jews, "now scattered and peeled, and expecting their Messiah," to use the phraseology of the bible, in a very different point of light, from that in which it has been customary to consider them? This has been dark indeed. They have been treated by the civilized nations as the offscouring of the earth -- despised, contemned and persecuted -- abused, reviled, and charged with the most abominable crimes, without evidence, unheard, and contrary to all 286 A STAR IN THE WEST. probability. Nay, they have been treated like the wild beasts of the forest -- have been proscribed, banished, murdered, or driven from one nation to another, but found safety in none. It is asserted by the best writers, that after the destruction of Jerusalem, in the time of Domition, multitudes of Jews who had survived the sad catastrophe of the destruction of their city and temple, sought an asylum in various parts of the world. Many retired into Egypt, where a Jewish colony had resided from the time of Alexander -- others fled to Cyrene -- a large number removed to Bablyon, and joined their brethren, who had remained in that country ever since the captivity -- some took refuge in Persia, and other eastern countries. They became divided into eastern and western Jews. The western included Egypt, Judea, Italy, and other parts of the Roman empire. The eastern were settled in Babylon, Chaldea, Assyria and Persia. This was about the second century; but previous to the destruction of the temple, those Jews who resided in the eastern countries, sent presents to Jerusalem; repaired thither from time to time to pay their devotions, and acknowledge the supreme authority of the high-priest. But after the ruin of their country, having no longer any bond of unity, which had before been formed by the high-priests and the temple, they elevated chiefs to preside over them, whom they styled princes of the captivity. -- Mod. Univ. Hist. vol. 13, page 156. In the year 130, Adrian, the Roman emperor, having provoked the Jews almost to madness and desperation, they took arms, headed by one Coziba, who took the name of BarChochebas, which signifies the son of the star, pretending to be the one prophesied of in the declaration of Balaam, "there shall A STAR IN THE WEST. 287 come a star out of Jacob," &c. After various and great successes, he was defeated and killed, and the town of Bither, where he had taken refuge, obliged to surrender. There were slain in battle five hundred and eighty thousand, besides a vast number who perished by sickness, fire, famine, and other calamities. Vast numbers were exposed to sale at the fair of Terebinth, at the price of horses, and dispersed over the face of the earth. In the year 1039, the sultan Gala Doullat, resolved to extirpate the Jews. For this purpose he shut up their academies, banished their professors, and slew the prince of the captivity, with his family. The persecution dispersed many into the desarts of Arabia, whilst others sought an asylum in the west. Benjamin of Tudela, found a prince of the captivity in Persia, in the twelfth century. In the time of the Crusaders, fifteen hundred were burnt at Strasburgh, and thirteen hundred at Mayence. According to the Jewish historians, five thousand, (but according to the christian writers, the number was three times greater) were either slaughtered or drowned. It is also said, that upwards of twelve thousand were slain in Batavia. In the year 1238, during the reign of St. Louis, of France, two thousand five hundred were put to death by the most cruel tortures. In 1240, the celebrated council of Lyons passed a decree, enjoining all christian princes who had Jews in their dominions, under penalty of excommunication, to compel them to refund to the crusaders all the money they had obtained by usury. This oppressed people were also prohibited from demanding debts due to them from the crusaders till their return. 288 A STAR IN THE WEST. In the time of Ferdinand, of Spain, and Pope Sixtus, the fourth, two thousand were put to death by the Inquisition. In 1192, Ferdinand and Isabella banished eight hundred thousand Jews from Spain. In 1349, a set of enthusiastic Catholics, called Flagellanti, incensed the populace against the Jews at Metz, and slew twelve thousand of them -- set fire to their houses, which were destroyed, with part of the town. -- Basnage, 686. But as it may tend to greater certainty, and really so fully confirms what is suggested in holy writ, the following quotation from a Jewish author, complaining of their hard treatment, though long, will be excused. It is taken from a work entitled "An Appeal to the justice of kings and nations," cited in the transactions of the Parisian Sanhedrim, page 64, and mentioned by Mr. Faber in his work on the prophecies. -- Vol. iii, 55, 58. "Soon after the establishment of christianity, the Jewish nation, dispersed since the second destruction of its temple, had totally disappeared. By the light of the flames, which devoured the monuments of its ancient splendour, the conquerors beheld a million of victims dead, or expiring on their ruins. "The hatred of the enemies of that unfortunate nation raged longer than the fire which had consumed its temple: active and relentless, it still pursues and oppresses them in every part of the globe, over which they are scattered. Their persecutors delight in their torments too much to seal their doom by a general decree of proscription, which at once would put an end to their burthensome and painful existence. It seems as if they were allowed to survive the destruction of A STAR IN THE WEST. 289 their country, only to see the most odious and calumnious imputations laid to their charge, to stand as the constant object of the grossest and most shocking injustice, as a mark for the insulting finger of scorn, as a sport to the most inveterate hatred; it seems as if their doom was incessantly to suit all the dark and bloody purposes which can be suggested by human malignity, supported by ignorance and fanaticism. -- Weighed down by taxes, and forced to contribute, more than christians, for the support of society, they had hardly any of the rights that it gives. If a destructive scourge happened to spread havoc among the inhabitants of a country, the Jews had poisoned the springs; or these men cursed by heaven, had, nevertheless, incensed it by their prayers against the nation, which they were supposed to hate. Did sovereigns want pecuniary assistance to carry on their wars? The Jews were compelled to give all those riches; in which they sought some consolation against the oppressing sense of their abject condition: as a reward for their sacrifices, they were expelled from the state, which they had supported; and were afterwards recalled to be stript again. Compelled to wear exteriorily the badges of their abject state, they were every where exposed to the insults of the vilest populace. "When, from his solitary retreat, an enthusiastic hermit preached the crusades to the nations of Europe, and a part of its inhabitants left their country to moisten with their blood the plains of Palestine, the knell of promiscuous massacre tolled before the alarm-bell of war. Millions of Jews were then murdered to glut the pious rage of the crusaders. It was by tearing the entrails of their brethren that these warriors sought to deserve the protection of heaven. Skulls of men 290 A STAR IN THE WEST. and bleeding hearts were offered as holo causts on the altars of that God, who has no pleasure even in the blood of the innocent lamb; and ministers of peace were thrown into an holy enthusiasm by those bloody sacrifices. It is thus that Basil, Treves, Coblentz and Cologn, became human shambles. It is thus that upwards of four hundred thousand victims, of all ages, and of both sexes, lost their lives at Alexandria and Cesaria. And is it, after having experienced such treatment, that they are reproached with their vices? Is it, after being for eighteen centuries the sport of contempt, that they are reproached with being no longer alive to it? Is it, after having so often glutted with their blood the thirst of their persecutors, that they are held out as enemies to other nations? Is it, that when they have been bereft of all means to mollify the hearts of their tyrants, that indignation is roused, if now and then they cast a mournful look towards the ruins of their temple, towards their country, where formerly happiness crowned their peaceful days, free from the cares of ambition and riches?" "By what crimes, have we, then, deserved this furious intolerance? What is our guilt? Is it that generous constancy which we have manifested in defending the laws of our fathers? But this constancy ought to have entitled us to the admiration of all nations, and it has only sharpened against us the daggers of persecution. Braving all kinds of torments, the pangs of death, the still more terrible pangs of life, we alone have withstood the impetuous torrent of time, sweeping indiscriminately in its course, nations, religions and countries. What is become of those celebrated empires, whose very name still excites our admiration by the ideas of splendid greatness A STAR IN THE WEST. 291 attached to them, and whose power embraced the whole surface of the known globe? They are only remembered as monuments of the vanity of human greatness. Rome and Greece are no more; their traditions, mixed with other nations, have lost even the traces of their origin; while a population of a few millions of men, so often subjugated, stands the test of thirty revolving centuries, and the fiery ordeal of fifteen centuries of persecution! We still preserve laws, which were given to us in the first days of the world, in the infancy of nature! The last followers of a religion which had embraced the universe, have disappeared these fifteen centuries, and our temples are still standing! We alone have been spared by the indiscriminating hand of time, like a colemn left standing amidst the wreck of worlds and the ruin of nature." While this picture gives another awful trait of the human character * and proves the degenerate state of man in his best natural state, and interests every feeling heart in the sufferings of this remarkable people. It also holds up, in a striking view, the threatenings of God's word and the literal fulfilment of them. -- It further shews, in the most unanswerable manner, the Jews themselves being both witnesses and judges, the truth of divine scriptures, and their strange blindness, until the end shall come, and the veil shall be taken from their eyes. Christians are assured by unerring truth, that it has been the obstinacy and idolatry of the tribes of Judah and Israel, that have caused the anger of the Almighty to be enkindled __________ * Had the Indians a faithful historian to write in their behalf, when their cruelties in battle were recorded in their worst colours, might they not refer to the facts set forth in the few foregoing pages, and point to them as a contrast to their conduct, and say, behold these were your civilized nations. 292 A STAR IN THE WEST. led against them, added to the awful invocation of Judah, that the blood of the Messiah, might rest on them and their children. Yet in the end, God will call their oppressors to a severe account for the unchristian manner in which they have carried the divine judgments into execution. Little of it has been done for the glory of God. Moses did solemnly forewarn the Jews, that all this would be the consequence of disobedience to the laws and statutes of Jehovah, and that at the very time that he encouraged them with a certainty of his special favours, in case of their obedience. The inspired language is exceedingly strong. "And it shall come to pass, if thou shalt hearken diligently unto the voice of the Lord thy God, to observe and do all his commandments which I command thee this day, that the Lord thy God will set thee on high above all nations of the earth, and all those blessings (before enumerated) shall come upon thee." "But it shall (also) come to pass, if thou wilt not hearken unto the voice of the Lord thy God to observe and do all his commandments and his statutes, which I command thee this day, that all those curses shall overtake thee. Cursed shalt thou be in the city, and cursed shalt thou be in the field." -- Deut. xxviii. 1, 2, 15, 16. The Lord shall bring thee and thy king into a nation, which neither thou nor thy fathers have known, and there shalt thou serve other Gods, wood and stone. And thou shalt become an astonishment, a proverb and a bye-word among all nations, whither the Lord shall lead thee." --Ibid 36, 37. "And they shall be upon thee for a sign and a wonder and upon thy seed forever," (or for ages.) -- Ibid 46. And thou shalt serve thine enemies, which the Lord shall send against thee, in hunger and thirst, and in nakedness, and in want of all things, A STAR IN THE WEST. 293 And he shall put a yoke of iron upon thy neck until he hath destroyed thee." -- Ibid 48. "f thou wilt not observe to do all the words of this law, that are written in this book that thou mayest fear, this glorious and fearful name, the Lord thy God." -- Ibid 58. "And the Lord shall scatter thee among all people, from one end of the earth to the other." -- Ibid 61. And among these nations thou shalt find no ease, neither shalt the sole of thy foot have rest, but the Lord shalt give thee a trembling of heart and failing of eyes and sorrow of mind." -- Ibid 65. "And thy life shall hang in doubt before thee, and thou shalt fear day and night, and shalt have none assurance of thy life." "And it shalt come to pass, when all these things are come upon thee, the blessing and the curse, which I have set before thee, and thou shalt call them to mind, among all the nations whither the Lord thy God hath driven thee, and shalt return unto the Lord thy God, and shalt obey his voice according to all that I command thee this day, thou and thy children, with all thy heart and with all thy soul, that then the Lord thy God will turn thy captivity and have compassion upon thee and will return and gather thee from all the nations, whither the Lord thy God hath scattered thee. If any of thine be driven out unto the uttermost parts of heaven, from thence will the Lord thy God gather thee, and from thence will he fetch thee. And the Lord thy God will bring thee unto the land which thy fathers possessed, and thou shalt possess it, and he will do thee good, and multiply thee above thy fathers. And the Lord thy God will circumcise thine heart and the hearts of thy seed, to love the Lord thy God, with all thine heart, and with all thy soul, that thou mayest live. And the Lord thy God will put all these curses upon 294 A STAR IN THE WEST. thine enemies, and on them who hate thee, who persecuted thee. And thou shalt return, and obey the voice of the Lord thy God and do all his commandments, which I command thee this day." -- Ibid xxx. 1, 8. Thus the Lord in the midst of the severest judgments remembered mercy for the descendants of Abraham, Isaac and Jacob: and these great encouragements to obedience, he frequently repeated by his prophets from time to time, as in Isaiah -- "For Jehovah will have compassion on Jacob and will yet choose Israel. And he will give them rest upon his own land -- and the stranger shall be joined to them and cleave unto the house of Jacob. And the nations shall take them and bring them in their own place; and the house of Jacob shall possess them into the land of Jehovah, as servants and as handmaids; and they shall take them captive, whose captives they were, and they shall rule over their oppressors." -- Lowth xiv, 1, 2. "Ho! land spreading wide the shadow of thy wings, * which art beyond the rivers of Cush, accustomed to send messengers by sea, even in bulrush vessels, upon the surface of the waters -- Go! swift messengers unto a nation dragged away and plucked; unto a people wonderful from the beginning hitherto." -- Chap. xviii. 1, 2. "At that season a present shall be led to the Lord of Hosts, a people dragged away and __________ * The translation of these verses, is taken from Mr. Faber, who quotes Bishop Horsley, in saying, "the shadow of wings is a very unusual image in prophetic language, for the protection afforded by the stronger, to the weak. God's protection of his servants is described by their being safe under the shadow of his wings. And in this passage, the broad shadowing wings may be intended to characterize some great people, who shall be famous for the protection they shall give to those whom they received into their alliance." "It is not impossible however, and certainly not incongruous with the figurative language of prophesy, that since the messengers described in this prediction, are plainly a maritime nation, the shadowy wings here spoken of may mean the sails of their ships." A STAR IN THE WEST. 295 plucked, even of a people wonderful from the beginning hitherto; a nation expecting, expecting, and trampled under foot, whose land rivers have spoiled, unto the place of the name of the Lord of Hosts, Mount Zion." -- Ibid 7. "For behold Jehovah shall come as a fire; and his chariot as a whirlwind; to breathe forth his anger in a burning heat, and his rebuke in flames of fire. For by fire shall Jehovah execute judgment, and by his sword upon all flesh; and many shall be slain of Jehovah." -- Ibid lxvi. 15, 16. Again in Jeremiah the subject is taken up. "For lo! the days come, saith the Lord, that I will bring again the captivity of my people Israel and Judah, and I will cause them to return to the land that I gave to their fathers and they shall possess it." -- Jerem. xxx. 3. "Therefore fear thou not O my servant Jacob, saith the Lord, neither be dismayed O Israel, for lo! I will save thee from afar, and thy seed from the land of their captivity; and Jacob shall return and shall be in rest and be quiet and none shall make him afraid. For I am with thee saith the Lord, to save thee; though I make a full end of all the nations whither I have scattered thee; yet will I not make a full end of thee; but I will correct thee in measure, and will not leave thee altogether unpunished." "Therefore all they who devour thee shall be devoured, and all thine adversaries, every one of them, shall go into captivity; and they who spoil thee, shall be a spoil; and all who prey upon thee, will I give for a prey." -- Verse 16. Remember this, and shew yourselves men: Reflect on it deeply, O ye apostates! -- I am God nor is there any thing like me. From the beginning, making known the end; And from early times, the things that are not yet done: 296 A STAR IN THE WEST. Saying my counsel shall stand, And whatever I have willed, I will effect. Calling from the east, the eagle, And from a land far distant, the man of my counsel: As I have spoken, so will I bring it to pass; I have formed the design, and I will execute it. (Loweth's Isaiah xlvi. 8, 11. "And this shall be the covenant that I will make with the house of Israel, after those days saith the Lord, I will put my law in their inward parts, and write it in their hearts, and will be their God and they shall be my people." -- Vide also xxxi. 1, 14. Joel also is very express on this subject. "For behold, says he, in those days, and in that time, when I shall bring again the captivity of Judah and Jerusalem, I will also gather all nations, and will bring them down into the valley of Jehoshaphat, and will plead with them there, for my people and for my heritage Israel, whom they have scattered among the nations, and parted my land. -- Chap. iii. i, 2. From all this it appears, with the greatest certainty, that in the latter day, the house of Israel shall be discovered, and brought from the land of their captivity afar off; to the city of God, the new Jerusalem, that shall be restored to more than its former glory. And that all those who have oppressed and despised them, wherever they are, will become subjects of the anger and fury of Jehovah their God. If then it is plain, that the Israelites have heretofore suffered the just indignation of the Almighty, for their sins and all his threatnings and fury have literally and most exactly been poured out upon them, according to the predictions of his servant Moses, what have not their enemies and oppressors to A STAR IN THE WEST. 297 fear, in the great day of God's anger, when he cometh to avenge his people, who have been dear to him as the apple of his eye? Is not the honor of God as much concerned in executing his threatnings on one as the other? Will it not be wise then to consider our ways betimes, and sincerely to repent of all improper conduct of oppression and destruction to any, who may turn out to have been the continual objects of God's regard, though suffering under his just displeasure. If his word has been yea and amen, in punishing the people of his choice, because of their disobedience, what hope can those gentiles have, who are found to continue in opposition to his positive commandments. Let all, then, carefully attend to the word of the Lord, as spoken by his prophets, and watch the signs of the times, seeking. to know the will of God, and what he expects from those who are awakened to see their error. Much is to be done when the signal is set up for the nations; and these children of God's watchful providence, shall be manifestly discovered. They are to be converted to the faith of Christ, and instructed in their glorious prerogatives, and prepared and assisted to return to their own land and their ancient city, even the city of Zion which shall become a praise in all the earth. Let not our unbelief, or other irreligious conduct, with a want of a lively, active faith in our Almighty Redeemer, become a stumbling block to these outcasts of Israel; wherever they may be. They will naturally look to the practice and example of those calling themselves christians for encouragement. Who knows but God has raised up these United States in these latter days, for the very purpose of accomplishing his will in bringing his beloved people to their own land. 298 A STAR IN THE WEST. We are a maritime people -- a nation of seafaring men. Our trade and commerce have greatly encreased for years past, except during our late troubles. We may, under God, be called to act a great part in this wonderful and interesting drama. And if not alone, we may certainly assist in a union with other maritime powers of Europe. The people of Great Britain are almost miraculously active in disseminating the gospel throughout the known world. The same spirit will carry them to accomplish the whole will of God. The time is hastening on, and if we have any understanding in the prophetic declarations of the Bible, it cannot be far off. "And I said, how long, O Jehovah! and he said, until cities be laid waste, so that there be no inhabitant and houses, so that there be no man; and the land be left utterly desolate, until Jehovah remove man far away, and there be many a deserted woman in the midst of the land. And though there be a tenth part remaining in it, even this shall undergo a repeated destruction. Yet as the ilex and the oak, though cut down, hath its stock remaining, a holy seed shall be the stock of the nation." Have not these wonderful things come to pass, and therefore have we not reason to believe the time of the end is near at hand. When Tiglah Pilnezer carried away the tribes from Samaria, he left about a tenth part of the common people behind. Salmanazer, his successor, some few years after, less than twenty, came and carried the rest into captivity, except a few stragglers about the country, and those who had taken refuge in Jerusalem. Even this small remnant were afterwards taken by Esarrhaddon and Nebuchadnezzar; and carried to Babylon, and the whole land left desolate, in strict A STAR IN THE WEST. 299 fulfillment of the divine word. And even yet a holy seed shall still appear to become the stock of the nation. What, then, is the use that christians ought to make of a discovery of this nature, should they be convinced of the truth of the proposition? First, To adore with humble reverence, the inscrutable riches of the grace of God, and his infinite wisdom in his conduct towards his servants, Abraham, Isaac and Jacob, and their posterity. Secondly, To rejoice in the absolute certainty of the fulfillment of the promises as well as the threatnings of his holy word -- "For though heaven and earth may pass away, yet not a tittle of his word shall pass away, but all shall be fulfilled." Thirdly, To enjoy the present benefit of the glorious hope set before them, even in the view of immediate death, knowing that when Christ shall come the second time, "in his own glory, and the glory of the Father, his saints shall come with him." -- Coloss. iii. 4. "For if we believe that Jesus died and rose again, even so, them also who sleep in Jesus, will God bring with him; for the Lord himself shall descend from heaven with a shout, with the voice of an arch-angel, and with the trump of God; and then shall christians be forever with the Lord." -- 1 Thess. iv. 14-17. Fourthly, This makes the grave the christian's privilege and consolation. As the scriptures positively declare, that flesh and blood cannot inherit the kingdom of heaven; this would have greatly weakened their faith and hope, had they not been assured, that they would leave their flesh and blood in the grave, and rise immortal and incorruptible through the power of the Redeemer, who had previously sanctified the grave by his own presence. 300 A STAR IN THE WEST. But after all, suppose we should be wholly mistaken in all our conjectures, and should treat these aborigines of this land with great kindness and compassion, under the mistaken opinion of their descent? Would any people have reason to repent acts of humanity and mercy to these wretched outcasts of society. Have not Europeans been the original cause of their sufferings? Are we not in possession of their lands. Have we not been enriched by their labours? Have they not fought our battles, and spilt their blood for us, as well as against us? If we speak as an European nation, has not a large proportion no their numbers perished in our wars, and by our means? Ought not we, then, now, at this day of light and knowledge, to think much of hearkening to the voice of mercy and the bowels of compassion in their behalf? But if it should turn out, that our conjectures are well founded, what aggravated destruction may we not avoid, by an obedient and holy temper, and exerting ourselves to keep the commands of the statutes of the God of Israel? "Behold, at that time, I will undo all who afflict thee: and I will save her who halteth, and gather her who is driven out. And I will get them fame and praise in every land, where they have been put to shame. At that time, I will bring you again, even in the time that I gather you, for I will make you a name and a praise among all people of the earth, when I turn back your captivity before your eyes, saith the Lord." -- Zeph. iii. 19-20. We are very apt, and indeed it is a common practice, to blame the Jews, and charge them with great perverseness, and call them an obstinate and stiff-necked race, when we read of the grace and mercy of Jehovah towards them, in the multiplied blessings promised on their obedience, and the awful A STAR IN THE WEST. 301 curses and severe threatnings in case of disobedience. We profess to be astonished at the hardness of their hearts and abominable wickedness of their conduct, committed in direct opposition to so much light and knowledge. Yet would not any impartial person, under a just view of our conduct to them since the discovery of this country, and the practices of a large majority of those who call themselves christians, draw a pretty certain conclusion that we had not much to insist on, in our favour -- That most certainly we have not done to them, as we should have expected from them, under a change of circumstances. We go on, under similar threatenings of the same Almighty Being. We shew much the same hardness of heart, under the like denunciations of vengeance, that he will afflict and destroy, without mercy, those nations who join in oppressing his people, without regard to his honour and glory. He will be found no respecter of persons; but will fulfill, not only his promised blessings, but will with equal certainty inflict all his threatened curses on obstinate offenders. "Who is wise, and he shall understand these things? Prudent, and he shall know them? For all the ways of the Lord are right, and the just shall walk in them; but the transgressors shall fall therein." -- Hosea xiv. 9. "And the Lord answered me and said, write the vision and make it plain upon a table, that he may run who readeth it -- For the vision is yet for an appointed time, but at the end it shall speak and not lie; though it tarry, wait for it, because it will surely come -- It will not tarry." -- Habakkuk ii. 2-3. 302 A STAR IN THE WEST. [This page is blank.] [ 303 ] APPENDIX. Historical Sketches of Louisiana. THE famous Ferdinand de Soto was sent by the Spaniards to succeed Narvaez, as governor of Florida. "He attacked the natives every where, and every where committed great slaughter; destroyed their towns, and subsisted his men on the provisions found in them. He crossed the Mississippi, explored the regions to the west of it, and in 1542 ended his days on Red River." -- Page 8. In 1562, the French growing jealous of the success of the Spaniards, admiral Coligni fitted out a fleet, with a colony of French protestants, under Rebaud. They landed in Florida, and planted the settlers about thirty miles from St. Augustine, where they erected a fort for their protection, and called it Fort Charles, in honour of Charles the 4th. Astonishment seized the Spaniards at this unexpected intrusion. However, the Spanish governor Menandez, after recovering from the first shock, assembled his forces, attacked Fort Charles, and carried it by storm. Those miserable French who escaped the sword, were doomed to the halter, with this label on their breasts: "Not as Frenchmen, but as heretics." -- Page 5. Of all the Indians known to the French, the Natchez were the most serviceable, and at the same time the most terrible. 304 A STAR IN THE WEST. Settlers at various times planted themselves among them, so as to become a large body. They were favourably received by the Natchez, who supplied them with provisions, assisted them in their tillage, and in building their houses, and indeed saved them from famine and death. They soon began to en croach on the rights of the Indians, and excited their jealousy. The Natchez possessed the strongest disposition to oblige, and would have continued eminently useful to the French settlers, if the commandant had not treated them with indignity and injustice. The first dispute was in 1723, when an old warrior owed a soldier a debt in corn. When payment was demanded, the warrior alledged that the corn was not ripe, but it should be delivered as soon as possible. They quarreled, when the sol dier cried murder. When the warrior left him to go to his village, a soldier of the guard fired at him and shot him. The commandant would not punish the offender. Revenge, the prominent passion of the Indians, drove them to arms. They attacked the French in all quarters but by the influence of a noted chief, peace was restored, which prevented the utter extermination of the settlers. Peace was made and duly rat ified by Mons. Branville ; yet he took advantage of it to inflict a sudden and dreadful blow on these innocent people. He privately brought seven hundred men he attacked the de fenceless Indians slaughtered them in their huts, and de manded the head of their chief; with which they were obliged to comply. This wanton slaughter lasted four days. A peace was then made, but confidence was destroyed. Shortly after, a French officer accidentally met a sachem, called the Sting- serpent, who appeared to avoid him. The officer said, why A STAR IN THE WEST. 305 do you avoid me, we were once friends ; are we so no longer ? The indignant chief replied why did the French come into our country ? We did not go to seek them. They asked us for land, and we told them to take it where they pleased ; there was enough for them and for us. The same sun ought to en lighten us both, and we ought to walk together as friends in the same path. We promised to give them food assist them to build and to lahour in the fields* We have done so. In 1729, the commandant of the fort had treated them so ill, that they obtained his being summoned to New-Orleans to answer* for his conduct. This gave much joy to the Indians. The officer found means to be sent back reinstated in his command. He now determined to indulge his malice against the Indians. He suddenly resolved to build a town oh the sc'ite of a village belonging to one of the sachems, which covered a square of three miles extent. He sent for the sun or chief, and directed him to clear the huts and remove to some other place. The chief replied, that their ancestors had lived there for many ages, and that it was good for their descendants to occupy the same ground. This dignified language served only to exas perate the haughty commandant. He declared, that unlcsd the village was abandoned in a few days, the inhabitants of it should repent their obstinacy ! The Indians finding a bloody conflict was inevitable, they laid their plans accordingly. They tried by the best excuses in their power to delay the execution of his plan but he treated all their proposals with disdain, and menaced immediate destruction if he was not gratified. The Indians ever fruitful in expedients* got per mission to wait till their harvest was got in. During this interval, short as it was, they formed their plan. They held 306 A STAR IN THE WEST. a council, and unanimously resolved to make one great effort to defend the tombs of their fathers. They proceeded with caution, yet one of their women betrayed them. The comman dant would not hearken to it, but punished the informant. Near the close of the last day of Nov. 1729, the Grand Sun, with some warriors, repaired to the fort with their tribute of corn and fowls agreed upon. They secured the gate and other passages, and instantly deprived the soldiers of the means of defence. So well was their plan laid, that all opposition was in vain. The massacre throughout the settlement, among the men, was general. The slaves, and some of the women, were spared. The chiefs and warriors, disdaining to stain their hands with the blood of the commander, he fell by the hands of one of the meanest of the Indians. In short, the whole set tlement, consisting of about seven hundred men, were wholly destroyed. They proceeded to two neighbouring settlements, at Yazous and Wastulu, which shared the same fate ; a very few escaped to carry the news to the capital. Pages 46 52. The governor of New-Orleans, persisting in destroying this nation? they fled over the Missisippi, and settled one hundred and eighty miles up the Red River, where they built a fort for their protection. After some time, the governor pursued them to this place with cannon, &c. besieged the fort, and they were obliged to surrender at discretion. The women and children were reduced to slavery, and scattered among the plantations. The men were sent to St. Domingo as slaves. Their villages at first consisted of twelve hundred souls. Of all the Indians, they were the most polished and civilized. . They had an established religion among them, in many par- , ticulars rational and consistent- as likewise regular orders of A STAR IN THE WEST. 307 priesthood. They had a temple dedicated to the great spirit, in which they preserved the eternal fire. No doubt these to kens of their religion were ever ohscured and perverted by tradition but this is rather the misfortune than the crime of the Indians. This remark is applicable to all the aborigines of America. Their civil polity partook of the refinement of a people apparently in some degree learned and scientific. They had kings or chiefs a kind of subordinate nobility and the usual distinctions created by rank were well under stood and preserved among them. They were just, generous and humane, and never failed to extend relief to the objects of distress and misery. They were well acquainted with the properties of medicinal plants, and the cures they performed, particularly among the French, were almost incredible. They were remarkable for not deeming it glorious to destroy the human species, and for this reason, seldom waged any other than defensive war. Pages 53 i. In short, the history of the European wars against the In dians, and particularly the Spanish, for more than, two centu ries, afford nothing but a series of complicated crimes, the black catalogue of which will continue to excite in every breast, the mingled. emotions of pity and indignation. They made war on defenceless nations without provocation spilt oceans of blood and involved millions of their fellow creatures in misery They trampled on all those laws deemed sacred by the civilized world, and their misdeeds find no other ex cuse than what is derived from the gratification of their ava rice Page 58. They not only enslaved the prisoners taken in battle, hut likewise those peaceable and effeminate people who submitted 308 A STAR IN THE WEST. themselves at discretion. They compelled them to labour in the mines of Hispaniola and Cuba, where vast numbers perish ed. The natives of Hispaniola, at Columbus' first arrival, amounted to more than a million of inhabitants fifteen years after they amounted to less than sixty thousand. In Cuba, upwards of five hundred thousand perisheda similar destruc tion took place on the 'continent. Page 56. The aborigines in general are extremely scrupulous in re gard to the fulfilment of national compacts ; though in their individual capacities they are less honest and more inclined to evade their engagements. Their want of faith in most instan ces, where it has been manifested may be traced either to the hard conditions imposed on them, or to the advantage taken of their ignorance. Whoever will attentively examine into the merits of the numerous quarrels between them and the whites, will be apt to find that the latter were almost uniform ly the aggressors. Page 64. A remarkable fact with respect to Florida. While it was in the hands of the English, a plan was concerted by Sir Wil liam Duncan and Dr. Turnbull, to entice a colony of Greeks to settle in this country. It was represented to them in the most favourable light. They were promised fertile fields and lands in abundance, and also transportation and subsistence. Fifteen hundred engaged in this undertaking but what was their surprise when they were ushered in to New-Smyrna, about seventy miles to the eastward of St. Augustine, which they found to be a desolate wilderness, without the means of support Instead of being proprietors of land, there was none for them, but iipon lease for ten years, and some could not obtain it on any terms. Hence they became labourers to the A STAR IN THE WEST. 309 planters as slaves, and suffered bunger and nakedness. Over seers were placed over them, who goaded them with the lash They were kept together and numbers were crouded to- gether in one mess The poor wretches were not allowed to procure fish for themselves, although plenty in the sea at their feet. People were forbidden to furnish them with vic tuals. Severe punishments were decreed against those who gave and those who received the charitable boon. Under this treatment many died, especially the old people. At length in 1769, seized with despair, they rose on their cruel tyrants and made themselves some small vessels But they were seized by the military, and five of the principal suffered death. This could scarcely be believed, considering the re puted humanity of the English, had it not been verified by / the solemn report of a British officer who was an eye witness, Page 121. Eraser's Key to the Prophecies. Speaking of the image of the beast, that it should speak, &c. &c. says, the Pope put to death in a variety of forms, such as dared to oppose him. He excluded from the privileges of civil society all such as did not submit to his claims and au thority. See the decree of Alexander 3d\$ in the Synod of Tours the bull of Martin against the errors of Wickliffe and Huss, annexed to the council of Constamce. There it is decreed that men of this sort be not permitted to have houses 310 A STAR IN THE WEST. & nbsp; to rear families, to make contracts, to carry on traffick or busi ness of any kind, or to enjoy the comforts of humanity, in common with the faithful." These are almost the words which prophecy has put into the mouth of the image. Sec the bull of Paul 3d, against Henry 8th, and that of Paul 5th, in the eleventh year of Queen Elizabeth. An energetical letter, dated London 19th January, 1791, signed by three vicars apostolic of England, expressly prohibits the Catholics of that kingdom to take an oath prescribed by government, though that oath contains nothing inconsistent with Catholic principles, but a renunciation of the Pope's su premacy in temporals. They express themselves, "The apostolical vicars, in the above mentioned energetical letter, (dated October 21, 1789) declared, that none of the faithful clergy or laity, ought to take any new oath or sign any new declaration or doctrinal matters, or subscribe any new instru ment wherein the interests of religion are concerned, without the previous approbation of their respective bishops, and they required submission to those determinations. The altered oath has not been approved by us, and therefore cannot be lawfully or conscientiously taken by any^ of the faithful of our districts." Here the lamb like beast speaks like a dragon Ten very respectable Catholics in England, met together as a committee, and protested against this letter, as inculcating principles hostile to the government, and contrary to the faith and moral character of the Catholics. Our adversaries account the visibility of their church as a community from the apostolic days, a demonstration of its being the true church, while they ask us with an air of triumph, where was your church before Luther? (In the widerness A STAR IN THE WEST. 311 where it yet is.) The prophecy furnishes a direct answer. The true church of christ ought to he invisible as a community for a period of twelve hundred and sixty years, and during all that time a harlot, pretending to he the spouse of Christ, and ought to propagate her idolatries successfully and extensively, throughout the world. The divisions among protestants have been urged hy their adversaries as an argument against them ; and the ineffectual efforts of learned and pious men to unite them into one com munity, have proved stumbling blocks to the faith of some of their friends. But by the prophetic representation, matters ought to be as they are. Had protestants united together into one society, the church of christ would have been visible as a community, which during the currency of twelve hundred and sixty years would flatly contradict the prophecy ; but the sev eral protcstant churches, having no connection with each other in government and ordinances like the ancient church, they constitute only individual members of the universal ehurch, which as a body politic is invisible now, as it was in the tenth century. While this view should reconcile us to a certain degree of separation among protestants during the currency of the twelve hundred and sixty years, it ought to remove wholly the violence of party spirit and every degree of bitterness and rancour which they have too frequently shewed to each other. Aviolent party spirit is founded on this principle, that those who possess it are the true church of christ. Hence they argue that those who separate from them are schismatics or heritics, and therefore ought to he treated as heathens and publicans. But the ground of their reasoning is false ; according to the prophecy no particular 312 A STAR IN THE WEST. church or party, now on earth, may claim the exclusive privileges of the universal church. Whoever does, acts the part of a daughter, usurping the place of the mother, and requiring that subjection of her sisters which the law of God does not require. -- Pages 134 - 5 - 162. FINIS.
Transcriber's Comments A Star in the West The Author, Elias Boudinot Elias Boudinot's A Star in the West is essentially an appendix to his 1815 book, The Second Advent. In the 1815 volume Mr. Boudinot argued that recent events in Europe indicated that the end of the world was at hand. He based this conclusion upon a literal interpretation of the Bible (especially of its books of Daniel and the Revelation) supplemented by his own explication of messianic prophecies from the books of Isaiah, Ezekiel, Zechariah, etc. In his 1815 book Boudinot is careful to reproduce and make use of numerous passages in reference to the "restoration" of scattered Israel and dispersed Judah. To author Boudinot it was important that those obscure passage should be brought to the forefront of his literalistic explanations and that his readers should understand clearly what he saw to be important in those ancient promises and predictions. Since Boudinot took seriously the idea that the nation of Israel had to be restored to its former privileged status, under God's protection, before Christ's second coming could occur, he felt it important to identify where the members of that long dispossessed could be found. It was easy to identify "dispersed Judah" with the Jews of the world, but "scattered Israel" was a people less apparent to the contemporary observer. In Second Advent Boudinot makes it clear that he suspects that the "lost tribes" of Israel have largely migrated to the Americas and are to be found there under the guise of American Indians. In A Star in the West he defends that conclusion by carefully building up what he feels to be evidence that the American Indians are the "lost tribes" of Israel. Before the second advent of Christ can commence, they, like their supposed brethren the Jews, must be restored to their ancient position among the nations, as a branch of God's chosen people. In fact, Boudinot seems to have felt that the popular identification of the Indians as the lost Israelites would bring with it a widespread realization that the Bible and its prophecies were true. When both the American Indians and the Jews of the world realized the great truth of their destiny, and converted to Christianity, the impact of that wondrous event upon the rest of the "heathen" world would be a profound one. Obviously, having seen ancient prophecy come true before their very eyes, so to speak, the heathen multitudes would rush to become Christians and the current luke-warm followers of Christ would be incited to spectacular new levels of zeal and devotion. All of this would set the stage for the "second advent" very nicely. Boudinot's selection of biblical proof texts, supporting his view of the promised restoration of ancient Israel, are much the same as those quoted by the Canadian "prophet" Richard Brothers, before him and by writers like the Rev. Ethan Smith in the years that followed. In fact, Ethan Smith's 1823 book, A View of the Hebrews, depends upon Boudinot's work for much of its biblical underpinnings. For more on Boudinot, his views on Christian Zionism, the origin of the American Indians, etc., see the comments accompanying the on-line text of his 1815 book.
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2019-03-21 03:54:49
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https://www.physicsoverflow.org/41859/how-can-you-explain-the-4-bell-states-in-quantum-physikcs
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# How can you explain the 4 Bell States in Quantum Physikcs?
+ 0 like - 1 dislike
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Im so confused. I'm trying to understand quantum mechanics for a school project about teleportation. I'm trying to understand how the 4 Bell states come to be. If we have two particles that can be either up or down, then, looking at both at the same time, they could be either both up/ down or one of them is up and one is down.
The 4 Bell states are
sqrt(1/2)*(up up + down down)
sqrt(1/2)*(up up - down down)
sqrt(1/2)*(up down -down up)
sqrt(1/2)*(up down+down up)
I don't know much about this topic so sorry if this question is dumb. I am just stuck trying to imagine the 4 states. What difference does the minus make? They are still either up and up or down and down regardless of the minus or plus right? But what is the difference between the states?
asked Dec 31, 2018 in Q&A
Those signs signify making superpositions of the sates. If you have a superposition $\psi=a\pm b$, then the square has an "interference term" $\psi^2=a^2 \pm ab + b^2$. Thus the observed probability $\propto \psi^2$ depends on the relative sign of the states in the superposition.
These 4 states are the conventional maximally entangled ones. For a given experiment, Alice and Bob chose and use only one of them.
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2019-07-22 18:28:44
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https://www.lesswrong.com/posts/FNK5Cw7d2i54kjpDT/quote-on-nate-silver-and-how-to-think-about-probabilities
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# 12
Personal Blog
From Ezra Klein:
If Mitt Romney wins on election day, it doesn’t mean Silver’s model was wrong. After all, the model has been fluctuating between giving Romney a 25 percent and 40 percent chance of winning the election. That’s a pretty good chance! If you told me I had a 35 percent chance of winning a million dollars tomorrow, I’d be excited. And if I won the money, I wouldn’t turn around and tell you your information was wrong. I’d still have no evidence I’d ever had anything more than a 35 percent chance.
Okay, technically, winning the money would be very weak Bayesian evidence that the initial probability estimate was wrong. Still a very good quote.
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If you told me I had a 35 percent chance of winning a million dollars tomorrow, I’d try to sell you my chance for 349 thousand dollars.
I'd first look for a multi-millionaire to whom to make the offer.
I'd undercut you! Sub-linear utility and all that.
I'd try to find out what my chance of winning really was, before attempting to trade.
I think the bigger question is what exactly it means for the probability estimate to be wrong. The best I can figure is that it's not whatever it's called where exactly x% of the predictions that you're x% sure of are correct. In that case, Romney winning is evidence that it's more extreme than it should be, and Obama winning is evidence that it's less extreme. Whether or not it's evidence that it's wrong depends on whether you thought it was more likely to be more or less extreme beforehand.
Usually probability prediction quality are broken into calibration and discrimination.
If you buy the Bayesian argument (e.g. in Jaynes) that there is a single correct Pr(A|I) where I is your state of information and A is any proposition in question, then p, an estimate of Pr(A|I), is wrong if and only if p != Pr(A|I). In practice, we virtually never know Pr(A|I), so we can't make this check. But as far as a conceptual understanding goes, that's it - if, as I said, you buy the argument.
In practice, we check the things gwern mentioned.
I am not seeing the rationality in Klein's analysis.
"If you told me I had a 35 percent chance of winning a million dollars tomorrow, I’d be excited." The difference is that presumably he [the speaker] could have a near infinite number of things happen to him. So picking one and giving it much greater odds than one could reasonably expect (given most contests that grant that kind of monetary rewards) does siginify a pretty unusual situation which he should be thrilled about.
However, for all intents and purposes, the winner of the election is a binary choice. (Let's say we have 999 of a 1000 units of probability to distribute between two candidates). I think it is a given that both candidates at the level of the general election are pretty excited to be there, given that they have radically greater odds than the rest of the eligible population.
But all that should be taken for granted by any adult with any familiarity with the system. Thus, a model that predicts one candidate over the other at 60/40, let alone 75/25, odds/confidence level is not any kind of good, exciting news for the other guy [Romney in Silver's model]. Of course Silver isn't ruling out the possibility. And the stakes are high, certainly greater than a million dollars. But nonetheless, 35% to win a million dollars when you didn't know the possibility existed is different than 35% to win the presidency when you are one of only two candidates.
Edit: Just thought of a better way to phrase the above--Whether news of 35% odds is good & exciting or bad & dispiriting depends on one's priors. I would assume the challenger in a pretty divided country would have had 40-45% odds to begin with and wouldn't be excited to update downwards.
It's not a question of whether it's "good news", but whether it's a plausible chance of occurring (or rather, whether it's a "big-enough-feeling" probability).
From the quote, it sounds like its a question of whether odds lower than one's prior should increase or decrease excitement when the stakes are high.
The quote seems to me to be about how, if you predict something is only about 35% likely, and that thing happens, that's not sufficient evidence to assume you predicted wrong or to throw out your methodology. The line about the million dollars looks like an example to back up the prior sentence, "that's a pretty good chance". Other than that example, the quote doesn't seem to be about excitement at all really.
Okay, that may be the intent of the argument. Not sure I agree with that, either, though. Silver's model is presumably built from several factors. If in the end it gives a prediction that doesn't come true, then there are likely factors that were considered incorrectly or left out. The "70 % odds" is basically saying "I'm 30% confident that the outcome of the model is wrong." If Obama ends up losing, that doesn't mean Silver knows nothing, but it is evidence that the model was flawed in some meaningful way, as he now suspects. That is, we should update on 'Silver's model was off' slightly more than 'Silver's model was accurate', despite the fact that he had less than 100% confidence in it.
Remember that Silver is running a monte-carlo type model. In his case, what his 'odds' mean are that when he runs the simulation N times, 70% or so of the times, Obama wins, 30% or so Romney wins. So its not "I'm 30% confident the outcome of the model is wrong" its that "30% of the time, the model outputs a Romney victory."
Okay (though to me that sounds like he has many related models that differ based on certain variables he isn't certain about... maybe that is being pointlessly pedantic) but would you agree that a R victory would be evidence that the model needs adjustment, stronger evidence than that the model was was reliable as is? If not, what if it was 99 to 1, instead of 60 to 40? Just trying to clarify my own thinking here.
I don't know much about the internals of his model, but I would say 'it depends.' I'm sure you can use his model to make predictions of the form 'given Romney victory, what should the final electoral map look like'?, etc, but I'm not sure if the public has that kind of access. Certainly questions like that can be used to probe the model after a Romney or Obama win. If either candidate wins 'in the wrong way' (i.e. carries the wrong states), its obviously stronger evidence the model is wrong than we could get from just Romney winning.
'given Romney victory, what should the final electoral map look like'?
He sometimes selected such maps in blog posts, and generally has one with all of the most likely outcomes, but these maps are not always-and-automatically publicly accessible, so far as I know.
Sure. But it's not a huge amount of evidence, Silver predicts that in 10 such elections there will be 3 such "surprises".
Better suited to the Quotes Thread?
Not being American, I don't know how much Ezra Klein is important in politics/media, but he has a page on Wikipedia, so I guess he has nonzero importance.
With this assumption, this is more than just a quote. Quote is just something someone once said or wrote. But this is an example that someone with nonzero importance understands what a mathematical concept means, and can explain it today in a context of political discussion.
Relatively to the usual low levels of the sanity waterline, this is pretty awesome. (Yeah, I wish we instead lived in a world where quotes like this don't deserve attention, but only "man, everybody knows that" reaction.)
If you told me I had a 35 percent chance of winning a million dollars tomorrow, I’d be excited.
If you told me I had a 1 percent chance of winning a million dollars tomorrow, I'd be excited.
If you told me I had a 1/3 chance of winning a million dollars tomorrow, I'd be excited and I'd spend at least 1/3 of the day working out plans for using that money, with a clear conscience.
(I can't find the Sequences post that needs to be linked here, the one about how a TV anchor should allocate the time writing about the two possible outcomes for an event that will be decided the next day.) ETA: that's it, thanks!
The fun idea for scoring guessing games is to wait till the result is known, and then give each contestant points according to the logarithm of the probability that they had previously given the to outcome that actually occurred.
(Hmm, all the scores are negative; some are more negative than others. Perhaps the score keeper needs to give log n - log p points if he wants the numbers to be human-friendly. (The point of log n - log p is that if there are n outcomes you can give each a probability of 1/n and that scores zero. You hope to do better and make a positive score, but you can do worse (if something you think really unlikely actually happens)))
The clever mathematics behind this is Gibbs' inequality. When you come up with your probability distribution q, your expect score (relevant if the game has many rounds) is $\\sum p\_i \\log q\_i$ where p is the true but unknown distribution. Gibbs' inequality tells you that this will always be less than $\\sum p\_i \\log p\_i$. In the short run you can get a higher score in a particular play of the game by combining overconfidence and good luck :-)
In the long run, you best hope for a high score is to do a good job of guessing p. Scoring log q sets up the game that way.
On the other end of the spectrum we have Elspeth Reeve coming to Nate Silver's defense while giving too much evidential weight to a Romney victory.
But Silver [takes a weighted average] because some pollsters have a better track record than others, and some have a clear partisan tilt, left or right. If his weighting is wrong, we'll know next week.
It's a little jarring, if not surprising, to see such a defense punctuated with such an off-base statement.
On a lighter note, Silver has publicly offered to bet Joe "It's A Tossup" Scarborough on the outcome of the election. I wish this sort of thing occurred more often.
The quote isn't talking about a Romney victory at all. If Silver's pollster weights are correct, then they will be negatively correlated with the pollsters' state-level absolute errors -- that is, more weight will correspond to smaller absolute errors. And since each swing state that a pollster surveys provides a datum, there will be quite a bit of data with which to estimate the correctness of Silver's weight scheme. In other words, if Silver's weighting is wrong, we'll know by next week.
(The partisan tilt thing is a distraction -- it's easy to correct so-called "house effects", leaving only whatever systematic bias affects the polls as a group.)
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2021-01-26 12:58:08
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https://codingbobby.xyz/blog/stereographic-animations/
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# Stereographic Animations
| 6 mins | Bob
tags: mathematics, coding, visualisation, science
Happy Pi-Day everyone!
When I recently found a postcard I got from a friend some years back, I was hooked. The front showed one of those magic-eye-autostereograms1 you might as well know. You don’t? Well, you have missed loads of joy!2
Anyway, because this year’s Pi-Day was just around the corner, I wanted to create some stereograms with $$\pi$$-inspired motifs. But heck, creating patterns that first of all look nice and secondly fit the magic requirements turned out to be very difficult3. I eventually gave up on that idea (so, no Pi things this time) but I was still struck by the amazingly simple working principle of stereograms.4
After I clearly understood how they work and what properties work better than others, I still needed good motifs. Those typical ones – cubes, hearts, trumpets, or similar simple objects – were too low level for my demands, especially if I don’t even have a pattern to hide them. But then I remembered that I always wanted to get a more profound feeling of the attractors5. I calculated them, I manipulated them, I interacted with them in 3D software and I even printed large renders of them onto paper – but none of that gave me a true feeling of their three differential equations.
So, here I ended up creating very non-magical-looking 3D images that are two 2D images unless you perform weird eye movements.
Here is a “training” image for you, if you are not familiar with how stereograms work:
All you have to do is to look through the screen as if you focus a distant object behind it. Make your eyes independently leave the single spot you currently look at somewhere on the screen and let them wander apart. Imagine firing lasers from your eyes, but those lasers have to shoot parallel to each other and not cross in front of the screen.
If you manage to do it, you get an odd feeling and see everything double at first. The two red dots will help you to move your eyes the right amount; each of the dots will split into two (now four dots in total) which then move further apart the more you “stretch” your eyes. The middle two dots will come closer to each other, while the outer ones move away. Now, try to align those two middle dots (right dot that emerged from the original left dot and the left dot which emerged from the right one). Aligning the dots can help to fixate your sight, and you can try to carefully look around without changing your eye’s relative position.
You are ready for what comes next if you can see that there now appear to be three dots in total – a middle one consisting of the two ones you aligned before, one further on the left, and one on the right of it. If you have problems fully aligning the dots, the image might be displayed too large, so scale down this website (about 10 cm between the dots is ideal); or your head is slightly tilted.
## Almost-Holograms of some Attractors
The following renders are not just images, but animations of some selected attractors rotating in space. In reality, the lines representing the trajectories should be very thin. I have given them thickness and an initial sense of depth by adding a shading depending on how far away the lines are. The two red dots are also included so that you get your eyes fixated easier. Please adjust the website’s scale if necessary.
If you do everything correct, you should be able to see the attractor that appears to pop out of the screen in the middle of each animation6. You should get a very strong, almost hologram-like feeling of three dimensions that is intensified by the slow rotation. Enjoy!
### Li
I hope this little series has given you goosebumps and joy just as it has me. It is a bit challenging to adapt the eye movements, but getting used to it is really rewarding. Finally seeing the attractors in actual 3D7 revealed their beauty in a very pure form.
There will probably be more animations and more attractor renders in the future, so I’m going to update the according page in my gallery here. There, you can also read more about chaos and find the precise equations and parameters I have used.
Have a great Pi-Day (or whatever-day you are reading this on) and take care, Bob.
1. Wikipedia has a detailed article about how they function and how clever patterns can increase the effects.
2. It might be the case that it is actually impossible for you to see the 3D images. If you only have one functioning eye, amblyopia, or if you are stereoblind because of other reasons, you are sadly not able to see the effect.
3. I could have taken the lazy route and use random dot patterns, but those are not very exciting and only act as a kind of magic curtain to hide the object from plain sight.
4. Jürgen Köller shows more viewing techniques and nicely explains, how you can even draw simple stereograms by hand.
5. My journey started here just over a year ago.
6. If you look at stereograms more frequently, you should be able to even focus correctly such that the attractors appear very sharp. Your brain is used to focus your lenses in accordance with your eye’s movements (the angle between the lasers you shoot), so focussing on the near screen whilst preserving an eye fixation that is normal for faraway objects can be hard.
7. Technically, it is the pretty much the same thing as if your brain would process what your eyes see when looking at a real object. The key here is to provide a slightly shifted and rotated image for the right eye. Virtual reality goggles do the same to trick your brain – but these autostereograms are even cooler, as you don’t need any additional tools!
Nice theme
<- 🕸 💍 ->
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2022-05-20 16:05:13
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https://quant.stackexchange.com/questions/19317/realized-volatility-vs-standard-deviation-of-log-returns
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# Realized Volatility vs. Standard deviation of log returns
I am interested in calculating high frequency 5-minute intraday volatility. I am going to use the standard Realized volatility which is the square root of the sum of squared log returns.
Given X is the log price of a stock the log return Y is calculated as $$log \ returns = Y = \frac{x_i+1}{x_i}$$
Then the realized variance is the sum of the squared log returns:
$$Realized\ Variance = \sum\limits_{i=1}^{n} (y_{t_{i}})^2$$
$$Realized \ Volatility = \sqrt{\sum\limits_{i=1}^{n} (y_{t_{i}})^2 }$$
For 5-minute realized volatility n = 78 (there are 6.5 hours in the NYSE trading day)
Now if Y is the log returns and the mean of Y is assumed to be zero you can also calculate a standard deviation
$$standard \ deviation = \sqrt{\frac{1}{N}\sum\limits_{i=1}^{N} (y_i)^2}$$
So you can see the only difference between the Realized Volatility of Y and the standard deviation of Y is the $$\frac{1}{N}$$ term in the standard deviation calculation.
Can you explain the significance of this? Why does realized variance not have 1/N and how can the 2 be interpreted?
It is all a matter of frequency. For instance if you want to get annual realized volatility you multiply your last expression by $\sqrt{(N*251)}$ or the second to last expression by $\sqrt{(251)}$.
In other words, your last expression is the 5-min realized volatility whereas the second to last expression is the daily realized volatility.
1. Your definition of a log return is wrong, it is $$y_i = \ln{(x_{i+1}/x_i)}$$
2. You have 5-min returns so you are going to first compute 5-min variance:
$$Variance = \frac{1}{N}\sum_{i=1}^N y_i^2 = \sum_{i=1}^N w_iy_i^2$$
Here we have $$w_i = 1/N$$. By doing this you assume that the $$y_i$$ are iid, a relatively strong assumption. The "identically distributed" means that variance is linearly distributed in time. If you don't suppose them to be identically distributed (for instance if you think that mid-day returns are less volatile than end of day returns), you can use a different weighting by changing the $$w_i$$.
1. You may want to incorporate the overnight return in your computation as well. It can matter or not, it depends on which purpose you are computing the variance for. If it is for an intraday purpose only then you don't need it.
If you want to work with volatilities longer than intraday, you should include the overnight move with an appropriate weighting $$w_i$$. For instance if you assume that the distribution of that overnight return is the same as that of a 5-min intraday return (probably a very bad assumption), you can also use $$w_i=1/N$$.
Other example: overnight time is $$24-6.5 = 17.5$$ hours, or $$17.5 * 60 / 5 = 210$$ periods of 5 minutes, so you can consider that the overnight move consists of 210 5-min returns. For instance 209 zeros and one $$y_i$$. Or 210 times $$y_i/210$$. It will obviously increase your number of observations $$N$$ by 210 for each overnight return you include.
1. Last, if you are not interested by 5-min variance (i.e. the variance of 5 min returns), you can scale it to obtain the variance for a different time horizon. Again this scaling is done under assumptions of how variance is distributed across time. If again we assume it is linearly distributed across time, you would have to scale your 5-min variance by 288 to get a daily variance, because there are 288 times 5-min in a day. And if you want to get a yearly variance from your daily variance, again you can multiply it by 365 (or 252, depending on how you account for non-business days...)
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2022-07-03 21:48:03
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http://portraitofacreative.com/books/gauge-theories-fundamental-interactions-and-rigorous-results-progress-in-physics
|
# Gauge Theories: Fundamental Interactions and Rigorous
Format: Hardcover
Language: English
Format: PDF / Kindle / ePub
Size: 9.35 MB
A wave can be described as a disturbance that travels through a medium from one location to another location. The eminent scientists Max Planck and Albert Einstein opened the twentieth century with Quantum Theory (1900) and the Theory of Relativity (1905), respectively. It was becoming cliché at this point, but there was an even stranger development still to come. Who are some of the most prominent physicists?
Pages: 385
Publisher: Birkhauser (September 1982)
ISBN: 3764330953
In particular, work is defined W = F · ∆x (8.28) where ∆x is now a vector displacement. The vector character of this expression yields an additional possibility over the one dimensional case, where the work is either positive or negative depending on the direction of ∆x relative to F. If the force and the displacement of the object on which the force is acting are perpendicular to each other, the work done by the force is actually zero, even though the force and the displacement both have non-zero magnitudes , source: http://petitions.pw/?freebooks/geometry-particles-and-fields-graduate-texts-in-contemporary-physics. The characters in the video game live in the explicate order. The computer program that describes the actions and world of the characters is the implicate order. The game player, with joystick in hand, manipulates the computer-generated world as the Super-Implicate order download online. You will find YouTube videos by me (I have also published a dozen scholarly articles in professional journals of physics and mathematics). Waves are everywhere in nature - and I mean everywhere. The surprising/astonishing/amazing thing is; all types of waves are almost identical in behaviour. If you truly understand the behaviour of water waves - the way they interact with each other, the exact way the ripples 'curve' around obstacles - then you can have no trouble understanding accoustics, the propagation of light, or even basic quantum mechanics http://portraitofacreative.com/books/singularities-and-oscillations-the-ima-volumes-in-mathematics-and-its-applications. You shoot one photon, it goes through the slit, and it hits the wall. It doesn't leave a "pattern" of any kind, just one little blotch download online. Hint: Use (1 − ǫ)x ≈ 1 − xǫ for small ǫ. 7. A mu particle normally lives about 2 × 10−6 sec before it decays. However, muons created by cosmic rays 20 km up in the atmosphere reach the Earth’s surface. The Stanford Linear Accelerator accelerates electrons to a speed such that the 3 km long accelerator appears to be 8 cm long to the electron, due to the Lorentz contraction buckscountyadventures.com. Office hours will be from 11:00am to 12 noon, Mondays and Fridays in the instructor's office (2103 Physics Building) http://portraitofacreative.com/books/acoustical-design-of-concert-halls-and-theatres-a-personal-account.
Second law: Appearance Semi-Quantitative comparison of the sum of the forces and the variation of the velocity vector of center of mass in a Galilean. Special case situations are located near the Earth. Conversion of potential energy into kinetic energy in the case of free fall , source: http://portraitofacreative.com/books/the-recursion-method-application-to-many-body-dynamics-lecture-notes-in-physics-monographs. The word quantum derives from quantity and refers to a small packet of action or process, the smallest unit of either that can be associated with a single event in the microscopic world , e.g. http://portraitofacreative.com/books/wave-propagation-in-a-turbulent-medium-dover-books-on-physics. Solitary wave: Solitary waves are hard to explain , e.g. http://lucassnell.com/ebooks/analysis-and-simulation-of-multifield-problems-lecture-notes-in-applied-and-computational. However, no experiment has yet found definitive proof that gravity waves exist , cited: download online. Consequently, particles behave like fermions or like bosons only if they are totally identical. The absolute identity of like particles is among the most mysterious aspects of quantum mechanics download.
For example, the collapse rule of textbook quantum theory blatantly violates Lorentz invariance. As a matter of fact, the intrinsic nonlocality of quantum theory presents formidable difficulties for the development of any (many-particle) Lorentz invariant formulation that avoids the vagueness of orthodox quantum theory (see Maudlin 1994). Bell made a somewhat surprising evaluation of the importance of the problem of Lorentz invariance , cited: funnyphotostoday.com. Topics include heat engines, kinetic theory, black-body radiation, ideal Fermi and Bose gases and phase transitions. William Bialek PHY 305Introduction to the Quantum Theory(STN)This course is a continuation of PHY 208. We will continue to develop the formalism of quantum mechanics and to explore its basis epub. But $\lambda$ is the wavelength and $p_0$ is the momentum, and in accordance with quantum mechanics, the wavelength times the momentum is Planck’s constant $h$ epub. To make it more confusing, Ψ can be negative or positive, but the probability is always positive—the probability only cares how far from zero Ψ is, it doesn't matter at all which direction http://inspireinfosol.com/library/numerical-solutions-of-the-n-body-problem-mathematics-and-its-applications! One way to test this idea is to look for quantum behaviour in larger and larger objects. If standard quantum theory is correct, there is no limit. And physicists have already carried out double-slit interference experiments with large molecules 12. But if collapse models are correct, then quantum effects will not be apparent above a certain mass download. Our deduction of the wave equation for sound has given us a formula which connects the wave speed with the rate of change of pressure with the density at the normal pressure: $$\label{Eq:I:47:21} c_s^2 = \biggl(\ddt{P}{\rho}\biggr)_0.$$ In evaluating this rate of change, it is essential to know how the temperature varies download epub. A wave travels along a rope to the right in the +x-direction (there is no reflected wave). At time t = 0, a plot of displacement y as a function of distance x along the rope looks like Fig. 3a below. Sketch a graph of displacement y as a function of x for t = -T/4, t = T/4, and t = T/2, where T is the period of the wave http://portraitofacreative.com/books/wave-propagation-in-a-turbulent-medium-dover-books-on-physics.
This is also the physics behind musical instruments. The purpose of the picture in one dimensional standing waves is to help you relate the wavelength of the wave to the physical parameters of the situation--for example, to the length of the musical instrument http://decopub-publicite.com/?freebooks/qed-the-strange-theory-of-light-and-matter-princeton-science-library. The first of the above equations shows that the tension in the string must be T = (M1 + M2 )g, (11.27) M1 d2 =. (11.28) M2 d1 Thus, the tension in the string is just equal to the weight of the masses attached to the balance beam, while the ratio of the two masses equals the inverse ratio of the associated beam arm lengths. while the second shows that Figure 11.8: Trajectory of a mass on a frictionless table attached to a string which passes through a hole in the table http://portraitofacreative.com/books/nematicons-spatial-optical-solitons-in-nematic-liquid-crystals. Another transformation could be lurking in the wings. Jennifer Ouellette on quantum interference, the wave-particle duality, and that “cheeky over-achieving upstart” Thomas Young online. Classes will meet on Mondays (10:00am to 10:50am), Wednesdays (10:00am and 11:50am) and Fridays (10:00am to 10:50am). Office hours will be from 11:00am to 12 noon, Mondays and Fridays in the instructor's office (2103 Physics Building) online. Hence, interference alters the brightness of light. Therefore, the principle of linear superposition can account for phenomena like double – slit and thin film interference , e.g. download here. The power in the Bohr model was its ability to predict the spectra of light emitted by atoms. In particular, its ability to explain the spectral lines of atoms as the absorption and emission of photons by the electrons in quantized orbits. Our current understanding of atomic structure was formalized by Heisenberg and Schroedinger in the mid-1920's where the discreteness of the allowed energy states emerges from more general aspects, rather than imposed as in Bohr's model portraitofacreative.com. A standing wave is formed by combining a wave to the right and to the left , e.g. http://buckscountyadventures.com/lib/advances-in-the-casimir-effect-international-series-of-monographs-on-physics. Despite remarkable partial gains, the problem is still far from a satisfactory solution. (Albert Einstein, 1950) Albert Einstein (1905) used Planck's relationship to explain the results of the photoelectric effect which showed that the energy E of ejected electrons was dependent upon the frequency f of incident light as described in the equation E=hf , e.g. http://funnyphotostoday.com/lib/electromagnetic-surface-waves-a-modern-perspective-elsevier-insights. This diagram is only an approximation, but it gives you some idea of the 'secondary' wavelength (the 'electromagnetic' wavelength of light) caused by the interactions of the In and Out-Waves of the two Electrons/Wave-Centers. Thus we realise that these different standing Wave patterns cause a cyclical oscillation in the Shape of the In and Out-Waves which describes a wave function that is ultimately the cause of the 'electromagnetic' wavelength and frequency of light pdf. Chain reaction: nuclear reaction in which neutrons are produced that can cause further reactions. Charged: object that has an unbalance of positive and negative electrical charges http://portraitofacreative.com/books/four-lectures-on-wave-mechanics-delivered-at-the-royal-institution-london-on-5-th-7-th-12-th-and. Now, if you put normalization, then the wave function will go to zero at infinity. AUDIENCE: Is normalization sufficient to ensure the derivative also goes to zero at infinity? AUDIENCE: Then why is integration by price generically valid buckscountyadventures.com?
Rated 4.5/5
based on 1610 customer reviews
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2017-08-19 01:53:09
|
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https://www.physicsforums.com/threads/finding-a-fourier-representation-of-a-signal.839595/
|
# Finding a Fourier representation of a signal
1. Oct 25, 2015
Given the following signal, find the Fourier representation, $V(jf)= \mathfrak{F}\left \{ v(t) \right \}$:
$v(t)=\left\{\begin{matrix} A, & 0\leqslant t\leqslant \frac{T}{3}\\ 2A, & \frac{T}{3}\leqslant t\leqslant T\\ 0, & Else \end{matrix}\right.$
Then sketch $V(jf)$.
2. Relevant equations:
I don't know whether in such a given form of a signal (apparently looks quite simple, involving constants only) I should use a Fourier SERIES representation or a Fourier TRANSFORM representation. If it's a Fourier Transform representation, then I don't know how would the signal look like in the frequency domain. I tried to Calculate the Transform according to its definition, yet couldn't quite get to anything "sketchable", at least not something I can see it.
3. The attempt at a solution:
Given below:
2. Oct 25, 2015
### Dr. Courtney
Hint: Is the function periodic in the time domain? What implications does that have for the answer?
3. Oct 25, 2015
It is periodic in the sense that it has a period $T$. However, It has a 0 value for time values larger than T, so that's a bit tricky to determine the signal's periodicity. But that's the first "mine" in finding my way here.
4. Oct 26, 2015
### rude man
Either it's periodic or it's zero for t > T. Can't have it both ways.
It sure looks like you're supposed to find the Fourier transform (integral).
Don't know about "graphing" V(jf). It may have real and imaginary parts; I don't have the time to look at your math in detail but if you did it right there are indeed real and imaginary parts to the transform. Which means I don't know about graphing the function. But maybe you messed up somewhere & there are only real or imaginary parts so then you could graph that.
Ther is an expression for the energy spectral density between two frequencies f1 and f2, given the Fourier integral of a pulse, but it's kind of advanced. But that can be graphed since it's a real number.
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2018-01-19 00:24:12
|
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https://squidpy.readthedocs.io/en/latest/external_tutorials/tutorial_nanostring.html
|
# Analyze Nanostring data
In this tutorial we show how we can use Squidpy and Scanpy for the analysis of Nanostring data.
[1]:
import scanpy as sc
import squidpy as sq
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from pathlib import Path
scanpy==1.9.0.dev183+gd74a0e68 anndata==0.8.0rc2.dev27+ge524389 umap==0.5.1 numpy==1.19.5 scipy==1.7.0 pandas==1.3.0 scikit-learn==0.24.2 statsmodels==0.12.2 python-igraph==0.9.6 pynndescent==0.5.4
Download the data from Nanostring FFPE Dataset. Unpack the .tar.gz file. Load the unpacked dataset into an anndata.AnnData object. The dataset used here consists of a non-small-cell lung cancer (NSCLC) tissue which represents the largest single-cell and sub-cellular analysis on Formalin-Fixed Paraffin-Embedded (FFPE) samples.
[2]:
# !mkdir tutorial_data
# !mkdir tutorial_data/nanostring_data
# !wget -P tutorial_data/nanostring_data https://nanostring-public-share.s3.us-west-2.amazonaws.com/SMI-Compressed/Lung5_Rep2/Lung5_Rep2+SMI+Flat+data.tar.gz
# !tar -xzf tutorial_data/nanostring_data/Lung5_Rep2+SMI+Flat+data.tar.gz -C tutorial_data/nanostring_data/.
[3]:
nanostring_dir = Path().resolve() / "tutorial_data" / "nanostring_data"
sample_dir = nanostring_dir / "Lung5_Rep2" / "Lung5_Rep2-Flat_files_and_images"
path=sample_dir,
counts_file="Lung5_Rep2_exprMat_file.csv",
fov_file="Lung5_Rep2_fov_positions_file.csv",
)
WARNING: FOV 31 does not exist, skipping it.
WARNING: FOV 32 does not exist, skipping it.
Make the variable names unique using the method anndata.var_names_make_unique. Obtain the mitochondrial genes using their names prefixed with “mt-”. Calculate the quality control metrics on the anndata.AnnData using scanpy.pp.calculate_qc_metrics.
[4]:
## Calculate quality control metrics
Plot the “total_counts” and “n_genes_by_counts” from adata.obs. The first subplot shows adata.obs["total_counts"], the second adata.obs["total_counts"] less than 10000. The third subplot displays adata.obs["n_genes_by_counts"] while the fourth displays the adata.obs["n_genes_by_counts"] less than 4000.
[5]:
fig, axs = plt.subplots(1, 4, figsize=(15, 4))
sns.distplot(
kde=False,
ax=axs[0],
)
sns.distplot(
kde=False,
bins=40,
ax=axs[1],
)
sns.distplot(
kde=False,
bins=60,
ax=axs[2],
)
sns.distplot(
kde=False,
bins=60,
ax=axs[3],
)
/Users/giovanni.palla/miniconda3/envs/squidpy/lib/python3.8/site-packages/seaborn/distributions.py:2557: FutureWarning: distplot is a deprecated function and will be removed in a future version. Please adapt your code to use either displot (a figure-level function with similar flexibility) or histplot (an axes-level function for histograms).
warnings.warn(msg, FutureWarning)
[5]:
<AxesSubplot:xlabel='n_genes_by_counts'>
Filter the cells based on the minimum number of counts required using scanpy.pp.filter_cells. Filter the genes based on the minimum number of cells required with scanpy.pp.filter_genes. The parameters for both were specified based on the plots above. They were set to filter out nearly 10% of the cells and genes with minimum counts and minimum cells respectively. It’s a conservative filtering, more relaxed ones could be also useful.
[6]:
Annotate the highly variable genes based on the count data by using scanpy.pp.highly_variable_genes with flavor="seurat_v3". Normalize counts per cell using scanpy.pp.normalize_total.
Logarithmize, do principal component analysis, compute a neighborhood graph of the observations using scanpy.pp.log1p, scanpy.pp.pca and scanpy.pp.neighbors respectively.
Use scanpy.tl.umap to embed the neighborhood graph of the data and cluster the cells into subgroups employing scanpy.tl.leiden.
You may have to install scikit-misc package for highly variable genes identification.
[8]:
# !pip install scikit-misc
[9]:
## Visualize annotation on UMAP and spatial coordinates
Subplot with scatter plot in UMAP (Uniform Manifold Approximation and Projection) basis. The embedded points were colored, respectively, according to the total counts, number of genes by counts and leiden clusters in each of the subplots. This gives us some idea of what the data looks like.
[10]:
sc.pl.umap(
color=[
"total_counts",
"n_genes_by_counts",
"leiden",
],
wspace=0.4,
)
Plot segmentation masks using squidpy.pl.spatial_segment. Specifically, the key library_id in adata.obs contains the same unique values contained in adata.uns["spatial"]. The "cell_ID" column is used to spot individual cells. Here, the images were colored in accordance with the intensity of the maximum pan-cytokeratin (CK) staining.
[12]:
sq.pl.spatial_segment(
color="Max.PanCK",
library_key="fov",
seg_cell_id="cell_ID",
)
The argument library_id can also be altered to obtain specific field(s) of view.
[13]:
sq.pl.spatial_segment(
color="Max.PanCK",
library_key="fov",
library_id=["12", "16"],
seg_cell_id="cell_ID",
)
There are several parameters that can be controlled. For instance, it is possible to plot segmentation masks as “contours”, in order to visualize the underlying image. The co-ordinates of the FOV can be specified using the argument crop_coord, to get the view of required section.
[14]:
sq.pl.spatial_segment(
color="Max.PanCK",
library_key="fov",
library_id="12",
seg_cell_id="cell_ID",
seg_contourpx=10,
crop_coord=[(0, 0, 700, 700)],
)
If groups of observations are plotted (as above), it’s possible to modify whether to “visualize” the segmentation masks that do not belong to any selected group. It is set as “transparent” by default (see above) but in cases where e.g. no image is present it can be useful to visualize them nonetheless.
A scale bar can also be added, where size and pixel units must be passed. The sizes of the scalebars for these examples are not real values and are purely for visualization purposes.
[15]:
sq.pl.spatial_segment(
color="Area",
library_key="fov",
library_id=["12", "16"],
seg_cell_id="cell_ID",
seg_outline=True,
cmap="plasma",
img=False,
scalebar_dx=1.0,
scalebar_kwargs={"scale_loc": "bottom", "location": "lower right"},
)
Plot the image, with an overlay of the Leiden clusters. Use squidpy.pl.spatial_segment for the same. The image is not visualized by specifying img=False.
One or multiple groups can also be used to overlay, by specifying the groups argument in squidpy.pl.spatial_segment as shown in the second subplot.
[20]:
fig, ax = plt.subplots(1, 2, figsize=(15, 7))
sq.pl.spatial_segment(
shape="hex",
color="leiden",
library_key="fov",
library_id="12",
seg_cell_id="cell_ID",
img=False,
size=60,
ax=ax[0],
)
sq.pl.spatial_segment(
color="leiden",
seg_cell_id="cell_ID",
library_key="fov",
library_id="16",
img=False,
size=60,
ax=ax[1],
)
## Building the spatial neighbors graphs
Spatial graph is a graph of spatial neighbors with observations as nodes and neighborhood relations between observations as edges. We use spatial coordinates of spots/cells to identify neighbors among them. Different approaches of defining a neighborhood relation among observations are used for different types of spatial datasets. We use squidpy.gr.spatial_neighbors to compute the spatial neighbors graph. We use this function for a non-grid dataset with coord_type = 'generic'.
Depending on the coord_type, n_neighs specifies the number of neighboring tiles if coord_type='grid' and when the coord_type is not a grid, n_neighs represents the number of neighborhoods. Moreover, radius is only available when coord_type='generic'.
Alternatively, delaunay = True can be used, for a Delaunay triangulation graph. This way, we can observe the difference in using K-nearest neighbors and Delaunay triangulation. You can appreciate that the neighbor graph is different than before.
[21]:
fig, ax = plt.subplots(1, 2, figsize=(15, 15))
sq.gr.spatial_neighbors(
n_neighs=10,
coord_type="generic",
)
idx = np.append(idx, 420)
sq.pl.spatial_scatter(
library_id="16",
color="leiden",
connectivity_key="spatial_connectivities",
size=3,
edges_width=1,
edges_color="black",
img=False,
title="K-nearest neighbors",
ax=ax[0],
)
sq.gr.spatial_neighbors(
n_neighs=10,
coord_type="generic",
delaunay=True,
)
idx = np.append(idx, 420)
sq.pl.spatial_scatter(
library_id="16",
color="leiden",
connectivity_key="spatial_connectivities",
size=3,
edges_width=1,
edges_color="black",
img=False,
title="Delaunay triangulation",
ax=ax[1],
)
/Users/giovanni.palla/Projects/anndata/anndata/compat/_overloaded_dict.py:106: ImplicitModificationWarning: Trying to modify attribute ._uns of view, initializing view as actual.
self.data[key] = value
/Users/giovanni.palla/Projects/anndata/anndata/compat/_overloaded_dict.py:106: ImplicitModificationWarning: Trying to modify attribute ._uns of view, initializing view as actual.
self.data[key] = value
The function builds a spatial graph and saves its adjacency matrix adata.obsp['spatial_connectivities'] and distances to adata.obsp['spatial_distances'] by default.
[22]:
[22]:
<89246x89246 sparse matrix of type '<class 'numpy.float64'>'
with 534344 stored elements in Compressed Sparse Row format>
[23]:
[23]:
<89246x89246 sparse matrix of type '<class 'numpy.float64'>'
with 534344 stored elements in Compressed Sparse Row format>
In order to get all spots within a specified radius (in units of the spatial coordinates) from each spot as neighbors, the parameter radius should be used. We can observe how this plot is unlike the above one.
[24]:
sq.gr.spatial_neighbors(
coord_type="generic",
)
idx = np.append(idx, 420)
sq.pl.spatial_scatter(
library_id="16",
color="leiden",
connectivity_key="spatial_connectivities",
size=3,
edges_width=1,
edges_color="black",
img=False,
)
/Users/giovanni.palla/Projects/anndata/anndata/compat/_overloaded_dict.py:106: ImplicitModificationWarning: Trying to modify attribute ._uns of view, initializing view as actual.
self.data[key] = value
[25]:
[25]:
<89246x89246 sparse matrix of type '<class 'numpy.float64'>'
with 1101570 stored elements in Compressed Sparse Row format>
## Compute centrality scores
This example shows how to compute centrality scores, given a spatial graph and cell type annotation.
The scores calculated are closeness centrality, degree centrality and clustering coefficient with the following properties: * closeness centrality - measure of how close the group is to other nodes. * clustering coefficient - measure of the degree to which nodes cluster together. * degree centrality - fraction of non-group members connected to group members.
All scores are descriptive statistics of the spatial graph.
This dataset contains Leiden cluster groups’ annotations in anndata.AnnData.obs, which are used for calculation of centrality scores.
First, we need to compute a connectivity matrix from spatial coordinates to calculate the centrality scores. We can use squidpy.gr.spatial_neighbors for this purpose. We use the coord_type="generic" based on the data and the neighbors are classified with Delaunay triangulation by specifying delaunay=True.
[26]:
)
Centrality scores are calculated with squidpy.gr.centrality_scores, with the Leiden clusters.
[27]:
The results were visualized by plotting the average centrality, closeness centrality, and degree centrality using squidpy.pl.centrality_scores.
[28]:
## Compute co-occurrence probability
This example shows how to compute the co-occurrence probability.
The co-occurrence score is defined as:
$$\frac{p(exp|cond)}{p(exp)}$$
where $$p(exp|cond)$$ is the conditional probability of observing a cluster $$exp$$ conditioned on the presence of a cluster $$cond$$, whereas $$p(exp)$$ is the probability of observing $$exp$$ in the radius size of interest. The score is computed across increasing radii size around each cell in the tissue.
We can compute the co-occurrence score with squidpy.gr.co_occurrence. Results of co-occurrence probability ratio can be visualized with squidpy.pl.co_occurrence. The ‘3’ in the $$\frac{p(exp|cond)}{p(exp)}$$ represents a Leiden clustered group.
[29]:
We can further visualize tissue organization in spatial coordinates with squidpy.pl.spatial_segment, with an overlay of the expressed genes which were colored in consonance with the Leiden clusters.
[30]:
sq.gr.co_occurrence(
cluster_key="leiden",
)
sq.pl.co_occurrence(
cluster_key="leiden",
clusters="3",
)
sq.pl.spatial_segment(
shape="hex",
color="leiden",
library_id="16",
library_key="fov",
seg_cell_id="cell_ID",
img=False,
size=60,
)
100%|████████████████████████████████████████████████| 1/1 [00:11<00:00, 11.68s/]
## Neighbors enrichment analysis
This example shows how to run the neighbors enrichment analysis routine.
It calculates an enrichment score based on proximity on the connectivity graph of cell clusters. The number of observed events is compared against $$N$$ permutations and a z-score is computed.
This dataset contains cell type annotations in anndata.Anndata.obs which are used for calculation of the neighborhood enrichment. We calculate the neighborhood enrichment score with squidpy.gr.nhood_enrichment.
[31]:
100%|██████████████████████████████████████████| 1000/1000 [00:15<00:00, 64.85/s]
The same can be done for a specific FOV, by creating a subset of the anndata.AnnData.
[32]:
100%|█████████████████████████████████████████| 1000/1000 [00:08<00:00, 123.89/s]
And visualize the results with squidpy.pl.nhood_enrichment.
[33]:
fig, ax = plt.subplots(1, 3, figsize=(22, 22))
sq.pl.nhood_enrichment(
cluster_key="leiden",
figsize=(3, 3),
ax=ax[0],
)
sq.pl.nhood_enrichment(
cluster_key="leiden",
figsize=(3, 3),
ax=ax[1],
)
sq.pl.spatial_segment(
shape="hex",
color="leiden",
library_id="16",
library_key="fov",
seg_cell_id="cell_ID",
img=False,
size=60,
ax=ax[2],
)
## Compute Ripley’s statistics
This example shows how to compute the Ripley’s L function.
The Ripley’s L function is a descriptive statistics generally used to determine whether points have a random, dispersed or clustered distribution pattern at certain scale. The Ripley’s L is a variance-normalized version of the Ripley’s K statistic. There are also 2 other Ripley’s statistics available (that are closely related): ‘G’ and ‘F’.
Ripley’s G monitors the portion of points for which the nearest neighbor is within a given distance threshold, and plots that cumulative percentage against the increasing distance radii.
For increasing separation range, Ripley’s F function assembles the percentage of points which can be found in the aforementioned range from an arbitrary point pattern spawned in the expanse of the noticed pattern.
We can compute the Ripley’s L function with squidpy.gr.ripley. Results can be visualized with squidpy.pl.ripley. The same was plotted for adata_subset. Other Ripley’s statistics can be specified using mode = 'G' or mode = 'F'.
[34]:
mode = "L"
fig, ax = plt.subplots(1, 2, figsize=(20, 6))
sq.pl.ripley(
cluster_key="leiden",
mode=mode,
ax=ax[0],
)
sq.pl.spatial_segment(
shape="hex",
color="leiden",
library_id="16",
library_key="fov",
seg_cell_id="cell_ID",
img=False,
size=60,
ax=ax[1],
)
## Compute Moran’s I score
This example shows how to compute the Moran’s I global spatial auto-correlation statistics.
The Moran’s I global spatial auto-correlation statistics evaluates whether features (i.e. genes) shows a pattern that is clustered, dispersed or random in the tissue are under consideration.
We can compute the Moran’s I score with squidpy.gr.spatial_autocorr and mode = 'moran'. We first need to compute a spatial graph with squidpy.gr.spatial_neighbors. We will also subset the number of genes to evaluate.
[42]:
sq.gr.spatial_autocorr(
mode="moran",
n_perms=100,
n_jobs=1,
)
100%|██████████| 100/100 [04:26<00:00, 2.67s/]
[42]:
I pval_norm var_norm pval_z_sim pval_sim var_sim pval_norm_fdr_bh pval_z_sim_fdr_bh pval_sim_fdr_bh
KRT19 0.774879 0.0 0.000076 0.0 0.009901 0.000193 0.0 0.0 0.01929
OLFM4 0.752305 0.0 0.000076 0.0 0.009901 0.000204 0.0 0.0 0.01929
CEACAM6 0.749981 0.0 0.000076 0.0 0.009901 0.000179 0.0 0.0 0.01929
KRT17 0.665890 0.0 0.000076 0.0 0.009901 0.000172 0.0 0.0 0.01929
S100A6 0.624956 0.0 0.000076 0.0 0.009901 0.000132 0.0 0.0 0.01929
TM4SF1 0.586031 0.0 0.000076 0.0 0.009901 0.000196 0.0 0.0 0.01929
EPCAM 0.556131 0.0 0.000076 0.0 0.009901 0.000171 0.0 0.0 0.01929
KRT8 0.528026 0.0 0.000076 0.0 0.009901 0.000132 0.0 0.0 0.01929
ANXA2 0.511835 0.0 0.000076 0.0 0.009901 0.000163 0.0 0.0 0.01929
PIGR 0.498941 0.0 0.000076 0.0 0.009901 0.000155 0.0 0.0 0.01929
We can visualize some of those genes with squidpy.pl.spatial_segment. We could also pass mode = 'geary' to compute a closely related auto-correlation statistic, Geary’s C. See squidpy.gr.spatial_autocorr for more information.
[43]:
sq.pl.spatial_segment(
|
2022-08-20 05:46:59
|
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|
https://www.circuitbread.com/textbooks/fundamentals-of-electrical-engineering-i/appendix/decibels
|
# Decibels
The decibel scale expresses amplitudes and power values logarithmically. The definitions for these differ, but are consistent with each other.
Here
and
represent a reference power and amplitude, respectively. Quantifying power or amplitude in decibels essentially means that we are comparing quantities to a standard or that we want to express how they changed. You will hear statements like "The signal went down by 3 dB" and "The filter's gain in the stopband is
" (Decibels is abbreviated dB.).
### Exercise
The prefix "deci" implies a tenth; a decibel is a tenth of a Bel. Who is this measure named for?
Alexander Graham Bell. He developed it because we seem to perceive physical quantities like loudness and brightness logarithmically. In other words, percentage, not absolute differences, matter to us. We use decibels today because common values are small integers. If we used Bels, they would be decimal fractions, which aren't as elegant.
The consistency of these two definitions arises because power is proportional to the square of amplitude:
Plugging this expression into the definition for decibels, we find that
Because of this consistency, stating relative change in terms of decibels is unambiguous. A factor of 10 increase in amplitude corresponds to a 20 dB increase in both amplitude and power!
Power RatiodB
10
1.5
23
5
46
57
89
1010
0.1−10
The accompanying table provides "nice" decibel values. Converting decibel values back and forth is fun, and tests your ability to think of decibel values as sums and/or differences of the well-known values and of ratios as products and/or quotients. This conversion rests on the logarithmic nature of the decibel scale. For example, to find the decibel value for
, we halve the decibel value for 2; 26 dB equals 10 + 10 + 6 dB that corresponds to a ratio of 10 × 10 × 4 = 400. Decibel quantities add; ratio values multiply.
One reason decibels are used so much is the frequency-domain input-output relation for linear systems:
. Because the transfer function multiplies the input signal's spectrum, to find the output amplitude at a given frequency we simply add the filter's gain in decibels (relative to a reference of one) to the input amplitude at that frequency. This calculation is one reason that we plot transfer function magnitude on a logarithmic vertical scale expressed in decibels.
|
2020-02-29 13:49:54
|
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|
https://www.gamedev.net/forums/topic/93071-newbie-question-camera-rotation-about-origin/
|
#### Archived
This topic is now archived and is closed to further replies.
# newbie question: camera rotation about origin
## Recommended Posts
riz 122
Hi all, I am having a bit of trouble figuring out a third person camera, and I'd appreciate it if someone could post a code sample on how to rotate the camera about the origin. By this, I am confused by the matrix multiplications... so far i have this:
D3DXMATRIX matTranslate, matRotate, matTranslateBack, camera;
D3DXMatrixTranslation(&matTranslate, -camera_loc_x, -camera_loc_y, -camera_loc_z);
D3DXMatrixRotationY(&matRotate, camera_yrot);
D3DXMatrixTranslation(&matTranslateBack, camera_loc_x, camera_loc_y, camera_loc_z);
D3DXMatrixMultiply(&camera, &matTranslate, &matRotate);
//D3DXMatrixMultiply(&camera, &camera, &matTranslateBack);
g_pd3dDevice->SetTransform( D3DTS_VIEW, &camera);
but this gives rotates the camera about ITS y-axis, but not about the origin's. BTW i know you are supposed to translate, rotate, and translate back, but if I uncomment the second line, then all goes haywire. Anyway, it's late and my brain is not working as it should, so if anyone has some pseudocode/code sample on simply implementing a 3d person camera about the origin, i would so greatly appreciate it! riz [edited by - riz on May 2, 2002 3:59:52 AM] [edited by - riz on May 2, 2002 4:00:17 AM]
|
2017-09-24 01:33:17
|
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|
http://www.r-bloggers.com/feature-selection-all-relevant-selection-with-the-boruta-package/
|
# Feature selection: All-relevant selection with the Boruta package
November 15, 2010
By
(This article was first published on CYBAEA Data and Analysis, and kindly contributed to R-bloggers)
Feature selection is an important step for practical commercial data mining which is often characterised by data sets with far too many variables for model building. There are two main approaches to selecting the features (variables) we will use for the analysis: the minimal-optimal feature selection which identifies a small (ideally minimal) set of variables that gives the best possible classification result (for a class of classification models) and the all-relevant feature selection which identifies all variables that are in some circumstances relevant for the classification.
In this article we take a first look at the problem of all-relevant feature selection using the Boruta package by Miron B. Kursa and Witold R. Rudnicki. This package is developed for the R statistical computing and analysis platform.
## Background
All-relevant feature selection is extremely useful for commercial data miners. We deploy it when we want to understand the mechanisms behind the behaviour or subject of interest, rather than just building a black-box predictive model. This understanding leads us to a better appreciation of our customers (or other subject under investigation) and not just how, but why they behave as they do, which is useful for all areas of the business, including strategy and product development. More narrowly, it also help us define the variables that we want to observe which is what will really make a difference in our ability to predict behaviour (as opposed to, say, run the data mining application a little longer).
I really like the theoretical approach that the Boruta package tries to implement. It is based on the more general idea that by adding randomness to a system and then collecting results from random samples of the bigger system, one can actually reduce the misleading impact of randomness in the original sample.
For the implementation, the Boruta package relies on a random forest classification algorithm. This provides an intrinsic measure of the importance of each feature, known as the Z score. While this score is not directly a statistical measure of the significance of the feature, we can compare it to random permutations of (a selection of) the variables to test if it is higher than the scores from random variables. This is the essence of the implementation in Boruta.
## The tests
This article is a first investigation into the performance of the Boruta package. For this initial examination we will use a test data sample that we can control so we know what is important and what is not. We will consider 200 observations of 20 normally distributed random variables:
run.name <- "feature-1"
library("Boruta")
set.seed(1)
## Set up artificial test data for our analysis
n.var <- 20
n.obs <- 200
x <- data.frame(V=matrix(rnorm(n.var*n.obs), n.obs, n.var))
Normal distribution has the advantage of simplicity, but for commercial application where highly non-normally distributed features like money spent are important may not be the best test. Nevertheless, we will use it for now and define a simple utility function before we get on to the tests:
## Utility function to make plots of Boruta test results
make.plots <- function(b, num,
true.var = NA,
main = paste("Boruta feature selection for test", num)) {
write.text <- function(b, true.var) {
if ( !is.na(true.var) ) {
text(1, max(attStats(b)$meanZ), pos = 4, labels = paste("True vars are V.1-V.", true.var, sep = "")) } } plot(b, main = main, las = 3, xlab = "") write.text(b, true.var) png(paste(run.name, num, "png", sep = "."), width = 8, height = 8, units = "cm", res = 300, pointsize = 4) plot(b, main = main, lwd = 0.5, las = 3, xlab = "") write.text(b, true.var) dev.off() } ### Test 1: Simple test of single significant variable For a simple classification based on a single variable, Boruta performs well: while it identifies three variables as being potentially important, this does include the true variable (V.1) and the plot clearly shows it as being by far the most significant. ## 1. Simple test of single variable y.1 <- factor( ifelse( x$V.1 >= 0, 'A', 'B' ) )
b.1 <- Boruta(x, y.1, doTrace = 2)
make.plots(b.1, 1)
Figure 1: Simple test of Boruta feature selection with single variable.
### Test 2: Simple test of linear combination of variables
With a test of a linear combination of the first four variables where the weights are decreasing from 4 to 1, we begin to get closer to the limitations of the approach.
## 2. Simple test of linear combination
n.dep <- floor(n.var/5)
print(n.dep)
m <- diag(n.dep:1)
y.2 <- ifelse( rowSums(as.matrix(x[, 1:n.dep]) %*% m) >= 0, "A", "B" )
y.2 <- factor(y.2)
b.2 <- Boruta(x, y.2, doTrace = 2)
make.plots(b.2, 2, n.dep)
Figure 2: Simple test of Boruta feature selection with linear combination of four variables.
The implementation correctly identified the first three variables (with weights 4, 3, and 2, respectively) as being important, but it had the fourth variable as possible along with the two random variables V.8 and V.9. Still, six variables are more approachable than twenty.
### Test 3: Simple test of less-linear combination of four variables
For this text and the following we consider less obvious combinations of the first four variables. If we just count how many of them are positive, then we get to a situation where Boruta excels (because random forests excel at this type of problem).
## 3. Simple test of less-linear combination
y.3 <- factor(rowSums(x[, 1:n.dep] >= 0))
print(summary(y.3))
b.3 <- Boruta(x, y.3, doTrace = 2)
print(b.3)
make.plots(b.3, 3, n.dep)
Figure 3: Simple test of Boruta feature selection counting the positives of four variables.
### Test 4: Simple test of non-linear combination
For a spectacular fail of the Boruta approach we will have to consider a classification in the hyperplane of the four variables. For this simple example, we simply count if there are an even or odd number of positive values among the first four variables:
## 4. Simple test of non-linear combination
y.4 <- factor(rowSums(x[, 1:n.dep] >= 0) %% 2)
b.4 <- Boruta(x, y.4, doTrace = 2)
print(b.4)
make.plots(b.4, 4, n.dep)
Figure 4: Simple test of Boruta feature selection with non-linear combination of four variables
Ouch. The package rejects the four known significant variables. It is too hard for the random forest approach. Increasing the number of observations to 1,000 does not help though at 5,000 observations Boruta identifies the four variables right.
## Limitations
Some limitations of the Boruta package are worth highlighting:
1. It only works with classification (factor) target variables. I am not sure why: as far as I remember, the random forest algorithm also provides a variable significance score when it is used as a predictor, not just when it is run as a classifier.
2. It does not handle missing (NA) values at all. This is quite a problem when working with real data sets, and a shame as random forests are in principle very good at handling missing values. A simple re-write of the package using the party package instead of randomForest should be able to fix this issue.
3. It does not seem to be completely stable. I have crashed it on several real-world data sets and am working on a minimal set to send to the authors.
But this is a really promising approach, if somewhat slow on large sets. I will have a look at some real-world data in a future post.
# You may also like these posts:
1. Feature selection is the data mining process of selecting the variables from our data set that may have an impact on the outcome we are considering. For commercial data mining, which is often characterised by having too many variables for model building, this is an important step in the analysis process. And since we often work on very large data sets the performance of our process is very important to us. Having looked at feature selection using the Boruta package and feature selection using the caret package separately, we now consider the performance of the two approaches. Neither approach is suitable out of the box for the sizes of data sets that we normally work with.
2. Feature selection is an important step for practical commercial data mining which is often characterised by data sets with far too many variables for model building. In a previous post we looked at all-relevant feature selection using the Boruta package while in this post we consider the same (artificial, toy) examples using the caret package. Max Kuhn kindly listed me as a contributor for some performance enhancements I submitted, but the genius behind the package is all his.
3. We continue working our way through the examples, case studies, and exercises of what is affectionately known here as “the two bears book” (Swedish björn = bear) and more formally as Non-Life Insurance Pricing with Generalized Linear Models by Esbjörn Ohl…
4. Insurance pricing is backwards and primitive, harking back to an era before computers. One standard (and good) textbook on the topic is Non-Life Insurance Pricing with Generalized Linear Models by Esbjorn Ohlsson and Born Johansson. We have been doing som…
5. I needed a fast way of eliminating observed values with zero variance from large data sets using the R statistical computing and analysis platform . In other words, I want to find the columns in a data frame that has zero variance. And as fast as possible…
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2014-10-25 07:09:52
|
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https://economics.stackexchange.com/tags/r/new
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# Tag Info
Standard error is not sd but se. Standard error is related to sd but they are not the same $\text{se}=\frac{sd}{\sqrt{n}}$. You can also confirm here that in the first case $1.41\approx 4.46/\sqrt{10}$ Confidence interval is, for 95% confidence and using t-statistics, $\pm t^* \cdot\text{se}$, not $\pm 1.96\cdot se$. The critical value of $t^*$ at 95% level ...
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2021-10-25 19:03:58
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https://tex.stackexchange.com/questions/175539/simulating-directly-typed-text-with-input
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Simulating directly typed text with \input
I currently have the following code written in Plain TeX:
\loadtoks\END
int main(void)
{
printf("Hello World! %!@#$%&*&^\n"); } \END {\tt\the\toks0} \bye The macro \loadtoks is something that I wrote myself. It sets the catcode of every ASCII character to 12 so that specials are now just regular characters, and then it scans ahead looking for the given \END token. As it scans, it appends every token that it finds to the \toks0 register. One could also do \loadtoks+Hello World+, for example, and in that case the end-marker would be +. It's a bit more complex than that internally, but that's basically the gist of it. Now, what I need to do is something like: \loadtoks\END \input file.txt \END {\tt\the\toks0} \bye Of course this simply typesets the words "\input file.txt", as opposed to reading in the file. But naturally what I need for it to do is to actually read in the file, and to behave as if the file's contents had actually been typed into the TeX source file directly (as in the first example). I've been scratching my head over this for a while now, but I can't seem to figure it out. Note I know about LaTeX's \verb, but I'm using strictly Plain TeX, and I'm curious to figure out a solution in terms of only TeX's plain and primitive macros. • You're painting yourself into a corner; if you don't have a character of category code 0, you can't type control sequences. The only way out is to reserve a character and give it category code 13, defining it to be \input. Or define a different macro, say \loadtoksinput. May 13, 2014 at 11:09 2 Answers I'll try to answer to the long unanswered question. The problem of this question was that the \loadtoks macro was not shown. This reminds us something about MWE mentioned very often at this site. I've tried to estimate your mystic and undisclosed macro \loadtoks. I mean that you set all catcodes to 12 (others) and the first parameter (\END in the example) is processed by \string and put as a separator. It means that this: \newcount\tmpnum \def\loadtoks#1{\bgroup \tmpnum=32 \loop\advance\tmpnum by1 \ifnum\tmpnum<128 \catcode\tmpnum=12 \repeat \obeylines \obeyspaces \expandafter\def\expandafter \loadtoksA\expandafter##\expandafter1\string#1{\egroup\toks0={##1}}% \loadtoksA } {\obeyspaces\global\let =\ } \loadtoks\END int main(void) { printf("Hello World! %!@#$%&*&^\n");
}
\END
{\tt\the\toks0}
\bye
works. Now, your question is about reading the external file. First, IMHO it is impossible to do it in traditional TeX by scanning whole file content as a parameter. The error File ended while scanning occurs and there is no possibilities to put something after the end of file as a separator. If you insist on traditional TeX, you need to use the loop with \read primitive and you must to read the file line per line with \ifeof test.
But, if you are using eTeX then you can put tokens after end of the file by the \everyeof tokens register. So, all what you need is to process the separator by \string in the \everyeof and to write some \expandafters:
\everyeof \expandafter{\string\END}
\input file.txt
{\tt\the\toks0}
\bye
Edit: one redundant \expandafter removed.
The contents of a file can be loaded into a macro with the help of package catchfile. The package requires e-TeX because of \everyeof.
Example with visible spaces:
\input catchfile.sty\relax
\CatchFileDef\MyFileContents{file.txt}{%
\def\do#1{\catcode#1=12 }%
\dospecials
\obeylines
}
{\tt\MyFileContents}
\bye
Example with invisible spaces:
\input catchfile.sty\relax
\CatchFileDef\MyFileContents{file.txt}{%
\def\do#1{\catcode#1=12 }%
\dospecials
\obeylines
\obeyspaces
}
\begingroup
\def\temp#1{\endgroup
\def\DefActiveSpace{\def#1}%
}%
\catcode\ =\active
\temp{ }
{\tt\DefActiveSpace{\ }\MyFileContents}
\bye
`
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2022-05-17 05:17:07
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https://www.openfoam.com/documentation/guides/latest/api/forceSuSp_8H_source.html
|
The open source CFD toolbox
forceSuSp.H
Go to the documentation of this file.
1/*---------------------------------------------------------------------------*\
2 ========= |
3 \\ / F ield | OpenFOAM: The Open Source CFD Toolbox
4 \\ / O peration |
5 \\ / A nd | www.openfoam.com
6 \\/ M anipulation |
7-------------------------------------------------------------------------------
8 Copyright (C) 2011-2016 OpenFOAM Foundation
9 Copyright (C) 2020 OpenCFD Ltd.
10-------------------------------------------------------------------------------
12 This file is part of OpenFOAM.
13
14 OpenFOAM is free software: you can redistribute it and/or modify it
16 the Free Software Foundation, either version 3 of the License, or
17 (at your option) any later version.
18
19 OpenFOAM is distributed in the hope that it will be useful, but WITHOUT
20 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
21 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
22 for more details.
23
24 You should have received a copy of the GNU General Public License
25 along with OpenFOAM. If not, see <http://www.gnu.org/licenses/>.
26
27Class
28 Foam::forceSuSp
29
30Description
31 Helper container for force Su and Sp terms.
32
33 F = Sp(U - Up) + Su
34
35 Explicit contribution, Su specified as a force
36 Implicit coefficient, Sp specified as force/velocity
37
38SourceFiles
39 forceSuSpI.H
40
41\*---------------------------------------------------------------------------*/
42
43#ifndef forceSuSp_H
44#define forceSuSp_H
45
46#include "Tuple2.H"
47#include "vector.H"
48
49// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
50
51namespace Foam
52{
53
54// Forward Declarations
55class forceSuSp;
56
57inline forceSuSp operator+(const forceSuSp& susp1, const forceSuSp& susp2);
58inline forceSuSp operator*(const scalar s, const forceSuSp& susp);
59
60
61/*---------------------------------------------------------------------------*\
62 Class forceSuSp Declaration
63\*---------------------------------------------------------------------------*/
65class forceSuSp
66:
67 public Tuple2<vector, scalar>
68{
69public:
70
71 // Constructors
72
73 //- Default construct
74 forceSuSp() = default;
75
76 //- Construct zero-initialized content
77 inline forceSuSp(const Foam::zero);
78
79 //- Construct given Tuple2
80 inline forceSuSp(const Tuple2<vector, scalar>& susp);
81
82 //- Construct given two components
83 inline forceSuSp(const vector& Su, const scalar Sp);
84
85 //- Construct from Istream
86 inline explicit forceSuSp(Istream& is);
87
88
89 // Member Functions
90
91 // Access
92
94 inline const vector& Su() const;
95
97 inline scalar Sp() const;
98
99
100 // Edit
101
102 //- Return reference to the explicit contribution
103 inline vector& Su();
104
105 //- Return reference to the implicit coefficient
106 inline scalar& Sp();
107
108
109 // Operators
110
112 inline void operator+=(const forceSuSp& susp);
113
114 //- Subtraction
115 inline void operator-=(const forceSuSp& susp);
116
117
118 // Friend Operators
119
121 friend inline forceSuSp operator*
122 (
123 const forceSuSp& susp1,
124 const forceSuSp& susp2
125 );
126
127 //- Multiplication
128 friend inline forceSuSp operator*
129 (
130 const scalar s,
131 const forceSuSp& susp
132 );
133};
134
135
136// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
137
138} // End namespace Foam
139
140// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
141
142#include "forceSuSpI.H"
143
144// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
145
146#endif
147
148// ************************************************************************* //
An Istream is an abstract base class for all input systems (streams, files, token lists etc)....
Definition: Istream.H:64
A 2-tuple for storing two objects of dissimilar types. The container is similar in purpose to std::pa...
Definition: Tuple2.H:58
Helper container for force Su and Sp terms.
Definition: forceSuSp.H:67
forceSuSp()=default
Default construct.
const vector & Su() const
Definition: forceSuSpI.H:61
void operator-=(const forceSuSp &susp)
Subtraction.
Definition: forceSuSpI.H:94
void operator+=(const forceSuSp &susp)
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2022-12-09 18:32:09
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https://math.stackexchange.com/questions/1688392/find-the-change-of-basis-matrix
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# Find the Change of Basis Matrix [duplicate]
Consider the 2-dimensional vector subspace $V = \{(x, y, z) : x + y + z = 0\}$ of $\mathbb{R}^3$ , and two bases: $\alpha_1 = (1,−1,0)$, $\alpha_2 = (1,0,−1)$, and $\beta_1 = (0,1,−1)$, $\beta_2 = (1,1,−2)$.
Find the change-of-basis matrix $A$.
I (foolishly) tried the standard way from linear algebra to find it and I found that $A$ is a 2*2 matrix $\begin{bmatrix} -1 &1 \\ 2&-1 \end{bmatrix}$, which doesn't make sense since the vectors in $V$ are of $\mathbb{R}^3$ any help?
## marked as duplicate by user99914, Claude Leibovici calculus StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Dec 16 '17 at 6:10
No, it does make sense.
Because the transformation matrix is not used to be multiplied by the basis vectors itselves, it is used for multiplication of coordinates of the point in the space with respect to the specified basis.
So in your example, say you have a point $P=(x,y,z)$ which satisfies $x+y+z=0$, then the coordinates of P wrt first and second basis are say, $(a,b)$ and $(c,d)$ respectively, that is, $P = a*\alpha_1 + b*\alpha_2 = c*\beta_1 + d*\beta_2$
then you have
$\begin{bmatrix} -1 &1 \\ 2&-1 \end{bmatrix}$ $\begin{bmatrix} a \\ b \end{bmatrix}$ = $\begin{bmatrix} c \\ d \end{bmatrix}$
finding change of coordinate from $\beta$ to $\alpha$ $[I_V]^\beta_\alpha$ \begin{aligned} \alpha_1&=b_1\beta_1+b_2\beta_2 =\begin{pmatrix}1 \\ -1 \\ 0 \end{pmatrix} = b_1 *\begin{pmatrix}0 \\ 1 \\ -1 \end{pmatrix} +b_2*\begin{pmatrix}1 \\ 1 \\ -2 \end{pmatrix} \end{aligned} So, $$\begin{bmatrix}1 \\-1\\0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 1 & 1 \\-1 & -2 \end{bmatrix} \begin{bmatrix} b_1 \\b_2\end{bmatrix}$$ we can append $\alpha_2$ we need the row reduced echolon form so
$$rref\left ( \begin{bmatrix} 0 & 1&|& 1 &1\\ 1 & 1 &|&-1 &0\\-1 & -2 &|&0&-1 \end{bmatrix} \right ) = \begin{bmatrix} 1 & 0&|& -2 &-1 \\ 0 & 1 &|&1 &1 \\0 & 0 &|&0&0 \end{bmatrix}$$ Making $$[I_V]^\beta _\alpha = \begin{pmatrix} -2& 1 &0 \\-1& 1 &0 \end{pmatrix}$$ $$rref(\alpha_1,\alpha_2:\beta_1,\beta2) =\begin{bmatrix} 1 &0 & | &-1 & -1 \\0& 1 &| &1 &2 \\ 0 &0 &| &0 &0 \end{bmatrix}$$ making $$[I_V]^\alpha_\beta= \begin{pmatrix} -1 &1 &0 \\ -1& 2& 0\end{pmatrix}$$ knwowing that can also help use other theorems and learn things about the problem
• How can a non-square matrix be the correct answer? – amd Mar 8 '16 at 19:04
• @amd does change of coordinate matrix have to be square? I'll rather get something wrong here than in an exam. – Tiger Blood Mar 9 '16 at 18:25
• It’s an endomorphism, i.e., a mapping from a vector space to itself, so its matrix is going to be square. What’s not entirely clear to me from the problem as it is posed (and all of the clones of it that regularly appear here) is whether they’re looking for a $2\times 2$ matrix or a $3\times 3$. See corbah’s answer for why a $2\times 2$ matrix makes sense here even though $\mathbb R^3$ is three-dimensional. – amd Mar 10 '16 at 2:11
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2019-08-25 14:24:04
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http://crypto.stackexchange.com/questions/13004/what-was-the-problem-in-the-ssl-version-2-0
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# What was the problem in the SSL version 2.0?
What is the main reason (attacks) the appearance of the protocol SSL version 2.0 to replace version 3.0?
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add comment
## 2 Answers
RFC 6176 lists four reasons why SSL 2.0 must not be used, in its section 2:
• Message authentication uses MD5 [MD5]. Most security-aware users have already moved away from any use of MD5 [RFC6151].
• Handshake messages are not protected. This permits a man-in-the- middle to trick the client into picking a weaker cipher suite than it would normally choose.
• Message integrity and message encryption use the same key, which is a problem if the client and server negotiate a weak encryption algorithm.
• Sessions can be easily terminated. A man-in-the-middle can easily insert a TCP FIN to close the session, and the peer is unable to determine whether or not it was a legitimate end of the session.
However not all reasons are equivalent in gravity. The main security issue with SSL 2.0 is that it has no notion of verified closure: the SSL 2.0 connection ends when the underlying TCP connection is closed. An active attacker can force that, and neither client nor server can know whether the closure is really what the peer wanted. This is a problem when the protocol within the SSL tunnel uses the closure in a semantic way; e.g., with HTTP when there is no Content-Length header or chunked encoding (a rarity nowadays).
The three other reasons are less imperious, and can be debated:
• Though MD5 is thoroughly broken with regards to collisions, it still is quite strong when used in other ways, e.g. to build a MAC (as long as you do it properly). You should not use MD5 in new designs, but that does not mean that MD5, as it is used in SSL 2.0, can trivially be broken (or even non-trivially).
• Through the lack of protection of the handshake messages, attackers can force client and server to use a weaker cipher suite than what they could have used -- but one could say that the real issue here is that the client and server agree to use a weak cipher suite at all.
• Indeed, when the negotiated encryption is weak (a 40-bit "export cipher"), the MAC key is equally weak. But, there again, if the client and server agree to use weak crypto, they only get what they deserve.
Apart from that, SSL 3.0 is more extensible and offers extra features:
• SSL 3.0 supports more generic key exchange mechanisms, such as Diffie-Hellman (SSL 2.0 is RSA-only). In particular, this opens the possibility of ephemeral Diffie-Hellman, which grants forward secrecy, a nice thing to have.
• SSL 3.0 allows for re-handshake: doing a new handshake within the context of the first. This allows for things such as requesting certificate-based client authentication conditionally, based on what was previously exchanged over the connection.
• SSL 3.0 supports optional compression (not always a good idea, but sometimes a nice thing to have).
• SSL 3.0 is extensible with room to put extensions in the ClientHello and ServerHello. This has been used extensively, e.g. to fully support elliptic-curve cryptography.
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add comment
Something tells me you didn't put much research effort into this one.
Let me quote part of a distantly related Crypto.SE question by @D.W.:
SSL 3 included special mitigations to prevent protocol downgrade attacks where a man-in-the-middle downgraded two SSL3-capable endpoints so they end up using SSL 2. This protection in SSL 3 was critical, because SSL 2 had some major problems, and if you could downgrade both endpoints to SSL 2, nasty attacks would have become possible. Happily, the SSL 3 designers anticipated this risk in advance and made sure to introduce a special mechanism in the protocol to prevent "downgrade-to-SSL2" attacks.
Can't add much to that one since it wraps it up rather nicely while skipping the need to bluntly list all individual v2 problems that were fixed by v3. Comparing both protocols will enable you to learn about the fixed issues yourself.
In fact, any search engine will enable you to find references to both protocols, as well as a truckload of papers related to discovered security weaknesses that v2 incorporates. Also, the answers to the linked question contain large chunks of usefull, related information you might want to check out.
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2014-04-24 22:34:06
|
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https://math.stackexchange.com/questions/3760767/mse-for-mle-of-normal-distributions-sigma2
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# MSE for MLE of normal distribution's ${\sigma}^2$
So I've known $$MLE$$ for $${\sigma}^2$$ is $$\hat{{\sigma}^2}=\frac{1}{n}\sum_{i=1}^{n} (X_{i} -\bar{X})^2$$, and I'm looking for $$MSE$$ of $$\hat{{\sigma}^2}$$. But I'm having trouble to get the result.
What I tried goes like below:
By definition, $$MSE$$ = $$E[(\hat{{\sigma}^2}$$ - $${\sigma}^2$$)$$^2$$], which is = $$Var(\hat{{\sigma}^2} - {\sigma}^2)+(E(\hat{{\sigma}^2} - {\sigma}^2))^2$$ = $$Var(\hat{{\sigma}^2})-Var({{\sigma}^2})+(E(\hat{{\sigma}^2} - {\sigma}^2))^2$$.
From here, I tried to find $$Var(\hat{{\sigma}^2})$$, which is = $$Var(\frac{1}{n}\sum_{i=1}^{n} (X_{i} -\bar{X})^2$$) = $$\frac{1}{n^2}Var(\sum_{i=1}^{n} X_{i}^2 -n\bar{X}^2)$$ = $$\frac{1}{n^2}(\sum_{i=1}^{n} Var (X_{i}^2) -n^2Var(\bar{X}^2))$$
But I'm not sure how to get $$Var (X_{i}^2)$$ and $$Var(\bar{X}^2)$$. I tried $$Var (X_{i}^2)$$ = $$E(X_i^4) - (E(X_i^2))^2$$, But I'm not quite sure what $$E(X_i^4)$$ would be.
Could anyone help me with this? Am I on the correct path to solve this? Thanks in advance!
With $$\displaystyle \widehat{\sigma^2} = \frac 1 n \sum_{i=1}^n \left( X_i - \overline X \right)^2$$ you have $$\displaystyle\frac{n\widehat{\sigma^2}}{\sigma^2} \sim \chi^2_{n-1},$$ so $$\operatorname{var}\left( \,\widehat{\sigma^2} \, \right) = \frac{\sigma^4}{n^2} \operatorname{var}(\chi^2_{n-1}) = \frac{\sigma^4}{n^2}\cdot 2(n-1).$$
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2021-05-17 03:31:28
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http://soscholar.com/domain/detail?domain_id=48a53803-27df-9471-091b-892130ab8729
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State Space 738 浏览 0关注
In the theory of discrete dynamical systems, a state space is a directed graph (mathematics)|graph where each possible state of a dynamical system is represented by a vertex, and there is a directed edge from a to b if and only if ƒ(a) = b where the function f defines the dynamical system. State spaces are useful in computer science as a simple model of machines. Formally, a state space can be defined as a tuple where: * N is a Set (mathematics)|set of states * A is a set of arcs connecting the states * S is a nonempty subset of N that contains start states * G is a nonempty subset of N that contains the goal states. A state space has some common properties: * complexity, where branching factor is important * structure of the space, see also graph theory: ** directionality of arcs ** tree ** rooted graph In a computer program, when the effective state space is small compared to all reachable state (computer science)|states, this is referred to as clumping....
相关概念
主要的会议/期刊 CDC ACC IEEE TR... ICRA Automat... CORR ISCAS ICASSP
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2019-06-16 17:17:06
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https://www.reproducibleassessment.com/practice-post/
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Practice Post
This is a first try at using Gabriel to create a practice Jekyll post for my Reproducible Assessment blog.
I will have to see how this works out. Here is a formula to check out the ability to move $\LaTeX$ code into posts as well. This seems to work ok to put symbols inline like alpha ($\alpha$) and others.
That should be enough for now.
Adding a little mermaid code too in order to try out a simple chart:
sequenceDiagram participant Alice participant Bob Alice->>John: Hello John, how are you? loop Healthcheck John->>John: Fight against hypochondria end Note right of John: Rational thoughts
prevail... John-->>Alice: Great! John->>Bob: How about you? Bob-->>John: Jolly good!
This is a Gantt chart, but I’m still fighting with the styling…
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2019-09-23 15:39:15
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https://socratic.org/questions/what-is-the-moment-of-inertia-of-a-pendulum-with-a-mass-of-5-kg-that-is-8-m-from
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# What is the moment of inertia of a pendulum with a mass of 5 kg that is 8 m from the pivot?
Feb 13, 2016
$320 k g . {m}^{2}$
#### Explanation:
Assuming simple pendulum, where the mass and dimensions are relatively small compared to the mass of the bob, we may use the approximation :
$I = M {R}^{2}$
$= 5 \times {8}^{2}$
$= 320 k g . {m}^{2}$.
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2019-11-12 20:38:21
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https://math.stackexchange.com/questions/354574/lorentz-space-property-left-f-right-lq-s-leq-lim-limits-n-to-inf
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# Lorentz Space property: $\left\| f \right\|_{L^{q,s}} \leq \lim\limits_{n\to\infty} \| f_n \|_{L^{q,s}}$
I would like to understand a statement similar to Fatou's Lemma in the Lorentz space setting. It is as follows.
Suppose $0 < q,s < \infty$ and $f_n,f$ are measurable functions on a measure space $(X,\mu)$ such that $f_n \to f$ in $L^{q,s}$. That is, $\|f_n - f\|_{L^{q,s}}\to 0$ as $n \to \infty$. Then $$\left\| f \right\|_{L^{q,s}} \leq \lim\limits_{n\to\infty} \| f_n \|_{L^{q,s}}$$
The source I have says it follows from the fact that $|f| \leq \liminf\limits_{n\to\infty} |f_n|$ $\mu$-a.e. implies $f^{*} \leq \liminf\limits_{n\to\infty} f_n^{*}$, where $f^{*}$ is the decreasing rearrangement of $f$. However, I don't see how this helps since $f_n$ may not convergence pointwise to $f$. Also, my attempts at simply trying to compute it directly using the definition below have failed. Any help is greatly appreciated! Some background material that may help:
$$d_{f}(\alpha) = \mu\big(\{x \in X : |f(x) | \geq \alpha\}\big)$$ $$f^{*}(t) = \inf \{\alpha > 0 : d_f(\alpha) \leq t\}$$ $$\|f\|_{L^{q,s}} = \left(\int_{0}^\infty \left(t^{\frac{1}{q}} f^{*}(t) \right)^s \frac{dt}{t}\right)^{1/s}$$
Edit: See my answer below.
Proof. From general theory (see for example Section 1.4 of Classical Fourier Analysis by L. Grafakos) we know that $$\sup_{\alpha > 0} \alpha \big(d_{f^n-f}(\alpha)\big)^{1/s} = \sup_{t > 0} t^{1/s} (f_n - f)^{*}(t) \leq \left(\dfrac{s}{q}\right)^{1/s} \| f_n - f\|_{L^{q,s}}.$$ and by assumption this goes to zero. Therefore, $f_n \to f$ in measure and so there is a subsequence $\{f_{n_k}\}$ of $\{f_n\}$ so that $f_{n_k} \to f$ $\mu$-a.e. Now the result mentioned above can be applied to conclude that $$f^{*} \leq \liminf_{n_k\to\infty} f_{n_k}^*.$$ Hence, this result and Fatou's lemma yields \begin{align*} \| f \|_{L^{q,s}} &= \left(\int_0^\infty \left(t^{1/q} f^*(t)\right)^{s} \frac{dt}{t} \right)^{1/s}\\ &\leq \left(\int_0^\infty \left(t^{1/q} \liminf_{n_k\to\infty}f_{n_k}^*(t)\right)^{s} \frac{dt}{t} \right)^{1/s} \quad \mbox{(result above)}\\ &\leq \liminf_{n_k\to\infty}\left(\int_0^\infty \left(t^{1/q} f_{n_k}^*(t)\right)^{s} \frac{dt}{t} \right)^{1/s} \quad \mbox{(Fatou's Lemma)}\\ &= \lim_{n_k \to \infty} \|f_{n_k}\|_{L^{q,s}} \\ &= \lim_{n\to\infty} \|f_{n}\|_{L^{q,s}} \end{align*} where the final equality follows from the fact that $L^{q,s}$ is a complete space.
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2019-11-20 01:33:39
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https://direct.mit.edu/adev/article/31/2/109/9884/The-Dragon-Is-Flying-West-Micro-level-Evidence-of
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## Abstract
Outward direct investment (ODI) from the People's Republic of China (PRC) is surging. A common perception is that it was driven by the country's resource-seeking and technology-seeking motives. Using a new, unique, and comprehensive dataset that covers close to 10,000 Chinese ODI deals from 1998 to 2009, we find that in contrast to the common perception, over half of the ODI deals are in service sectors, with many of them appearing to be export-related. In addition to documenting the pattern and trend of the PRC's ODI, we empirically examine both the determinants and effects of ODI at the firm level. We find that ex ante larger, more productive, and more export-intensive firms are more likely to start investing abroad. Using matching estimation techniques, we find that ODI is associated with better firm performance, including higher total factor productivity, employment, and export intensity, and greater product innovation. To assess the relative contributions of technology transfer, export promotion, and resource seeking to the positive effects of ODI, we use ODI data merged with customs transaction-level trade data. We find that firms’ ODI participation is associated with significantly better trade performance, measured by export and import volumes, export and import unit values, and number of export destinations. Contrary to perceived technology-seeking and resource-seeking motives, we find no evidence that ODI firms import more capital or intermediate inputs compared to non-ODI firms.
## I. Introduction
The People's Republic of China (PRC) is the world's fifth largest source of foreign direct investment in 2010 (in terms of flow), after the US, France, Germany, and Japan.1He et al. (2012) predict that the PRC's cumulative outward direct investment (ODI) would probably exceed $5 trillion in 2020, increasing from a mere$3 billion in 2010. Given the PRC's sheer size, the volume of its ODI may be expected; but considering its relatively early stage of economic development, its recent surge in ODI is surprising to many. While it is still a fairly new phenomenon, reports about Chinese ODI often hit news headlines, such as Lenovo's acquisition of IBM PC units, CNOOC's rejected acquisition of Unocal, Huawei's investment in 3Leaf System, and Dalian Wanda Group's acquisition of AMC Theaters. The target sectors are widespread, and even as mundane a product as pork has attracted tremendous media attention recently due to Shuanghui's acquisition of Smithfield. Tensions in developed countries towards Chinese ODI are rising, similar to the 1980s when Japanese firms were making high-profile acquisitions.
Despite the rising concerns, existing studies about Chinese ODI are either descriptive in nature or based on aggregate data. Among the recent studies that use micro data, the focus has been on understanding the motives of ODI, with the primary goal to verify the media hype about the PRC's attempt to control natural resources and technology around the world.2 Little research has been done about which firms are engaged in ODI and how ODI may enhance their performance.
This paper has two goals. It first documents several stylized facts about Chinese ODI. A point of departure from all existing studies is that we document our facts based on the most comprehensive micro-level data on Chinese ODI. The dataset, which was made available by the PRC's Ministry of Commerce, covers close to 10,000 ODI deals of over 7,000 firms in all sectors over the period of 1998–2009. Consistent with the existing literature, we find that the motives of Chinese ODI can be broadly categorized into three types—resource seeking, technology seeking, and market seeking (export promotion). In contrast with the common perception, both the aggregate statistics and our micro data lend no support for the popular speculation that the recent rise of Chinese ODI is driven by resource seeking. Instead, we find that business services and wholesale/retail trade have accounted for a large and increasing share of Chinese ODI in terms of the number of deals as well as the volume of flows. The presence of private firms in Chinese ODI is also increasing. Half of the top 20 destinations of its ODI are in Asia.
The second goal of the paper is to analyze the firm-level determinants and effects of ODI, which have implications for other emerging countries. To obtain a long list of firm performance measures, we rely on manufacturing firms’ survey data from the PRC's National Bureau of Statistics, which we merge with the ODI firm list.3 By estimating a probit model of ODI participation, we find that more productive (measured by total factor productivity), larger (measured by employment), and more export-intensive firms are more likely to invest abroad. These findings lend support to the studies that typically assume higher fixed costs of horizontal foreign direct investment (FDI) compared to that of exporting. We also find that relative to domestic private firms, state-owned enterprises (SOEs) are more likely to invest abroad, consistent with the conventional view that the PRC's government is behind a lot of the country's ODI flows. In contrast, foreign firms are less likely to undertake ODI.
We then apply the propensity-score matching techniques commonly used in the program evaluation literature to assess the average treatment effects of ODI on the treated firms’ performance. We find that ODI has a positive effect on a wide range of firms’ performance measures including value added, employment, productivity, export intensity, R&D intensity, and the propensity to innovate new products.
Since the positive effects of ODI on firm performance can be due to technology transfer, resource seeking, or export promotion, we use customs transaction-level trade data merged with our ODI list to shed light on the relative contributions of the three channels. By employing propensity-score matching techniques again to establish causality, we find that firms’ ODI participation is associated with a significant improvement in their trade performance, measured by export and import volumes, export and import unit values, and number of export destinations. To the extent that unit value proxies for the quality of goods, these results imply that ODI induces quality upgrading of both imports and exports. In other words, these results show that horizontal FDI from the PRC complements rather than substitutes firms' trade. These findings are consistent with the idea that exporting entails high fixed costs, such as marketing and information signaling, which can be reduced by ODI. Finally, we find no evidence based on the composition of firms’ imports and exports that ODI is associated with technology or resource seeking.
In summary, our paper shows that export-promoting ODI from emerging countries can potentially raise and sustain the benefits of exporting, which in turn contribute to the countries’ structural transformation from low-skill manufacturing to high-skill manufacturing, and eventually from manufacturing to high-skill services. Our findings have important policy implications for countries beyond the PRC, which have been experiencing rising labor costs after years of FDI and export-promotion policies.
The paper proceeds as follows. Section II reviews the related literature. Section III describes our three data sources. Section IV uses the new ODI data to describe overall patterns of ODI firms. Section V presents the characteristics of ODI from the PRC. Section VI examines the determinants and the effects of ODI at the firm level. Section VII focuses on the export-facilitation motive and examines how ODI is related to firms’ trade patterns and performance. The final section concludes with some policy discussions.
## II. Literature Review
Our paper is related to various strands of literature. First, it relates to the classical theory of multinational enterprises (MNEs) about how firms use their capabilities and resources to generate competitive advantage over indigenous firms in host countries (Caves 1971, Hymer 1976, Kindleberger 1969 and 1970). More recent studies show that in addition to facilitating foreign sales, firms undertake ODI to acquire resources, assets and technology to develop their competitive advantage (Child and Rodrigues 2005, Makino et al. 2002, Mathews 2006).4
Second, our paper contributes to the growing literature on Chinese ODI. Most of the earlier studies were descriptive in nature, sometimes relying on case studies (e.g., Deng 2003 and 2004, Wu and Chen 2001). Cai (1999) proposes that Chinese firms invest overseas mainly to seek markets, natural resources, technology, managerial skills, and financial capital.5 More recent studies focus on the empirical examination of the determinants of Chinese ODI (e.g., Buckley et al. 2007), but most of these studies rely on aggregate data for analysis. There are a few notable exceptions that use micro-level data. For instance, Luo et al. (2011) show empirically that ODI by private Chinese firms had been prompted to exploit firm-specific advantages as well as to tackle market imperfections due to the underdevelopment of the PRC's domestic institution. Other studies on Chinese overseas mergers and acquisitions (M&As) support the resource-seeking and technology-seeking motives (Antkiewicz and Whalley 2007, Rui and Yip 2008). Using aggregate data, Cheng and Ma (2007) and Cheung and Qian (2009) show that the PRC's investment was motivated by both market seeking and resource seeking. However, they find no evidence that its investment in Africa and other oil-producing countries account for the rise. In addition, they find that the PRC's international reserves and exports to developing countries tended to complement ODI. Our findings based on firm-level data are largely consistent with the macro patterns they document.
Based on detailed firm-level data from Zhejiang province, Huang and Wang (2013) empirically identify export facilitation as the third motive, which is as important as the other two emphasized by earlier studies. Our paper finds supporting evidence but is unique in two respects. We use a much more comprehensive micro dataset from the PRC, which covers all industries and provinces. We merge our ODI data with customs transaction-level data and manufacturing survey data so that we can assess the effects of ODI on firm performance. In particular, we examine how exporters and importers benefit from ODI.
Third, our paper contributes to the large literature on the relation between FDI and trade. Besides the early theoretical literature (Krugman 1980, Helpman 1984), there is an extensive empirical literature on the relation between FDI and trade. On the one hand, there are studies showing substitution between FDI and exports (Brainard 1997; Markusen and Venables 2000; and Helpman, Melitz, and Yeaple 2004). The key idea is the proximity concentration trade-off (i.e., a trade-off between transportation costs and firm level returns to scale). These models are explicitly designed for horizontal FDI. On the other hand, some studies show that FDI and exports can be complements (Lipsey and Weiss 1981 and 1984; Yamawaki 1991, Clausing 2000). By using Japanese product-level data on foreign production in the US and exports to the US, Blonigen (2001) finds both substitution and complementarity effects of FDI on exports. Substitution is likely to be found for final goods exports, while complementarity is likely to be found for intermediate inputs and finished products. A more recent strand of literature studies the complex interactions between ODI and exports by highlighting the export-platform type of exports by multinational firms (Antràs 2003; Grossman, Helpman, and Szeidl 2006; Ekholm, Forslid, and Markusen 2007; Yeaple 2003; Conconi et al. 2013).6 Our paper finds that FDI and trade are complements in the PRC.
## III. Data
We use data on ODI's by Chinese companies provided by the Chinese Ministry of Commerce (MOFCOM). The dataset covers all ODI transactions that were approved by the MOFCOM between January 1, 1998 and December 31, 2009. For each ODI deal, the dataset reports the name of the investing firm, the firm's sector of business, the province of origin, and the recipient country of the ODI flow. There is, however, no information on the amount of the deal or the name of the target for M&As. There are altogether 9,744 deals from 7,202 unique firms for the 12-year period (1998–2009) included in the dataset. Since all Chinese firms need to be approved by MOFCOM for each cross-border deal, this data source is the most official and comprehensive among all other firm-level sources that have been used. To verify the representativeness of our data, we compare the number of deals in our data with those studied by Huang and Wang (2013). Our dataset covers 90% of the deals from Zhejiang, the province they focus on over the same sample period.7
The second data source is the Annual Survey of Industrial Enterprises, conducted by the PRC's National Bureau of Statistics (NBS) over the period of 1998–2009. The survey includes all industrial firms that are either state owned or non-state owned with sales above CNY5 million (around \$600,000 during the sample period). The survey covers all manufacturing, mining, and utilities sectors. The number of firms covered in this data set ranges from around 150,000 in 1998 to 431,000 in 2007. The dataset contains information on ownership structure, tangible assets, number of employees, research and development (R&D), advertising, value added, sales, new product sales, and exports. Readers are referred to Ma et al. (2014) for a more detailed description.
The third data source is the transaction-level trade data from the PRC's customs over the period of 2000–2006. This dataset contains information on values (in US dollars), quantities, and prices of all import and export transactions between the PRC and over 200 destination countries at the HS 6-digit level (over 5,000 products).8 This level of disaggregation is the finest for empirical studies in international trade—i.e., transactions at the firm-product-country-month level. For each trading firm, the dataset also provides information on ownership type (state, private, foreign) and customs regime (processing and non-processing).9 Mainly based on firm names, we merge the ODI data with the firm-level manufacturing data and the transaction-level trade data, respectively. More details will be described below.
## IV. Overall Patterns of ODI Firms
Before analyzing the three micro-level datasets, let us highlight an overlooked pattern simply based on aggregate data. Using sector-level data on Chinese ODI reported by MOFCOM for the period of 2006–2010, Figure 1 reveals that the “mining” sector used to account for about 40% of total Chinese ODI flows in 2006, followed by “leasing and business services” which contributed about 21% of the total. Since then, the share of “mining” in ODI flows declined gradually, while that of “leasing and business services” increased continuously until it became the most prevalent sector in terms of Chinese ODI flows (44% of the total). Together with “wholesale and retail trade,” these two broad sectors accounted for over half of the aggregate volume of the PRC's ODI in 2010, compared to 27% in 2006. Mining and banking, on the other hand, accounted for only 8% and 13% of the PRC's total ODI flows in 2010, respectively. These findings, based on official statistics, do not support the common perception that the rising ODI from the PRC is due to rising financial outflows or resource seeking.10 Instead, these aggregate patterns and trends suggest that the recent rise in Chinese ODI could be related to its continuous growth in exports. Motivated by these aggregate patterns, we will verify how firms’ ODI are related to their overall and export performance.
Figure 1.
Share of ODI Flows (2006–2010)
Figure 1.
Share of ODI Flows (2006–2010)
The aggregate patterns outlined above say nothing about which firms are engaged in ODI, where they invest, and how ODI may enhance their performance. In the rest of the paper, we will use our firm-level ODI data along with official micro-level balance sheet and trade data to analyze the determinants and effects of ODI. Our dataset contains 9,744 deals conducted by 7,202 unique companies that were approved by the PRC's MOFCOM between 1998 and 2009. Table 1 reports the distribution of the deals per year during our sample period of 1998–2009.11 As expected, the number of ODI deals increased significantly from 19 deals in 1998 to 3,060 deals in 2009. The increase is particularly sharp in 2005, when the number of deals increased from 244 to 1,091 (over a 300% increase). Table A2 in the appendix shows that most of the increase is due to the massive liberalization of ODI by domestic private firms.12
Table 1.
ODI Deals Breakdown, by Year
YearFrequencyPercent
1998 19 0.19
1999 0.09
2000 20 0.21
2001 21 0.22
2002 66 0.68
2003 79 0.81
2004 244 2.50
2005 1,091 11.20
2006 1,412 14.49
2007 1,632 16.75
2008 2,091 21.46
2009 3,060 31.40
Total 9,744 100.00
YearFrequencyPercent
1998 19 0.19
1999 0.09
2000 20 0.21
2001 21 0.22
2002 66 0.68
2003 79 0.81
2004 244 2.50
2005 1,091 11.20
2006 1,412 14.49
2007 1,632 16.75
2008 2,091 21.46
2009 3,060 31.40
Total 9,744 100.00
ODI = outward direct investment.
Source: The PRC's Ministry of Commerce.
Table 2 tabulates the distribution of Chinese ODI deals by host country in our data. Between 1998 and 2009, Hong Kong, China appears as the major recipient of ODI from the PRC, accounting for close to 20% of total deals. One may argue that it may not be the final destination of Chinese ODI, as there can be a lot of transit or round-trip FDI. First, firms in the PRC may take advantage of the low tax regime and more developed legal and financial institutions in Hong Kong, China to raise funds. Second, many firms in the PRC may choose to set up subsidiaries and even headquarters to channel capital to a third country or even back to the PRC. Both transit and round-trip FDI through Hong Kong, China are well-known. A drawback of our dataset is that we have no information to separate both types of ODI from genuine ODI to Hong Kong, China. We will check the robustness of our main results by excluding Hong Kong, China as the host country of ODI.
Table 2.
Top 20 Destinations of the PRC's ODI
CountryFrequencyPercent
Hong Kong, China 1,946 19.97
United States 918 9.42
Russian Federation 551 5.65
Viet Nam 464 4.76
United Arab Emirates 370 3.80
Japan 360 3.69
Korea, Rep. 299 3.07
Germany 270 2.77
Lao PDR 267 2.74
Australia 236 2.42
Indonesia 180 1.85
Singapore 167 1.71
Thailand 143 1.47
Nigeria 137 1.41
United Kingdom 134 1.38
India 128 1.31
Mongolia 102 1.05
Kazakhstan 101 1.04
Malaysia 95 0.97
CountryFrequencyPercent
Hong Kong, China 1,946 19.97
United States 918 9.42
Russian Federation 551 5.65
Viet Nam 464 4.76
United Arab Emirates 370 3.80
Japan 360 3.69
Korea, Rep. 299 3.07
Germany 270 2.77
Lao PDR 267 2.74
Australia 236 2.42
Indonesia 180 1.85
Singapore 167 1.71
Thailand 143 1.47
Nigeria 137 1.41
United Kingdom 134 1.38
India 128 1.31
Mongolia 102 1.05
Kazakhstan 101 1.04
Malaysia 95 0.97
ODI = outward direct investment.
Source: The PRC's Ministry of Commerce.
After Hong Kong, China, the US comes as the second most important recipient of ODI, accounting for 9.4% of the total number of deals. Following the US are the Russian Federation and Viet Nam, respectively. Interestingly, the United Arab Emirates (UAE) is the fifth important recipient country. To the extent that UAE is a major oil exporter, the high ranking of UAE as a major recipient of the PRC's ODI provides some support for the resource-seeking hypothesis (Antkiewicz and Whalley 2007, Rui and Yip 2008). It is worth noting that out of the top 20 Chinese ODI destinations (in terms of the number of deals), 12 are in Asia. The prevalence of Asian countries among the top hosts is consistent with the sectoral pattern that horizontal ODI (leasing and business services, along with wholesale and retail trade) accounts for the majority of ODI flows in recent years, rather than technology-seeking or resource-seeking ODI as commonly speculated.
Table 3 shows the numbers of deals by regions (e.g, Asian versus non-Asian, OECD versus non-OECD, and so on) in our sample. The average fraction of Chinese firms investing in OECD countries across all years (1998–2009) is only 30% (last row). Among the non-OECD countries, Asian countries accounted for about 80% (55.75/69.83). After 2004, Asian countries consistently accounted for over 60% of Chinese ODI deals, while OECD countries never accounted for more than 40% again. Sub-Saharan Africa rarely accounted for more than 10% of the total Chinese ODI deals over the sample period. Just by considering the number of deals across host countries, the relatively small fractions of ODI to OECD countries and the concentration of ODI in Asia lend little support to the hypothesis that technology seeking or resource seeking are the main drivers of the recent rise of ODI from the PRC. We are aware of the fact that some of the resource-seeking deals, for example those in Sub-Saharan Africa, are much larger in monetary value than the export-related deals in Asia. However, the trends in shares shown in Figure 1 imply that the relatively large resource-seeking deals are unlikely to overturn the conclusion based on the number of deals.
Table 3.
Fraction of ODI Deals, by Region and Year
YearNon-OECDOECDNon-AsiaAsiaNon-SSASSATotal (No.)
1998 95 21 79 95 19
1999 56 44 89 11 78 22
2000 75 25 45 55 85 15 20
2001 71 29 29 71 100 21
2002 70 30 58 42 88 12 66
2003 58 42 61 39 91 79
2004 69 31 41 59 90 10 244
2005 69 31 38 62 93 1,091
2006 64 36 37 63 94 1,411
2007 69 31 38 62 91 1,632
2008 72 28 35 65 92 2,091
2009 70 30 39 61 91 3,058
Average 69.83 30.17 44.25 55.75 90.69 9.31 9,741
YearNon-OECDOECDNon-AsiaAsiaNon-SSASSATotal (No.)
1998 95 21 79 95 19
1999 56 44 89 11 78 22
2000 75 25 45 55 85 15 20
2001 71 29 29 71 100 21
2002 70 30 58 42 88 12 66
2003 58 42 61 39 91 79
2004 69 31 41 59 90 10 244
2005 69 31 38 62 93 1,091
2006 64 36 37 63 94 1,411
2007 69 31 38 62 91 1,632
2008 72 28 35 65 92 2,091
2009 70 30 39 61 91 3,058
Average 69.83 30.17 44.25 55.75 90.69 9.31 9,741
ODI = outward direct investment, OECD = Organisation for Economic Co-operation and Development, SSA = Sub-Saharan Africa.
Note: Numbers are in % in the first eight columns, while they are in whole numbers in the last column.
Source: The PRC's Ministry of Commerce ODI data (1998–2009).
Next, we turn to analyzing the distribution of ODI deals across industries. Consistent with Figure 1 that shows shares in total flows, Table 4 shows that a majority of the PRC's ODI deals belong to the service sectors. In particular, based on a sample pool of observations from all years, “business services” and “wholesale trade” stand out as the top two sectors in which most ODI deals are found. Together, they account for 5,235 deals and thus, over half of the country's total. The third largest ODI sector in terms of the number of deals is “building and civil engineering,” but it accounts for only 3% of the total. The sectors that are often suspected as the main drivers of the rise in the PRC's ODI—“nonferrous metals mining and dressing,” “nonmetal mineral products,” and “geologic prospecting”—together account for less than 6% of the total, consistent with the continuous decline in the share of mining in the country's aggregate ODI flow depicted in Figure 1. One can argue that some of the firms in the mining sector can invest in other industries abroad. Moreover, mining-related ODI could induce other types of ODI, such as “waterway transport.” However, given that the “business services” and “wholesale trade” account for the bulk of ODI deals, the required complementary effects of ODI from mining to other sectors will need to be very large to support the hypothesis that the PRC's ODI is ultimately driven by resource seeking but not export promotion. In sum, over half of the PRC's ODI deals are in the service sectors. ODI in manufacturing, mining, and high-tech sectors have not been rising as has been postulated by many.
Table 4.
Industry Breakdown of ODI (Top 20 Only)
IndustryFrequencyPercent
Building and civil engineering 285 2.93
Nonferrous metals mining and dressing 212 2.18
Nonmetal mineral products 202 2.08
Garments, shoes, and caps manufacturing 189 1.94
Forestry 181 1.86
Real estate 169 1.74
Electric equipment and machinery 162 1.66
R&D 159 1.63
Geologic prospecting 157 1.61
Other financial activities 143 1.47
Metal products 135 1.39
Transport equipment 118 1.21
Food production 106 1.09
Water way transport 101 1.04
Agriculture 86 0.88
Ordinary machinery 86 0.88
Software 84 0.86
IndustryFrequencyPercent
Building and civil engineering 285 2.93
Nonferrous metals mining and dressing 212 2.18
Nonmetal mineral products 202 2.08
Garments, shoes, and caps manufacturing 189 1.94
Forestry 181 1.86
Real estate 169 1.74
Electric equipment and machinery 162 1.66
R&D 159 1.63
Geologic prospecting 157 1.61
Other financial activities 143 1.47
Metal products 135 1.39
Transport equipment 118 1.21
Food production 106 1.09
Water way transport 101 1.04
Agriculture 86 0.88
Ordinary machinery 86 0.88
Software 84 0.86
ODI = outward direct investment.
Note: Industry classification is based on ODI firms' description of the main business scope.
Source: The PRC's Ministry of Commerce OFDI data (1998–2009).
In Table A2 in the appendix, we also show the distribution of the origin of ODI across provinces in the PRC. The origins tend to be concentrated in coastal provinces (e.g., Zhejiang, Jiangsu, Shandong, Guangdong, and Shanghai). These findings are consistent with the common perception that the PRC's engagement in globalization started in coastal provinces and is still largely concentrated there.
## V. Characteristics of ODI Firms
The ODI dataset does not contain balance sheet information. To study the relationship between the causes and effects of ODI at the firm level, we merge the ODI data with the PRC's NBS manufacturing firm survey data. Since there is no common firm identifier in the two datasets, the merging is done based on firm names. The statistics of the merging is reported in Table A5 in the appendix. The NBS data are available for the period of 1998–2009. On average, about 35% of the ODI deals can be merged to a firm in the NBS data, with the success rate ranging from 11% (in 1999) to 55% (in 2002).13 We present the list of challenges we face when merging the two datasets in online appendixes.14 Besides the imperfect match, another drawback of using the merged dataset is that all “services” firms will be excluded from our sample.
Before dealing with selection and endogeneity issues, let us simply compare the means of several key variables between firms that conduct ODI (after they got at least one deal approved) and those that do not. Table 5 reports the results. Compared to non-ODI firms, ODI firms are significantly larger (in terms of sales, value added, or employment). Specifically, the log difference in sales, value added, and employment between ODI and non-ODI firms are 2.3, 2.5, and 1.6, respectively.
Table 5.
The t-test of Key Characteristics between ODI and non-ODI
ODInon-ODIDiff
Size
ln(Sales) 12.32 10.006 2.314***
No. or s.e. 7464 2,345,223 (0.017)
No. or s.e. 3,323 1,705,234 (0.026)
ln(Employment) 6.339 4.726 1.613***
No. or s.e. 7,561 2,711,011 (0.014)
Ownership Type
Foreign 0.119 0.034 0.085***
No. or s.e. 10,418 7,696,402 (0.002)
HKG; TAP; and Macau, China invested firms 0.112 0.099 0.013***
No. or s.e. 7,566 2,744,253 (0.003)
SOE 0.04 0.12 –0.08***
No. or s.e. 7,566 2,744,253 (0.004)
General Performance Measures
ln(Labor Productivity) 5.425 4.39 1.035***
No. or s.e. 7,539 2,524,320 (0.017)
Export/Sales 0.393 0.144 0.249***
No. or s.e. 7,464 2,345,223 (0.004)
No. or s.e. 3,299 1,548,148 (0.002)
R&D/Sales (multiplied by 1,000) 0.0211 0.00501 0.000***
No. or s.e. 3,241 899,075 (0.00)
Raw materials/Sales 0.618 0.616 0.002
No. or s.e. 3,384 1,171,453 (0.003)
ODInon-ODIDiff
Size
ln(Sales) 12.32 10.006 2.314***
No. or s.e. 7464 2,345,223 (0.017)
No. or s.e. 3,323 1,705,234 (0.026)
ln(Employment) 6.339 4.726 1.613***
No. or s.e. 7,561 2,711,011 (0.014)
Ownership Type
Foreign 0.119 0.034 0.085***
No. or s.e. 10,418 7,696,402 (0.002)
HKG; TAP; and Macau, China invested firms 0.112 0.099 0.013***
No. or s.e. 7,566 2,744,253 (0.003)
SOE 0.04 0.12 –0.08***
No. or s.e. 7,566 2,744,253 (0.004)
General Performance Measures
ln(Labor Productivity) 5.425 4.39 1.035***
No. or s.e. 7,539 2,524,320 (0.017)
Export/Sales 0.393 0.144 0.249***
No. or s.e. 7,464 2,345,223 (0.004)
No. or s.e. 3,299 1,548,148 (0.002)
R&D/Sales (multiplied by 1,000) 0.0211 0.00501 0.000***
No. or s.e. 3,241 899,075 (0.00)
Raw materials/Sales 0.618 0.616 0.002
No. or s.e. 3,384 1,171,453 (0.003)
*** = , HKG = Hong Kong, China, ODI = outward direct investment, TAP = Taipei,China.
Note: Data on value added are only available from 1998–2007. Data on R&D are only available from 2002–2007. Standard errors in parentheses.
Source: Authors' computations using manufacturing survey data (1998–2009).
Interestingly, proportionately more ODI firms are foreign firms, including those that have investors from Hong Kong, China; Macau, China; and Taipei,China. Against the common perception that a lot of the ODI deals are initiated by the state, we find proportionately fewer ODI firms that are SOEs. It is possible that the government does not need to invest in the ODI firms directly in order to influence it. What they need to do is provide capital and other types of support to firms that invest abroad.
ODI firms are also on average more productive (in terms of labor productivity) and more export-intensive. Specifically, the average export-to-sales ratio of ODI firms is 0.25 higher than that of non-ODI firms. These findings are consistent with the export-promotion motive of ODI firms, which we will further confirm using transaction-level trade data below. ODI firms are also on average more R&D-intensive (measured by the ratio of R&D expenses to total sales) and have a slightly lower value-added/sales ratio. This finding is consistent with the theory of horizontal FDI that firms may offshore the most downstream part of global supply chains (e.g., marketing) to foreign affiliates.
As Table 4 already showed, ODI deals are unevenly distributed across sectors. Other unobserved factors may shape the revealed differences in observables between ODI and non-ODI firms. Besides the simultaneity bias, there could be selection bias behind the observed differences in the means reported in Table 5. Suppose more productive firms choose to undertake ODI overseas, which would be the case based on Helpman, Melitz, and Yeaple (2004) who emphasize higher fixed cost for horizontal FDI than that for exporting, the observed superior performance among ODI firms could be driven by selection. Without a feasible instrument in the dataset, we will rely on matching techniques (i.e., Heckman et al. 1997 and subsequent studies) to identify the effects of ODI on firm performance, relative to the control group that shares similar ex ante characteristics.
Before introducing the matching estimation results, we estimate the following linear specification, which fully controls for firm-specific, time-invariant determinants of post-ODI performance:
1
where and stand for firm and year fixed effects, and is the regression residual. is the measure of firm performance, including (log) sales, (log) value added, (log) employment, (log) total factor productivity (TFP), export to sales ratio, value added to sales ratio, R&D to sales ratio, new output sales to total sales ratio, and new product dummy, and material to sales ratio. Notice that any sector-level and province-level effects are already absorbed by firm fixed effects.15 The ODI dummy equals 1 in and after the year the firm reported positive ODI, 0 otherwise. By including firm fixed effects, we are identifying the within-firm relationship between ODI and firm performance. In addition to all non-ODI firms, in the control group, we also include observations of ODI firms before their engagement in ODI. Thus, the coefficients on the ODI dummy should be interpreted as the difference-in-difference in the average outcomes between ODI and non-ODI firms.16
Table 6 reports the results. Standard errors are clustered at the 2-digit industry level. Controlling for firm and year fixed effects, we find that engaging in ODI increases firms’ employment and propensity to innovate new products. The effects on sales and exports are also positive, but only marginally significant. We cannot find supporting evidence for a positive effect on R&D activities or productivity. This is inconsistent with the idea that ODI from emerging markets transfers technology from their affiliates in advanced economies. Although we are still far from establishing any causal relationship or tackling the selection bias, the regression results provide some preliminary evidence that whenever a significant effect of ODI on firm performance is identified, it is positive.
Table 6.
ODI Effects on Firm Performance (FE Regressions)
Dependentln(Value
ODI 0.047* 0.071 0.118*** –0.001 0.019
(0.022) (0.037) (0.016) (0.033) (0.012)
Firm FE Yes Yes Yes Yes Yes
Year FE Yes Yes Yes Yes Yes
R-squared 0.883 0.851 0.904 0.787 0.8639
No. of obs. 2,419,825 1,719,528 2,400,966 1,713,660 2,445,197
Dependent New Product New Product
Variable: VA/Sales R&D/Sales Sales Share Dummy Materials/Sales
ODI –0.005 0.001 0.013* 0.034*** 0.003
(0.005) (0.001) (0.006) (0.007) (0.005)
Firm FE Yes Yes Yes Yes Yes
Year FE Yes Yes Yes Yes Yes
R-squared 0.628 0.702 0.598 0.484 0.609
No. of obs. 1,706,349 857,519 1,845,020 2,445,197 1,640,541
Dependentln(Value
ODI 0.047* 0.071 0.118*** –0.001 0.019
(0.022) (0.037) (0.016) (0.033) (0.012)
Firm FE Yes Yes Yes Yes Yes
Year FE Yes Yes Yes Yes Yes
R-squared 0.883 0.851 0.904 0.787 0.8639
No. of obs. 2,419,825 1,719,528 2,400,966 1,713,660 2,445,197
Dependent New Product New Product
Variable: VA/Sales R&D/Sales Sales Share Dummy Materials/Sales
ODI –0.005 0.001 0.013* 0.034*** 0.003
(0.005) (0.001) (0.006) (0.007) (0.005)
Firm FE Yes Yes Yes Yes Yes
Year FE Yes Yes Yes Yes Yes
R-squared 0.628 0.702 0.598 0.484 0.609
No. of obs. 1,706,349 857,519 1,845,020 2,445,197 1,640,541
* = , ** = , *** = , ODI = outward direct investment, FE = fixed effects.
Note: ODI = 1 for all firm-years when and after a firm reported overseas investment, 0 otherwise. The number of observations fluctuates because data for some variables are not available in all years (e.g., R&D only for 2003–2005). Standard errors, clustered at the industry level (2-digit), are in brackets.
Source: Authors’ computations.
The next step is to implement the propensity-score matching methods to deal with the selection bias, which potentially drives the results reported so far. To this end, we will need to estimate propensity scores for each firm so that we can match ODI with similar non-ODI firms. We estimate a probit model, using a dummy for the firm's first year of ODI as the dependent variable. Specifically, we estimate the following specification:
2
where i, s, p, and t stand for the firm, industry (2-digit, 29 categories), province (30), and year (12), respectively. Sector fixed effects (29 categories), , and province fixed effects (30 categories), , are always included to capture all regional (e.g., ODI promotion policies) and sectoral unobserved determinants (e.g., comparative advantage) of ODI participation.
equals 1 if a firm starts engaging in ODI in year t, 0 otherwise. Notice that an ODI firm will only appear once in the sample, and firms that never report any ODI can appear multiple times in the sample. is a vector of (lagged) firm characteristics that are suspected to affect a firm's participation in ODI. Based on previous models on FDI and exports (e.g., Helpman, Melitz, and Yeaple 2004), we include firm TFP and employment as regressors.17 To capture the idea that exporters may have stronger incentive to invest overseas to facilitate trade, we include the ratio of exports to total sales. Specific to the institutional background of the PRC, where foreign firms and SOEs have better financial access (e.g., Zhu 2012) and even preferential policy treatments (Huang and Tang 2012), we include three firm ownership type dummies to indicate SOEs, foreign-owned (both wholly-owned and joint ventures) firms, and firms owned by investors from Hong Kong, China; Macau, China; and Taipei,China (i.e., domestic private firms are the excluded firm group). Moreover, to account for ODI that is driven by resource or technology seeking, we include firm-level measures of material and capital intensities, respectively.
Table 7 reports the probit estimation results. Similar to our explanations for the t-test and the regression results, we find that ex ante (lagged by one year) more productive (measured by TFP) and larger (measured by employment) firms are more likely to start investing in foreign markets. More export-intensive firms are also more likely to undertake ODI.
Table 7.
Participation in ODI Based on Manufacturing Firm Characteristics (Probit)
Dependent VariableODI Dummy
Sample 1 year before ODI 2 years before ODI
ln(TFP) 0.270*** 0.255***
(0.014) (0.014)
ln(Employment) 0.114*** 0.106***
(0.011) (0.011)
Export intensity 0.424*** 0.368***
(0.028) (0.027)
Capital intensity 0.409*** 0.354***
(0.052) (0.050)
Material intensity 0.691*** 0.616***
(0.093) (0.088)
SOE 0.0744** 0.0837**
(0.027) (0.026)
HKG; TAP; and Macau, China invested firms –0.0868** –0.0385
(0.033) (0.032)
Foreign –0.105*** –0.0513
(0.031) (0.030)
Industry FE Yes Yes
Province FE Yes Yes
No. of obs. 1,075,673 877,378
Dependent VariableODI Dummy
Sample 1 year before ODI 2 years before ODI
ln(TFP) 0.270*** 0.255***
(0.014) (0.014)
ln(Employment) 0.114*** 0.106***
(0.011) (0.011)
Export intensity 0.424*** 0.368***
(0.028) (0.027)
Capital intensity 0.409*** 0.354***
(0.052) (0.050)
Material intensity 0.691*** 0.616***
(0.093) (0.088)
SOE 0.0744** 0.0837**
(0.027) (0.026)
HKG; TAP; and Macau, China invested firms –0.0868** –0.0385
(0.033) (0.032)
Foreign –0.105*** –0.0513
(0.031) (0.030)
Industry FE Yes Yes
Province FE Yes Yes
No. of obs. 1,075,673 877,378
* = , ** = , *** = , FE = fixed effects, HKG = Hong Kong, China, ODI = outward direct investment, TAP = Taipei,China, TFP = total factor productivity, SOE = state-owned enterprises.
Note: The ODI dummy is equal to 1 for a firm in the year when it reports positive ODI, 0 for the same firm otherwise. ODI is equal to 0 for all observations of firms that never conducted any ODI during the sample period. All independent variables are lagged by one year in column 1, and by two years in column 2. Standard errors in parentheses.
Source: Authors’ computations, based on Manufacturing Firm Survey from the PRC's National Bureau of Statistics (NBS).
We also find that compared to domestic private firms, SOEs are more likely to undertake ODI, consistent with the conventional view that the PRC's ODI has a strong government backing. The first finding implies that the t-test results reported in Table 5 are pure correlation and cannot be inferred as a rejection that SOEs are less likely to invest abroad.18 Foreign firms and firms with major investors from Hong Kong, China; Macau, China; and Taipei,China are less likely to invest in a third market. These findings are consistent with the idea that foreign firms (e.g., Foxconn which assembles all products for Apple) tend to outsource assembly and processing tasks to the PRC and import the finished products back to the headquarters or export them directly to a third market. If these are their incentives to conduct ODI, they tend to initiate the investment directly from the headquarters, rather than doing it through their processing plants in the PRC. Column 2 shows that the results remain robust to using the same set of regressors lagged by two years instead of one year.
Before moving to the next section about the effects of ODI on firm and export performance, a final remark is in order. All results from Tables 5 to 7 are robust to the exclusion of firms that had ODI in Hong Kong, China. This eliminates the concern that some of the documented patterns are an artifact of investment intermediation in Hong Kong, China. In other words, our claim that a majority of ODI projects belong to the service sectors is robust to excluding the main tax haven for the PRC's ODI.
## VI. The Effects of ODI on Firm Performance
We use the concept of the average treatment effect on the treated (ATET) to gauge the effects of ODI on firm performance. To this end, we use the propensity-score matching methods, proposed by Rosenbaum and Rubin (1983) and applied by Heckman, Ichimura, and Todd (1997) in the program evaluation literature, to compare the post-ODI average outcomes of ODI firms with ex ante similar non-ODI firms.19
We first obtain the propensity score from each firm by estimating the probit model as specified in eq. (2). We then compute the average effect of ODI based on Rosenbaum and Rubin (1983), in which the authors propose reweighting estimators using propensity scores. Specifically, the estimator for the ATET is
3
The first term is just the mean of the outcomes for the ODI firms (i.e., when ). The second term is the weighted average of the outcomes of the control units, i.e., firms that do not conduct ODI, where the weights have been normalized by dividing each of them by the sum of all individual weights, so that they add up to one. A firm that is more likely to conduct ODI receives a larger weight by virtue of reweighting the propensity score with the probability of being a control unit. For instance, for a firm with zero probability of treatment, the control unit gets a weight of 0 (before normalization) because it is always observed as a control unit. In contrast, a control unit with a probability of treatment of 0.9, for instance, gets its outcome divided by 0.1 (before normalization) to reflect the fact that we observe only 1 in 10 of such units as control units. Thus, control units with higher probabilities of treatment receive more weight since they resemble the treated units more. Propensity score reweighting has the advantage of avoiding the bandwidth selection problem, as well as the need to decide what type of kernel to use or how many neighbors to select.
As with many two-step estimation procedures, using the simple formula for the variance of the estimator is incorrect. We adjust the standard errors in the second step by bootstrapping to account for measurement errors from the first stage estimation.20 Table A7 in the appendix shows the balancing test results for the matching. It clearly shows significant reductions in the differences in the average ex ante characteristics between ODI and non-ODI firms after matching.
Table 8 reports the ATET estimation results, using the same set of dependent variables from Table 6. In general, we find statistically more significant effects of ODI on firm performance. For a firm that invests abroad (including the year of investment), we find positive ATET of ODI on the firm's value added (0.29 log points), employment (0.42 log points), and TFP (0.16 log points). All these results are statistically significant at the 0.1% level. Compared to the matched non-ODI firms, ODI firms derive on average a slightly larger share of their sales from exports (a 0.02 log-point increase; significant at the 1% level). They also spend slightly more on R&D (a 0.2% higher share in total sales; significant at the 5% level), create new products, and derive a larger portion of sales from new products (significant at the 1% level). All results remain robust to the exclusion of Hong Kong, China as the host economy of ODI. These results are consistent with the hypotheses that ODI transfers technology or complements sales abroad by decreasing fixed cost of exporting. We will provide more evidence to disentangle these two channels in the following section.
Table 8.
ODI Effects on Firm Performance (Based on Propensity-score Matching 1 Year before ODI)
Dependentln(Value
ATET 0.288*** 0.257*** 0.067*** 0.163*** 0.021***
(0.026) (0.036) (0.016) (0.027) (0.005)
No. of obs. 1,145,251 934,158 1,140,946 934,689 1,148,377
Dependent New Output New Output
Variable: VA/Sales R&D/Sales Share Dummy Materials/Sales
ATET –0.008* 0.002* 0.018** 0.028** 0.005
(0.004) (0.001) (0.007) (0.009) (0.004)
No. of obs. 929,661 553,322 864,991 1,148,377 899,973
Dependentln(Value
ATET 0.288*** 0.257*** 0.067*** 0.163*** 0.021***
(0.026) (0.036) (0.016) (0.027) (0.005)
No. of obs. 1,145,251 934,158 1,140,946 934,689 1,148,377
Dependent New Output New Output
Variable: VA/Sales R&D/Sales Share Dummy Materials/Sales
ATET –0.008* 0.002* 0.018** 0.028** 0.005
(0.004) (0.001) (0.007) (0.009) (0.004)
No. of obs. 929,661 553,322 864,991 1,148,377 899,973
* = , ** = , *** = , ODI = outward direct investment, TFP = total factor productivity, VA = value added, ATET = average treatment effect on the treated.
Note: ODI = 1 for all firm-years when and after a firm reported overseas investment. The number of observations fluctuates because data for some variables are not available in all years (e.g., R&D only for 2003–2005). Bootstrapped standard errors are in brackets.
Source: Authors’ computations.
## VII. The Effects of ODI on Firms’ Trade Performance
The positive effects of ODI on firm performance documented in the previous section can be due to technology transfer or market expansion. For instance, the finding that a firm tends to create more products after ODI can be induced by new ideas or market expansion, which makes innovative activities profitable. In this section, we focus on the market-seeking (export-promotion) motive of ODI and examine how ODI affects a firm's export performance, and through the export channel enhances firm performance as documented above. Since the ODI dataset has no information on exports and imports, we merge the ODI data with the customs transaction-level trade data by firm names. Table A5 in the appendix shows the fractions of firms in the ODI data that can be merged to the customs data. Notice that the customs transaction-level data are only available for the year 2000–2006 (7 years).
The match success rate is fairly high for the last two years of the customs sample (2005–2006). Around 40% of the deals in our ODI dataset can be matched with an observation in the customs trade dataset. Table A4.2 in the appendix shows the distribution of the successfully matched observations across industries. The industrial distribution of deals in the matched sample is very close to the ones in the original ODI sample, providing some support that the matched observations are systematically unbiased across sectors. The challenges that arise for this merging are very similar to those encountered when we merge the ODI data with the customs data.21
Similar to the analysis of the ODI effects on firms’ overall performance, we apply the matching techniques outlined in Section 5 again to assess the ATET of ODI on firms’ export performance. To implement the matching estimation exercise, we need to first obtain the propensity score for each exporter (or importer), which requires an estimation of the ODI participation equation using probit again. While we try to include regressors as close as possible to those from manufacturing survey data, customs trade data only include information related to firms’ trade and we are restricted to use proxies. To proxy for TFP and firm size, we use the exporter's total export value (to the rest of the world). To proxy for material intensity (or reliance on imported inputs), we include the exporter's ratio of imports to exports. Similar to Table 7, we include a set of ownership type dummies, with private firms being the excluded group with no dummy included.
Consistent with Table 7, we find that larger (or more productive) exporting firms are more likely to start investing abroad. Similarly, compared to domestic private exporters, foreign exporters are less likely to undertake ODI. SOEs are also less likely to conduct ODI, compared to domestic private exporters. This result should not be taken as a rejection of the earlier finding that SOEs are more likely to invest abroad, as here we focus on a subset of firms—only those that export. Finally, in column 2, we show that the results remain robust to using the same set of regressors lagged by two years instead of one year.
Next we use the propensity scores estimated from Table 9 to assess the ATET of ODI on firms export performance. The dependent variables include ODI firm's export volume, export unit value, number of products (HS6) exported, and number of foreign countries served. All these variables are in log. We also conduct the same regression analysis by using the same four measures but for imports. By matching ODI exporters with non-ODI exporters based on ex ante characteristics, we aim to tackle the bias due to firms’ selection into ODI.
Table 9.
Participation in ODI for Customs (Probit)
Dependent VariableODI Dummy
Sample 1 year before ODI 2 years before ODI
Export 0.114*** 0.121***
(0.007) (0.007)
SOE –0.0801** –0.090**
(0.036) (0.040)
Foreign –0.565*** –0.574***
(0.035) (0.039)
Collective 0.085* 0.123**
(0.047) (0.050)
Import/Export –0.075 –0.0764058
(0.052) (0.057)
Industry FE Yes Yes
Province FE Yes Yes
No. of obs. 366,566 289,344
Dependent VariableODI Dummy
Sample 1 year before ODI 2 years before ODI
Export 0.114*** 0.121***
(0.007) (0.007)
SOE –0.0801** –0.090**
(0.036) (0.040)
Foreign –0.565*** –0.574***
(0.035) (0.039)
Collective 0.085* 0.123**
(0.047) (0.050)
Import/Export –0.075 –0.0764058
(0.052) (0.057)
Industry FE Yes Yes
Province FE Yes Yes
No. of obs. 366,566 289,344
* = , ** = , *** = , ODI = outward direct investment, SOE = state-owned enterprises, FE = fixed effects.
Note: The ODI dummy is equal to 1 for a firm in the year when it reports positive ODI, 0 for the same firm otherwise. ODI is equal to 0 for all observations of firms that never conducted any ODI during the sample period. Industry is an HS2 category. Domestic private firms are the excluded ownership type. Standard errors in parentheses.
Source: Authors’ computations, based on the PRC's customs transaction-level trade data.
Table 10 reports the matching estimation results. We find evidence that after investing overseas, existing exporters’ total export volume (in US dollars), export unit value, and number of destinations all increase. In particular, ODI exporters on average export about 0.6 log points more than non-ODI. Their unit value of the same product (a HS6 category) is 0.4 log points higher, while the number of export destinations increases by 0.2 log points. To the extent that unit value proxies for quality, we postulate that ODI can lead to quality upgrading, but higher unit values can also arise from more effective marketing. In the presence of fixed exporting costs, the increase in the number of export destinations after ODI suggests that ODI may be associated with an across-the-board reduction in those fixed costs. Collectively, these results confirm our conjecture that ODI from the PRC has been mostly related to export promotion. We find no effect in terms of the number of exported products after ODI.
Table 10.
Export Performance (Based on Propensity-score Matching 1 Year before ODI)
SampleAll firms (ODI = 0 for non-ODI firms and observations before ODI)
Dependent Variable:Exp ValueExp Unit ValNo. of HS6 ExpNo. of Exp Countries
ATET 0.586*** 0.396*** 0.031 0.239***
(0.058) (0.075) (0.051) (0.047)
No. of obs. 314,240 314,240 316,011 316,011
Dependent Variable: Imp Value Imp Unit Val No. of HS6 Imp No. of Exp Countries
ATET 0.363*** 0.286*** –0.070 0.057
(0.084) (0.010) (0.052) (0.040)
No. of obs. 307,119 307,119 307,119 310,766
SampleAll firms (ODI = 0 for non-ODI firms and observations before ODI)
Dependent Variable:Exp ValueExp Unit ValNo. of HS6 ExpNo. of Exp Countries
ATET 0.586*** 0.396*** 0.031 0.239***
(0.058) (0.075) (0.051) (0.047)
No. of obs. 314,240 314,240 316,011 316,011
Dependent Variable: Imp Value Imp Unit Val No. of HS6 Imp No. of Exp Countries
ATET 0.363*** 0.286*** –0.070 0.057
(0.084) (0.010) (0.052) (0.040)
No. of obs. 307,119 307,119 307,119 310,766
* = , ** = , *** = , ODI = outward direct investment, ATET = average treatment effect on the treated.
Note: The ODI dummy is equal to 1 for a firm in the year when it reports positive ODI, 0 for the same firm otherwise. ODI is equal to 0 for all observations of firms that never conducted any ODI during the sample period. Bootstrapped standard errors reported in brackets.
Source: Authors’ computations, based on the PRC's customs transaction-level trade data.
In the lower panel of Table 10, we repeat the same exercises but for importers. We find that importers that invest abroad have higher import volumes and unit values for a given product. These results show that ODI serves not only as a platform for exports, but also for imports, an aspect of ODI that has not received its deserved attention in the literature. However, there is no effect on import variety or the number of source countries for imports.
While Table 10 shows very strong export promotion effects of ODI, we still cannot rule out technology transfer or resource seeking as the source of the positive effects. To this end, we rely on firms’ imports to provide indirect evidence. If technology and resource seeking are important, we should expect ODI firms to import more capital goods and intermediate inputs, compared to non-ODI firms. To verify these speculations, we repeat the same estimation as in Table 10 but with dependent variables replaced by the shares of capital goods and intermediate inputs (materials) in firms’ exports and imports, respectively.
To classify a product (HS6) as capital good, raw material, and others, we use the list from the United Nations Broad Economic Categories (UN BEC) classification.22 If the increase in import volume documented in Table 10 is really associated with technology transfer, we should observe an increase in the share of capital goods in imports. The findings about the share of raw materials will then inform us about whether the PRC's ODI could be associated with resource seeking.
Table 11 reports the results. The first four columns report the results regarding the share of capital and materials in firms’ imports, in terms of total value or the total number of imported varieties. The last four columns report the results regarding those shares in exports. We find no evidence for a higher import share or fraction of capital goods in total imports by ODI firms. There is also no significant effect of ODI on imports of material. Based on trade of tangible goods, we find no evidence that Chinese ODI is technology seeking. However, it is worth noting that there can still be transfer of intangible asset from foreign affiliates to the headquarters in the PRC that are not observed in trade data, as pointed out by Atalay et al. (2014).
Table 11.
Capital Goods and Raw Materials in Exports and Imports and ODI
SampleAll Firms (ODI = 0 for non-ODI firms and observations before ODI)All Firms (ODI = 0 for non-ODI firms and observations before ODI)
Dep.Share ofShare ofShare ofShare of
VariableCapital GoodsMaterialsCapital GoodsMaterials
Firm No. of Firm No. of Firm No. of Firm No. of
Import Imported Import Imported Export Exported Export Exported
Volume Goods Volume Goods Volume Goods Volume Goods
ATET 0.027 –0.004 0.014 –0.003 0.042*** –0.003 –0.009 0.001
(0.015) (0.005) (0.012) (0.005) (0.013) (0.005) (0.006) (0.003)
No. of obs. 301,043 301,043 301,043 301,043 309,817 309,817 309,817 309,817
SampleAll Firms (ODI = 0 for non-ODI firms and observations before ODI)All Firms (ODI = 0 for non-ODI firms and observations before ODI)
Dep.Share ofShare ofShare ofShare of
VariableCapital GoodsMaterialsCapital GoodsMaterials
Firm No. of Firm No. of Firm No. of Firm No. of
Import Imported Import Imported Export Exported Export Exported
Volume Goods Volume Goods Volume Goods Volume Goods
ATET 0.027 –0.004 0.014 –0.003 0.042*** –0.003 –0.009 0.001
(0.015) (0.005) (0.012) (0.005) (0.013) (0.005) (0.006) (0.003)
No. of obs. 301,043 301,043 301,043 301,043 309,817 309,817 309,817 309,817
* = , ** = , *** = , ODI = outward direct investment, ATET = average treatment effect on the treated.
Note: The ODI dummy is equal to 1 for a firm in the year when it reports positive ODI, 0 for the same firm otherwise. ODI is equal to 0 for all observations of firms that never conducted any ODI during the sample period. Bootstrapped standard errors reported in brackets.
Source: Authors’ computations, based on the PRC's customs transaction-level trade data.
For completeness, we also examine the ODI effects on firms’ composition of exports. Interestingly, as reported in the last four columns, we find a significantly positive effect on firms’ capital export share, consistent with the export promotion or quality upgrading effects reported in Table 10. However, there is no effect when it is measured as a fraction of total export varieties. There is no evidence of an effect on exports of materials.
## VIII. Conclusion
Using a new panel dataset of Chinese multinational firms that covers close to 10,000 deals from all provinces and industries from 1998 to 2009, we find that over half of the ODI deals are in service sectors, with many of them appearing to be related to export promotion. In addition to documenting the pattern and trend of the PRC's ODI firms, this paper empirically examines both the determinants and effects of the PRC's ODI at the firm level.
We find that ex ante larger, more productive, and more export-intensive firms are more likely to start engaging in ODI. Using matching estimation techniques, we find that ODI enhances firm performance in terms of TFP, export intensity, product creation, and employment. To shed light on the relevant importance of technology transfer and export promotion of ODI, we use customs transaction-level trade data merged with the ODI firm list for analysis. We find that firms’ ODI participation is associated with better performance in both exports (in terms of volume, unit value, and number of destination countries) and imports (in terms of volume and unit value). We find no evidence of technology upgrading and resource seeking based on the pattern of imported products.
What lessons do we learn from the PRC about development strategies that are applicable for other developing nations and emerging markets? One of the intriguing findings in the literature about the PRC is its fast transition from processing exports, which mostly originate from foreign-invested exporting firms, to non-processing exports by indigenous Chinese firms. Our findings on the export promotion effects of ODI in the PRC imply that ODI may have played an important role in driving this transition.
It has been shown that inward FDI into the PRC has transferred know-how, technology, and management skills to the country. However, the benefits of promoting exports and inward FDI are diminishing for the PRC, about 20 years after the country's economic integration with the rest of the world, initiated by Deng's famous southern trip in 1992. This phenomenon is not specific to the PRC and has been or will be faced by many developing countries that lose comparative advantage in labor-intensive sectors due to increasing labor costs. When the average wage level of low-skilled workers continues to increase, a country will have to transition to more skill-intensive and capital-intensive sectors. While this transition can happen naturally (with some adjustment cost), there could be room for policies to make the transition smoother. In Chen and Tang (2013), we find evidence of skills upgrading and capital deepening through ODI, as revealed in the pattern of exported products from the PRC's exporters that engage in ODI. In summary, our paper shows that export-promoting ODI can potentially raise and sustain the benefits of exporting, which may in turn contribute to a country's structural transformation from low-skill manufacturing to high-skill manufacturing, and eventually from manufacturing to high-skill services.
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## Appendix
Table A1.
Industry Breakdown
IndustryFreq.Percent
Building and civil engineering 285 0.029
Nonferrous metals mining and dressing 212 0.022
Nonmetal mineral products 202 0.021
Garments, shoes, and caps manufacturing 189 0.019
Forestry 181 0.019
Real estate 169 0.017
Electric equipment and machinery 162 0.017
R&D 159 0.016
Geologic prospecting 157 0.016
Other financial activities 143 0.015
Metal products 135 0.014
Transport equipment 118 0.012
Food production 106 0.011
Waterway transport 101 0.010
Agriculture 86 0.009
Ordinary machinery 86 0.009
Software 84 0.009
Plastic products 82 0.008
Professional and technical services 82 0.008
Timber processing, bamboo, cane, palm fiber, and straw products 82 0.008
Food processing 77 0.008
Textile industry 75 0.008
Telecom and other information transmission 67 0.007
Securities 61 0.006
Leather, furs, down, and related products 58 0.006
Medical and pharmaceutical products 57 0.006
Raw chemical materials and chemical products 57 0.006
Telecom, computer, and other electronic equipment 57 0.006
Ferrous metals mining and dressing 55 0.006
Cultural, educational, and sports goods 53 0.005
Instruments, meters, cultural, and clerical machinery 48 0.005
Waste materials recycling and reprocessing 48 0.005
Catering 47 0.005
Smelting and pressing of ferrous metals 44 0.005
Computer services 42 0.004
Special purposes equipment 42 0.004
Smelting and pressing of nonferrous metals 41 0.004
Art and craft, and other manufacturing 34 0.003
Other services 34 0.003
Building installation 33 0.003
Science and technology exchange, and promotion services 33 0.003
Petroleum and natural gas extraction 32 0.003
Nonmetal minerals mining and dressing 29 0.003
Fishing 28 0.003
Furniture manufacturing 26 0.003
Rubber products 25 0.003
Air transport 22 0.002
Papermaking and paper products 22 0.002
Leasing 20 0.002
Railway transport 18 0.002
Chemical fiber 17 0.002
Building decoration 15 0.002
Animal husbandry 14 0.001
Beverage production 13 0.001
Hotels 13 0.001
Services to households 13 0.001
Storage 13 0.001
Culture and art 12 0.001
Highway transport 12 0.001
Production and supply of power, steam, and electricity 11 0.001
Services for agriculture, forestry, animal husbandry, and fishing 10 0.001
Urban public transport 10 0.001
Education 0.001
Other construction 0.001
Banking 0.001
Broadcasting, television, film, and audio 0.001
Health 0.001
News and publishing industry 0.001
Petroleum refining, coking, and nuclear energy 0.001
Printing and record medium reproduction 0.001
Tobacco processing and production 0.001
Post 0.001
Entertainment industry 0.001
Production and supply of water 0.001
Production and supply of gas 0.000
Coal mining and processing 0.000
Pipeline transport 0.000
Sports 0.000
Management of public facilities 0.000
Management of environment 0.000
Other 13 0.001
Total 9,744 1.000
IndustryFreq.Percent
Building and civil engineering 285 0.029
Nonferrous metals mining and dressing 212 0.022
Nonmetal mineral products 202 0.021
Garments, shoes, and caps manufacturing 189 0.019
Forestry 181 0.019
Real estate 169 0.017
Electric equipment and machinery 162 0.017
R&D 159 0.016
Geologic prospecting 157 0.016
Other financial activities 143 0.015
Metal products 135 0.014
Transport equipment 118 0.012
Food production 106 0.011
Waterway transport 101 0.010
Agriculture 86 0.009
Ordinary machinery 86 0.009
Software 84 0.009
Plastic products 82 0.008
Professional and technical services 82 0.008
Timber processing, bamboo, cane, palm fiber, and straw products 82 0.008
Food processing 77 0.008
Textile industry 75 0.008
Telecom and other information transmission 67 0.007
Securities 61 0.006
Leather, furs, down, and related products 58 0.006
Medical and pharmaceutical products 57 0.006
Raw chemical materials and chemical products 57 0.006
Telecom, computer, and other electronic equipment 57 0.006
Ferrous metals mining and dressing 55 0.006
Cultural, educational, and sports goods 53 0.005
Instruments, meters, cultural, and clerical machinery 48 0.005
Waste materials recycling and reprocessing 48 0.005
Catering 47 0.005
Smelting and pressing of ferrous metals 44 0.005
Computer services 42 0.004
Special purposes equipment 42 0.004
Smelting and pressing of nonferrous metals 41 0.004
Art and craft, and other manufacturing 34 0.003
Other services 34 0.003
Building installation 33 0.003
Science and technology exchange, and promotion services 33 0.003
Petroleum and natural gas extraction 32 0.003
Nonmetal minerals mining and dressing 29 0.003
Fishing 28 0.003
Furniture manufacturing 26 0.003
Rubber products 25 0.003
Air transport 22 0.002
Papermaking and paper products 22 0.002
Leasing 20 0.002
Railway transport 18 0.002
Chemical fiber 17 0.002
Building decoration 15 0.002
Animal husbandry 14 0.001
Beverage production 13 0.001
Hotels 13 0.001
Services to households 13 0.001
Storage 13 0.001
Culture and art 12 0.001
Highway transport 12 0.001
Production and supply of power, steam, and electricity 11 0.001
Services for agriculture, forestry, animal husbandry, and fishing 10 0.001
Urban public transport 10 0.001
Education 0.001
Other construction 0.001
Banking 0.001
Broadcasting, television, film, and audio 0.001
Health 0.001
News and publishing industry 0.001
Petroleum refining, coking, and nuclear energy 0.001
Printing and record medium reproduction 0.001
Tobacco processing and production 0.001
Post 0.001
Entertainment industry 0.001
Production and supply of water 0.001
Production and supply of gas 0.000
Coal mining and processing 0.000
Pipeline transport 0.000
Sports 0.000
Management of public facilities 0.000
Management of environment 0.000
Other 13 0.001
Total 9,744 1.000
Note: Industry classification is based on NBS 4-digit code.
Sources: PRC's Ministry of Commerce, PRC National Bureau of Statistics, and authors’ own calculation.
Table A2.
Distribution across Provinces
ProvinceFreq.PercentCum.
Zhejiang 1,993 20.45 20.45
Shandong 996 10.22 30.68
Jiangsu 938 9.63 40.30
Guangdong 920 9.44 49.74
Central Enterprises 568 5.83 55.57
Shanghai 508 5.21 60.79
Beijing 489 5.02 65.80
Fujian 410 4.21 70.01
Liaoning 341 3.50 73.51
Heilongjiang 302 3.10 76.61
Hunan 295 3.03 79.64
Tianjin 253 2.60 82.24
Yunnan 201 2.06 84.30
Henan 171 1.75 86.05
Hebei 170 1.74 87.80
Jilin 164 1.68 89.48
Sichuan 159 1.63 91.11
Guangxi 139 1.43 92.54
Xinjiang 123 1.26 93.80
Anhui 103 1.06 94.86
Chongqing 95 0.97 95.83
Jiangxi 89 0.91 96.75
Hubei 73 0.75 97.50
Shaanxi 73 0.75 98.25
Shanxi 72 0.74 98.98
Gansu 35 0.36 99.34
Hainan 28 0.29 99.63
Ningxia 13 0.13 99.76
Guizhou 12 0.12 99.89
Qinghai 0.09 99.98
Xizang 0.02 100.00
Total 9,744 100.00
ProvinceFreq.PercentCum.
Zhejiang 1,993 20.45 20.45
Shandong 996 10.22 30.68
Jiangsu 938 9.63 40.30
Guangdong 920 9.44 49.74
Central Enterprises 568 5.83 55.57
Shanghai 508 5.21 60.79
Beijing 489 5.02 65.80
Fujian 410 4.21 70.01
Liaoning 341 3.50 73.51
Heilongjiang 302 3.10 76.61
Hunan 295 3.03 79.64
Tianjin 253 2.60 82.24
Yunnan 201 2.06 84.30
Henan 171 1.75 86.05
Hebei 170 1.74 87.80
Jilin 164 1.68 89.48
Sichuan 159 1.63 91.11
Guangxi 139 1.43 92.54
Xinjiang 123 1.26 93.80
Anhui 103 1.06 94.86
Chongqing 95 0.97 95.83
Jiangxi 89 0.91 96.75
Hubei 73 0.75 97.50
Shaanxi 73 0.75 98.25
Shanxi 72 0.74 98.98
Gansu 35 0.36 99.34
Hainan 28 0.29 99.63
Ningxia 13 0.13 99.76
Guizhou 12 0.12 99.89
Qinghai 0.09 99.98
Xizang 0.02 100.00
Total 9,744 100.00
Sources: PRC's Ministry of Commerce and authors’ own calculations.
Table A3.
Number of ODI Firms by Ownership Type
YearState-ownedDomestic PrivateForeign
1998
1999
2000
2001
2002 28
2003 22
2004 39 13
2005 20 273 82
2006 21 366 79
2007 23 366 93
2008 12 335 79
2009 26 336 110
YearState-ownedDomestic PrivateForeign
1998
1999
2000
2001
2002 28
2003 22
2004 39 13
2005 20 273 82
2006 21 366 79
2007 23 366 93
2008 12 335 79
2009 26 336 110
ODI = outward direct investment.
Source: PRC's Ministry of Commerce.
Table A4.1.
Breakdown of Industries by NBS Classifications of NBS ODI Dataset
SectorFreq.Percent
Manufacturing 2,544 98.76
Mining 19 0.74
Power, gas, and water 13 0.50
SectorFreq.Percent
Manufacturing 2,544 98.76
Mining 19 0.74
Power, gas, and water 13 0.50
NBS = National Bureau of Statistics, ODI = outward direct investment.
Sources: PRC's Ministry of Commerce, PRC National Bureau of Statistics, and authors’ own calculation.
Table A4.2.
Breakdown of Industries by ODI Assigned Classifications of Customs ODI Dataset
IndustryFreq.PercentCum.
74 Business services 718 39.56 39.56
63 Wholesale trade 421 23.20 62.75
18 Garments, shoes, and caps manufacturing 46 2.53 65.29
47 Building and civil engineering 45 2.48 67.77
31 Nonmetal mineral products 32 1.76 69.53
39 R&D 31 1.71 71.24
75 Electric equipment and machinery 31 1.71 72.95
65 Retail trade 28 1.54 74.49
14 Food production 25 1.38 75.87
34 Metal products 23 1.27 77.13
19 Transport equipment 22 1.21 78.35
37 Leather, furs, down, and related products 22 1.21 79.56
30 Plastic products 20 1.10 80.66
60 Telecom and other information transmission 19 1.05 81.71
24 Cultural, educational, and sports goods 18 0.99 82.70
Forestry 17 0.94 83.64
20 Timber processing, bamboo, cane, palm fiber, and straw products 17 0.94 84.57
35 Ordinary machinery 17 0.94 85.51
Nonferrous metals mining and dressing 16 0.88 86.39
17 Textile industry 15 0.83 87.22
40 Telecom, computer, and other electronic equipment 14 0.77 87.99
78 Geologic prospecting 14 0.77 88.76
41 Instruments, meters, cultural, and clerical machinery 13 0.72 89.48
13 Food processing 12 0.66 90.14
71 Other financial activities 11 0.61 90.74
Agriculture 10 0.55 91.29
36 Special purposes equipment 10 0.55 91.85
26 Raw chemical materials and chemical products 0.50 92.34
27 Medical and pharmaceutical products 0.50 92.84
72 Real estate 0.44 93.28
76 Professional and technical services 0.44 93.72
Software 0.39 94.10
29 Waterway transport 0.39 94.49
32 Rubber products 0.39 94.88
54 Smelting and pressing of ferrous metals 0.39 95.26
62 Ferrous metals mining and dressing 0.39 95.65
22 Papermaking and paper products 0.33 95.98
33 Waste materials recycling and reprocessing 0.33 96.31
43 Smelting and pressing of nonferrous metals 0.33 96.64
69 Securities 0.33 96.97
83 Other services 0.28 97.25
Services to households 0.22 97.47
42 Fishing 0.22 97.69
61 Computer services 0.22 97.91
82 Art and craft, and other manufacturing 0.22 98.13
Building installation 0.17 98.29
21 Science and technology exchange, and promotion services 0.17 98.46
28 Furniture manufacturing 0.17 98.62
48 Catering 0.17 98.79
58 Chemical fiber 0.17 98.95
67 Storage 0.17 99.12
77 Petroleum and natural gas extraction 0.17 99.28
10 Highway transport 0.11 99.39
16 Nonmetal minerals mining and dressing 0.11 99.50
52 Tobacco processing and production 0.11 99.61
15 Air transport 0.06 99.72
55 Post 0.06 99.78
57 Leasing 0.06 99.83
59 Beverage production 0.06 99.89
73 Culture and art 0.06 99.94
90 Services for agriculture, forestry, animal husbandry, and fishing 0.06 100.00
Total 1,815 100
IndustryFreq.PercentCum.
74 Business services 718 39.56 39.56
63 Wholesale trade 421 23.20 62.75
18 Garments, shoes, and caps manufacturing 46 2.53 65.29
47 Building and civil engineering 45 2.48 67.77
31 Nonmetal mineral products 32 1.76 69.53
39 R&D 31 1.71 71.24
75 Electric equipment and machinery 31 1.71 72.95
65 Retail trade 28 1.54 74.49
14 Food production 25 1.38 75.87
34 Metal products 23 1.27 77.13
19 Transport equipment 22 1.21 78.35
37 Leather, furs, down, and related products 22 1.21 79.56
30 Plastic products 20 1.10 80.66
60 Telecom and other information transmission 19 1.05 81.71
24 Cultural, educational, and sports goods 18 0.99 82.70
Forestry 17 0.94 83.64
20 Timber processing, bamboo, cane, palm fiber, and straw products 17 0.94 84.57
35 Ordinary machinery 17 0.94 85.51
Nonferrous metals mining and dressing 16 0.88 86.39
17 Textile industry 15 0.83 87.22
40 Telecom, computer, and other electronic equipment 14 0.77 87.99
78 Geologic prospecting 14 0.77 88.76
41 Instruments, meters, cultural, and clerical machinery 13 0.72 89.48
13 Food processing 12 0.66 90.14
71 Other financial activities 11 0.61 90.74
Agriculture 10 0.55 91.29
36 Special purposes equipment 10 0.55 91.85
26 Raw chemical materials and chemical products 0.50 92.34
27 Medical and pharmaceutical products 0.50 92.84
72 Real estate 0.44 93.28
76 Professional and technical services 0.44 93.72
Software 0.39 94.10
29 Waterway transport 0.39 94.49
32 Rubber products 0.39 94.88
54 Smelting and pressing of ferrous metals 0.39 95.26
62 Ferrous metals mining and dressing 0.39 95.65
22 Papermaking and paper products 0.33 95.98
33 Waste materials recycling and reprocessing 0.33 96.31
43 Smelting and pressing of nonferrous metals 0.33 96.64
69 Securities 0.33 96.97
83 Other services 0.28 97.25
Services to households 0.22 97.47
42 Fishing 0.22 97.69
61 Computer services 0.22 97.91
82 Art and craft, and other manufacturing 0.22 98.13
Building installation 0.17 98.29
21 Science and technology exchange, and promotion services 0.17 98.46
28 Furniture manufacturing 0.17 98.62
48 Catering 0.17 98.79
58 Chemical fiber 0.17 98.95
67 Storage 0.17 99.12
77 Petroleum and natural gas extraction 0.17 99.28
10 Highway transport 0.11 99.39
16 Nonmetal minerals mining and dressing 0.11 99.50
52 Tobacco processing and production 0.11 99.61
15 Air transport 0.06 99.72
55 Post 0.06 99.78
57 Leasing 0.06 99.83
59 Beverage production 0.06 99.89
73 Culture and art 0.06 99.94
90 Services for agriculture, forestry, animal husbandry, and fishing 0.06 100.00
Total 1,815 100
ODI = outward direct investment.
Sources: PRC's Ministry of Commerce, PRC National Bureau of Statistics, and authors’ own calculation.
Table A5.
Success Rates of Matching between ODI and Customs Data, and ODI and Manufacturing Survey Data
CustomsCustoms
YearNBS MatchesMatches ExportMatches Import
1998 0.16
1999 0.11
2000 0.35 0.01 0.01
2001 0.29 0.01 0.01
2002 0.55 0.06 0.06
2003 0.47 0.05 0.05
2004 0.32 0.09 0.08
2005 0.47 0.39 0.37
2006 0.44 0.41 0.42
2007 0.39
2008 0.34
2009 0.27
Average 0.35 0.15 0.14
CustomsCustoms
YearNBS MatchesMatches ExportMatches Import
1998 0.16
1999 0.11
2000 0.35 0.01 0.01
2001 0.29 0.01 0.01
2002 0.55 0.06 0.06
2003 0.47 0.05 0.05
2004 0.32 0.09 0.08
2005 0.47 0.39 0.37
2006 0.44 0.41 0.42
2007 0.39
2008 0.34
2009 0.27
Average 0.35 0.15 0.14
NBS = National Bureau of Statistics, ODI = outward direct investment.
Sources: PRC's Ministry of Commerce, PRC National Bureau of Statistics, PRC Customs, and authors’ own calculation.
Table A6.
ODI Effects on Export Performance (FE Regressions)
SampleAll Firms (ODI = 0 for non-ODI firms and observations before ODI)
Dependent Var:Export Total ValueExp Unit Value# of Exp HS6Exp # of Country
ODI 0.017 0.103 0.079** 0.067**
(0.069) (0.057) (0.027) (0.023)
Firm FE Yes Yes Yes Yes
Year FE Yes Yes Yes Yes
R-squared 0.727 0.806 0.839 0.838
No. of obs. 717,355 717,355 751,589 751,588
Dependent Var: Imp Total Value Imp Unit Value Imp # of HS6 Value Imp # of Country
ODI 0.630*** 0.047 0.301*** 0.064*
(0.074) (0.083) (0.034) (0.030)
Firm FE Yes Yes Yes Yes
Year FE Yes Yes Yes Yes
R-squared 0.826 0.795 0.835 0.716
No. of obs. 659,392 659,392 659,392 677,740
SampleAll Firms (ODI = 0 for non-ODI firms and observations before ODI)
Dependent Var:Export Total ValueExp Unit Value# of Exp HS6Exp # of Country
ODI 0.017 0.103 0.079** 0.067**
(0.069) (0.057) (0.027) (0.023)
Firm FE Yes Yes Yes Yes
Year FE Yes Yes Yes Yes
R-squared 0.727 0.806 0.839 0.838
No. of obs. 717,355 717,355 751,589 751,588
Dependent Var: Imp Total Value Imp Unit Value Imp # of HS6 Value Imp # of Country
ODI 0.630*** 0.047 0.301*** 0.064*
(0.074) (0.083) (0.034) (0.030)
Firm FE Yes Yes Yes Yes
Year FE Yes Yes Yes Yes
R-squared 0.826 0.795 0.835 0.716
No. of obs. 659,392 659,392 659,392 677,740
* = , ** = , *** = , FE = fixed effects, NBS = National Bureau of Statistics, ODI = outward direct investment.
Note: ODI = 1 for all firm–years when and after a firm reported overseas investment. All dependent variables are in log form. All custom firms and treated firms prior to ODI are included in the control group. Robust standard errors reported in brackets.
Source: Authors’ computations.
Table A7.
Balancing Test of Matching ODI and non-ODI (NBS Sample)
Mean%Reduction
VariableSampleTreatedControl%BiasBiastp>t
Export Intensity Unmatched 0.441 0.202 62.1 21.89 0.000
Matched 0.441 0.443 −0.6 99.1 −0.05 0.962
Ln Employment Unmatched 5.722 4.703 85.8 32.65 0.000
Matched 5.716 5.736 −1.7 98.1 −0.01 0.989
Capital Intensity Unmatched 0.217 0.204 8.5 2.78 0.000
Matched 0.217 0.213 2.9 65.4 0.34 0.735
Material Intensity Unmatched −0.248 −0.279 11.4 3.20 0.001
Matched −0.248 −0.262 5.1 55.0 0.21 0.834
TFP Unmatched 6.258 5.477 72.3 26.02 0.000
Matched 6.254 6.298 −4.1 94.4 0.03 0.976
SOE Unmatched 0.252 0.176 18.5 6.64 0.000
Matched 0.251 0.244 1.8 90.0 0.13 0.893
HKG; TAP; and Macau, Unmatched 0.133 0.128 1.3 0.45 0.653
China invested firms Matched 0.133 0.134 −0.3 74.3 −0.01 0.990
Foreign Unmatched 0.169 0.119 14.4 5.23 0.000
Matched 0.169 0.176 −2.0 86.1 −0.46 0.643
Mean%Reduction
VariableSampleTreatedControl%BiasBiastp>t
Export Intensity Unmatched 0.441 0.202 62.1 21.89 0.000
Matched 0.441 0.443 −0.6 99.1 −0.05 0.962
Ln Employment Unmatched 5.722 4.703 85.8 32.65 0.000
Matched 5.716 5.736 −1.7 98.1 −0.01 0.989
Capital Intensity Unmatched 0.217 0.204 8.5 2.78 0.000
Matched 0.217 0.213 2.9 65.4 0.34 0.735
Material Intensity Unmatched −0.248 −0.279 11.4 3.20 0.001
Matched −0.248 −0.262 5.1 55.0 0.21 0.834
TFP Unmatched 6.258 5.477 72.3 26.02 0.000
Matched 6.254 6.298 −4.1 94.4 0.03 0.976
SOE Unmatched 0.252 0.176 18.5 6.64 0.000
Matched 0.251 0.244 1.8 90.0 0.13 0.893
HKG; TAP; and Macau, Unmatched 0.133 0.128 1.3 0.45 0.653
China invested firms Matched 0.133 0.134 −0.3 74.3 −0.01 0.990
Foreign Unmatched 0.169 0.119 14.4 5.23 0.000
Matched 0.169 0.176 −2.0 86.1 −0.46 0.643
HKG = Hong Kong, China, NBS = National Bureau of Statistics, ODI = outward direct investment, TAP = Taipei,China, TFP = total factor productivity, SOE = state-owned enterprises.
Source: Authors’ computations.
Table A8.
Balancing Test of Matching ODI and non-ODI (Customs Sample)
Mean%Reduction
VariableSampleTreatedControl%BiasBiastp>t
Import Share Unmatched 0.280 0.404 −41.6 −10.36 0.000
Matched 0.280 0.283 −1.1 97.4 −0.19 0.850
Export Unmatched 15.324 13.467 76.9 18.63 0.000
Matched 15.324 15.261 2.6 96.6 0.48 0.629
SOE Unmatched 0.329 0.136 46.9 13.58 0.000
Matched 0.329 0.331 −0.5 99.0 −0.07 0.946
POE Unmatched 0.445 0.166 63.6 18.13 0.000
Matched 0.445 0.427 4.2 93.5 0.63 0.532
Foreign Unmatched 0.226 0.687 −104.5 −24.04 0.000
Matched 0.226 0.241 −3.5 96.7 −0.61 0.541
Mean%Reduction
VariableSampleTreatedControl%BiasBiastp>t
Import Share Unmatched 0.280 0.404 −41.6 −10.36 0.000
Matched 0.280 0.283 −1.1 97.4 −0.19 0.850
Export Unmatched 15.324 13.467 76.9 18.63 0.000
Matched 15.324 15.261 2.6 96.6 0.48 0.629
SOE Unmatched 0.329 0.136 46.9 13.58 0.000
Matched 0.329 0.331 −0.5 99.0 −0.07 0.946
POE Unmatched 0.445 0.166 63.6 18.13 0.000
Matched 0.445 0.427 4.2 93.5 0.63 0.532
Foreign Unmatched 0.226 0.687 −104.5 −24.04 0.000
Matched 0.226 0.241 −3.5 96.7 −0.61 0.541
ODI = outward direct investment, SOE = state-owned enterprises, POE = privately-owned enterprises.
Source: Authors’ computations.
## Notes
*
The authors thank the editor, two anonymous referees, and all participants at the Asian Development Bank Conference for valuable comments. Chen would like to thank the Institute for International Economic Policy for financial support. Shan Li provided excellent research assistance.
1
UNCTAD (http://unctadstat.unctad.org/ReportFolders/reportFolders.aspx). The PRC's ODI flow rank is 17th in 2006, 12th in 2008, 5th in 2009, and 11th in 2011.
3
Given that a large fraction of the ODI firms in the PRC are non-manufacturing, the drawback of using manufacturing survey is that all ODI firms in the service sectors are dropped in our analysis. Notice that a firm can be classified as a service firm in the ODI list but can still be merged with the NBS data, as long as it has some businesses in manufacturing.
4
Here, technology is broadly defined to include production technology, management skills, and brand names.
5
Deng (2004) identified two additional motives: strategic assets (e.g., brands, marketing networks) and diversification. The focus of our paper focuses on the nonfinancial type of ODI. Clearly, because the PRC was itself a low-cost production base, cost minimization was not a major motivation of Chinese ODI.
6
By considering a dynamic model with uncertainty and learning, Conconi et al. (2013) show that ODI and export are substitutes in the short run but can be complements in the long run.
7
Liao and Tsui (2012) compare the aggregate ODI data from the PRC's Ministry of Commerce (MOFCOM) and the transaction-level data put together by the Heritage Foundation and show that the former dataset systematically underreports the PRC's ODI in mining. Their main explanation for the discrepancy is that MOFCOM did not track the final destination of ODI that went through tax havens (e.g., Hong Kong, China; Virgin Islands, etc.). While we verify that the overall patterns and the regression results remain robust to the exclusion of tax havens—in particular Hong Kong, China—two more remarks are in order. First, their datasets begin in 2005, which make the comparison between ours and theirs difficult. Second, it is not clear why the Heritage Foundation dataset provides a more comprehensive coverage of the PRC's ODI transactions compared to MOFCOM data. Selection could be an issue in the Heritage Foundation data as well.
8
Example of a product: 611241 – Women's or girls’ swimwear of synthetic fiber, knitted, or crocheted.
9
Readers are referred to Fernandes and Tang (2013) for details of this dataset.
10
There are concerns that the MOFCOM dataset is not representative. We will discuss the quality of the data and other related research in Section 3.
11
Cheng and Ma (2007) pointed out that the gap between official statistics and figures found in news reports appears to be big. We therefore focus mostly on the distribution of ODI across sectors and countries, and their associated impact, rather than the actual amount of ODI when reporting our summary statistics.
12
According to Cheng and Ma (2007), the Ministry of Commerce along with the All-China Federation of Industry and Commerce started a discussion on policy reforms that encourage private firms to go overseas. A draft document surfaced in 2006, which called for stronger support for domestic private and foreign firms in the areas of taxation, finance, insurance, and foreign exchange.
13
These success rates are in the same order of magnitude of merging the PRC's customs data with NBS data done by other scholars (e.g., Ma, Tang, and Zhang 2014; Manova and Yu 2013).
14
Appendixes A and B, available at http://www.hwtang.com/adb_appendix.html.
15
In unreported results, we verify that a majority of firms in the sample are single-plant firms.
16
The first difference is the difference from firms’ means (across the sample periods). The second difference is the difference from the non-ODI firms’ demeaned average within each year.
17
Since data on firms’ value added and thus TFP are only available for 1998–2007, the last two years of the sample 2008–2009 are automatically dropped.
18
During the sample period, the PRC's central government embarked on an active privatization program (Zhu 2012). The fraction of SOEs in the total number of enterprises dropped significantly, which may explain the seemingly contrasting results about SOEs’ likelihood to invest abroad between Tables 5 and 7.
19
Previous studies that have used the matching approach to search for causal effects of exporting on productivity include Girma, Greenaway, and Kneller (2003), Konings and Vandenbussche (2005), and De Loeker (2007), among others.
20
The implementation of the propensity score reweighting is closely based on the inverse probability regression as proposed in Brunell and DiNardo (2004). We use the Stata routine treatrew following Cerulli (2012).
21
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2021-04-10 19:18:27
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https://socratic.org/questions/what-is-the-derivative-of-y-sin-tan-5x
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# What is the derivative of y = sin(tan(5x))?
Dec 16, 2015
$y ' = 5 {\sec}^{2} \left(5 x\right) \cos \left(\tan \left(5 x\right)\right)$
#### Explanation:
Use the chain rule. The first overriding issue is the tangent function inside the sine function.
$y ' = \cos \left(\tan \left(5 x\right)\right) \frac{d}{\mathrm{dx}} \left[\tan \left(5 x\right)\right]$
Next, use chain rule again to find the derivative of the tangent function.
$\frac{d}{\mathrm{dx}} \left[\tan \left(5 x\right)\right] = {\sec}^{2} 5 x \frac{d}{\mathrm{dx}} \left[5 x\right] = 5 {\sec}^{2} \left(5 x\right)$
Multiply this back in to find $y '$.
$y ' = 5 {\sec}^{2} \left(5 x\right) \cos \left(\tan \left(5 x\right)\right)$
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2019-10-20 05:48:23
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|
https://email.esm.psu.edu/pipermail/macosx-tex/2010-November/045716.html
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# [OS X TeX] strange problem
Herbert Schulz herbs at wideopenwest.com
Thu Nov 11 08:05:08 EST 2010
On Nov 11, 2010, at 4:30 AM, claudio Procesi wrote:
>
>> dear colleaguesI have a very strange problem developed with Texshop. Often the produced PDF file does not appear although in the windows panel it seems to exist. If I make a new document with new name then it may appear in the new document. If I do a "save as" it usually disappears.
>
Howdy,
When you do the Save As... are you sure you are saving the file into the same directory you are working in? If you compile the file you shouldn't have to do a Save As.. since the pdf file is automatically produced in the same directory as the source file (or root file if you are using \include commands).
> When I close Texshop and then open it again I get the same problem and the PDF does not appear.
>> I tried to repair the disk and similar actions to no avail. Any suggestions?
>
I'm not sure I understand what you are saying here. When you say Close and the Open TeXShop are you double-clicking a source file which has the pdf file with the same base name proesent in the same directory?
This really sounds like a file system problem. What version of the OS are you using? How full is your Hard Drive?
Good Luck,
Herb Schulz
(herbs at wideopenwest dot com)
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2018-10-20 08:07:58
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https://forum.allaboutcircuits.com/threads/solved-nrf24l01.181739/
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# [SOLVED] nRF24L01
#### wernerdejong1972
Joined Sep 10, 2021
12
Hi all
I´m trying to get to arduinos to communicate together througt a nRF24L01. On the receiver side I want to turn the builtin LED to turn on every time I press a button on the transmitter. Furthermore I want to show a message on the Serial monitor that tells me the state off the button.
The part with the serial monitor works as planned but the builtin LED doesn´t turn on. What am I doing wrong?
C:
//RECEIVER
#include <SPI.h>
#include <nRF24L01.h>
#include <RF24.h>
RF24 radio(9, 10); // CE, CSN
boolean button_state = 0;
void setup() {
pinMode(LED_BUILTIN, OUTPUT);
Serial.begin(9600);
radio.setPALevel(RF24_PA_MIN); //You can set this as minimum or maximum depending on the distance between the transmitter and receiver.
}
void loop()
{
if (radio.available()) //Looking for the data.
{
char text[32] = ""; //Saving the incoming data
if (button_state == HIGH)
{
Serial.println(text);
delay(5);
digitalWrite(LED_BUILTIN, HIGH);
}
else
{
Serial.println(text);
delay(5);
digitalWrite(LED_BUILTIN, LOW);
}
}
delay(5);
}
Moderator edit: added code tags like this [code] ... your code... [/code]
#### MrChips
Joined Oct 2, 2009
27,651
Welcome to AAC!
Please post the code for the transmitter.
#### wernerdejong1972
Joined Sep 10, 2021
12
Sure
C-like:
// Transmitter
#include <SPI.h>
#include <nRF24L01.h>
#include <RF24.h>
RF24 radio(9, 10); // CE, CSN
const byte address[6] = "00001"; //Byte of array representing the address. This is the address where we will send the data. This should be same on the receiving side.
int button_pin = 2;
boolean button_state = 0;
void setup() {
pinMode(button_pin, INPUT);
radio.setPALevel(RF24_PA_MIN); //You can set it as minimum or maximum depending on the distance between the transmitter and receiver.
radio.stopListening(); //This sets the module as transmitter
}
void loop()
{
if(button_state == HIGH)
{
const char text[] = "Your Button State is HIGH";
}
else
{
const char text[] = "Your Button State is LOW";
}
delay(1000);
}
#### wernerdejong1972
Joined Sep 10, 2021
12
When I remove:
digitalWrite(LED_BUILTIN, HIGH);
in setup, the builtin LED turn on. Otherwise it´s impossible to get the LED to turn on.
#### Ya’akov
Joined Jan 27, 2019
6,853
Why are your delays between the serial write and the digital write?
#### wernerdejong1972
Joined Sep 10, 2021
12
Yaakov: I don´t really know, somebody recommended it. It has no function what so ever. It har no effect when I remove the delay.
#### Ya’akov
Joined Jan 27, 2019
6,853
Try adding a delay at the end of the on button state block, just to check.
C++:
{
Serial.println(text);
digitalWrite(LED_BUILTIN, HIGH);
delay(1000);
}
#### wernerdejong1972
Joined Sep 10, 2021
12
Nope, no succes....
#### Ya’akov
Joined Jan 27, 2019
6,853
OK. More quick troubleshooting. Comment out the serial write, just leave the digital write code in the conditional block.
#### wernerdejong1972
Joined Sep 10, 2021
12
Do you mean this?
C-like:
if (button_state == HIGH)
{
//Serial.println(text);
digitalWrite(LED_BUILTIN, HIGH);
delay(1000);
}
else
{
//Serial.println(text);
digitalWrite(LED_BUILTIN, HIGH);
delay(1000);
}
}
(didn´t work either...)
Last edited by a moderator:
#### Ya’akov
Joined Jan 27, 2019
6,853
Yes, that's what I meant. So I have this straight, when the serial code is uncommented, it does write to the serial monitor, correct?
#### wernerdejong1972
Joined Sep 10, 2021
12
yes, serial monitor works but no LED,
#### Ya’akov
Joined Jan 27, 2019
6,853
Which pins are the nRF24L01 wired to?
#### wernerdejong1972
Joined Sep 10, 2021
12
8 - 13
#### wernerdejong1972
Joined Sep 10, 2021
12
#### Ya’akov
Joined Jan 27, 2019
6,853
Pin 13 is probably your LED. You need to put an external one on a different pin, if so.
#### wernerdejong1972
Joined Sep 10, 2021
12
" SPI uses pin 13, so you can’t really use the built in LED while using SPI. "
could this be the solution? I´m not sure, because I´ve also tried pin3 without luck.
#### Ya’akov
Joined Jan 27, 2019
6,853
" SPI uses pin 13, so you can’t really use the built in LED while using SPI. "
could this be the solution? I´m not sure, because I´ve also tried pin3 without luck.
See above. pin 13 is hardwired to the builtin LED. You will ahve to put an external led on a different pin to make it work.
|
2023-01-28 07:29:12
|
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|
https://plainmath.net/secondary/geometry/high-school-geometry/similarity
|
# Get help with geometry similarity problems
Recent questions in Similarity
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### Based on the drawing below, in order for $\mathrm{△}ABC$ to be similar to $\mathrm{△}DEF$ by SAS similarity, which of the following needs to be true? A.$m\mathrm{\angle }D={38}^{\circ }$ B.$m\mathrm{\angle }D={52}^{\circ }$ C.$m\mathrm{\angle }F={38}^{\circ }$ D.$m\mathrm{\angle }F={52}^{\circ }$
smileycellist2 2021-08-11 Answered
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2022-05-29 08:13:54
|
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|
https://www.mybitcoin.com/cryptocurrency-profile-bitcoin/
|
Connect with us
# Cryptocurrency Profile: What Is Bitcoin, How It Works & Why BTC Now?
Published
on
Let’s face it: the only way you haven’t heard of Bitcoin (BTC) so far is if you’ve lived under a rock for the past few years. From tech-obsessed gimmick to global powerhouse and #1 investment in 2017, Bitcoin has indeed gone a long way. Although lots of people have now heard of it, or even invested in it, some mystery still revolves around the digital currency. Will it have the same incredible results in 2018, or is it just a bubble waiting to burst? Is this a good time to invest in Bitcoin? And at the end of the day, what IS Bitcoin?
## What is Bitcoin?
Bitcoin a cryptocurrency – a digital coin and payment system. It is one of the over 1300 in use today, but it is the first one that appeared. It was created as a decentralized payment solution since the entire network works without an administrator or single point of contact.
The technology based on which the peer-to-peer decentralized network works is called a blockchain, and many other coins (or altcoins, as the industry calls any other coin except Bitcoin and Ethereum) use this technology. And although the Bitcoin is the leader in the cryptocurrency industry, both in terms of price and overall market cap, the future might not look so good. If 2017 was the year in which cryptocurrencies hit the mainstream, 2018 would be all about practical solutions.
### What problems does it solve/What benefits does it bring?
At its origins, Bitcoin was designed as a decentralized, private network which users could use cheaply, quickly and anonymously. At first glance, the solution seemed near-perfect. No more banking fees, long wait times or being forced into sending money during work days. You now had a comfortable, convenient and secure way to send money to anyone around the globe, if you knew their Bitcoin address. The initial people who believed in Bitcoin used it to send vast amounts between them all over the world, back when the currency valued almost nothing and mining it was a breeze.
## Final Thoughts On Bitcoin
At the start of 2018, investing in Bitcoin can be seen in two ways. If you’re a skeptic, it is a train which is probably too late to be boarded, and will only bring disappointment – case in which you should look towards other currencies, like Ethereum, Ripple, Dash or Neo, all coins which will have their own profiles written shortly.
However, if you’re an enthusiast, investing in Bitcoin is like buying blue-chip stocks: sure, it probably won’t give you the same return it had last year, but it’s the biggest thing in the crypto business. And being so large means that’s it’s probably fail-proof. Regardless of what side you’re on, you should follow its price quite carefully if you want to invest in cryptocurrencies in 2018. It’s still big enough to take the entire market with it, regardless of the path it chooses.
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# Initial Coin Offering Investing: Top 7 ICO Research Factors To Study Before Buying
Published
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### 2. Transactions Are Traceable And Untraceable
Depending on who you ask and how much information you know. Bitcoin has a public permanent blockchain ledger that details how much a bitcoin wallet or public address is holding.
But, that’s all the information there is about that. You cannot find an associated name, address or phone number of the individual. Now you understand why we said it can be both traceable and untraceable at the same time.
Before anyone can determine your public address, they’ll need to know that it’s your first. That’s the only way they can determine that you own the wallet. This is how the FBI caught the founder of Silk Road, the world’s erstwhile biggest online drugs market.
### 3. Bitcoin Mining Is Getting Harder
Some people prefer mining bitcoin as a means of generating passive income. In the heydays, that used to be a moneymaker. Nowadays, not so much.
Because there’s a limit to the amount of bitcoins that can be in circulation -21 million- it’s getting harder to mine bitcoins. As a result, you have many people jumping ship and mining other less limiting cryptocurrencies.
Eventually, chances are people would mine that amount of bitcoins, resulting in max supply. When that happens, bitcoin mining as an income generating strategy will be retired.
### 4. 10,000 BTC For Two Pizzas
One of the most popular facts about bitcoin is, about a year after it was created, someone paid 10,000 BTC for two pizzas on May 22, 2010 –this is why Bitcoin Pizza day is marked on the same date every year.
### 6. Bitcoin’s Network Is Incredibly Powerful
You know how the world’s best supercomputers are thought of as something of a legend? Well, turns out bitcoin’s network is 500 times more powerful. This is because the network has a computing power of 2,046,364 Pflops/s. This is more power than 500 of the world’s fastest and most powerful supercomputers.
### 7. Transactions Are Irreversible
Here’s why you need to be careful about the address you’re sending money to. If you send some funds to the wrong address, you better pray real hard that the receiver is kind enough to send your money back.
And if you send it to a nonexistent address, your funds are gone all the same. In conventional payment networks, you just have to request a refund if you send money to a wrong account.
With bitcoin, that money is gone. And it’s worse because there’s no way to reach the receiver because, that information is not public at all. So, be careful with sending funds.
### 8. Bitcoin Investment Can Be Financially Rewarding
Early adopters of bitcoin made a fortune from the cryptocurrency. Even those who invested early 2016, made at least 6 times their money. The returns from bitcoin is singlehanded responsible for the term “cryptomillionaire”, as it has made quite a few guys millionaires.
If you bought just 100 BTC in 2013 for about $400, that 100 BTC would be valued at$750,000 today. And if you had sold in December, that’d have been an easy $1.8 million. While the cost of getting into bitcoin is pretty high now, it still has the potential to grow over the next few years. There are speculations that it could it the$100k mark within a few years.
So, if you’ve got the cash, bitcoin is still the most financially rewarding and stable cryptocurrency to invest in.
### 9. Lost Wallets Are Irretrievable
Retrieving lost wallets is almost impossible. The same goes for losing your private keys or password. There’s no way to request a password reset or get your private keys back.
This is why you need to keep your private keys, passwords and wallets safe. Many people who bought bitcoins in the early days and promptly forgot about it, or were unable to retrieve their details have regretted that decision.
Don’t be one of those people. Keep those wallets really close.
### 10. Costs Nothing To Send Funds
You know how you would have to pay some fees to send funds through your conventional financial institutions?
Well, that’s nonexistent with bitcoin transactions. You can send any amount of bitcoins you want without paying a dime in network fees. Now, sending funds was really fast in the past –more like instant.
But with the increased transaction volumes, your transactions can take some time to reflect.
# Cryptocurrencies Taxes Guide: How Bitcoin & IRS Reporting Works?
Published
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Tax season is right around the corner. That means, filing taxes, lots of paperwork and everything that makes us all dread the tax season. As if it’s not enough that you already have to deal with so much, you now need to worry about the role that cryptocurrencies play in your taxes.
For many people, the question often is, are cryptos taxable?
Should I include them when I’m filing my taxes?
Does the IRS even recognize cryptos as an investment or financial instrument, considering that even the SEC hasn’t really decided on what to make of it?
Well, the answer is yes, unfortunately.
You see, the IRS has determined that cryptocurrencies are in the same category as property. The document outlining this is the Notice 2014-21, which is accessible on the IRS website.
In summary, the document highlights the IRS’s recognition of cryptos as property and refers to it as a Convertible Virtual Currency. This means that it can be exchanged for FIAT currencies like the dollar.
What they didn’t highlight however, is how to go about filing the taxes for cryptos. It is assumed that the process will be in a manner similar to that of properties.
Whatever the case, the IRS considers every action involving the buying, selling and exchange of cryptocurrencies as taxable events. So, if you were to buy some Tether (USDT) with cash, that’s a taxable event.
If you exchange that for Ethereum (ETH), that’s another taxable event. However, if you move the Ethereum from the exchange e.g. Binance to your personal Myetherwallet, that’s not considered a taxable event.
Bottom line, any FIAT to Cryptocurrency or Cryptocurrency to Cryptocurrency exchange you make within a fiscal year, will be taxable. So, as long as it implied reward of some sort –whether you profit from it or not is moot- you will pay taxes on it.
However, if you hold it in keeping without selling or exchanging, you will not be taxed –this is even more so when you move from one of your wallets to another.
### So, How Do You Determine How Much To Pay The IRS?
Well, since cryptocurrencies are taxed pretty much the same way as your property, assets, investments, stocks or bonds, you will need to factor in your capital gains and losses before paying the taxes.
This means that to determine the correct amount you need to pay, you’ll have to first determine the following:
• Date of initial cryptocurrency purchase or exchange
• The exact costs involved in the transaction (this also includes any costs, charges or fees accrued in the process; not just the cost of the cryptocurrency you purchased)
• Date of cryptocurrency exchange for cash or sale
• Exact price at which cryptocurrency was sold and total income/revenue from the sale (this also includes charges and fees)
While it’s true that tracking your trades might be somewhat overwhelming particularly if you do a lot of trading, the reality is this might be one of the best ways to protect yourself from the IRS.
The IRS will not accept negligence or ignorance as an excuse. So, you should actively seek to document all your trades. Also, there’s the added advantage of tax reports provided by certain exchanges.
These exchanges understand that there could be tax problems associated with their traders’ accounts. So, they have taken the initiative of making tax reports of their trades available on the exchange. Check your exchange to see if they offer such services –usually comes free of charge too.
### How About Capital Gains and Losses For Taxes?
When it comes to cryptocurrency tax filings, there are different types of gains and losses. For the capital gains, there’s the
• Short term gains
• Long term gains
Short term gains are usually placed in the same category as income. So, whatever you make from your trades will most likely be subject to tax at your current tax bracket. Long term gains however, are taxed differently, and are usually lower than your short term capital gains. Long term capital gains often refer to the profits you make from “hodling” your cryptocurrency.
So, if you bought 10 ETH when it was priced at $7.95 in November 2016, and then stored it until December 2017, when it was valued at over$1,300, you will pay taxes on the profit made from holding the cryptocurrency. In this instance, that would be the $1,292.05 you made per ETH after subtracting the$7.95 from the $1,300. Losses on the other hand is how much you lost from the investment. If, using the earlier example of 10 ETH, you bought at$1,300 in December 2017, and sold recently at $800, your capital losses would be valued at$500 per ETH.
You’ll need to report this too in your tax filing and deduct it from your taxes. Please note however, that you can only claim up to \$3,000 in losses.
Also, with the Tax Cuts and Jobs Act kicking in this year, this might be the last time to claim tax losses on property –remember cryptocurrencies are categorized as property. So, if your ethereum wallet was hacked or you lost your cryptocurrency, this is the right time to claim those losses. You won’t get that opportunity in 2019.
## Cryptocurrencies Taxes Guide Conclusion
If you had a great year with cryptocurrencies last year, chances are you might be tempted to completely skip the inclusion of your cryptocurrency gains and proceeds from your tax declarations and filings.
Well, we wouldn’t recommend that. The IRS is pretty serious about cryptocurrency proceeds tax as evidenced in their requisitioning of date from Coinbase. It’s only a matter of time before they do the same for all other popular exchanges. Once they do, you could potentially be in trouble.
It’s best to just file the taxes and be done with it. Better to be free and solvent so you can take advantage of the opportunities that 2018 will present, than be in jail because you omitted your crypto portfolio.
#### Trending
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2022-12-09 13:11:53
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https://plotly.net/reference/plotly-net-traceobjects-error.html
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## Error Type
Error type inherits from dynamic object
### Constructors
Constructor Description Error() Full Usage: Error() Returns: Error Returns: Error
### Static members
Static member Description Error.init (?Visible, ?Type, ?Symmetric, ?Array, ?Arrayminus, ?Value, ?Valueminus, ?Traceref, ?Tracerefminus, ?Copy_ystyle, ?Color, ?Thickness, ?Width) Full Usage: Error.init (?Visible, ?Type, ?Symmetric, ?Array, ?Arrayminus, ?Value, ?Valueminus, ?Traceref, ?Tracerefminus, ?Copy_ystyle, ?Color, ?Thickness, ?Width) Parameters: ?Visible : bool - Determines whether or not this set of error bars is visible. ?Type : ErrorType - Determines the rule used to generate the error bars. If "constant, the bar lengths are of a constant value. Set this constant in value. If "percent", the bar lengths correspond to a percentage of underlying data. Set this percentage in value. If "sqrt", the bar lengths correspond to the square of the underlying data. If "data", the bar lengths are set with data set array. ?Symmetric : bool - Determines whether or not the error bars have the same length in both direction (top/bottom for vertical bars, left/right for horizontal bars. ?Array : seq<'a> - Sets the data corresponding the length of each error bar. Values are plotted relative to the underlying data. ?Arrayminus : seq<'b> - Sets the data corresponding the length of each error bar in the bottom (left) direction for vertical (horizontal) bars Values are plotted relative to the underlying data. ?Value : float - Sets the value of either the percentage (if type is set to "percent") or the constant (if type is set to "constant") corresponding to the lengths of the error bars. ?Valueminus : float - Sets the value of either the percentage (if type is set to "percent") or the constant (if type is set to "constant") corresponding to the lengths of the error bars in the bottom (left) direction for vertical (horizontal) bars ?Traceref : int - ?Tracerefminus : int - ?Copy_ystyle : bool - ?Color : Color - Sets the stoke color of the error bars. ?Thickness : float - Sets the thickness (in px) of the error bars. ?Width : float - Sets the width (in px) of the cross-bar at both ends of the error bars. Returns: Error Returns a new Error object with the given styling. ?Visible : bool Determines whether or not this set of error bars is visible. ?Type : ErrorType Determines the rule used to generate the error bars. If "constant, the bar lengths are of a constant value. Set this constant in value. If "percent", the bar lengths correspond to a percentage of underlying data. Set this percentage in value. If "sqrt", the bar lengths correspond to the square of the underlying data. If "data", the bar lengths are set with data set array. ?Symmetric : bool Determines whether or not the error bars have the same length in both direction (top/bottom for vertical bars, left/right for horizontal bars. ?Array : seq<'a> Sets the data corresponding the length of each error bar. Values are plotted relative to the underlying data. ?Arrayminus : seq<'b> Sets the data corresponding the length of each error bar in the bottom (left) direction for vertical (horizontal) bars Values are plotted relative to the underlying data. ?Value : float Sets the value of either the percentage (if type is set to "percent") or the constant (if type is set to "constant") corresponding to the lengths of the error bars. ?Valueminus : float Sets the value of either the percentage (if type is set to "percent") or the constant (if type is set to "constant") corresponding to the lengths of the error bars in the bottom (left) direction for vertical (horizontal) bars ?Traceref : int ?Tracerefminus : int ?Copy_ystyle : bool ?Color : Color Sets the stoke color of the error bars. ?Thickness : float Sets the thickness (in px) of the error bars. ?Width : float Sets the width (in px) of the cross-bar at both ends of the error bars. Returns: Error Error.style (?Visible, ?Type, ?Symmetric, ?Array, ?Arrayminus, ?Value, ?Valueminus, ?Traceref, ?Tracerefminus, ?Copy_ystyle, ?Color, ?Thickness, ?Width) Full Usage: Error.style (?Visible, ?Type, ?Symmetric, ?Array, ?Arrayminus, ?Value, ?Valueminus, ?Traceref, ?Tracerefminus, ?Copy_ystyle, ?Color, ?Thickness, ?Width) Parameters: ?Visible : bool - Determines whether or not this set of error bars is visible. ?Type : ErrorType - Determines the rule used to generate the error bars. If "constant, the bar lengths are of a constant value. Set this constant in value. If "percent", the bar lengths correspond to a percentage of underlying data. Set this percentage in value. If "sqrt", the bar lengths correspond to the square of the underlying data. If "data", the bar lengths are set with data set array. ?Symmetric : bool - Determines whether or not the error bars have the same length in both direction (top/bottom for vertical bars, left/right for horizontal bars. ?Array : seq<'a> - Sets the data corresponding the length of each error bar. Values are plotted relative to the underlying data. ?Arrayminus : seq<'b> - Sets the data corresponding the length of each error bar in the bottom (left) direction for vertical (horizontal) bars Values are plotted relative to the underlying data. ?Value : float - Sets the value of either the percentage (if type is set to "percent") or the constant (if type is set to "constant") corresponding to the lengths of the error bars. ?Valueminus : float - Sets the value of either the percentage (if type is set to "percent") or the constant (if type is set to "constant") corresponding to the lengths of the error bars in the bottom (left) direction for vertical (horizontal) bars ?Traceref : int - ?Tracerefminus : int - ?Copy_ystyle : bool - ?Color : Color - Sets the stoke color of the error bars. ?Thickness : float - Sets the thickness (in px) of the error bars. ?Width : float - Sets the width (in px) of the cross-bar at both ends of the error bars. Returns: Error -> Error Returns a function that applies the given style parameters to an Error object ?Visible : bool Determines whether or not this set of error bars is visible. ?Type : ErrorType Determines the rule used to generate the error bars. If "constant, the bar lengths are of a constant value. Set this constant in value. If "percent", the bar lengths correspond to a percentage of underlying data. Set this percentage in value. If "sqrt", the bar lengths correspond to the square of the underlying data. If "data", the bar lengths are set with data set array. ?Symmetric : bool Determines whether or not the error bars have the same length in both direction (top/bottom for vertical bars, left/right for horizontal bars. ?Array : seq<'a> Sets the data corresponding the length of each error bar. Values are plotted relative to the underlying data. ?Arrayminus : seq<'b> Sets the data corresponding the length of each error bar in the bottom (left) direction for vertical (horizontal) bars Values are plotted relative to the underlying data. ?Value : float Sets the value of either the percentage (if type is set to "percent") or the constant (if type is set to "constant") corresponding to the lengths of the error bars. ?Valueminus : float Sets the value of either the percentage (if type is set to "percent") or the constant (if type is set to "constant") corresponding to the lengths of the error bars in the bottom (left) direction for vertical (horizontal) bars ?Traceref : int ?Tracerefminus : int ?Copy_ystyle : bool ?Color : Color Sets the stoke color of the error bars. ?Thickness : float Sets the thickness (in px) of the error bars. ?Width : float Sets the width (in px) of the cross-bar at both ends of the error bars. Returns: Error -> Error
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2022-08-16 07:00:32
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https://www.tutorialspoint.com/working-principle-of-star-delta-starter
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# Working Principle of Star-Delta Starter
Digital ElectronicsElectronElectronics & Electrical
## Circuit Diagram and Working Principle of Star-Delta Starter
The figure shows the connection diagram of a 3-phase induction motor with a star-delta starter. The star-delta starter is a very common type of starter and is extensively used for starting the squirrel cage induction motors. It is used for starting a squirrel cage induction motor which is designed to run normally on delta connected stator winding.
When the switch S is in the START position, the stator windings are connected in star. When the motor attains a speed about 80 % of rated speed, then the changeover switch S is thrown to the RUN position which connects the stator windings in delta.
By connecting the stator windings, first in star and then in delta, the line current drawn by the motor at starting is reduced to one-third than the starting current with the windings connected in delta. At the time starting, when the stator windings are connected in star, then each phase gets a voltage equal to 𝑉𝐿/√3, where VL is the line voltage.
Since the torque developed by an induction motor is directly proportional to the square of the applied voltage, the star-delta starter reduces the starting torque to one-third as compared to the torque obtained with the direct-delta starting.
## Theory of Star-Delta Starter
At starting, the stator windings are connected in star fashion, thus the voltage across each phase winding is
$$\mathrm{𝑉_1 =\frac{𝑉_𝐿}{\sqrt{3}}}$$
Where,
• VL = Line voltage
• V1 = Stator phase voltage
Let,
𝐼𝑠𝑡.𝑌.𝑝ℎ = Starting current per phase with stator windings connected in star
𝐼𝑠𝑡.𝑌.𝐿 = Starting line current with stator windings connected in star
Also, for star connection, the line current is equal to the phase current, i.e.,
$$\mathrm{𝐼_{𝑠𝑡.𝑌.𝐿} = 𝐼_{𝑠𝑡.𝑌.𝑝ℎ}}$$
Again,
𝐼𝑠𝑡.∆.𝑝ℎ = Starting current per phase with stator windings connected in delta
𝐼𝑠𝑡.∆.𝐿 = Starting line current with stator windings connected in delta
𝐼𝑠𝑐.∆.𝑝ℎ = Short circuit phase current with stator windings connected in delta
𝑍𝑒10 = Equivalent impedance per phase of motor referred to stator at standstill
Therefore, the starting current per phase with starting windings connected in star is given by,
$$\mathrm{𝐼_{𝑠𝑡.𝑌.𝑝ℎ} =\frac{𝑉_1}{𝑍_{𝑒10}}=\frac{𝑉_𝐿}{\sqrt{3} 𝑍𝑒10}… (1)}$$
And the starting current per phase with starting windings connected in delta is
$$\mathrm{𝐼_{𝑠𝑡.∆.𝑝ℎ} =\frac{𝑉_𝐿}{𝑍_{𝑒10}}… (2)}$$
For the delta connection,
$$\mathrm{Line \:current = \sqrt{3} × Phase\: current}$$
$$\mathrm{⇒ 𝐼_{𝑠𝑡.∆.𝐿} = \sqrt{3} × 𝐼_{𝑠𝑡.∆.𝑝ℎ} =\frac{\sqrt{3} × 𝑉_𝐿}{𝑍_{𝑒10}}… (3)}$$
Now, taking the ratio of starting line current with star-delta starting to the starting line current with the direct delta switching is
$$\mathrm{\frac{𝐼_{𝑠𝑡.𝑌.𝐿}}{𝐼_{𝑠𝑡.∆.𝐿}}=\frac{𝐼_{𝑠𝑡.𝑌.𝑝ℎ}}{𝐼_{𝑠𝑡.∆.𝐿}}=\frac{(𝑉_𝐿/\sqrt{3}𝑍_{𝑒10})}{\sqrt{3}(𝑉_𝐿/𝑍_{𝑒10})}=\frac{1}{3}}$$
⇒ 𝐼_{𝑠𝑡.𝑌.𝐿} = 𝐼_{𝑠𝑡.𝑌.𝑝ℎ} =\frac{1}{3}× 𝐼_{𝑠𝑡.∆.𝐿} … (4)
Hence, with the star-delta starter, the starting current from the main supply is one-third of that with the direct delta switching.
Also,
$$\mathrm{\frac{Starting\:torque\:with \:Y − ∆ \:starter}{Starting torque with direct delta switching} =\frac{(𝑉_𝐿/\sqrt{3})^2}{𝑉_𝐿^2} =\frac{1}{3}… (5)}$$
Thus, with the star-delta starting, the starting torque is reduced to one-third times of the starting torque obtained with the direct delta switching.
Published on 28-Aug-2021 14:30:04
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2022-05-21 14:40:52
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http://mathcentral.uregina.ca/QQ/database/QQ.09.11/h/reuchen1.html
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SEARCH HOME
Math Central Quandaries & Queries
Question from Reuchen, a student: Find equations of the lines tangent to y^2=4x and containing (-2,1).
Hi Reuchen,
Suppose that $P$ is a point on the curve where the tangent line to the curve at $P$ passes through $(-2, 1).$ Let $P$ have coordinates $(a, b)$ then since $P$ is on the curve we know that $b^2 = 4a.$
1. Find the slope of the line containing $P = (a, b)$ and $(-2, 1).$
2. Differentiate both sides of $y^2 = 4 x$ with respect to $x$ and solve for $y'.$ Evaluate $y'$ at $P.$
The value of $y'$ at $P$ and the slope you found in part 1. are both the slope of the tangent line to the curve at $P.$ Substitute for $a$ using $b^2 = 4a$ and solve for b.
Penny
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
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2019-02-20 05:45:53
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https://socratic.org/questions/how-do-you-write-6000-in-scientific-notation#641347
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How do you write 6000 in scientific notation?
Jul 17, 2018
$6 \times {10}^{3}$
Explanation:
In science the first number must be between 1 and 10 (excluding 10).
The following is in the form of ${10}^{x}$
x represents number of zeros.
Therefore your number is $6 \times {10}^{3}$
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2021-10-16 03:53:54
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https://www.zbmath.org/?q=an%3A1153.11058
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# zbMATH — the first resource for mathematics
New families of quadratic almost perfect nonlinear trinomials and multinomials. (English) Zbl 1153.11058
The authors present two families of APN functions on certain finite binary fields, one on fields of order $$2^{2k}$$ for $$k$$ not divisible by 2, and the other on fields of order $$2^{3k}$$ for $$k$$ not divisible by 3. The polynomials they present are specific in terms of their roots and number of terms. The polynomials in the first family have between three and $$k+2$$ terms, the second family’s polynomials have three terms.
##### MSC:
11T06 Polynomials over finite fields 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 94A60 Cryptography
Full Text:
##### References:
[1] Budaghyan, L.; Carlet, C., Classes of quadratic APN trinomials and hexanomials and related structures, preprint, available at · Zbl 1177.94134 [2] Budaghyan, L.; Carlet, C.; Felke, P.; Leander, G., An infinite class of quadratic APN functions which are not equivalent to power mappings, () [3] L. Budaghyan, C. Carlet, G. Leander, A class of quadratic APN binomials inequivalent to power functions, preprint · Zbl 1177.94135 [4] Budaghyan, L.; Carlet, C.; Pott, A., New constructions of almost bent and almost perfect nonlinear functions, IEEE trans. inform. theory, 52, 3, 1141-1152, (2006) · Zbl 1177.94136 [5] C. Carlet, Boolean functions for cryptography and error correcting codes, in: P. Hammer, Y. Crama (Eds.), Boolean Methods and Models, Cambridge Univ. Press, a chapter of the monography, in press · Zbl 1209.94035 [6] Carlet, C.; Charpin, P.; Zinoviev, V., Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. codes cryptogr., 15, 2, 125-156, (1998) · Zbl 0938.94011 [7] J. Dillon, Slides from talk given at “Polynomials over Finite Fields and Applications,” held at Banff International Research station, November 2006 [8] Edel, Y.; Kyureghyan, G.; Pott, A., A new APN function which is not equivalent to a power mapping, IEEE trans. inform. theory, 52, 2, 744-747, (2006) · Zbl 1246.11185 [9] Nyberg, K., Differentially uniform mappings for cryptography, (), 55-64 · Zbl 0951.94510 [10] Rothaus, O., On bent functions, J. combin. theory ser. A, 20, 181-199, (1976) · Zbl 0336.12012 [11] J.F. Voloch, Symmetric cryptography and algebraic curves, preprint · Zbl 1151.14319
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2021-04-20 06:52:28
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https://physics.stackexchange.com/questions/233266/how-can-i-solve-this-quantum-mechanical-paradox/233271
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# How can I solve this quantum mechanical "paradox"?
Let a (free) particle move in $[0,a]$ with cyclic boundary condition $\psi(0)=\psi(a)$. The solution of the Schrödinger-equation can be put in the form of a plane wave. In this state the standard deviation of momentum is $0$, but $\sigma_x$ must be finite. So we find that $\sigma_x\sigma_p=0$. Is something wrong with the uncertainty principle?
This is what happens if one cares not for the subtlety that quantum mechanical operators are typically only defined on subspaces of the full Hilbert space.
Let's set $$a=1$$ for convenience. The operator $$p =-\mathrm{i}\hbar\partial_x$$ acting on wavefunctions with periodic boundary conditions defined on $$D(p) = \{\psi\in L^2([0,1])\mid \psi(0)=\psi(1)\land \psi'\in L^2([0,1])\}$$ is self-adjoint, that is, on the domain of definition of $$p$$, we have $$p=p^\dagger$$, and $$p^\dagger$$ admits the same domain of definition. The self-adjointness of $$p$$ follows from the periodic boundary conditions killing the surface terms that appear in the $$L^2$$ inner product $$\langle \phi,p\psi\rangle - \langle p^\dagger \phi,\psi\rangle = \int\overline{\phi(x)}\mathrm{i}\hbar\partial_x\psi(x) - \overline{\mathrm{i}\hbar\partial_x\phi(x)}\psi(x) = 0$$ for every $$\psi\in D(p)$$ and every $$\phi\in D(p^\dagger) = D(p)$$, but not for $$\phi$$ with $$\phi(0)\neq\phi(1)$$.
Now, for the question of the commutator: the multiplication operator $$x$$ is defined on the entire Hilbert space, since for $$\psi\in L^2([0,1])$$ $$x\psi$$ is also square-integrable. For the product of two operators $$A,B$$, we have the rule $$D(AB) = \{\psi\in D(B)\mid B\psi\in D(A)\}$$ and $$D(A+B) = D(A)\cap D(B)$$ so we obtain \begin{align} D(px) & = \{\psi\in L^2([0,1])\mid x\psi\in D(p)\} \\ D(xp) & = D(p) \end{align} and $$x\psi\in D(p)$$ means $$0\cdot \psi(0) = 1\cdot\psi(1)$$, that is, $$\psi(1) = 0$$. Hence we have $$D(px) = \{\psi\in L^2([0,1])\mid \psi'\in L^2([0,1]) \land \psi(1) = 0\}$$ and finally $$D([x,p]) = D(xp)\cap D(px) = \{\psi\in L^2([0,1])\mid \psi'\in L^2([0,1])\land \psi(0)=\psi(1) = 0\}$$ meaning the plane waves $$\psi_{p_0}$$ do not belong to the domain of definition of the commutator $$[x,p]$$ and you cannot apply the naive uncertainty principle to them. However, for self-adjoint operators $$A,B$$, you may rewrite the uncertainty principle as $$\sigma_\psi(A)\sigma_\psi(B)\geq \frac{1}{2} \lvert \langle \psi,\mathrm{i}[A,B]\rangle\psi\rvert = \frac{1}{2}\lvert\mathrm{i}\left(\langle A\psi,B\psi\rangle - \langle B\psi,A\psi\rangle\right)\rvert$$ where the r.h.s. and l.h.s. are now both defined on $$D(A)\cap D(B)$$. Applying this version to the plane waves yields no contradiction.
• But how do you recover Heisenberg's uncertainty principle then? Feb 3, 2016 at 8:53
• @AndreaDiBiagio: It is the last equation in my post. For plane waves, the r.h.s. reads zero, so $\sigma_p = 0$ is not a contradiction. For functions in $D([x,p])$, it is the usual $\sigma_p\sigma_x\geq\hbar/2$. I'm not sure what you think there needs to be recovered. Feb 3, 2016 at 14:18
• @ACuriousMind You answer is detailed and very informative. However, it seems to me to miss OP's question's spirit, which is about the HUP understood colloquially as the product of the uncertainty about the position and momentum of a particle is always larger than some non-zero bound. To better you answer you could perhaps consider adding a less mathematical punch-line. Feb 3, 2016 at 15:40
Notice that $\psi(x)$ is defined on a circle of circumference $a$. Multiplying $x$ on this circle is really multiplying a periodic extension of $x$, i.e., the sawtooth function $x - a\lfloor x/a\rfloor$, where $\lfloor y\rfloor$ means the largest integer not greater than $y$. So, the commutator of the position and momentum operators involves the derivative of not only $x$ but also the discontinuous part $-a\lfloor x/a\rfloor$. Therefore, $$\sigma_{x} \sigma_p \geq \frac{1}{2}\Big|\langle \psi|\,[\hat{x},\hat{p}]\,|\psi\rangle\Big| = \frac{\hbar}{2}\Bigg|\Big\langle\psi\,\Big|\frac{d}{dx}\big(x - a\lfloor x/a\rfloor\big)\Big|\,\psi\Big\rangle\Bigg| = \frac{\hbar}{2}\Big|1-a|\psi(0)|^{2}\Big|.$$ For a plane wave $\psi(x) = e^{ikx}/\sqrt{a}$, the above reduces to $\sigma_{x} \sigma_p\ge0$, as desired.
• This is a nonrigorous version of more mathematical answers other users might provide. Feb 2, 2016 at 9:38
• You don't need to modify the position operator, it is perfectly self-adjoint without the sawtooth function added to it. What you need to do is think about the domain of validity of the uncertainty principle, in particular the commutator. Feb 2, 2016 at 15:01
• @ACuriousMind You are right, but my purpose is to keep the discussion at the level of undergraduate QM. Considering domains of operators and distinguishing between Hermiticity and self-adjointness are somewhat too mathematically involved for many physicists including myself. Feb 2, 2016 at 15:32
• @higgsss from what you wrote, doesn't the expression in the absolute value read $1-a/|\sqrt{a}|^2 = 0$ for a plane wave? Feb 2, 2016 at 20:21
• @AndreaDiBiagio Yes it does. So we have $\sigma_x \sigma_p\ge 0$ for a plane wave. Feb 2, 2016 at 21:04
There are two ways to interpret the boundary conditions you are imposing.
The first case is that of a system which is infinite in extent, but has a periodic regularity. This is like an electron in an idealised 1D crystal, where the periodic boundary condition is imposed by the presence of nuclei regularly spaced. In this case, the plane wave solution has $\sigma_p$ = 0 but $\sigma_x$ is infinite.
The second case, is that of a particle in a ring. In this case, you can imagine the particle as being constrained within the ring by a infinitely deep potential well. The system is not actually 1D, it is 2D. Now you have to consider both $\sigma_x \sigma_{p_x}$ and $\sigma_y \sigma_{p_y}$, and even though $\sigma_x = \sigma_y \sim a$, the uncertainty in momentum will be imposed by the thickness of the ring. The plane wave solution will in fact represent angular momentum eigenstates.
• If one embeds the ring in 2D (realized by a ring-shaped potential well), the "plane wave along the ring" will become an angular momentum eigenstate. The "paradox" will persist because in an angular momentum eigenstate, the uncertainty of the angle is $2\pi$, while that of the angular momentum is 0. Feb 3, 2016 at 19:22
• But $2\pi$ corresponds to total uncertainty, so the "colloquial" HUP holds. @higgsss Feb 3, 2016 at 22:48
• "Colloquially", we have $\Delta L_z \Delta\varphi \ge \hbar/2$, and this doesn't hold for angular momentum eigenstates. Feb 4, 2016 at 2:52
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2022-06-29 21:48:15
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https://math.stackexchange.com/questions/2641333/proving-disproving-that-%E2%88%9A7-%E2%88%9A2-is-irrational?noredirect=1
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# Proving/disproving that √7 - √2 is irrational [duplicate]
It's been proven that √7 and √2 are irrational.
However, I am not sure how to go about proving that √7 - √2. Is it an acceptable proof to just solve the equation which would prove/disprove the equation or as should the proof be done as a contrapositive, similar to how √7 and √2 are proven to be irrational.
What would a valid proof/disproof of irrationality look like in this case?
## marked as duplicate by Hans Lundmark, Chase Ryan Taylor, Parcly Taxel, The Phenotype, Arnaud MortierFeb 8 '18 at 13:18
• One way is to find the equation with integer coefficients that $\,\sqrt{7}-\sqrt{2}\,$ satisfies, then prove that it has no rational roots. – dxiv Feb 8 '18 at 5:42
• not quite sure what you mean by integer coefficients, could you provide an example? – Jessica Tiberio Feb 8 '18 at 5:44
• Please use MathJax to format your posts. – Chase Ryan Taylor Feb 8 '18 at 6:15
Suppose $\sqrt{7}-\sqrt{2}$ were rational; that is, suppose $$\sqrt{7}-\sqrt{2}=\frac{a}{b},$$ where $\text{gcd}(a,b)=1$. Multiply both sides of the equation by $\sqrt{7}+\sqrt{2}$ to obtain $$5=7-2=(\sqrt{7}-\sqrt{2})(\sqrt{7}+\sqrt{2}) = \frac{a}{b}(\sqrt{7}+\sqrt{2}).$$ Since $\frac{5b}{a}\in\mathbb{Q}$, $\sqrt{7}+\sqrt{2}$ is also a rational number. Since the sum of two rational numbers is rational, $$(\sqrt{7}-\sqrt{2}) + (\sqrt{7}+\sqrt{2}) = 2\sqrt{7}$$ is rational. So $\sqrt{7}$ is rational. This is a contradiction.
Hint: suppose $\,\sqrt{7}-\sqrt{2}\,$ were rational, then so would be $\,\dfrac{5}{\sqrt{7}-\sqrt{2}}=\sqrt{7}+\sqrt{2}\,$, then so would be their difference $\,2 \sqrt{2}\,$.
[ EDIT ] Following up on the previous comment: let $\,x=\sqrt{7}-\sqrt{2}\,$, then $\,x^2=9-2\sqrt{14}\,$, then $(x^2-9)^2=4 \cdot 14 \iff x^4 - 18 x^2 + 25 = 0\,$. But the latter equation has no rational roots, since by the rational root theorem the only such roots could be $\,\pm1, \pm5, \pm25\,$ which none work.
• Not really a hint :) – user223391 Feb 8 '18 at 5:46
• Your alt hint is not really a hint either :) – user223391 Feb 8 '18 at 5:49
• @ZacharySelk Right about the alt, just dropped the "hint" misnomer. The first one still leaves at least something to work out, though ;-) – dxiv Feb 8 '18 at 5:51
Suppose $\sqrt{a}-\sqrt{b} = r$ is rational.
Squaring this, $a+b-2\sqrt{ab} = r^2$, so $\sqrt{ab}$ is rational.
If $\sqrt{ab}$ is irrational, this can not hold.
Therefore, if $\sqrt{ab}$ is irrational, so is $\sqrt{a}-\sqrt{b}$.
Since $\sqrt{14}$ is irrational, so is $\sqrt{7}-\sqrt{2}$.
Note that this works for $\sqrt{a}+\sqrt{b}$ also.
Note 2: There are many proofs here that if $n$ is not a perfect square then $\sqrt{n}$ is irrational.
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2019-05-27 07:05:48
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http://pub.acta.hu/acta/showCustomerArticle.action?id=2248&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=40a2a842e2a9149&style=
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ACTA issues
## When dim$=$const implies subspace$=$const
J. M. Szucs
Acta Sci. Math. (Szeged) 66:3-4(2000), 541-552
2750/2009
Abstract. If the dimension of the space spanned by the vectors $\langle f_{1}^{(s)}(x)$, $\ldots$, $f_{n}^{(s)}(x)\rangle$, $s=0,1,\ldots,k$, of $n$ real-valued functions $f_{1},\ldots,f_{n}$ and of their first $k$ derivatives is independent of $x\in I$ (an interval $\subseteq{\msbm R}{}$) and is at most $k$, then the space itself is independent of $x\in I$. This was proved by Curtiss and Moszner assuming the continuity of $f_{1}^{(k)},\ldots,f_{n}^{(k)}$. Their proofs are simplified and extended to operator-valued maps. The extension relies on this generalization of a theorem of Peano: Let $T\colon I\to L(V,W)$ be a differentiable map from a nondegenerate interval $I\subseteq{\msbm R}$ to the space $L(V,W)$ of linear operators from a real finite-dimensional vector space $V$ to another such space $W$. Then $\mathop{\rm range}T(x)$, $x\in I$, is constant if and only if $\mathop{\rm range}T^{\prime }(x)\subseteq\mathop{\rm range}T(x)$, $\dim\mathop{\rm range}T(x)=\mathop{\rm const}$, $x\in I$. AMS Subject Classification (1991): 15A15, 26A06, 15A04 Received June 23, 1999, and in revised form December 2, 1999. (Registered under 2750/2009.)
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2020-10-25 02:26:33
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https://www.physicsforums.com/threads/finding-the-hamiltonian-if-im-given-the-lagrangian.516484/
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# Finding the Hamiltonian if I'm given the Lagrangian
1. Jul 23, 2011
### fluidistic
1. The problem statement, all variables and given/known data
Determine the Hamiltonian corresponding to the an-harmonic oscillator having the Lagrangian $L(x,\dot x )=\frac{\dot x ^2}{2}-\frac{\omega ^2 x^2}{2}-\alpha x^3 + \beta x \dot x ^2$.
2. Relevant equations
$H(q,p,t)=\sum p_i \dot q _i -L$.
$p _i=\frac{\partial L}{\partial \dot q _i}$.
3. The attempt at a solution
Using the equations in 2), I get $p_x= \dot x +2 \beta x \dot x$.
So that $p_x ^2= \dot x ^2 (1+4 \beta x +4 \beta ^2 x^2)$.
Now if I can write the following function in function of only p, q and t then it would be the Hamiltonian, but I couldn't do it.
Here's the function: $\dot x^2 \left ( \frac{1}{2} +\beta x \right ) +\omega ^2 \frac{x^2}{2}+\alpha x^3$.
I see absolutely no way to get rid of the x's terms (or q in this case).
Any help is appreciated. Thanks!
2. Jul 23, 2011
### Pengwuino
I don't see what the issue is here. You already know how to exchange all $\dot{x}$ for momenta based off your first calculation. So just do that.
3. Jul 23, 2011
### mathfeel
Why do you want to get rid of $x$? The term you want to get rid of are $\dot{x}$, with the help of the definition of momenta.
But you are on the right track.
4. Jul 23, 2011
### fluidistic
Whoops guys you are right. This is so new to me that I don't have almost any intuition in this stuff yet.
I get $H(p,q,t)= p_x ^2 \frac{\left (\frac{1}{2}+\beta x \right ) }{1+4 \beta x +4 \beta ^2 x^2}+\omega ^2 \frac{x}{2}+ \alpha x^3$. I hope it's right and it's the answer the exercise is looking for.
By the way so far it seems like that the Hamiltonians I've seen so far depend on $p _i ^2$ rather than $p_i$. Is there any example of a system whose Hamiltonian depends on $p_i$ rather than $p_i ^2$?
5. Jul 23, 2011
### jambaugh
The problem there is that to carry out the Legendre transformation from $L(q,\dot{q})$ to $H(q,p)[itex], the mapping must be invertible. You need to be able to express [itex]\dot{q} = F(q,p)[itex] where [itex] p \equiv \partial L / \partial \dot{q}$.
In particular the matrix of elements $M_{ij} = \frac{\partial^2 }{\partial \dot{q}_i\partial \dot{q}_j}L$ must be invertible.
Given it is symmetric it can be diagonalized via a linear transformation on the q's. To be invertible none of the diagonal terms can be zero.
From there I think you can get to a definite No to your question.
Now you can get only mixed quadratic factors. For $L = \dot{x}\dot{y}$ you'll get $H = p_x p_y$.
6. Jul 24, 2011
### fluidistic
Thank you very much. I must admit it's over my head for now; I hope I'll be able to fully understand this after taking the mathematical methods used in physics course.
But the answer is good to know for me, at least I can know I obtained an error when I get that the Hamiltonian of a system depends on p_i alone.
By the way, it seems what you said concern only systems where the energy is not dissipated, i.e. the Lagrangian and Hamiltonian do not depend explicitly on time. Would the conclusion be the same if $L(q,\dot q ,t)$ and $H(p,q,t)$ rather than just $L(q, \dot q)$ and $H(p,q)$? I'm guessing that it's an obvious "yes", but I just want to be 100% sure.
Thanks.
7. Jul 24, 2011
### mathfeel
Yes, acoustic phonon (quantum of lattice vibration) in solid has low energy effective Hamiltonian $H = v |p|$, where $v$ is the speed of sound in the solid.
You can't really have $H \propto p$ because then you can keep have larger negative momentum and having lower and lower energy, which is strange. Well not completely, but that's a whole different discussion involving special relativity.
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2017-09-21 01:21:05
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http://tex.stackexchange.com/questions/145944/make-symbol-index-with-description
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Make symbol index with description
I'm starting to make a symbol and subject index with the package multind. In the symbol index, I would like to group the symbols by thematics, e.g., write set theory notation and have all the symbols below the description, after write operators and have all the notation for the operators, etc.
Does anyone knows how to do this?
I would like to obtain something like the following
Can anyone help me with a minimal working example? Preferably changing the section with page number.
-
The package multind is very old and unmaintained. I suggest you to use either imakeidx or splitidx to generate multiple indexes. – karlkoeller Nov 20 '13 at 6:01
This seems more a work for the glossaries package. – egreg Nov 20 '13 at 8:49
You can use glossaries to do this. I assume you either want to sort your symbols according to definition or to usage since symbols are difficult to sort alphabetically (and your sample isn't alphabetically sorted). With glossaries you can use the package option sort=def to sort in order of definition or sort=use to sort according to usage. In the example below, I've used sort=def (change \setglossarystyle to \glossarystyle if you are using a version of glossaries prior to version 4.0):
\documentclass{report}
\usepackage{amsfonts}
\usepackage[nomain,section,sort=def]{glossaries}
\usepackage{glossary-mcols}
% (Pre glossaries v4.0 requires \glossarystyle rather than
% \setglossarystyle)
\setglossarystyle{mcolindex}
\renewcommand{\glspostdescription}{\dotfill}
\newglossary[op-glg]{operators}{op-gls}{op-glo}{Operators}
\newglossary[cn-glg]{constants}{cn-gls}{cn-glo}{Constants}
\makeglossaries
% Define a command to define operators
% Syntax: \newoperator[options]{label}{operator symbol}
\newcommand*{\newoperator}[3][]{%
\newglossaryentry{#2}{type=operators,%
name={$#3$},text={#3},description={},#1}%
}
% Similarly for constants
\newcommand*{\newconstant}[3][]{%
\newglossaryentry{#2}{type=constants,%
name={$#3$},text={#3},description={},#1}%
}
% Define operators
\newoperator{Mop}{\mathcal{M}}
\newoperator{Aalpha}{A_\alpha}
\newoperator{nablaOmega}{\nabla_\Omega}
\newoperator{T}{T}
\newoperator{I2}{I_2}
\newoperator{Jz}{J_z}
\newoperator{Gz}{G_z}
% Define constants
\newconstant{pdash}{p'}
\newconstant{jnu}{j_\nu}
\newconstant{Mcn}{\mathfrak{M}}
\newconstant{Dpq}{D_{p,q}}
\begin{document}
\chapter{Sample Chapter}
Some sample usage of constants:
$\gls{pdash}, \gls{Mcn}, \gls{jnu}$
Some sample usage of operators:
$\gls{Mop}, \gls{Aalpha}, \gls{nablaOmega}$
\newpage
More sample usage:
$\gls{T}, \gls{I2}, \gls{Jz}, \gls{Gz}, \gls{Mcn}, \gls{Dpq}, \gls{jnu}$
\chapter*{Index}
\printglossaries
\end{document}
In order to build the PDF you need to do:
1. Run pdflatex
2. Run makeglossaries
3. Run pdflatex
The above example produces:
Page 1:
Page 2:
Page 3:
In your example image, you have some lines with multiple entries (for example, c, c_1, c_2, \ldots, v_n). There are various ways to deal with these. You could designate the first in the list (e.g, c) as the main parent entry, and the remainder as sub-entries, and then define a glossary style that puts the sub-entry names after the parent entry name, but this will complicate the location list. However, I think the easiest solution is probably to define an entry with the specified list and then use \glsdisp or \glslink whenever you use it.
For example:
\newconstant{c}{c, c_1, c_2, \ldots, v_n}
Then in the document:
In-line: $\glslink{c}{c}$ or $\glslink{c}{c_1}$ or $\glslink{c}{c_2}$ etc.
-
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2016-07-23 09:24:33
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http://buycheapfollower.askbot.com/question/1448/online-divorce-jacksonville/
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# Online Divorce Jacksonville
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2019-02-22 06:35:26
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http://topaz.codeplex.com/workitem/12
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1
Vote
# Path too long
## description
Hi,
when I add the TOPAZ nuget package for Azure Storage I end up with the dll in a folder similar to
packages\EnterpriseLibrary.TransientFaultHandling.WindowsAzure.Storage.6.0.1304.1\lib\NET45/Microsoft.Practices.EnterpriseLibrary.TransientFaultHandling.WindowsAzure.Storage.dll
That is 178 characters, what is about 2/3 of the maximum allowed path length in windows. That imposes a serious restriction on the folder structure that can be used for the project under development. Is there any chance you can shorten that name so that it still leaves room for development folder structures. Sure, we can subst or mklink, but that's additional effort that could easily be avoided. Since you already coined the abbreviation topaz, why not go for that?
Thanks and regards
Florian
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2017-11-22 03:53:52
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https://www.jiskha.com/questions/755965/integral-e-3x-dx-a-e-3x-c-b-1-3e-3x-c-c-e-4x-c-d-1-4e-4x-c-evaluate-integral
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Calculus
(integral) e^3x dx
A. e^3x+C
B. 1/3e^3x+C
C. e^4x+C
D. 1/4e^4x+C
Evaluate (integral) dx/(Square root 9-8x-x^2)
A. sin^-1(x+4/5)+C
B. sin^-1(x-4/5)+C
C. Ln(Square root 9-8x-x^2)+C
D. Ln(-8-2x)+C
1. 👍 0
2. 👎 0
3. 👁 80
1. if u = 3x you have integral e^u du/3, so (B)
since 9-8x-x^2 = 25 - (x+4)^2, and integral du/sqrt(a^2-u^2) = arcsin(u/a), you have arcsin u/5, so (A)
1. 👍 0
2. 👎 0
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2020-06-05 12:32:06
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http://www.maa.org/programs/faculty-and-departments/course-communities/demonstration-for-curvature-and-the-osculating-circle
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# Demonstration for Curvature and the Osculating Circle
This applet shows a curve in the $xy$-plane, the vector from the origin to a point on the curve, and the osculating circle. Also displayed is the curvature. The student can use a slider to move the point along the curve and view the changing osculating circle and curvature. There is also a button that activates the animated motion of the point and the corresponding osculating circle and curvature.
Identifier:
http://web.monroecc.edu/manila/webfiles/calcNSF/JavaCode/CalcPlot3D-CurvatureOscCircle.htm
Rating:
Creator(s):
Paul Seeburger
Cataloger:
Larry Green
Publisher:
Exploring Multivariable Calculus
Rights:
Paul Seeburger
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2016-12-08 07:17:58
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https://math.stackexchange.com/questions/1745469/what-happens-when-we-pick-a-random-sample
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# What happens when we pick a Random sample?
Let $(\Omega, \mathcal{F} , \mathbb{P})$ be a probability space and $X:\Omega \to \mathbb{R}$ be a random variable.
When we simulate or pick a random sample of size $n$ from $X$, are we picking values $X (\omega_1), X (\omega_2), X (\omega_3), \dots, X (\omega_n)$ where each $\omega_i\in \Omega$, or are we picking values $X_1(\omega_1), X_2 (\omega_2), X_3(\omega_3), \dots, X_n (\omega_n)$ where the $X_i$s are iid random variables with the same distribution as $X$?
• – Artem Mavrin Apr 16 '16 at 23:56
• The right option is a third one: we are picking values $X_1(\omega)$, $X_2 (\omega_)$, $X_3(\omega)$, $\dots$, $X_n (\omega)$, where the $X_i$s are i.i.d. random variables with the same distribution as $X$. – Did Apr 17 '16 at 7:49
• Thanks guys. @Did, which space would that 'omega' belong to? Is it the product space ? – SuperM Apr 17 '16 at 15:41
• No, to the set Omega each random variable is defined upon. – Did Apr 17 '16 at 21:11
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2019-09-19 15:11:05
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https://math.stackexchange.com/questions/576325/on-the-notion-of-pro-category
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# On the notion of pro-category
I can't understand the notion of pro-category and ind-category.
According to [Kashiwara-Schapira, Categories and Sheaves], an ind-object of category $C$ is a contravariant functor $F\in Set^{C^{op}}$ which is isomorphic to a filtered colimit of representable functors, and $Ind(C)$ denotes their full subcategory of $Set^{C^{op}}$. A pro-object and $Pro(C)$ is also defined similarly. However, I can't understand what this category means.
I thought this is a certain completion or cocompletion of categories. Because the category of finite sets does not have arbitrary colimits and its indization $Ind(Set_f)$ is isomorphic to the category of sets. However, according to nlab it seems that the category of groups $Grp$ is cocomplete but its indization $Ind(Grp)$ is not isomorphic to $Grp$.
Furthermore, in [Artin-Mazur, Etale homotopy], homology and cohomology functor is extended to pro-categories. In detail, let $H_{CW}$ be a homotopy category of CW complexes and denote its homology and cohomology functor by $H_*:H_{CW}\to Ab, H^*:H_{CW}^{op}\to Ab$. Then, these are extended to pro-category $H_*:Pro(H_{CW})\to Pro(Ab), H^*:Pro(H_{CW}^{op})\to Pro(Ab)$. However, the image of latter functor is actually an abelian group, while the former is just a pro-abelian group. I don't understand this difference.
To sum up, what I want to ask is
• What is the motivation of this concept? Although I know every presheaf is canonically isomorphic to a colimit of representable functors, to restrict its index category to be filtered is seemed to be unnatural.
• Are there any criterion for when the category $C$ and its indization $Ind(C)$ are isomorphic?
• What is the difference of homology and cohomology in the above case? And when the image of the pro-extension $F:Pro(C)\to Pro(D)$ of a functor $F:C\to D$ is always in $D$?
• Why this concept is important? I want to know some applications.
There are excellent reasons to look at ind-completions: for one thing, we have a complete axiomatisation of all categories equivalent to $\mathbf{Ind}(\mathcal{C})$ for a locally small category $\mathcal{C}$ – they are precisely the class $\aleph_0$-accessible categories. A special case are the $\aleph_0$-accessible categories, and very many of the categories of algebraic structures are examples of this: sets, groups, rings, modules, fields, chain complexes of abelian groups, presheaves, (small) categories etc. On the other hand $\mathbf{Pro}(\mathcal{C})$ is simply $\mathbf{Ind}(\mathcal{C}^\mathrm{op})^\mathrm{op}$. There are not as many examples of such categories, but one example is the category of Stone spaces (a.k.a. profinite sets).
Another thing I should point out is that $\mathbf{Ind}(\mathcal{C})$ is almost never equivalent to $\mathcal{C}$, let alone isomorphic to $\mathcal{C}$, because it is a free completion – emphasis on free! This is not unlike the fact that $\mathbb{C} \otimes_\mathbb{R} M$ is rarely isomorphic to $M$ even when $M$ is a $\mathbb{C}$-module.
• You mean $Set, Grp, Mod$ etc are the indization of $FinSet, FinGrp, FinMod$? Even if it is true, I can't still understand why such a "finite" description is so useful? – Fujita Tomomi Nov 21 '13 at 21:53
Indization provides a $2$-functor which is left adjoint to the forgetful $2$-functor from the $2$-category of categories with directed colimits with functors preserving directed colimits to the $2$-category of all categories with functors. Thus, the idea is to adjoin directed colimits to a given category freely. We consider the given category only as a category, any additional random properties such as the existence of certain colimits are ignored. Thus, even if it already has directed colimits, the indization will change the category, since it simply doesn't know that directed colimits are already there. At least this is my intuition. One might ask if it is possible to remember the colimits which already exist, but I doubt that this has a universal solution. Perhaps the confusion is also connected to the following: When one thinks of other adjunctions, for example that of torsion-free abelian groups and abelian groups, the left adjoint which mods out the torsion subgroup of course doesn't change torsion-free groups. More generally, if $F$ is left adjoint to $G$ and $G$ is fully faithful, then the counit of the adjunction $FG \to 1$ is an isomorphism. But this is not the case when $G$ is only faithful, i.e. corresponds to the inclusion of a category which is not assumed to be full and therefore may have less morphisms. The same remark applies to $2$-adjunctions, and in particular to the one mentioned above: Between categories with directed colimits, we consider only those functors preserving directed colimits.
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2021-04-23 09:03:46
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https://math.stackexchange.com/questions/2916718/calculating-the-square-root-of-2/2919953
|
# Calculating the square root of 2
Since $\sqrt{2}$ is irrational, is there a way to compute the first 20 digits of it?
What I have done so far
I started the first digit decimal of the $\sqrt{2}$ by calculating iteratively so that it would not go to 3 so fast. It looks like this:
\begin{align} \sqrt 2 & = 1.4^{2} \equiv 1.96\\ \sqrt 2 & = 1.41^{2} \equiv 1.9881\\ \sqrt 2 & = 1.414^{2} \equiv 1.999396\\ & \ldots \end{align}
First I tell whether it passes such that $1.x^{2}$ would be not greater than 3.
If that passes, I will add a new decimal to it. Let's say $y.$ $1.xy^{2}$
If that y fails, I increment $y$ by 1 and square it again.
The process will keep repeating. Unfortunately, the process takes so much time.
• You can go on trying to compute the square of $1.414x$, where $x$ is a number between $0$ and $9$. The greatest number between $1.4140$ and $1.4149$ such that its square is less then $2$ is your next candidate to repeat the process. – Gibbs Sep 14 '18 at 12:14
• – lhf Sep 14 '18 at 12:17
• @Gibbs I tried that so far. But the reason is that it takes more time to compute it. – MMJM Sep 14 '18 at 12:21
• – user202729 Sep 14 '18 at 15:22
• @Gibbs Please don't post answers as comments. – David Richerby Sep 14 '18 at 18:44
Calculating the square root of a number is one of the first problems tackled with numerical methods, known I think to the ancient Babylonians. The observation is that if $$x,\,y>0$$ and $$y\ne\sqrt{x}$$ then $$y,\,x/y$$ will be on opposite sides of $$\sqrt{x}$$, and we could try averaging them. So try $$y_0=1,\,y_{n+1}=\frac12\left(y_n+\frac{x}{y_n}\right)$$. This is actually the Newton-Raphson method 5xum mentioned. The number of correct decimal places approximately doubles at each stage, i.e. you probably only have to go as far as $$y_5$$ or so.
• Definitely one of the fastest methods: $$y_0 = 1.\color{tan}{0};\\ y_1 = 1.\color{tan}{5};\\ y_2 = 1.41\color{tan}{666666666666666666666666666...};\\ y_3 = 1.41421\color{tan}{568627450980392156862745...};\\ y_4 = 1.41421356237\color{tan}{468991062629557889...};\\ y_5 = 1.41421356237309504880168\color{tan}{962350...};\\ \cdots$$ – Oleg567 Sep 14 '18 at 12:26
• @Oleg567 We could go even faster with post-Newton Householder methods, but the individual steps become more computationally complex. BTW the calculator you used to check that probably also used Newton-Raphson for the division. – J.G. Sep 14 '18 at 12:30
• The beauty of this method is that the initial estimate can be way off and the method will converge quickly anyway. of course, making an educated guess to pick the initial estimate helps to reduce the number of iterations. – Vasya Sep 14 '18 at 12:32
• Love the intuitive explanation for it! – dbx Sep 14 '18 at 12:52
• @Paul Since it's Newton-Raphson it'll be about $2^n$ of them, but a more detailed answer than that can't be obtained without careful analysis of the specifics of the problem. However, if you look at how which digits have "gotten stuck", you can be confident from the shrinking error terms that they won't change. See the black digits in Oleg567's comment for an example. – J.G. Sep 14 '18 at 22:04
Here's the way I learnt to obtain decimal digit after decimal digit when I began middle school:
\begin{array}{lcl} 2&\big( &\color{red}1.414\,2\dots \\[1ex] 1\,00&& 24\times \color{red}4=96<100\\ -96\,&& 25\times5=125>100\\[1ex] \phantom{-0}4\,00&&281\times\color{red}1<400\\ \;\:-2\,81&&282\times2>400\\[1ex] \phantom{-0}119\,00&&2824\times\color{red}4<11900\\ \phantom{0}{-}112\,96&&2825\times5>11900 \\[1ex] \phantom{00\;}604\,00&&28282\times\color{red}2 < 60400 \\ &&28283\times3> 60400 \end{array} &c.
Let me explain the procedure on the first two steps. It relies on a clever use of the identity $(x+y)^2=x^2+2xy+y^2$. Suppose more generally we want to find the square root of a number $a$.
1. We first find the greatest natural number $n$ such that $n^2\le a$.
2. If $a$ is not a perfect square, i.e. if $n^2<a$, let $d$ be the first decimal digit of the square root. This is the greatest digit such that $\;\Bigl(n+\frac d{10}\Bigr)^2\le a$. We'll transform this inequality into a more easy-to-use test: \begin{align} \Bigl(n+\frac d{10}\Bigr)^2\le a&\iff \frac{2n}{10}d+\frac{d^2}{100}<a -n^2\\ &\iff (10\times 2n+d)\times d\le (a-n^2)\times 100 \end{align} In practice, this means, we calculate the difference $a-n^2$ and add two 0s. Then we double $n$, add a digit d (this is the result of calculating $10\times 2n+d$) and multiply what we obtain by this digit. Last, we test whether the result is less than $100(a-n^2)$, and retain the largest possible digit.
• Looks interesting, can you talk us through it a bit? I don't really get it. e.g. where does 100 come from? – goblin Sep 15 '18 at 1:57
• @goblin, There are some references for this method at math.stackexchange.com/a/538055/117057 and math.stackexchange.com/q/376365/117057 – shoover Sep 15 '18 at 5:21
• @goblin: I've added an explanation for the first two steps. The following stepsruns along te same lines, only the first step is different. Hope this will make it clear. – Bernard Sep 15 '18 at 9:24
• @Bernard, thanks. – goblin Sep 15 '18 at 9:39
• @goblin You start off with 1 because 1 is the largest integer whoose square is less than 2. Then extend 1 by the next two digits, 00, to get 100. Now double the 1 just obtained and find the largest digit such that 2x times x is less than 100. – Paul Evans Sep 15 '18 at 9:41
The number $\sqrt{2}$ is the solution to the equation $x^2-2=0$, so any method for numerically approximating the roots of an equation (such as the Newton method) will be able to approximate $\sqrt{2}$.
• I don't see how this qualifies as an answer. It is just a general statement. – M. Wind Sep 16 '18 at 5:29
On a similar note to the answer by R. Romero: in the special case of taking the square root of an integer $N$, it is fairly straightforward to calculate the continued fraction representation of $\sqrt{N}$.
In the particular case $N=2$, we have: $$\sqrt{2} = 1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \ddots}}}.$$ (This follows from the fact that if $x = \sqrt{2}-1$, then $x = \sqrt{2}-1 = \frac{1}{\sqrt{2}+1} = \frac{1}{2+x}$.)
Now, from this we can calculate subsequent rational approximations to $\sqrt{2}$:
$$\begin{matrix} & & 1 & 2 & 2 & 2 & 2 & 2 & \cdots \\ 0 & 1 & 1 & 3 & 7 & 17 & 41 & 99 & \cdots \\ 1 & 0 & 1 & 2 & 5 & 12 & 29 & 70 & \cdots \end{matrix}$$ So, for example $\frac{99}{70} \approx 1.4142857$ whereas $\sqrt{2} \approx 1.4142136$.
(It also happens that this procedure generates solutions to Pell's equation $a^2 - 2 b^2 = \pm 1$; for example, $99^2 - 2 \cdot 70^2 = 1$. The connection is: if $a^2 - 2 b^2 = \pm 1$ then $a - b \sqrt{2} = \pm \frac{1}{a + b \sqrt{2}}$; so if $a$ and $b$ are large positive integers satisfying Pell's equation, then $a - b\sqrt{2} \approx \pm\frac{1}{2a}$ which implies $\frac{a}{b} - \sqrt{2} \approx \pm\frac{1}{2ab} \approx \pm\frac{1}{a^2\sqrt{2}}$.)
• Is there somewhere I can read more about this, especially the connection between continued fractions and Pell's equation? – goblin Sep 15 '18 at 2:00
• Once you see the first few rational approximations it's easy to guess and prove the recursion for $p/q$, namely, $p_n = p_{n-1} + 2q_{n-1}$, $q_n = p_{n-1} + q_{n-1}$..See en.wikipedia.org/wiki/…, en.wikipedia.org/wiki/Pell%27s_equation – Ethan Bolker Sep 15 '18 at 13:13
Okay, I searched through the answers, but none seems to mention this one: long quadratic root calculation.
From the name it is obvious that it resembles long division, like this:
\begin{align} \sqrt{2.00\;00\;00\;00\;..} \end{align}
Notice how they are grouped into tuples. Now estimate the first digit, namely $1$:
\begin{align} &~~~1.\\ 1&\sqrt{2.00\;00\;00\;00\;..}\\ &~~~1\\ &~~~\overline{1\,00} \end{align}
We calculate $1\times1=1$, write it down, and calculate the "remainder", just like divisions. Notice that we append 2 digits behind instead of 1.
Next, double the number on the top, and write it on the left of $1\,00$:
\begin{align} &~~~1.\;*\\ 1&\sqrt{2.00\;00\;00\;00\;..}\\ &~~~1\\ 2*&\,\,|\overline{1\,00} \end{align}
Now we estimate the next digit, *. It is written both on the top and to the left. Of course, we know that it is 4, so:
\begin{align} &~~~1.\;4\;\;\;*\\ 1&\sqrt{2.00\;00\;00\;00\;..}\\ &~~~1\\ 24&\,\,|\overline{1\,00}\\ &\,\,|\,\,\,\,96\\ &2\overline{8{*}|\,4\,00} \end{align}
We double the numbers on the top again to get $28*$, and repeat the process:
\begin{align} &~~~1.\;4\;\;\;1\\ 1&\sqrt{2.00\;00\;00\;00\;..}\\ &~~~1\\ 24&\,\,|\overline{1\,00}\\ &\,\,|\,\,\,\,96\\ &2\overline{8{1}|\,4\,00}\\ &\,\,\,\,\,\,\,\,\,|\,2\,81 \end{align}
I found a picture, but not of $\sqrt{2}$:
This is extremely inefficient for computers, but great for manual calculation. After all, we don't do multiplication through fast Fourier transforms!
Also, this method is developed in ancient China.
Suppose you want to find the square root of $p$ and suppose your initial guess is $x/y$:
Let $\mathbf M=\begin{bmatrix} 1 & p \\ 1 & 1 \end{bmatrix}$ and $\mathbf q=\begin{pmatrix} x \\ y \end{pmatrix}$ Then $\mathbf M\mathbf M\mathbf M...\mathbf q$ gives a numerator and denominator the ratio of which converges to the square root of $p$. This gives an approximation to the square root of $2$ as fast as the other methods but with no floating point arithmetic until the final division.
Performs well for calculation tools optimized for Matrix arithmetic. This also gives you solutions for Pell's equation for $p=2$ as mentioned by Daniel Schepler.
In this answer, there is a method using continued fraction approximations for $\sqrt2$ and the generating function for the central binomial coefficients to get some very quickly convergent series for $\sqrt2$. For example, $$\sqrt2=\frac75\sum_{k=0}^\infty\binom{2k}{k}\frac1{200^k}\tag1$$ and $$\sqrt2=\frac{239}{169}\sum_{k=0}^\infty\binom{2k}{k}\frac1{228488^k}\tag2$$
For example, summing to $k=4$ in $(2)$ gives $$\sqrt2=1.414213562373095048801688$$ which is accurate to $23$ places.
• (+) This method could be used to calculate millions of $\sqrt{2}$ digits (especially when notice that the series has rational terms, and apply en.wikipedia.org/wiki/Binary_splitting technique). – Oleg567 Sep 24 '18 at 4:44
• Good description of the elementary school method. You are not likely to succeed unless you do at least one square root every day. After age 75, you have to do more than one every day. That is why there are calculators. – richard1941 Sep 24 '18 at 23:53
Binary search for it.
Since $1 < 2 < 4$, we must have $\sqrt{1} < \sqrt{2} < \sqrt{4}$, so $\sqrt{2} \in (1,2)$. Now repeatedly: find the midpoint, $m$, of the current interval, $(a,b)$, square $m$ and compare with $2$, and if $2 = m^2$ declare that $m = \sqrt{2}$, or if $2 < m^2$, make the new interval $(a,m)$, otherwise make the new interval $(m,b)$. This process halves the size of the interval on each step. Since $\log_2(10^{-20}) = -66.438\dots$, after 67 doublings, the error in taking any value from the interval is $<10^{-20}$ (but, if the interval straddles a digit change, you may have to perform additional steps to find out on which side of the change is $\sqrt{2}$).
This process is shown in the table below. Each decimal number is computed to $21$ digits and has trailing zeroes stripped. If there are still $21$ digits, a space is inserted between the $20^\text{th}$ and $21^\text{st}$.
\begin{align} \text{step} && \text{interval} && m && m^2 \\ 1 && (1., 2.) && 1.5 && 2<2.25 \\ 2 && (1., 1.5) && 1.25 && 1.5625<2 \\ 3 && (1.25, 1.5) && 1.375 && 1.890625<2 \\ 4 && (1.375, 1.5) && 1.4375 && 2<2.06640625 \\ 5 && (1.375, 1.4375) && 1.40625 && 1.9775390625<2 \\ 6 && (1.40625, 1.4375) && 1.421875 && 2<2.021728515625 \\ 7 && (1.40625, 1.421875) && 1.4140625 && 1.99957275390625<2 \\ 8 && (1.4140625, 1.421875) && 1.41796875 && 2<2.0106353759765625 \\ 9 && (1.4140625, 1.41796875) && 1.416015625 && 2<2.005100250244140625 \ \\ 10 && (1.4140625, 1.416015625) && 1.4150390625 && \ 2<2.00233554840087890625 \\ 11 && (1.4140625, 1.4150390625) && 1.41455078125 && \ 2<2.00095391273498535156\ 3 \\ 12 && (1.4140625, 1.41455078125) && 1.414306640625 && \ 2<2.00026327371597290039 \\ 13 && (1.4140625, 1.414306640625) && 1.4141845703125 && \ 1.99991799890995025634\ 8<2 \\ 14 && (1.4141845703125, 1.414306640625) && 1.41424560546875 && \ 2<2.00009063258767127990\ 7 \\ 15 && (1.4141845703125, 1.41424560546875) && 1.414215087890625 && \ 2<2.00000431481748819351\ 2 \\ 16 && (1.4141845703125, 1.414215087890625) && 1.4141998291015625 && \ 1.99996115663088858127\ 6<2 \\ 17 && (1.4141998291015625, 1.414215087890625) && 1.41420745849609375 && \ 1.99998273566598072648<2 \\ 18 && (1.41420745849609375, 1.414215087890625) && \ 1.414211273193359375 && 1.99999352522718254476\ 8<2 \\ 19 && (1.414211273193359375, 1.414215087890625) && \ 1.4142131805419921875 && 1.99999892001869739033\ 3<2 \\ 20 && (1.4142131805419921875, 1.414215087890625) && \ 1.41421413421630859375 && 2<2.00000161741718329722 \end{align}\begin{align} 21 && (1.4142131805419921875, 1.41421413421630859375) && \ 1.41421365737915039062\ 5 && 2<2.00000026871771297010\ 1 \\ 22 && (1.4142131805419921875, 1.41421365737915039062\ 5) && \ 1.41421341896057128906\ 2 && 1.99999959436814833679\ 8<2 \\ 23 && (1.41421341896057128906\ 2, 1.41421365737915039062\ 5) && \ 1.41421353816986083984\ 4 && 1.99999993154291644259\ 5<2 \\ 24 && (1.41421353816986083984\ 4, 1.41421365737915039062\ 5) && \ 1.41421359777450561523\ 4 && 2<2.00000010013031115363\ 4 \\ 25 && (1.41421353816986083984\ 4, 1.41421359777450561523\ 4) && \ 1.41421356797218322753\ 9 && 2<2.00000001583661290993\ 6 \\ 26 && (1.41421353816986083984\ 4, 1.41421356797218322753\ 9) && \ 1.41421355307102203369\ 1 && 1.99999997368976445422\ 1<2 \\ 27 && (1.41421355307102203369\ 1, 1.41421356797218322753\ 9) && \ 1.41421356052160263061\ 5 && 1.99999999476318862656\ 8<2 \\ 28 && (1.41421356052160263061\ 5, 1.41421356797218322753\ 9) && \ 1.41421356424689292907\ 7 && 2<2.00000000529990075437\ 4 \\ 29 && (1.41421356052160263061\ 5, 1.41421356424689292907\ 7) && \ 1.41421356238424777984\ 6 && 2<2.00000000003154468700\ 1 \\ 30 && (1.41421356052160263061\ 5, 1.41421356238424777984\ 6) && \ 1.41421356145292520523 && 1.99999999739736665591\ 7<2 \\ 31 && (1.41421356145292520523, 1.41421356238424777984\ 6) && \ 1.41421356191858649253\ 8 && 1.99999999871445567124\ 2<2 \\ 32 && (1.41421356191858649253\ 8, 1.41421356238424777984\ 6) && \ 1.41421356215141713619\ 2 && 1.99999999937300017906\ 8<2 \\ 33 && (1.41421356215141713619\ 2, 1.41421356238424777984\ 6) && \ 1.41421356226783245801\ 9 && 1.99999999970227243302\ 1<2 \\ 34 && (1.41421356226783245801\ 9, 1.41421356238424777984\ 6) && \ 1.41421356232604011893\ 3 && 1.99999999986690856000\ 8<2 \\ 35 && (1.41421356232604011893\ 3, 1.41421356238424777984\ 6) && \ 1.41421356235514394938\ 9 && 1.99999999994922662350\ 4<2 \end{align}\begin{align} 36 && (1.41421356235514394938\ 9, 1.41421356238424777984\ 6) && \ 1.41421356236969586461\ 8 && 1.99999999999038565525\ 2<2 \\ 37 && (1.41421356236969586461\ 8, 1.41421356238424777984\ 6) && \ 1.41421356237697182223\ 2 && 2<2.00000000001096517112\ 7 \\ 38 && (1.41421356236969586461\ 8, 1.41421356237697182223\ 2) && \ 1.41421356237333384342\ 5 && 2<2.00000000000067541319\ 0 \\ 39 && (1.41421356236969586461\ 8, 1.41421356237333384342\ 5) && \ 1.41421356237151485402\ 1 && 1.99999999999553053422\ 1<2 \\ 40 && (1.41421356237151485402\ 1, 1.41421356237333384342\ 5) && \ 1.41421356237242434872\ 3 && 1.99999999999810297370\ 5<2 \\ 41 && (1.41421356237242434872\ 3, 1.41421356237333384342\ 5) && \ 1.41421356237287909607\ 4 && 1.99999999999938919344\ 7<2 \\ 42 && (1.41421356237287909607\ 4, 1.41421356237333384342\ 5) && \ 1.41421356237310646974\ 9 && 2<2.00000000000003230331\ 9 \\ 43 && (1.41421356237287909607\ 4, 1.41421356237310646974\ 9) && \ 1.41421356237299278291\ 2 && 1.99999999999971074838\ 3<2 \\ 44 && (1.41421356237299278291\ 2, 1.41421356237310646974\ 9) && \ 1.41421356237304962633 && 1.99999999999987152585<2 \\ 45 && (1.41421356237304962633, 1.41421356237310646974\ 9) && \ 1.41421356237307804804 && 1.99999999999995191458\ 5<2 \\ 46 && (1.41421356237307804804, 1.41421356237310646974\ 9) && \ 1.41421356237309225889\ 5 && 1.99999999999999210895\ 2<2 \\ 47 && (1.41421356237309225889\ 5, 1.41421356237310646974\ 9) && \ 1.41421356237309936432\ 2 && 2<2.00000000000001220613\ 5 \\ 48 && (1.41421356237309225889\ 5, 1.41421356237309936432\ 2) && \ 1.41421356237309581160\ 8 && 2<2.00000000000000215754\ 3 \\ 49 && (1.41421356237309225889\ 5, 1.41421356237309581160\ 8) && \ 1.41421356237309403525\ 2 && 1.99999999999999713324\ 7<2 \\ 50 && (1.41421356237309403525\ 2, 1.41421356237309581160\ 8) && \ 1.41421356237309492343 && 1.99999999999999964539\ 5<2 \\ 51 && (1.41421356237309492343, 1.41421356237309581160\ 8) && \ 1.41421356237309536751\ 9 && 2<2.00000000000000090146\ 9 \\ 52 && (1.41421356237309492343, 1.41421356237309536751\ 9) && \ 1.41421356237309514547\ 5 && 2<2.00000000000000027343\ 2 \\ 53 && (1.41421356237309492343, 1.41421356237309514547\ 5) && \ 1.41421356237309503445\ 2 && 1.99999999999999995941\ 4<2 \\ 54 && (1.41421356237309503445\ 2, 1.41421356237309514547\ 5) && \ 1.41421356237309508996\ 3 && 2<2.00000000000000011642\ 3 \\ 55 && (1.41421356237309503445\ 2, 1.41421356237309508996\ 3) && \ 1.41421356237309506220\ 8 && 2<2.00000000000000003791\ 8 \\ 56 && (1.41421356237309503445\ 2, 1.41421356237309506220\ 8) && \ 1.41421356237309504833 && 1.99999999999999999866\ 6<2 \\ 57 && (1.41421356237309504833, 1.41421356237309506220\ 8) && \ 1.41421356237309505526\ 9 && 2<2.00000000000000001829\ 2 \\ 58 && (1.41421356237309504833, 1.41421356237309505526\ 9) && \ 1.41421356237309505180\ 0 && 2<2.00000000000000000847\ 9 \\ 59 && (1.41421356237309504833, 1.41421356237309505180\ 0) && \ 1.41421356237309505006\ 5 && 2<2.00000000000000000357\ 3 \\ 60 && (1.41421356237309504833, 1.41421356237309505006\ 5) && \ 1.41421356237309504919\ 7 && 2<2.00000000000000000111\ 9 \\ 61 && (1.41421356237309504833, 1.41421356237309504919\ 7) && \ 1.41421356237309504876\ 4 && 1.99999999999999999989\ 3<2 \\ 62 && (1.41421356237309504876\ 4, 1.41421356237309504919\ 7) && \ 1.41421356237309504898 && 2<2.00000000000000000050\ 6 \\ 63 && (1.41421356237309504876\ 4, 1.41421356237309504898) && \ 1.41421356237309504887\ 2 && 2<2.00000000000000000019\ 9 \\ 64 && (1.41421356237309504876\ 4, 1.41421356237309504887\ 2) && \ 1.41421356237309504881\ 8 && 2<2.00000000000000000004\ 6 \\ 65 && (1.41421356237309504876\ 4, 1.41421356237309504881\ 8) && \ 1.41421356237309504879 && 1.99999999999999999996\ 9<2 \\ 66 && (1.41421356237309504879, 1.41421356237309504881\ 8) && \ 1.41421356237309504880\ 4 && 2<2.00000000000000000000\ 8 \\ 67 && (1.41421356237309504879, 1.41421356237309504880\ 4) && \ 1.41421356237309504879\ 8 && 1.99999999999999999998\ 9<2 \\ 68 && (1.41421356237309504879\ 8, 1.41421356237309504880\ 4) && \ 1.41421356237309504880\ 1 && 1.99999999999999999999\ 8<2 \\ 69 && (1.41421356237309504880\ 1, 1.41421356237309504880\ 4) && \ 1.41421356237309504880\ 3 && 2<2.00000000000000000000\ 3 \end{align}
Using the fact that $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, then we have to find $2 \sin \frac{\pi}{4}$.
We can approximate $\sin x$ using the Taylor series to three terms:
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} + O(x^6),$$
so we have:
$$\sin \frac{\pi}{4} \approx \frac{\pi}{4} - \frac{(\pi/4)^3}{3!} + \frac{(\pi/4)^5}{5!} .$$
If we approximate $\pi$ as $\frac{22}{7}$, then we have $\frac{\pi}{4} = \frac{11}{14}$, then we have:
$$\sin \frac{\pi}{4} \approx\frac{11}{14} - \frac{(11/14)^3}{3!} + \frac{(11/14)^5}{5!},$$
which when you multiply by $2$ to get $\sqrt{2}$, gives $1.4147$, while the actual value is $1.4142$.
If we expand the Taylor series to more terms, or improve the approximation of $\pi$ (such as $\frac{355}{113}$), then we can get to $20$ correct digits.
• Don’t you need pi to nearly 20 digits for this to work? – JoeTaxpayer Sep 15 '18 at 2:10
There's a general method that converges about as quickly as Newton-Raphson but is somewhat more general. It's based off of Continued Fractions:
Suppose you want to find the square root of $N$. Let $a+b = N$ where $b$ has an easy to calculate square root.
let $y_{n+1} = \sqrt b + \frac{a}{ \sqrt b + y_n}$
$y_{n+1}$ converges to $\sqrt N$.
Start with an initial guess $x$ for the square root of $2$. Then add a correction term $y$. Write down $(x+y)^2 - 2 = 0$. Solve this equation for $y$ by expanding it up to third order in the difference $(2-x^2)$. This is a straightforward calculation. Combining all contributions, the result is elegant:
$$x + y = (x^4+12x^2+4)/(4x^3+8x)$$
For a rational initial guess $x$ the result $(x + y)$ is also rational, but much closer to the desired value.
For example if we take $x = 3/2$, then $(x +y)=577/408$, which differs from the square root of 2 by a factor 1.0000015. If we start with $x = 7/5$, the result is $19601/13860$, which differs from the square of root of $2$ by a factor $1.0000000013$
• Please show what happens with 140/99. I find the error to be 1.2 10^-18 on my WP-34s iPhone emulator in double precision mode (good to at least 30 digits). If you recycle 577/408, you get an error 9.0 10^-25. That meets to goal of 20 digits. Recycling 19601/13860 gives an error of absolute zero (on the calculator). – richard1941 Dec 21 '18 at 17:50
• Thanks! The initial values $99/70$ and $140/99$ both result in $768398401/543339720$. – M. Wind Dec 23 '18 at 19:39
You can compute it manually using the algorithm:
1. $p=0$, $r=0$, $i=0$
2. Split the number into sections of two digits
3. Take i'th section $n_i$, let $k=100t+n_i$
4. Find the greatest number $x$, such that $$y=x(20p+x)\leq k$$
5. Assign $p=10 p + x$, $i=i+1$, if the accyracy of the result is not satisfied, then return to 3.
Example:
02.00 00 00 00 00
• $n_0 = 2$, $k=2$, therefore for $x=1$: $y=1$ and $p=1$
• $n_1=0$, $k=100$, so for $x=4$: $y=24*4=96<100$ and $p=14$
• $n_2=0$, $k=400$, so for $x=1$, $y=281*1=281<400$ and $p=141$
• $n_3=0$, $k=11900$, so for $x=4$, $y=2824*4=11296<11900$ and $p=1414$
• $n_4=0$, $k=60400$, so for $x=2$, $y=28282*2=56564<60400$ and $p=14142$
• $n_5=0$, $k=383600$, so for $x=1$, $y=282841*1=282841<383600$ and $p=141421$
• ...
After all just remember to point the comma in place, where it should be, ie. after first number (it depends how many sections were there on the left side of our number), so you'll have: $$\sqrt{2}\approx 1.41421$$
To obtain accuracy of 20 numbers after the comma, you should append 20 sections of 00 in the step 2. , ie.:
02.00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
Newton-Rhapson is a good idea because of the convergence rate. However, I am more of a fan of using Taylor's expansions here since it is super easy to derive on the go to give fairly ok estimates in quite a reasonable time. So, the way to go to find $\sqrt{x}$ is to find first the closest integer which approximates $\sqrt{x}$ and call this $a$, then apply Taylor to $a^2$. Then Taylor says $$\sqrt{x} \approx a + (x-a^2)\cdot \frac{1}{2 a} - (x-a^2)^2/2 \cdot \frac{1}{4 a^3} + \cdots.$$ The thing that is nice here is that you also get bounds on the error you make. So, denote $f(x) = \sqrt{x}$, then the error of a $n$th order approximation (i.e., going as far as $(x-a^2)^n/n! \cdot f^{(n)}(a^2)$ in the approximation above) is given by $$(x-a)^{n+1}/(n+1)! \cdot f^{(n+1)}(\xi)$$ for a certain $\xi$ between $a^2$ and $x$. This can be estimated quite easily since this $f^{(n+1)}$ is monotone around $x$. Thus look at the boundaries of the domain of $\xi$ and find the 'best' maximal value which you can calculate without a calculator.
Example for $x=2$. Apparently $1$ is the closest integer to $\sqrt{2}$ and thus we will take $a=1$. Then, let's take a second order approximation $$\sqrt{2} \approx 1 + (2-1)\cdot \frac{1}{2} - (2-1)^2/2\cdot \frac{1}{4} = 1 + 0.5 - 0.125 = 1.375$$ and the absolute error is given by $$E=\left|(2-1)^3/3!\cdot \frac{3}{8 \cdot \xi^2\sqrt{\xi}}\right| = \frac{1}{16} \cdot \frac{1}{|\xi^2\sqrt{\xi}|}$$ for a certain $\xi$ between $1$ and $2$. Since this is a decreasing function on $(1,2)$. The maximum is attained at $1$ and hence the error is bounded by $$E \leq \frac{1}{16}$$ which seems to be a good estimate since $E = 0.039\dots$ and $1/16 = 0.0625$.
Edit As some of you noted this method 'looks' more difficult than Newton-Rhapson and the convergence is slower. The last part is obviously true and I would answer this question with: How quick do you need it to be and do you want to calculate it in your head or do you have a computer? Do you need to have a quick guess which is approximately equal to the value of $\sqrt{2}$ or do you need a precise estimate. If you don't have a computer but pen and paper, the best method is Newton-Rhapson.
I would argue that my method is better if you don't have pen and paper or a computer and you are asked to give an estimation of $\sqrt{10}$ on the go (especially for $\sqrt{x}$ with $x$ big, the Taylor approximation is better since the $\sqrt{\bullet}$ function becomes more linear as $x$ grows).
I agree that my method looks way more difficult but it isn't if you get more familiar with it. Also, this method is super quick in terms of calculation time in your head and if you practice a little with it, it becomes way easier. Also, this method works particularly nice for $\sqrt{x}$ where $x$ differs one from a perfect square because then the $(x-a^2)^n$ term will always be one.
Let's look at an example here. Suppose you need to calculate $\sqrt{122}$, then first order approximation of my method gives $$\sqrt{122} \approx 11 + \frac{1}{2\cdot 11}.$$ It took me less than one second to find this approximation and the second order approximation works almost as quick here. You just need to add $\frac{-1}{8\cdot 11^3}$. Please note that the error of the first order approximation here is approximately equal to $10^{-4}$.
If you apply Newton-Rhapson here you get the same approximation after one step if you choose $x_0=11$. The only thing is that I always forget what the exact form is of Newton-Rhapson. So when I want to apply it, I have to think about it where I could have immediately applied Taylor but I would say that is just my particular preference.
• I'd say this is more difficult, less precise, and not as generally applicable as Newton-Raphson. – leftaroundabout Sep 14 '18 at 14:48
• I would say it is less difficult since when you apply Newton-Rhapson you always have to find the exact algorithm and this method can be applied to find $\sqrt{2.243}$ also quite quickly. – Stan Tendijck Sep 14 '18 at 15:38
• I agree with @leftaroundabout, but perhaps if you edit into your post an illustration of how this method could be used by hand to compute rad 2 to high accuracy, it would appear simpler. Right now, it looks much more difficult. – Wildcard Sep 14 '18 at 18:17
• Taylor's converges much more slowly than Newton Raphson. Note the second order term starting with initial guess 1 is 1.4166.... already correct to two digits behind the decimal. You might get an additional correct digit at each step of the calculation heavy Taylor series. The accuracy doubles per step for Newton Raphson without the difficulty of calculating the Taylor coefficients. There might be ways to patch it up. There's an alternative series to the Taylor series for arctan that converges much faster than Taylor. – TurlocTheRed Sep 15 '18 at 2:09
I came up with an interesting, but terribly inefficient method.
Consider the sequence {$$x_n$$}: $$1,\frac{1}{2},\frac{1}{2},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}$$, ...
Suppose you want k digits of the square root of 2. Then add up the first $$100^k$$ terms and then divide the sum by $$10^k$$.
I know an easy way to calculate the binary digits of $$\sqrt{2}$$. Take the ordered pair (1, 2) $$1^2$$ is less than 2 and $$2^2$$ is more than 2. Calculate the square of the average $$1.5^2$$ in base 2. The square of the average is just the average of the squares minus $$\frac{1}{4}$$. The result expressed in binary is 10.01 so the first binary digit after the decimal is 0. Take the next ordered pair to be (1, 1.5) and calculate the square of its average which is the average of its squares minus $$\frac{1}{16}$$. The result expressed in binary is 1.1001 so the next binary digit is 1.
Towers' bisection method above is similar to your own approach, but more efficient. Another method that is not as good as binary search, but is better than your own method, is to increment the last digit in bigger steps. I would try incrementing by 3. The worst case is that you reach the correct digit in 5 steps instead of 9.
My favorite method for mental approximation is to find the next lowest square, determine the error, and add to its square root the error divided by double the guess. For sqrt(200), the lowest square is 196. The error is 4, so my mental estimate is 14 + 4/14 = 14.142857...
I apologize for off-topic, but note that square roots can be used to calculate logarithms by a process similar to bisection. I suspect that is how it was done in the late 16th century, as they did not yet have calculus. In our times, there are extremely accurate formulas for logarithm that still require square roots. This exercise should make you appreciate the power of a square root button on a calculator, even if you have no "scientific" functions.
## protected by cactus314Sep 30 '18 at 16:59
Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
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2019-06-26 17:58:39
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https://www.ingentaconnect.com/content/bsc/jace/2012/00000095/00000010/art00047
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# Stability of Monosulfate in the Presence of Iron
\$52.00 + tax
Monosulfate (3CaO·(Al x Fe1x )2O3·CaSO4·12H2O) is an AFm phase that can be formed during the hydration of cement. Fe‐containing monosulfate and (Al,Fe) mixed monosulfate were synthesized and characterized. Fe‐monosulfate is composed of a positively charged main layer [Ca2Fe(OH)6]+ and a negatively charged interlayer [1/2SO4·3H2O], crystallizes in the trigonal $\mathbit{R}\overline{\mathbf{3}}$ symmetry, and is isotypic with Al‐monosulfate. The solubility product at 25°C was determined to be −31.57. The formation of solid solution due to AlFe substitution in the main layer structure of monosulfate was observed. Based on the evolution of the unit cell parameters and the thermodynamic investigations, a presence of solid solution from 0.0 to 0.45 Al/(Al + Fe) ratio and a miscibility gap in the range 0.45 < Al/(Al + Fe) ratio <0.95 is suggested.
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Document Type: Research Article
Publication date: October 1, 2012
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2019-06-17 15:01:48
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http://superwinds.blogspot.com/2010/02/interesting-astrophysics-feb-01-to-feb.html
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## Friday, February 19, 2010
### Interesting Astrophysics: Feb 01 to Feb 19, 2010
I have no time to provide commentary, so without further interruption here are the papers and preprints that I consider interesting that have appeared over the period Feb 01-19, 2010.
Galaxies and Starbursts
The AGN properties of the starburst galaxy NGC 7582
T.V. Ricci, J.E. Steiner, R.B. Menezes, A. Garcia-Rissmann, R. Cid Fernandes,
Comments: 1 page, 2 figures, to be published in the Proceedings of the IAU Symposium no. 267
The stellar populations of the AGN/Starburst galaxy NGC7582
T.V. Ricci, J.E. Steiner, R.B. Menezes, A. Garcia-Rissmann, R. Cid Fernandes,
Comments: 2 pages, 2 figures, to be published in the Proceedings of the IAU Symposium no. 262
Their abstract: "NGC 7582 is defined as a Starburst/AGN galaxy, since its optical and X-Ray spectra reveal both characteristics. In this work, we show the results of a stellar population modeling in a datacube taken with the Gemini South telescope. We found that $\sim$ 90% of the light in the field of view is emitted by stars that are less than 1 billion years old. A strong burst occurred about $\sim$ 6 million years ago and has nearly solar metallicity. We also found a Wolf-Rayet cluster."
Star Formation in the Outer Filaments of NGC 1275
R.E.A. Canning, A.C. Fabian, R.M. Johnstone, J.S. Sanders, C.J. Conselice, C.S. Crawford, J.S. Gallagher III, E. Zweibel,
Comments: Accepted for publication in MNRAS, 14 pages, 14 figures, 3 tables
A panoramic view of the Milky Way analogue NGC 891
M. Mouhcine, R. Ibata, M. Rejkuba,
Comments: 2 figures, Apj Letter, in press
A panoramic view of M81: New stellar systems in the debris field
M. Mouhcine, R. Ibata,
Evidence of Early Enrichment of the Galactic Disk by Large-Scale Winds
T. Tsujimoto, J. Bland-Hawthorn, K.C. Freeman,
Comments: 9 pages including 5 figures, accepted for publication in PASJ
Starburst evolution: free-free absorption in the radio spectra of luminous IRAS galaxies
M. S. Clemens, A. Scaife, O. Vega, A. Bressan,
Comments: 13 pages, 5 figures, accepted by MNRAS
Metal-line emission from the warm-hot intergalactic medium: II. Ultraviolet
Serena Bertone, Joop Schaye, C.M. Booth, Claudio Dalla Vecchia, Tom Theuns, Robert P.C. Wiersma,
Comments: 21 pages, 22 figures. Submitted to MNRAS
Their abstract: "Approximately half the baryons in the local Universe are thought to reside in the warm-hot intergalactic medium (WHIM). Emission lines from metals in the UV band are excellent tracers of the cooler fraction of this gas. We present predictions for the surface brightness of a sample of UV lines that could potentially be observed by the next generation of UV telescopes at z<1. We use a subset of simulations from the OWLS project to create emission maps and to investigate the effect of varying the physical prescriptions for star formation, supernova and AGN feedback, chemodynamics and radiative cooling. Most models produce results in agreement within a factor of a few, indicating that the predictions are robust. Of the lines we consider, C III is the strongest line, but it typically traces gas colder than 10^5 K. The same is true for Si IV. The second strongest line, C IV, traces circum-galactic gas with T~10^5 K. O VI and Ne VIII probe the warmer (T~10^5.5 K and T~10^6 K, respectively) and more diffuse gas that may be a better tracer of the large scale structure. N V emission is intermediate between C IV and O VI. The intensity of all emission lines increases strongly with gas density and metallicity, and for the bright emission it is tightly correlated with the temperature for which the line emissivity is highest. In particular, the C III, C IV, Si IV and O VI emission that is sufficiently bright to be potentially detectable in the near future (>10^3 photon/s/cm^2/sr), comes from relatively dense (rho>10^2 rho_mean) and metal rich (Z>0.1 Z_sun) gas. As such, emission lines are highly biased tracers of the missing baryons and are not an optimal tool to close the baryon budget. However, they do provide a powerful means to detect the gas cooling onto or flowing out of galaxies and groups. (Abridged)"
Black Holes and AGN
Timing the starburst-AGN connection
Vivienne Wild, Timothy Heckman, Stephane Charlot,
Comments: 16 pages, 12 figures, accepted for publication in MNRAS
Interstellar Medium
Metal-Ion Absorption in Conductively Evaporating Clouds
Orly Gnat, Amiel Sternberg, Christopher F. McKee,
ITERA: IDL Tool for Emission-line Ratio Analysis
Brent Groves, Mark Allen,
Stars, Supernove and Planets
The Birth Environment of the Solar System
Comments: 58 pages including 7 figures, to appear in Annual Reviews of Astronomy and Astrophysics (2010, Vol. 48)
Absract in full: "This paper reviews our current understanding of the possible birth environments of our Solar System. Since most stars form within groups and clusters, the question becomes one of determining the nature of the birth aggregate of the Sun. This discussion starts by reviewing Solar System properties that provide constraints on our environmental history. We then outline the range of star-forming environments that are available in the Galaxy, and discuss how they affect star and planet formation. The nature of the solar birth cluster is constrained by many physical considerations, including radiation fields provided by the background environment, dynamical scattering interactions, and by the necessity of producing the short-lived radioactive nuclear species inferred from meteoritic measurements. Working scenarios for the solar birth aggregate can be constructed, as discussed herein, although significant uncertainties remain."
A scenario of planet erosion by coronal radiation
J. Sanz-Forcada, I. Ribas, G. Micela, A. M. T. Pollock, D. Garcia-Alvarez, E. Solano, C. Eiroa,
Comments: Accepted by A&A Letters (8 Feb. 2010). 4 pages, 3 figures, 1 online table (included)
On the Disruption of Star Clusters in a Hierarchical Interstellar Medium
Bruce G. Elmegreen, Deidre A. Hunter,
Comments: ApJ vol. 712, March 20, 2010, 33 pages 15 figures
Do Wolf-Rayet stars have similar locations in hosts as type Ib/c supernovae and long gamma-ray bursts?
G. Leloudas, J. Sollerman, A. J. Levan, J. P. U. Fynbo, D. Malesani, J. R. Maund,
Comments: Under review in A&A. The present version includes suggestions from the referee
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2017-10-22 01:05:15
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https://indico.fnal.gov/event/16384/session/12/contribution/50
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# New Perspectives 2018
18-19 June 2018
Fermilab, Wilson Hall
US/Central timezone
Home > Timetable > Session details > Contribution details
# Contribution Oral Presentation
Fermilab, Wilson Hall - One West
# Repurposing MINOS Scintillator Modules for the Short Baseline Neutrino Program Far Detector (ICARUS) Cosmic Ray Tagger
## Speakers
• Mr. Christopher HILGENBERG
## Content
The ICARUS T600 liquid argon time-projection chamber will be the far detector for the short baseline neutrino program. The detector will operate at shallow depth and therefore be exposed to the full surface flux of cosmic rays, which poses a problematic background to the electron neutrino appearance analysis. A direct way to remove this background is to utilize a detector external to the liquid argon active volume capable of tagging thoroughgoing cosmic muons with high efficiency. Ideally, this cosmic ray tagger (CRT) would provide full geometric coverage of the T600 amounting to about 1000$m^2$. This is achieved through adopting a system based on extruded organic scintillator, wavelength-shifting fibers, and silicon photomultipliers. Due to the large area, the CRT is broken into 3 subsystems: the top portion will be new construction, the side coverage will be provided by salvaged MINOS scintillator modules, and the bottom will be covered by Double Chooz veto modules. To cope with high rates of cosmic muons, the MINOS system requires a new optical readout and front-end electronics. Here, I present results from the research and development of this new readout scheme and testing of the salvaged modules.
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2019-12-07 19:14:25
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https://www.springerprofessional.de/design-of-a-passive-gait-based-ankle-foot-exoskeleton-with-self-/18094796?fulltextView=true
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main-content
Weitere Artikel dieser Ausgabe durch Wischen aufrufen
01.12.2020 | Original Article | Ausgabe 1/2020 Open Access
Design of a Passive Gait-based Ankle-foot Exoskeleton with Self-adaptive Capability
Zeitschrift:
Chinese Journal of Mechanical Engineering > Ausgabe 1/2020
Autoren:
Xiangyang Wang, Sheng Guo, Bojian Qu, Majun Song, Haibo Qu
1 Introduction
Legs are important for locomotive ability, and enable us to move and travel in our daily life. Enduring walking and lower energy costs can significantly expand the range of human activities. In some rough terrains, walking is the only option because wheeled vehicles are unable to transport people. Thus, improved walking economy is beneficial to everyone. Although the walking behavior of humans is well-tuned under natural selection [ 1], there is still room for improvement, especially when exoskeleton devices are developed and enter into service.
Since half of the required mechanical power output is generated by the ankle during push-off [ 2], the ankle-foot exoskeleton has been extensively studied over the past two decades. Many powered and unpowered devices have been proposed to assist in the push-off process, reducing the energy cost to the human body [ 3]. Although ankle-foot orthoses (AFOs) are also capable of providing assistance during walking, they are often designed for people suffering from a leg pathology to increase their ambulatory ability, such as the MIT AFO [ 4].
Unlike AFOs, the ankle-foot exoskeleton (AFE) is a device that augments the performance of ankle movements by an able-bodied wearer [ 5]. In general, such devices can be divided into two categories according to the actuation mode: active ankle-foot exoskeleton (AAFE) and passive ankle-foot exoskeleton (PAFE).
AAFEs employ an external power source to provide biologically equivalent levels of joint power, and reduce the essential amount of energy required from human muscles [ 6]. Some clever mechanisms are designed so that the driving force of the motor can be converted into an assistive torque about the ankle joint. For example, Mooney et al. designed an autonomous exoskeleton that adopts a triangular structure formed by struts. Driven by a winch actuator, the device converts the pulling force into a torque about the ankle joint [ 7, 8]. Liu et al. [ 9] proposed a lightweight exoskeleton with an elastic spring that can continuously store the energy injected by a lightweight motor, and release it quickly to provide high-powered assistance. To achieve linear motion, Meijneke et al. [ 10] presented a series elastic actuator (SEA) consisting of an electric motor and a ball-screw gear to control the distance between two rigid endpoints on the shank and foot, respectively. Compared with electric motors, lightweight and the structural similarity with a muscle-tendon unit, greatly allow for the extensive application of pneumatic muscles in the design of the AAFE, such as the exoskeletons developed for the elderly by Galle [ 11, 12], and Ferris [ 1316]. Although the mechanical structures of the devices mentioned above are sufficiently light, the energy cost of the combined mass of actuators and power sources is more than four times that of the same mass attached to the waist [ 17]. Therefore, the strategy of placing the actuators on the waist or back has been employed to achieve better walking economy. One example is the AAFE designed by Jackson et al. [ 18, 19], which is powered by a flexible Bowden cable and a motor. Based on this device, Zhang [ 20, 21] proposed a control method to realize the optimization of assistance during walking.
Although AAFEs can provide as much energy as required, there are still some disadvantages. One major concern is that the force profiles and onset timing are gait-based. Therefore, accurate detection of the gait stages is critical. Electronic sensors must be used to detect all kinds of gait events. In addition, the exoskeleton must be sufficiently compliant to adapt to the full ankle-joint range of motion without impeding normal human movement during the swing phase. The inability to dorsiflex freely could impose a significant metabolic penalty [ 22]. The distal mass is another problem. Circuit-based sensor-control systems and power sources can result in additional system mass and lead to reduced wearing comfort. Shock while walking is also a side factor that may damage the circuit. Because of the above reasons, purely mechanical exoskeletons (PAFEs) have been developed as an alternative.
Totally different from the powered exoskeleton, PAFEs have recently been studied by some scholars. Previous work revealed the advantages of such devices, e.g., these require no power source and have a circuit-based control system. Moreover, since all the functions are realized mechanically, these devices are highly reliable and can provide continuous assistance. Elastic elements (e.g., springs and elastic cables) are usually employed to store energy from the human body, and then release this energy to make users move more quickly. As a result, walking efficiency is noticeably increased [ 23]. Farris et al. [ 24] designed an exoskeleton with a suspended spring parallel to the calf muscles. However, owing to the lack of gait identification, the resting length of the spring must be adjusted for each participant.
To identify the gait period without using electronic sensors, smart clutches [ 1, 22, 2527] have been designed to mechanically control the engagement of the elastic components and provide assistance at the proper time. The clutch ensures that assistance is always provided at the right time without impeding the free movement of the ankle joint during the swing phase. Collins et al. [ 1, 22] designed a clutch based on the ratchet-pawl mechanism so that the spring linkage can be controlled by setting the timing of pawl latch and release. However, this device lacks adaptability as it needs to be customized for individuals. In another research by his team, an electro-adhesive clutch [ 25, 26] was presented to achieve the same purpose, which was actually quasi-passive. Recently, Yandell et al. [ 27] proposed a low-profile PAFE with a clutch that fits under clothes. However, the clutch may fail to be clutched if the stiffness of the spring is large enough. To solve these problems, a novel two-input clutch was presented in our previous work [ 28] with springs triggering the clutched state.
However, our previous design has a weakness. A time- delay problem occurs during state-switching due to the deformation of the trigger spring. The clutch only works normally when the users walk at a low speed. The problem tends to become more obvious as walking speed increases. In this paper, we perform an iteration based on our previous design. Walking trials are carried out based on the prototype to evaluate and verify the assistive performance of the newly proposed ankle-foot exoskeleton. The new clutch uses rigid instead of soft triggers to achieve faster state-switching. Compared with the existing clutch structure, the proposed clutch is suitable for almost all users and is reliable in maintaining a clutched state even though a large tensile force is generated in the rope attached to the clutch.
An energy conversion model is also developed to theoretically validate the feasibility of metabolic cost reduction when the PAFEs are used. This part of the work has not been verified by mathematics in the past. Although many walking trials in previous research have shown that human metabolic cost is reduced by wearing a PAFE, a theoretical model must be established and validated before the design process.
2 Biomechanics and Energetics
2.1 Biomechanics during Human Walking
A complete gait cycle can be divided into four gait stages according to the contact states between the shoe sole and the ground: the heel strike (HS), flat foot (FF), push-off (PO), and swing. For the ankle joint, it generates positive or negative mechanical work during different gait periods (e.g., negative work during FF, and positive work during PO [ 28]).
Both positive and negative work impose an energy cost on the human body [ 29]. Positive work is usually performed to assist the trunk as it vaults over the stance limb [ 30], whereas negative work is done with mechanical energy (ME) converted into other forms of energy. The reason for the negative part can be the energy stored in the tendons or dissipated owing to the damped motion of fat, viscera, and muscle [ 31]. Hence, only a part of the energy can be recycled during normal walking, with most of the energy wasted. Consequently, humans need to consume their biomass energy continuously to maintain their walking speed and compensate for the energy loss in the body due to dissipation.
Since energy stored in an elastic component can be returned almost without any loss, using a spring to reduce the dissipated part of the energy consumed by human tissues may be a good way to reduce human energy expenditure.
2.2 Energy Conversion Model
To demonstrate the feasibility of energy cost reduction using elastic components, a mathematic model is developed to compare the cases of normal walking and assisted walking with flexible components spanning the ankle joint. As shown in Figure 1, the skeletons in cases (a) and (b) undergo the same walking process (i.e., the initial ME at the initial state, final ME at the end state; the joints’ trajectories are the same in both cases). A spring parallel to the calf muscles is attached to the human body with its two ends anchored at the shank and the foot. Our model focuses on the ankle dorsi/plantar flexion muscle group. The energy consumed in other segments of the human body is assumed the same in both cases.
One of the legs is considered, where negative work is done as the ankle dorsiflexes from the HS to the end stance. Part of the initial ME, which has a total amount of x, is converted and goes to two places, i.e., converted into elastic potential energy in the series elastic muscle-tendon mechanics [ 32], or dissipated in biological tissue. Bioenergy is continuously consumed from human body and injected into the kinematic system (KS) so that the final ME will not decrease gradually with step-to-step transitions.
Plantar flexor muscles (i.e., soleus, gastrocnemius, tibialis anterior) are the main muscle group that contributes to walking. Assuming that the proportional relationship between stored and dissipated energy in the plantar flexor muscles is fixed, for every unit of negative work done by the ankle joint, the stored part is p(0 < p < 1), and there is a (1 − p) unit of energy dissipation. In case (a), the ME of a total amount of q 1 is converted as the ankle outputs negative work, with $$x - q_{ 1}$$ left in the KS. The amount of energy stored in the muscle-tendon is $$q_{{_{ 1} }} \times p$$, while the rest $$q_{ 1} \times (1 - p)$$ dissipates simultaneously. During the PO, the stored energy is returned. Additional energy in the amount of $$m - x + q_{ 1} (1 - p)$$ must be injected into the KS from the human body, so that the desired movement can be achieved.
For case (b), in addition to the energy stored in the muscle-tendon unit, the spring is also stretched with quantities of ME converted into strain energy. The stiffer the spring, the more ME is converted. The total amount of $$(q_{ 2} - n) \times p + n$$ is thus recycled during PO. It can be derived that the total amount of $$m - x + (q_{ 2} - n)(1 - p)$$ must be additionally provided by the human body. Compared with case (a), case (b) would save more energy when the following condition is satisfied:
$$q_{ 2} - q_{ 1} < n.$$
(1)
Since part of the ME that should have been stored in the muscle-tendon unit is stored in the spring instead, as shown in the blue block in both cases in Figure 1, this implies
$$q_{ 1} \times p > (q_{ 2} - n) \times p,$$
(2)
Eq. ( 1) is thus satisfied.
Compared with normal walking, walking with elastic elements can save the human energy cost of $$(q_{ 1} - q_{ 2} + n)(1 - p)$$. The amount relies on the proportional factor p and the spring stiffness. Usually, it is impossible to alter the factor p since it is a physiological property of individuals. However, we can increase the proportion of energy stored in the spring, i.e., the factor n, so that the metabolic cost to the human body can be reduced.
It’s worth noting that n must be less than x. In general, the ME conversion process during dorsiflexion will slow the pendular motion of the human body. A spring that is too stiff will impede the normal walking process and result in reduced wearing comfort.
3 Design Overview
3.1 Goals
The main purpose of this research is to design a PAFE with a smart clutch based on our previous work [ 28]. The device is intended to address the problem of time-delay, and perform the function of automatic state-switching based on mechanical identification of the current gait stages. The engagement of the elastic spring relies on the clutch state, and controls the energy storage and release process without impeding the free ankle motion during the swing phase. The device aims to reduce the metabolic expenditure of human body without any power sources, motors, or electronic sensors.
3.2 General Overview
Figure 2 presents an overview structure of the PAFE with the newly designed clutch. It consists of four parts: the clever clutch, an extension spring, a shank brace, and a shoe. The clutch is fixed at the outer side of the shoe sole and mounted on a 10 mm thick Al-alloy plate. The Al-alloy plate is bolted with a shoe sole, of which half the heel is cut off from the backside for the Al-alloy plate so that wearers can keep their foot flat when standing normally.
One end of the wire rope is attached to a pulley in the clutch, whereas the other end is routed to the back of the human foot via three U-groove bearings and connected to the lower end of the extension spring. The shank brace comprises a plastic shell and strap. The inside of the plastic shell is filled with sponges to improve wearing comfort. The strap is made of nylon to prevent deformation when force is applied. The proposed PAFE has no hinge-like artificial joint, and nonsagittal plane motion can be achieved without restraints.
The upper end of the extension spring is attached to an anchoring point at the backside of the shank brace. The spring is supposed to be engaged at the beginning of the HS when the clutch is clutched and stretched to its maximum length at the terminal stance with a large restoring force. During the PO, the spring releases the stored energy and provides assistive ankle moment, propelling the human body to move forward, thus reducing muscle force. During the swing phase, the extension spring is supposed to be disengaged when the clutch is switched to an unclutched state, which ensures the free movement of the ankle joint.
3.3 Clutch Design
Considering that the foot is “rolling” from the rear foot to the forefoot (see Figure 3), we propose a double-trigger method to distinguish corresponding gait stages. A trigger rod and trigger button are separately distributed at the foresole and rearsole. They can be moved under the ground reaction force (GRF), pushing other components in the clutch to achieve mechanical control of the clutch states.
The clutch is the core component of the exoskeleton. The proposed clutch has a small size and consists of many tiny components. Figure 4a shows an enlarged view of the internal structure of the clutch. Shaft seat (a) and (b) are mounted on the plate. The pulley and connecting rod (b) are concentric with the shaft (a) passing through their center holes. The lever near the connecting rod (b) is threaded onto the lever holder that is bolted to the shaft seat (a). The return spring (c) is used to help the lever revert to its original position when the human foot is off the ground. At the original position, there is no contact between the rubber block and pulley. The connecting rod (b) and lever are connected by the connecting rod (a). A four-bar mechanism is thus formed.
When the foresole touches the ground, the forced motion of the trigger rod can be transmitted by a transmission mechanism and the four-bar mechanism, resulting in contact between the flange of the pulley and the rubber block attached to the lever. The transmission mechanism consists of an upper rod and a lower rod used for length adjustment so that the GRF can be transmitted into the clutch as the bottom gripper is compressed into the U-shape slot at the foresole (see Figure 4b). Wearing comfort is thus improved.
Similarly, the trigger button pushes the pin (a) to move upward when the rearfoot touches the ground. This leads to the clockwise motion of the lever about the lever holder and results in contact between the rubber block and pulley. The return spring (a) is placed outside the trigger button with its two ends keeping contact with the shaft shoulder of the trigger button and the inner shoulder of the bottom block, respectively, so that the trigger button can be moved to its original position when no GRF is applied.
Friction is generated between the rubber block and pulley flange as contact occurs, resisting the clockwise rotation of the pulley. Deformation of the rubber block occurs under the action of friction and results in a much larger friction force due to the increased normal stress perpendicular to the contact surface. The clutch is therefore clutched with the clockwise rotation limited. The two inputs, represented as red arrows in Figure 5, are independent in triggering the clutched state of the clutch.
As the clutch is clutched, the wire rope cannot be dragged out from the pulley. Consequently, the lower end of the extension spring is fixed, and the spring can be stretched as the foot dorsiflexes during the FF with energy stored. Until the foot is off the ground, the clutch is unclutched, and the pulley is allowed to rotate in both directions. The rope can be freely dragged in and out of the pulley without impeding the free rotation of the biological ankle joint.
To keep the cord tensioned throughout the gait cycle, a constant force spring is placed on the spring holder. It is attached to the edge of the pulley and always generates a constant counterclockwise torque, which can apply a constant tensile force on the rope. When the rope is slack, or the operating length (distance between the points A and C in Figure 2) decreases, the excess rope will be dragged into the pulley immediately.
3.4 Prototype Specification
A prototype has been manufactured and assembled. The thickness of the pulley flange is 1 mm, and the constant spring can provide a pulling force of 8 N. The metal plate and linkage mechanisms are made of 6061 Al-alloy, while all the shafts are made of stainless steel. The material of the shoe is rubber. The shank brace was made from a leg guard often used by football players as it is lightweight, and its curved surface fits the users’ leg size. The total weight of our prototype is 765.5 g (shoe included), of which 349 g is due to the Al-alloy plate, 42 g is due to the shank brace, and 46.2 g is due to the extension spring. A comparison was made between our PAFE and the previous ones listed in Table 1. It shows that AAFE is generally heavier than those without power sources. Among the passive devices, our prototype seems slightly heavier than those developed by Collins and Yandell. However, the weight of our prototype can be further reduced when lightweight materials are used. The Al-alloy plate is 10 mm thick, and occupies a large proportion (45.6%) of the overall mechanical device.
Table 1
Mass comparison on recent ankle-foot exoskeletons (some data are collected by Yandell et al. [ 27])
Mooney et al. [ 7]
Witte et al. [ 19]
Collins et al. [ 1]
Ferris et al. [ 14]
Meijneke et al. [ 10]
Yandell et al. [ 27]
Wang et al.
Mass per leg (g)
2300
835
875
408–503
1700
1500
598
765.5
Mass at waist/back (g)
1700
0
0
0
0
5200
0
0
Powered
Yes
No
No
No
Yes
Yes
No
No
4 Walking Trials
4.1 Performance Evaluation
To evaluate the assistance provided by our PAFE to the plantarflexion muscles, five volunteers participated in walking trials (all males aged 22–46 years old). All procedures in this study were carried out in accordance with the protocol approved by the Institutional Review Board of Beijing Jiaotong University. All the human subjects gave their written informed consent prior to participation.
First, subjects walked on a treadmill in casual shoes at 4 km/h for 1 min without a break. Then, the speed was increased to 5 km/h, and the walking test was repeated. The subjects were required to rest for 10 min between consecutive sets of gait tests. Both tests served as control. Then, they walked while wearing the PAFE on their right leg and repeated the tests at the same speeds. Since walking speed relies on the stride length and gait frequency, the subjects were required to keep their walking pace constant (i.e., frequency is almost fixed) during the test, and their stride length consequently varied. Two sets of spring stiffness were used in the assistive walking to verify the performance (low stiffness: 1.58 N/mm and high stiffness: 9.05 N/mm). The electromyography (EMG) of the soleus of each subject was recorded during 1 min of walking. Two electrodes were attached to the soleus (see Figure 6), whereas the reference electrode was attached to the patella, where the muscle activity is extremely weak so that the calibration process is easier. After low-pass filtering, rectification, and amplification, the EMG signal could be read directly.
Figure 7 shows the soleus activity of an arbitrary three strides of one subject in the walking test. Compared with normal walking (without assistance), the maximum instantaneous muscle activity during the PO decreased when the PAFE was worn. At 4 km/h, however, the soleus was more active during the FF compared with normal walking when the low-stiffness spring was employed. This could be attributed to the additional mass attached to the human body and very limited assistance provided by the exoskeleton. In contrast, the spring with high stiffness provided relatively larger assistance. Consequently, muscle activity remained low throughout the gait cycle. In the swing phase, there was little difference in activity among these cases, since the spring was disengaged and had almost no influence on muscle activity.
As the walking speed increased, muscle activity increased accordingly as more energy is required from the human body to maintain a faster walking speed. Table 2 shows the comparison of the peak value of the muscle activity at different walking speeds. The magnitudes of the activities were averaged across three continuous strides. Comparatively speaking, the exoskeleton with high stiffness provides more assistance and results in a remarkable reduction in soleus activity. When a low-stiffness spring was used, less assistance was provided. The results also show increased percentage reduction of muscle activity under the same spring stiffness condition as the walking speed increased, because a longer step length results in larger spring stretching and restoring force.
Table 2
Data comparison on EMG signals and spring forces
Normal walking
Low stiffness
Reduction
High stiffness
Reduction
4 km/h
E: 804.8
E: 699.0
T: 29.7 N
E: 13%
E: 269.6
T: 101 N
E: 66.5%
5 km/h
E: 1256.7
E: 1050.6
T: 30.6 N
E: 16.4%
E: 349.4
T: 107.3 N
E: 72.2%
Note: E: EMG signal; T: tensile force
In addition to the EMG signal, the assistive tensile force was also measured by a load cell serially connected with the spring. Figure 8a shows the measured tensile forces averaged across three gait cycles. The tensile force during swing is due to the pulling force (approximately 8 N) of the constant force spring within the clutch. From the HS, the spring is continuously stretched, with energy slowly stored in the spring, and then released stably and quickly during the PO. The maximum spring forces measured (also known as the tensile force) are also listed in Table 2.
However, according to our calculations based on the data from the motion capture system (Cortex, Motion Analysis Co., USA), the operating length increased by 25 mm (at 5 km/h) and 24.3 mm (at 4 km/h) from its initial length at the HS to its maximum value as the ankle dorsiflexed. The increment is equal to the stretched length of the spring, of which the maximum restoring force can be calculated based on Hooke’s law:
$$F_{\text{max} } = k \cdot \Delta l,$$
(3)
where k denotes the stiffness of the spring.
The measured spring force differs from the calculated spring force $$F_{\text{max} }$$ (defined as the reference spring force) under all stiffness conditions, as shown in Figure 8a. This can be attributed to the reduced operating length resulting from the deformation of biological tissues at the interface between the shank and brace. Since the spring force is held by the shank brace, slippage from its original position occurs. The actual stretched distance is smaller than the expected value. The stiffer the spring, the larger the generated deformation. Walking speed may also affect the slipping distance.
To verify this, two markers were attached to points A and C, respectively. The practical operating length can be measured with the motion capture system during the walking trials at different walking speeds. As shown in Figure 8b, the operating length decreased by approximately 50% when the high-stiffness spring was used, and by 7.0%–20.2% in the low-stiffness case as the walking speed gradually increased from 4 km/h to 5 km/h. This is because a faster walking speed leads to a larger step length and larger ankle angle variation.
Slipping is a critical problem that leads to less assistance and benefits from the PAFE. We attempted to solve this problem by using more straps to affix the plastic shell of the shank brace. The slipping was reduced, and the generated spring force was comparatively larger. However, this caused lower wearing comfort. This problem can be further addressed by designing an improved human-machine interface; for example, a shank frame.
4.2 Effects on Normal Gait
It has been shown that the proposed PAFE can reduce the soleus force to a certain extent, but its influence on the walking habit is another concern.
With the help of the motion capture system, we also examined the potential influence of the PAFE on human walking habits. Kinematic information was collected when the PAFE with high-stiffness spring was worn on the right foot. The left foot was in a casual shoe and regarded as a control group. A comparison can be made between the walking behavior of the two legs.
Figure 9 shows the plotted trajectories of the markers at the heel of both legs. Owing to the additional mass of the exoskeleton, the subject tends to consciously raise his right foot higher than his left foot. Also, the landing point A of the right foot is always further forward than the left foot, which is due to the introduced inertia of the exoskeleton. During the PO, the heel of the right foot is raised earlier than the left foot. As a result, the point B of the right foot, where the heel begins to leave the ground, is always ahead of the one of the left foot. This is due to the generated assistive torque about the ankle joint.
The ankle angles of both legs are shown in Figure 10. At 4 km/h, the average variations of the dorsiflexion angle $$\alpha$$ (defined in Figure 9) during the FF are 24.9° (for normal walking) and 14.9° (for assisted walking). At 5 km/h, the angle variations are 25.6° and 19.4°, respectively. With PAFE, the foot dorsiflexion was attenuated, which also led to a shorter stride length. During the swing phase, the ankle angle of normal walking was always less than that of assisted walking, with an average angle difference of 6.3°. An obvious “foot-drop” occurred when the PAFE was worn on the foot due to the additional gravitational torque generated by the device.
Subjectively speaking, the subject did not feel discomfort nor was he uncoordinated with the device. If a lighter material is used, then the influence on the normal gait can be further reduced.
5 Limitations
Although the proposed PAFE can provide favorable assistance during walking, the upper stiffness limit of the spring was not determined. The stiffness of the spring should be properly chosen so that the COM velocity of the human body will not drop to zero when it performs the pendular motion. A further experiment can be carried out to find a subjectively suitable stiffness range for users with different body weights and walking speeds. Another limitation is the slipping of the shank brace, which results in reduced tensile force. In our walking tests, the tensile force decreased by as much as 55.2% when the high-stiffness spring was used. The interface between the shank and shank brace must be redesigned to address this problem.
6 Conclusions
In this paper, we proposed a novel PAFE that is lightweight and can assist in the walking process by the mechanical identification of current gait stages, which is completely passive and suitable for nearly all users. Compared with our previous work, the newly designed clutch adopted rigid triggers so that the clutch state can be immediately changed as the GRF is transmitted into the clutch without any delay.
An energy conversion model was developed to validate the feasibility of our design in theory. Unlike previous research, a clever clutch was designed with two inputs that can identify the gait stages based on the contact status between the shoe sole and the ground. By introducing special mechanical constraints, the clutch can stay latched despite the application of an extremely large pulling force. The switching between the clutched and unclutched states is comparatively more reliable.
A prototype was produced based on the design. Walking trials were also carried out to evaluate the assistance provided by the PAFE under different speed and stiffness conditions. The muscle activity and spring forces were compared, which showed the benefit of the exoskeleton and its potential use in walking assistance. The influence of the PAFE on walking habits was also studied. With the assistance, the stride length was shortened along with possible foot-drop problems. However, these side effects can be addressed by practice walking and further reducing the device mass.
Acknowledgements
All the authors would like to thank thank Yang Du for providing equipment support (Zebris FDM-T, Zebris Medical GmbH, Germany).
Competing Interests
The authors declare no competing financial interests.
Literatur
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2021-01-16 18:17:54
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http://mathonline.wikidot.com/the-completion-of-a-normed-algebra
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The Completion of a Normed Algebra
# The Completion of a Normed Algebra
Definition: Let $\mathfrak{A}$ be a normed algebra. A Banach algebra $\mathfrak{B}$ is said to be a Completion of $\mathfrak{A}$ if there exists an isometric isomorphism $T$ from $\mathfrak{A}$ onto a dense subalgebra of $\mathfrak{B}$.
The following theorem tells us that every normed algebra has a completion.
Theorem 1: If $\mathfrak{A}$ is a normed algebra, then $\mathfrak{A}$ has a completion.
• Proof: We know that as a normed space, $\mathfrak{A}$ has a completion $\mathfrak{B}$, i.e., a Banach space such that there exists an isometric isomorphism $T$ from $\mathfrak{A}$ onto a dense subspace of $\mathfrak{B}$. All that remains to show is that with the additional structure of product on $\mathfrak{A}$, that $\mathfrak{B}$ also has an analogous product.
• Let $b, b’ \in \mathfrak{B}$. Since $T(\mathfrak{A})$ is assumed to be dense in $\mathfrak{B}$, there exists sequences $(a_n), (a_n)’ \subset \mathfrak{A}$ such that:
(1)
\begin{align} b = \lim_{n \to \infty} T(a_n) \quad \mathrm{and} \quad b’ = \lim_{n \to \infty} T(a_n’) \end{align}
• Since $(T(a_n))$ converges to $b$, $(T(a_n))$ is a Cauchy sequence in $\mathfrak{B}$. That is, for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $m, n \geq N$ then $\| T(a_m) - T(a_n) \| < \epsilon$. But $T$ is an isometric isomorphism. So for all $m, n \in \mathbb{N}$, $\| T(a_m) - T(a_n) \| = \| T(a_m - a_n) \| = \| a_m - a_n \|$. Thus, for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $m, n \geq N$ then $\| a_m - a_n \| < \epsilon$, and so $(a_n)$ is a Cauchy sequence in $\mathfrak{A}$. A similar argument shows that since $(T(a_n’))$ converges to $b’$, we have that $(a_n’)$ is a Cauchy sequence in $\mathfrak{A}$ as well.
• Also, since $T(a_n)$ converges to $b$, we have that $(T(a_n))$ is a bounded sequence, i.e., there exists an $M > 0$ such that $\| T(a_n) \| \leq M$ for all $n \in \mathbb{N}$. Again, since $T$ is an isometry, this implies that $(a_n)$ is bounded with $\| a_n \| \leq M$ for all $n \in \mathbb{N}$. A similar argument shows that there exists an $M’ > 0$ such that $\| a_n’ \| \leq M’$ for all $n \in \mathbb{N}$.
• Consider the sequence $(a_na_n’) \subset \mathfrak{A}$. For each $m, n \in \mathbb{N}$, we have that:
(2)
\begin{align} \| a_ma_m’ - a_na_n’ \| &= \| a_ma_m’ - a_ma_n’ + a_ma_n’ - a_na_n’ \| \\ &= \| a_m(a_m’ - a_n’) \| + \| (a_m - a_n)a_n’ \| \\ &= \| a_m \| \| a_m’ - a_n’ \| + \| a_m - a_n \| \| a_n’ \| \\ & \leq M \| a_m’ - a_n’ \| + M’ \| a_m - a_n \| \end{align}
• Given $\epsilon > 0$, since $(a_n)$, $(a_n’)$ are both Cauchy, there exists $N_1, N_2 \in \mathbb{N}$ such that if $m, n \geq N_1$ then $\| a_m - a_n \| < \frac{\epsilon}{M’}$ and if $m, n \geq N_2$ then $\| a_m’ - a_n’ \| < \frac{\epsilon}{M}$. Then $N := \max \{ N_1, N_2 \}$ is such that if $m, n \geq N$ then $\| a_ma_m’ - a_na_n’ \| < \epsilon$ from the above inequality. So $(a_na_n’)$ is a Cauchy sequence.
• For a third time, since $T$ is an isomorphism, this implies that $(T(a_na_n’))$ is a Cauchy sequence in $\mathfrak{B}$. Since $\mathfrak{B}$ is complete, this sequence converges to some $c \in \mathfrak{B}$. So define the multiplication on $\mathfrak{B}$ by:
(3)
$$bb’ := c$$
• Then the normed space isomorphism $T$ extends to an algebra isomorphism onto a dense subalgebra of $\mathfrak{B}$, since for each $a, a' \in \mathfrak{A}$, let $(a_n), (a_n') \subset \mathfrak{A}$ be the constant sequences with $a_n = a$ and $a_n' = a'$ for all $n \in \mathbb{N}$. Then:
(4)
\begin{align} \quad T(aa') = \lim_{n \to \infty} T(a_na_n') = c = bb' = \lim_{n \to \infty} T(a_n) \lim_{n \to \infty} T(a_n') = T(a)T(a') \end{align}
• It is clear that this multiplication on $\mathfrak{B}$ satisfies the multiplication axioms, and the norm on $\mathfrak{B}$ becomes an algebra norm too. So the normed algebra $\mathfrak{A}$ has a completion to a Banach algebra $\mathfrak{B}$. $\blacksquare$
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2019-12-16 04:54:35
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http://in4mationflow.com/Latex/LaTeX-command-itemize
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## LaTeX command itemize
The itemize command is used in LaTeX to create an itemized list. The elements of the list are introduced using the \item command, which may have an optional parameter.
itemize is used whenever a list of a few items is needed. Each element is introduced by a bullet symbol, which can be reproduced using the $\bullet$ command.
To customize the output of the itemize command, you need to change a few parameters of lists, such as the amount of horizontal skip at the beginning of the list, the vertical distance between list elements, etc. The interface to change these lengths uses the \setlength command in LaTeX
An example of usage of the itemize environment is the following:
\begin{itemize}
\item The first element of the list.
\item The second element of the list.
\item The third element of the list.
\end{itemize}
Article created on 2008-12-05 22:03:27
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2017-12-16 09:03:14
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http://crypto.stackexchange.com/questions?page=106&sort=votes
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# All Questions
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### Comparing n-multiple amount of merkle trees?
I would like to know what best steps are there in finding differences between a high order amount of merkle tree's or if a better associative structure should be used. I ask this questions because ...
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2016-05-27 06:38:11
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http://www.stmzc.com/
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## How to prove the identity $\sum_{n=1}^{\infty} \dfrac{{H_{n}}^2}{n^2} = \dfrac{17}{360} {\pi}^4$ February 14
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## Show given relation R is equivalence relation on S February 14
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for examplesequence: $1/(2^n), \qquad n\ge 0$sequence for the series: $1, 1.5, 1.75, 1.875, \ldots$and if so, does that mean you can use/extend sequence theorems for series?Yes in fact that is what a series is considered to be. When you ask about con
## Calculating a growing series in Spreadsheet February 14 1
I've got a spreadsheet, where I'm trying to calculate the amount of retained users over time for a subscription based service. https://docs.google.com/spreadsheet/ccc?key=0ArE8l42n3soadGJGRmtZMWdpeFZKSTQycEtKRXE5ZFE#gid=0I can get the totals but it's
## How do you derive a function that describes a series February 14 2
It's been a really long time since I've done calculus or any other kind of math beyond tip calculation. I was given a spreadsheet that calculates and plots a growth curve over time based on a handful of inputs. It goes out for one year. I'd like to b
## Cartesian Product converted into Summation February 14
I am looking at the proof of Maximum Likelihood Estimator and So let's get to it: first take the $\log$ of the equation: $$\log(P(\text{DATA}))=\log\prod i=1N(PX_i(1−P)1−X_i)$$Since $$\log(a b)=\log(a)+\log(b)$$then all the terms of the product becom
## The exact binomial test formula February 14 2
Could you help me find the one-sided exact binomial test formula?I use this statistical test in R-language, but I can't find the formula for it.Eng Wikipedia (https://en.wikipedia.org/wiki/Binomial_test) and RLang help gives me only examples without
## Inverse Gaussian, Limiting Distributions February 14 1
I'm trying to understand the nature of limiting distributions and distributions, specifically$1/Z_n \longrightarrow ~?$ where $Z_n\longrightarrow Z -Gaussian(0,1)$I understand that the gamma distribution converges to the gaussian for a large enough $You Might Also Like • I need some help! Thank you in advance.Let K$^{n*n}$& M$^{n*n}$be two square matrices, and K$\cdot$M=\ ... • How should I solve this second order, nonlinear ODE?: $$\left(\frac{f''(x)}{B}\right)^n=-(f(x)-a_0-a_1x-\cdo ... • I have these two definitions of span:Span: Su ... • Suppose U = \{(x, y, x+y, x -y, 2x) \in \Bbb F^5 : x, y \in \Bbb F\}. Find three subspaces W_1, W_2, W_3 ... • It would be helpful if I can get some comparison between these three books,T. Tao, An epsilon of room, I, Gr ... • In H. S. Wilf's generatingfunctionology, (1.6.8) describes:$$ A_n(y) = \sum_k \begin{Bmatrix}n-1\\k-1\end{Bm ... • I've been told that strong induction and weak induction are equivalent. However, in all of the proofs I've s ... • Please forgive my lack of maths knowledge,It is my understanding that:Standard Deviation is the average dist ... • Prove$x^{2 \over 3} \ln(x)$is uniformly continuous in$(1,\infty)$To my understanding I need to show the d ... • I have the following integral involving a confluent hypergeometric function:$\$\int_{0}^{\infty}x^3e^{-ax^2}{ ...
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2016-02-14 22:53:59
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|
http://connection.ebscohost.com/c/articles/79824203/algebraic-geometry-over-algebraic-structures-ii-foundations
|
TITLE
# Algebraic geometry over algebraic structures. II. Foundations
AUTHOR(S)
Daniyarova, E.; Myasnikov, A.; Remeslennikov, V.
PUB. DATE
September 2012
SOURCE
Journal of Mathematical Sciences;Sep2012, Vol. 185 Issue 3, p389
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
In this paper, we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove so-called unification theorems that describe coordinate algebras of algebraic sets in several different ways.
ACCESSION #
79824203
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2017-12-14 03:20:06
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https://tjmachinelearning.com/lectures/1718/rf/
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# Overfitting
Before we cover random forests, we will firstly describe overfitting. The goal of our models is to learn the generalize pattern in data. However, by nature, our data has some random noise. If we look at the image above, we can clearly see the black line represents the semantic separation of the data but some points are misclassified. By contrast, the green line perfectly separates the data, but clearly is abusing the data points that are shifted due to noise. Our goal is to prevent overfitting, which is the green line and maintain our separation on the black line.
## Identifying overfitting
We can identify overfitting by seeing increases in loss / error for validation data. If the training loss continues to go down, but testing for the generalizable pattern (the validation data) goes up, we have a problem!
# Regularization
The most common technique used to prevent overfitting involves regularization. Essentially, some random noise is added to our model in order to prevent it from being overly tuned to the data. When we adjust parameters in the model, some adjustment is also made that is mostly random. In decision trees, this involves selecting random thresholds and selecting the best one of this set instead of calculating the ideal threshold. This accelerates training and minimizes cases where a threshold is picked to fit a specific case. However, this is often too much noise so isn’t commonly used.
## Restricting Depth
In decision trees, the most common approach involves limiting the depth of the trees. For example, it is likely that the last node in the tree focuses on an ad hoc case where one data point causes issues. As seen in the image, we see that the decision tree creates odd lines for specific points. By restricting the depth, we prevent these ad hoc nodes.
# Introduction
Random forests are an ensemble of decision trees, combining many weak learners to build a robust, strong learner that generalizes better to unseen data than the individual weak learners.
# Building a Random Forest
1. Randomly choose $$n$$ samples from the training set with replacement (i.e. draw a bootstrap sample of size $$n$$).
2. Build a decision tree using the bootstrap sample. At each node:
1. Randomly select $$d$$ features without replacement.
2. Split the node using the feature among the $$d$$ features previously selected that provides the best split by maximizing the information gain.
3. Repeat steps 1 to 2 $$k$$ times
4. Aggregate the prediction by each tree by majority vote to assign the final class label.
Note that when building the decision trees, instead of evaluating all features to find the best split, we select the best feature among only a randomly chosen subset of those features. This creates more diversity among the trees, helping to prevent overfitting. Because of these overfitting-preventative measures, it is not necessary to prune the trees. Individually, each decision tree would perform poorly, but the aggregate of many of these trees leads to a model with a much higher performance.
## Hyperparameters
The most important hyperparameter to be optimized here is $$k$$, the number of decision trees the random forest uses. Although more trees increase the performance of the classifier as a whole, they also increase the computational expense. The hyperparameters $$n$$ and $$d$$ can also be optimized. The bootstrap sample size $$n$$ correlates with the degree of overfitting; larger values of n decrease the randomness, increasing the likelihood of overfitting while smaller values of n increase the randomness at the cost of the model’s performance. A good balance is to set $$n$$ to the size of the training set. Similarly, the size of the feature subset $$d$$ must be less than the number of features in order to promote diversity among the trees, but must be large enough to not reduce the model’s performance. A common convention is to set $$d$$ to the square root of the number of features.
## Extra-Trees
Instead of searching for the best threshold at each split when building each decision tree, another option is to randomly choose thresholds for each feature of the chosen feature subset and choose the best one from these. This further prevents overfitting at the cost of performance. A random forest built this way is called an Extremely Randomized Trees, or Extra-Trees.
# Feature Importance
Random Forests can be helpful in ranking and evaluating feature importance. In an individual decision tree, intuitively, the most important features are split near the root of the tree while less important features are either split near the leaf nodes or are not split at all. In other words, the depth of a feature in a decision tree gives a measure of the importance of that feature. In a random forest, one can take advantage of this property by taking the average depth of a feature across all the individual trees to measure its importance. Here is an example of a random forest ranking the importance of pixels in a face recognition problem where the brighter pixels are more important:
← Back to lectures
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2023-03-24 22:29:26
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https://kristerw.blogspot.com/2016/12/pointer-comparison-invalid-optimization.html
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## Thursday, December 15, 2016
### Pointer comparison — an invalid optimization in GCC
I wrote in an earlier blog post that GCC has a somewhat aggressive interpretation of the standard so that it compiles p == q to false if p and q are pointers derived from different objects. I'm now convinced GCC is wrong.
#### The problem
C does not permit pointers to point at any memory — a pointer must point to an address within an object or on the address immediately following the object, and pointer arithmetic cannot make it point outside that range. When comparing pointers, the C standard says (emphasis added)
Two pointers compare equal if and only if both are null pointers, both are pointers to the same object (including a pointer to an object and a subobject at its beginning) or function, both are pointers to one past the last element of the same array object, or one is a pointer to one past the end of one array object and the other is a pointer to the start of a different array object that happens to immediately follow the first array object in the address space.
That is, the comparison in the printf call
#include <stdio.h>
int main(void)
{
int x[4], y[4];
int *p = &x[4];
int *q = &y[0];
printf("%p %p %d\n", (void*)p, (void*)q, p == q);
return 0;
}
evaluates to true if the compiler happens to place y directly after x in memory. GCC does, however, substitute address comparison to false if the pointers are derived from different objects, and the program above may give output such as
0x7fff66339290 0x7fff66339290 0
where the comparison evaluates to false even though both pointers are the same. The argument in bug 61502 is essentially that the standard is unclear what "immediately follow" means, and applications cannot rely on memory layout anyway (so there is no real use case). I believe both are wrong.
#### Use case
One use case for this kind of comparison in firmware for small embedded devices — you have a framework that supports different hardware configurations, but you want to avoid adding an #ifdef for every optional functionality. Instead, each implemented functionality exports a structure containing configuration data, and the linker collects them into an address range and generates a symbol __start for the start address, and __end for the address immediately following the array. The good thing about this is that the array contains exactly what is present at link time — elements are added to the array by just adding files to the link command, and functionality can be included/excluded by just relinking, without any need for additional configuration.
C does not have a convenient way of representing the __end label, so it is typically expressed as
extern struct my_struct __start[];
extern struct my_struct __end[];
where the linker guarantees that __end is immediately following __start. The array is processed as
void foo(void)
{
for (struct my_struct *x = __start; x != __end; x++)
do_something(x);
}
which is compiled to an infinite loop if GCC optimizes comparisons of pointers derived from two objects as always being different. The workaround suggested in the GCC mailing lists is to insert some inline assembly to confuse the compiler
void foo(void)
{
struct my_struct *x = __start;
struct my_struct *y = __end;
asm("":"+r"(x));
asm("":"+r"(y));
for (; x != y; x++)
do_something(x);
}
but this should not be needed for the natural reading of the standard excerpt quoted above.
This particular use case seems to work now as a side-effect by the "missing" optimization of global constant addresses, so the nasty workaround is not needed. But some GCC developers still claim the code is invalid...
#### The meaning of "following"
Comment #3 of the bug report argues that GCC is correct:
Except within a larger object, I'm not aware of any reason the cases of two objects following or not following each other in memory must be mutually exclusive. (If the implementation can track the origins of bit-patterns and where copies of those bit-patterns have got to, it might have a compacting garbage collector that relocates objects and changes what's adjacent to what, for example — I think such implementations are within the scope of what the C standard is intended to support. Or if you're concerned about how this changes bit-patterns of pointers, imagine that a C pointer is a (object key, offset) pair, and that comparison first converts the C pointer into a hardware address, where it's the mapping from object keys to hardware addresses that changes as a result of garbage collection rather than anything about the representation of the pointer.)
I do not think this is a correct interpretation. For example, DR #077 asks the question "Is the address of an object constant throughout its lifetime?", and the committee answers (emphasis added)
The C Standard does not explicitly state that the address of an object remains constant throughout the life of the object. That this is the intent of the Committee can be inferred from the fact that an address constant is considered to be a constant expression. The framers of the C Standard considered that it was so obvious that addresses should remain constant that they neglected to craft words to that effect.
That is, compacting garbage collection is not within the scope of what the C standard supports. I'm even less convinced by the key/offset example — this is the same situation as "normal" hardware, as you can view the virtual address as a pair (page, offset), and it is still the case that x[16] is immediately following x[15] for an array
uint32_t x[32];
even if x crosses a page boundary so that x[15] and x[16] are placed on different pages.
#### 8 comments:
1. I don't believe this is common knowledge, so it is worth noting that &x[4] is not undefined behavior due to §6.5.3.2 which has the following note(not normative):
"Thus, &*E is equivalent to E (even if E is a null pointer), and &(E1[E2]) to ((E1)+(E2))."
but the normative text agrees: http://stackoverflow.com/a/21247407/1708801
2. But how do you know that it is placed directly adjacent in the memory? The only way you could tell within the C++ memory model is by comparing the pointers - which is undefined behavior in and of itself. Casting them to intptr_t won't do the trick, either - the number that you get is just some number that can be converted back into a pointer at a later point, and there are no other guarantees about its meaning. Same thing with printing them - what you see printed is just some abstract representation, it doesn't carry any guarantee wrt physical layout (that it does in practice is immaterial - you cannot rely on that to make any judgment over said layout).
Thus, per the as-if rule, gcc is correct, because they behave as-if those objects were not adjacent - and there's no way for you to tell otherwise. The address may be constant, but you don't have any means of determining if one address is adjacent to another that does not trigger U.B. (and so the result of the test is meaningless, and the compiler is not bound to respect it in any way).
So far as I can tell, the only way you could really guarantee adjacency is by placing two arrays next to each other in a struct, like so:
struct {
int x[4], y[4];
} foo;
int* p = &foo.x[4];
int* q = &foo.y[0];
Does GCC still give you the "wrong" result in this case?
1. GCC give the "correct" result for your code.
When it comes to pointer comparison, it is defined behaviour in C. And I would claim that the text in the standard that says that they compare equal "if and only if they are immediately following in the address space" means the real layout should be taken into account when applying the as-if rule (i.e. I read the text as a (misguided) mechanism to let the application inspect the data layout).
But I see no real use case for this on stack variables, so it does probably not matter... 😀
2. > When it comes to pointer comparison, it is defined behaviour in C.
Two arbitrary pointers can only be compared via == and !=. But to determine which object is higher or lower in memory than the other one, you need to be comparing them using < and > - and those are undefined if you compare two pointers derived from different objects. This is from ISO C99 spec:
"When two pointers are compared, the result depends on the relative locations in the address space of the objects pointed to. If two pointers to object or incomplete types both point to the same object, or both point one past the last element of the same array object, they compare equal. If the objects pointed to are members of the same aggregate object, pointers to structure members declared later compare greater than pointers to members declared earlier in the structure, and pointers to array elements with larger subscript values compare greater than pointers to elements of the same array with lower subscript values. All pointers to members of the same union object compare equal. If the expression P points to an element of an array object and the expression Q points to the last element of the same array object, the pointer expression Q+1 compares greater than P. In all other cases, the behavior is undefined."
So C doesn't provide you the means to determine the relative layout of two random objects, with the exception of that "one past the last" rule. So whatever answer == gives to you, that is the definitive answer from the perspective of the C memory model for a given implementation.
> means the real layout should be taken into account
The question here becomes what "real layout" is. Remember that this phrase has to be interpreted in the context of the C memory model, not the actual OS and architecture. So you can only use the facilities provided to you by the language to establish said layout. And then the point above applies - the answer that == gives you is the only thing you can rely on to test. If it returns false, then the objects are not adjacent as far as your program is concerned, even if they're actually implemented as adjacent in the physical or virtual address space.
Otherwise, the only way to test for adjacency is to set the code up in such a way that other language rules guarantee that it is (or isn't) adjacent in advance, like struct fields - then you can predict what == returns.
3. That is a strained interpretation of the standard that has the effect of limiting C utility. char *x = malloc(100); char *y = x+50; while(x < y) *x++ = 0;
3. Related topic: https://gcc.gnu.org/bugzilla/show_bug.cgi?id=78420
4. The quoted response to DR#077 hints at an important principle some compiler writers fail to recognize when they focus too much upon the wording of the Standard--that the Committee didn't always feel a need to say things it thought were obvious. That point needs to be applied, however, not just to the narrow issue of whether objects' addresses are constant throughout their lifetime, but rather to something much broader that should be obvious but isn't: the Standard makes no attempt to forbid all unreasonable behaviors, but instead relies upon people to use common sense and sound judgment in deciding what behaviors are reasonable and unreasonable for compilers targeting various platforms and application fields.
From the point of view of the Standard, the question here is whether pointer comparisons are guaranteed to behave as an equivalence relation for the lifetime of the objects involved, or whether there may be certain combinations of pointer values where equality comparisons would yield Unspecified results without invoking Undefined Behavior. I don't think the Standard would forbid such a thing, but I also don't think compiler writers should care.
Some application fields like systems programming need certain guarantees about pointer behavior that aren't needed in other fields like high-end number crunching. If making certain assumptions about pointers would make a compiler unsuitable for some purposes but improve its performance for some others, a compiler which is intended to serve both purposes should provide an option to enable or disable such assumptions, regardless of whether the mode enabled thereby is non-conforming, or is conforming but is nonetheless unsuitable for system programming.
C could evolve from being a good language to being a really great one if compiler writers could re-recognize the principle that quality compiler suitable for particular target platforms and application fields should behave in a fashion which is reasonable for those platforms and fields, whether the Standard requires it or not, and programmers should be entitled to expect such behavior.
1. The compiler writers appear to consider themselves adversaries of application programmers.
|
2020-05-26 17:45:44
|
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|
http://events.berkeley.edu/?event_ID=108980&date=2017-05-01&tab=all_events
|
## $K$-theory seminar: The Chern Character in twisted $K$-theory
Seminar | May 1 | 11:10 a.m.-12:30 p.m. | Evans Hall, 740 and then 959
German Stefanich, UC Berkeley
Department of Mathematics
We will recall the construction of the (equivariant) Chern character in $K$-theory and proceed to construct its twisted version.
teleman@math.berkeley.edu
|
2017-05-25 21:54:28
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https://msp.org/involve/2016/9-4/p02.xhtml
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#### Vol. 9, No. 4, 2016
Recent Issues
The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Author Index Coming Soon Other MSP Journals
Rings of invariants for the three-dimensional modular representations of elementary abelian $p$-groups of rank four
### Théo Pierron and R. James Shank
Vol. 9 (2016), No. 4, 551–581
##### Abstract
We show that the rings of invariants for the three-dimensional modular representations of an elementary abelian $p$-group of rank four are complete intersections with embedding dimension at most five. Our results confirm the conjectures of Campbell, Shank and Wehlau (Transform. Groups 18 (2013), 1–22) for these representations.
##### Keywords
modular invariant theory, elementary abelian $p$-groups
Primary: 13A50
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2021-03-04 19:10:03
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