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 xdoc and xparse Lars Hellström <[log in to unmask]> Thu, 3 Aug 2000 23:12:44 +0200 text/plain (113 lines) ```To begin with, I should explain that I have written a package xdoc2 (second prototype for xdoc) which reimplements quite a lot of the features defined in the doc package and adds a few of its own; the most important of the new features is probably the class writer's interface for defining new commands analogous to \DescribeMacro and new environments analogous to the macro environment. xdoc2 can be found on CTAN in macros/latex/exptl/xdoc. This is said more as a bibliographic reference than an announcement (although I suspect that this list might have the largest concentration of people who might be interested in this package that you can find anywhere on the net), as my real reason for writing this mail is to share some ideas cencerning xparse that I've got from developing xdoc---thus continuing the debate on xparse of September--October last year. The reason there is any relation at all is that I have included an argument-grabbing mechanism very similar to that of xparse in xdoc2. This is not as strange as it might seem, since many of the basic doc commands actually do grab their arguments in various special ways and setting up a mechanism for doing this is a more concise solution than leaving it as in doc; there \macro and \environment contain nine tokens each, the first eight of which are identical! I didn't bother implementing an argument specification interpreter; it's not essential for the functionality and the argument grabbing should probably be implemented using the argument grabber in xparse eventually anyway. (What's more, I have no idea about what would be a reasonable specifier for some of the argument types I've used.) Most of the variation between the argument types defined in xparse lies in what is considered to be the argument---is it a primitive TeX undelimited argument, is it bracket-delimited, is it parenthesis-delimited, or is it just some given character?---but the commands in the doc package show little variation in that respect; all arguments are mandatory. Instead there is another factor which is of utmost importance for many doc commands: that they can control the catcodes that are in force when the argument is grabbed. In doc v2.0h I count 18 user level commands/environments that take arguments, and 8 of these modify the catcodes for some argument. Is catcode-changing hard to incorporate in the xparse argument grabber, then? Not at all. If the list of argument grabbers contains something like   \begingroup\MakePrivateLetters\expandafter\endgroup\@ddc@m rather than a simple \@ddc@m, then it will grab one mandatory argument whilst having catcodes temporarily changed as indicated by \MakePrivateLetters. How such a catcode-changing argument should be specified to \DeclareDocumentCommand is of course another matter. One possibility would be as   M{} I cannot recall ever seeing any case of an optional or other special form argument in which the catcodes were being changed (maybe because it is trickier to implement), so I suspect that it would be sufficient to provide this feature for mandatory arguments. It is not clear that such an argument type should be available by default in xparse just because a couple of doc commands has such arguments, but there are some commands in the LaTeX kernel which also have arguments of this type---namely \index, \glossary, \DeclareFontShape, and \DeclareFontEncoding (while searching for such commands I also noticed that \ProvidesFile actually is an example of a command that does catcode trickeries for its optional argument). This argument type might also be called in question on the grounds that it might encourage poor programming style---e.g. constructing commands so that they only work if the arguments are tokenized with special catcodes, and thus are useless for basing other commands on---but I cannot see anything bad in the use of catcode-changing in the \index command. In xdoc2 I also found it useful to feed the argument that was grabbed through some function before it is contributed to the list of arguments, i.e., instead of a grabber \@ddc@m whose pseudocode is    \@ddc@m GRABBERS \toks@ ARG == BEGIN        \toks@ := \toks@ * {ARG}        GRABBERS \toks@      END I use a grabber (\@ddc@mf say) whose pseudocode is    \@ddc@mf FUNC GRABBERS \toks@ ARG == BEGIN        TEMP := FUNC(ARG)        \toks@ := \toks@ * {TEMP}        GRABBERS \toks@      END where `FUNC' is some function. If the argument is expected to be for example a number, dimen, or glue one could let FUNC consist of evaluating this quantity, e.g.    \reserved@skip=#2\relax    \toks@=\expandafter{\the\expandafter\toks@       \expandafter{\the\reserved@skip}}%    #1\toks@ The FUNC function could alternatively be an "evaluate calc-style expression". By doing such evaluations in the argument grabber, one can use temporary variables in the arguments without having to worry about whether they are used in the definition of the command you are using as well. It would also provide consistency; if you define a command whose replacement text is something like     ...          ...          ...          ... then you usually expect #1 to have the same value in both places. In xdoc2 the function FUNC is rather something called \MakeHarmless which takes a piece of TeX code and breaks it down to (a robust and makeindex-friendly representation of) the characters TeX's eye would have scanned to produce that TeX code. That is probably not something which should be incorporated in xparse; it rather illustrates the usefulness of allowing using package-defined functions for processing arguments. I have no idea how an argument like that should be specified, though. Lars Hellström ```


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## A cyclist riding in fog overtakes a man walking...

tagged by: BTGmoderatorLU

This topic has 2 expert replies and 1 member reply

### Top Member

BTGmoderatorLU Moderator
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#### A cyclist riding in fog overtakes a man walking...

Mon Mar 05, 2018 1:37 pm
A cyclist riding in fog overtakes a man walking at the speed of 3 km/h. If the man could see the cyclist for 4 minutes an up to a distance of 100m, what is the speed of the cyclist in km/h?

A. 1.5
B. 2.5
C. 4.5
D. 5
E. 6

The OA is C.

I try to solve it as follow,
$$Dist=100m\ =\frac{1}{10}km.$$
$$Time=4\min=\frac{1}{15}hours.$$
$$rate=rate\ of\ cyclits\left(x\right)-rate\ of\ walking=x-3.$$
$$Time=\frac{Dist}{Speed}\ \Rightarrow \frac{\frac{1}{10}}{x-3}=\frac{1}{15}\ \Rightarrow x=4.5\.$$

### GMAT/MBA Expert

mbawisdom Senior | Next Rank: 100 Posts
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Mon Mar 05, 2018 1:44 pm
LUANDATO wrote:
A cyclist riding in fog overtakes a man walking at the speed of 3 km/h. If the man could see the cyclist for 4 minutes an up to a distance of 100m, what is the speed of the cyclist in km/h?

A. 1.5
B. 2.5
C. 4.5
D. 5
E. 6

The OA is C.

I try to solve it as follow,
$$Dist=100m\ =\frac{1}{10}km.$$
$$Time=4\min=\frac{1}{15}hours.$$
$$rate=rate\ of\ cyclits\left(x\right)-rate\ of\ walking=x-3.$$
$$Time=\frac{Dist}{Speed}\ \Rightarrow \frac{\frac{1}{10}}{x-3}=\frac{1}{15}\ \Rightarrow x=4.5\.$$
Absolutely perfect - love the way to set up the parameters, wrote down your equation given the speed of the walker and derived the solution. I would have done it in the same way.

As a challenge, what would be the answer if the person is walking towards the cyclist as opposed to the same direction?

### Top Member

swerve Master | Next Rank: 500 Posts
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Tue Mar 06, 2018 7:59 am
Hi,

In 4 minutes, the cyclist gains 100m on the person walking.

So let's see how much does the man travel in 4 minutes at 3 km/h = (3000/60)*4 = 200m.

If the man has traveled 200m, then the cyclist travels = 200m + 100m = 300m.

So, his speed = (300/4)*(60/1000) = 4.5 km/h. Option C.

Regards!

### GMAT/MBA Expert

Jeff@TargetTestPrep GMAT Instructor
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Thu Mar 08, 2018 4:38 pm
LUANDATO wrote:
A cyclist riding in fog overtakes a man walking at the speed of 3 km/h. If the man could see the cyclist for 4 minutes an up to a distance of 100m, what is the speed of the cyclist in km/h?

A. 1.5
B. 2.5
C. 4.5
D. 5
E. 6
We see that the rate of the cyclist exceeds the rate of the man by 100 m in 4 minutes, or 25 m/min. We need to convert this into km/h:

25 m/min x 1 km/1000 m x 60 min/h = 1500/1000 km/h = 1.5 km/h

If we let r = the rate of the cyclist in km/h, then

r - 3 = 1.5

r = 4.5

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# What happens if you strip everything but the “between” relation in metric spaces

Given a metric space $$(X,d)$$ and three points $$x,y,z$$ in $$X$$, say that $$y$$ is between $$x$$ and $$z$$ if $$d(x,z) = d(x,y) + d(y,z)$$, and write $$[x,z]$$ for the set of points between $$x$$ and $$z$$.

Obviously, we have

1. $$x,z\in[x,z]$$;
2. $$[x,z]=[z,x]$$;
3. $$y \in [x,z]$$ implies $$[x,y] \subseteq [x,z]$$;
4. $$w,y \in [x,z]$$ implies: $$w\in [x,y]$$ iff $$y \in [w,z]$$.

# My question:

Has the family of objects with such an axiomatic “interval structure” $$[\bullet,\bullet]:X \times X \to \mathcal{P}(X)$$ satisfying approximately the above conditions been studied somewhere?

• There is a bit of that in Busemann's Geometry of Geodesics because of its relation to inverse problems: determine all metrics which have the same geodesics (or same "between" structure: set of triplets $(x,y,z) \in X \times X \times X$ such $y$ is between $x$ and $z$). If you allow more structure (manifolds, foliations, etc.) this is basically the study of path geometries. Mar 31 '20 at 13:39
• You might find Knuth's Axioms and Hulls of interest. Not only betweenness on an interval but also cyclic order as well. For betweenness the main idea is ordered geometry. Mar 31 '20 at 18:18
• I don't think this is exactly what you are looking for but this reminds me a lot of median spaces/median algebra. You might be interested in looking at Notes on coarse median spaces by Bowditch
– user35370
Apr 1 '20 at 0:38
• Since Bowditch has been mentioned you may well be aware of this, but a special case of these axioms is given by pretrees. These can be defined by the axioms: (a) $[x,y]=[y,x]$; (b) $y\in[x,z]$ and $z\in[x,y]$ implies $y=z$; and (c) $[x,z]\subseteq[x,y]\cup[y,z]$. The last axiom ensures the structure is treelike, and is of course not shared by most metric spaces. See Bowditch's Memoir of the AMS Treelike structures arising from continua and convergence groups and Adeleke and Neumann's Memoir Relations related to Betweenness. Apr 6 '20 at 8:50
• @user148575 Pretrees aren't necessarily metrisable, at least not if the distances are required to be real numbers. Any linearly ordered set admits a natural pretree structure, so sufficiently large losets are counterexamples. If you are prepared to generalise the definition of metric space to allow distances in a pre-assigned linearly ordered abelian group, then every pretree admits a metric where each $[x,y]$ consists of the points between $x$ and $y$ in the sense of the question. Apr 6 '20 at 16:25

There is a wide body of work on this in connection with the classic De Bruijn–Erdős theorem.

De Bruijn–Erdős Theorem. Every set of $$n$$ points in the plane (not all lying on the same line) determine at least $$n$$ lines.

There is a beautiful conjecture of Chen and Chvátal that the De Bruijn–Erdős theorem actually holds in every metric space, where lines are defined using the notion of betweenness that you describe. That is, given a metric space $$M$$ and two points $$a,b \in M$$, the line determined by $$a$$ and $$b$$ is the set of points $$c$$ such that $$c$$ is between $$a$$ and $$b$$, or $$a$$ is between $$c$$ and $$b$$, or $$b$$ is between $$a$$ and $$c$$. Note that this definition reduces to the usual notion of lines if we use the Euclidean distance.

Chen-Chvátal Conjecture. Every set of $$n$$ points in a metric space (not all of which lie on the same line) determine at least $$n$$ lines.

If you Google 'Chen-Chvátal Conjecture' you will find many results, including some that focus on the combinatorial aspects of betweenness.

I'm reminded of W.A. Coppel's book which looks at these kinds of structures from a slightly different vantage point, namely closure systems. I can't actually find the book right now, but here's a quick synopsis of what's going on.

Given a set $$X$$ together with an arbitrary function $$[\bullet,\bullet] : X^2 \rightarrow \mathcal{P}(X),$$ let's call $$A \subseteq X$$ closed if and only if for all $$a,b \in A$$, we have that $$[a,b] \subseteq A$$. It's easy to show that an arbitrary intersection of closed sets is closed, or in other words the collection of closed sets forms a Moore family. This gives us a closure operator on $$\mathcal{P}(X)$$, which makes $$(X,\mathcal{P}(X))$$ into a closure system. This means that the collection of closed sets forms a complete lattice given by $$\bigwedge_{i \in I} C_i := \bigcap_{i \in I} C_i, \qquad \bigvee_{i \in I} C_i := \mathrm{cl}\left(\bigcup_{i \in I} C_i\right).$$

If we furthermore assume that $$[a,a] = \{a\},$$ which is true in an arbitrary metric space (though I'm not sure whether this follows from your axioms), then we can prove that singletons are closed sets. Thus we're allowed to consider the expression $$\{a\} \vee \{b\}$$ in the aforementioned lattice. This set clearly includes $$[a,b]$$, and I think your axioms probably imply that these sets are equal. Ergo the Moore family alone allows us to recover the original between-relation.

I think that if you take the first three axioms you've written down, and add in the axiom $$[a,a] = \{a\},$$ the resulting axiom list should provide a complete characterization of closure systems $$X$$ in which firstly, every singleton is closed, and secondly, for all $$A \subseteq X$$, we have that $$A$$ is closed if and only if for all $$a,b \in A$$, we have $$\{a\} \vee \{b\} \subseteq A$$. But you should check this explicitly rather than taking my word for it, because right now this is just a hunch.

I'll also remark that you might learn interesting things by looking at partially ordered sets and defining $$[a,b] := \{x \in X : a \leq x \leq b\} \cup \{x \in X : b \leq x \leq a\}.$$

• Ah, this seems actually closer to what I had in mind, thank you! The axiom $[a,a] = {a}$ is indeed natural and, combined with my third one, implies the first, so should probably be used instead. Apr 1 '20 at 5:58
• @user148575, glad I was able to help. These are exactly the kinds of questions I love by the way; looking at old structures (e.g. metric spaces) via new categories (e.g. between-relation spaces), fresh points of view on familiar ideas, and the axiomatic method at its simplest and most appealing. This is how things get better. Apr 1 '20 at 6:01

This is more of an extended remark than a full answer. My message is: The property $$[x,x]=\{x\}$$ does not necessarily hold, but you can always make it hold by taking a natural quotient.

Your axioms don't imply $$[x,x]=\{x\}$$, so you have to add it if you want it. A pseudometric (e.g. a seminorm) will satisfy your axioms, and $$[x,x]$$ is the set of points distance zero from $$x$$, and this set can have any size.

In a pseudometric space you can take the quotient by the relation $$x\sim y\iff d(x,y)=0$$ and you get a metric space. It turns out the same works with the interval systems.

We can define a relation $$\sim$$ on $$X$$ so that $$x\sim y\iff x\in[y,y]$$. This is an equivalence relation:

• Reflexivity: Axiom 1.
• Symmetry: By axioms 1 and 2 $$x,y\in[y,x]$$. Thus axiom 4 says $$x\in[y,y]\iff y\in[x,x]$$.
• Transitivity: Suppose $$x\sim y\sim z$$. By axiom 3 $$x\sim y$$ and $$y\sim z$$ imply $$[y,x]\subset [y,y]$$ and $$[z,y]\subset [z,z]$$. By axioms 2 and 3 $$[y,y]\subset[z,y]$$. Thus $$x\in[y,x]\subset[y,y]\subset[z,y]\subset[z,z]$$.

Then we can take the quotient space $$X/{\sim}$$. To avoid confusion, let's denote by $$(x)$$ the quotient set of $$x\in X$$. (In fact, $$(x)=[x,x]$$, but I find it cleaner to keep the separate notation.) Then declaring $$(x)\in[(y),(z)]\iff x\in[y,z]$$ defines an interval structure on $$X/{\sim}$$. To see this, we need to check two things:

1. $$[x,y]\subset[x',y]$$ when $$x\sim x'$$. (The reverse inclusion and varying both $$x$$ and $$y$$ follow easily.)

Proof: Since $$x\in[x',x']\subset[x',y]$$, we have indeed $$[x,y]\subset[x',y]$$.

2. $$x\in[z,y]\implies[x,x]\subset[z,y]$$.

Proof: Since $$x\in[z,y]$$ and $$x\in[x,y]$$, applying axiom 3 twice gives $$[x,x]\subset[x,y]\subset[z,y]$$.

In the quotient structure we evidently have $$[(x),(x)]=\{(x)\}$$.

• Interesting, thanks for the remark! Apr 2 '20 at 15:51


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Is there an elegant algebraic proof of this formula for quadratic field discriminants? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T17:53:51Z http://mathoverflow.net/feeds/question/64461 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64461/is-there-an-elegant-algebraic-proof-of-this-formula-for-quadratic-field-discrimin Is there an elegant algebraic proof of this formula for quadratic field discriminants? Frank Thorne 2011-05-10T05:00:53Z 2011-05-13T17:59:54Z
Consider the Dirichlet series counting discriminants of real quadratic fields. Quadratic field discriminants are "basically" squarefree integers, so the associated Dirichlet series $\sum D^{-s}$ is "basically" $\zeta(s)/\zeta(2s)$. However, there is the funny business at 2, and one derives the formula

$\sum D^{-s} = \frac{1}{2} \big( 4^{-s} - 1 \big) \frac{\zeta(s)}{\zeta(2s)} + \frac{1}{2} \big(1 - 4^{-s} \big) \frac{L(s, \chi_4)}{L(2s, \chi_4)},$

which is a wee bit messy. (This formula, and all the subsequent ones, include 1 as a "quadratic field discriminant" for convenience.)

However, I was reading a fantastic paper by David Wright, where he considers positive and negative discriminants together, in which case you have the much nicer formula

$\sum |D|^{-s} = \big(1 - 2^{-s} + 2 \cdot 4^{-s} \big) \frac{\zeta(s)}{\zeta(2s)}.$

This formula is easy to derive from scratch, but he derives it as a consequence of the beautiful formula

$\sum |D|^{-s} = \prod_p \Big( \frac{1}{2} \sum_{[K_v : \mathbb{Q}_p] \leq 2} |\text{Disc}(K_v)|^s_p \Big).$

He uses the parameterization of quadratic fields by $\mathbb{Q}^{\times} / (\mathbb{Q}^{\times})^2$, which may be thought of as $\text{GL}_1$-orbits on a one-dimensional prehomogeneous vector space, where $\text{GL}_1$ acts by $t(x) = t^2 x$ rather than the usual $t(x) = tx$. He then analyzes these orbits by means of an adelic zeta function; note that with the usual action you recover Tate's thesis.

These formulas generalize quite a bit, with some complications, to $n$th-root extensions of any global field (with some restrictions on the characteristic). They also allow for twisting by characters, allowing (for example) a nice formula for $\sum \text{sgn}(D) |D|^{-s}.$ Note that the first formula considered looks nicer when viewed as a linear combination of $\sum |D|^{-s}$ and $\sum \text{sgn}(D) |D|^{-s}$.

The MathSciNet review says that "similar results, however, can be obtained by class field theory", and this is also hinted at in Wright's paper, but the details aren't worked out. My first question is the following.

Is there an elegant algebraic proof of the above identity for $\sum |D|^{-s}$ and its generalizations?

And my second question, which is essentially a vaguer version of the first one, is:

What is the best way to think of these formulas?

I quite like the prehomogeneous vector space approach, but I imagine there might be a nice algebraic proof as well, and in particular some kind of local-to-global'' principle for quadratic discriminants. I am no expert in class field theory, and I am curious if any of these identities look simple and natural when viewed in the correct light.

Thank you!
 http://mathoverflow.net/questions/64461/is-there-an-elegant-algebraic-proof-of-this-formula-for-quadratic-field-discrimin/64592#64592 Answer by Franz Lemmermeyer for Is there an elegant algebraic proof of this formula for quadratic field discriminants? Franz Lemmermeyer 2011-05-11T10:50:19Z 2011-05-13T17:59:54Z
The discriminant $D$ of a quadratic number field can be written uniquely as a product of prime discriminants, namely $-4$, $\pm 8$, and $p^* = (-1)^{(p-1)/2}p$ for odd primes $p$. The Dirichlet series for odd discriminants therefore simply is $$\sum_{D \text{ odd}} |D|^{-s} = \prod_{p \text{ odd}} (1 + p^{-s}),$$ and the contribution of the even prime discriminants is taken care of by the factor $$1 + 4^{-s} + 2 \cdot 8^{-s}.$$ Both the beautiful as well as the much nicer formula now follow immediately.

Factorization of quadratic discriminants into prime discriminants holds over totally real algebraic number fields with class number $1$ in the strict sense (see e.g. L. Goldstein, On prime discriminants, Nagoya Math. J. 45, 119-127 (1972); J. Sunley, Remarks concerning generalized prime discriminants, Boulder 1972; J. Sunley, Prime discriminants in real quadratic fields of narrow class number one, Carbondale 1979); a weaker version good enough for the purpose of counting discriminants works if the class number in the strict sense is odd.


text:
# Introduction

Forecasting time series data can be useful in many applications, be it simply for acquiring an overview of expected behavior or as incorporated in more complex anomaly detection systems. While machine-learning forecasting methods have been shown to produce excellent results in this regard, more traditional statistical modeling approaches are still very much competitive, providing the benefits of being simple yet effective, computationally inexpensive, and explainable [1].

In this application note, the popular Holt-Winters model has been implemented and used for decomposing and forecasting seasonal transformer temperature data. The following sections explain step-by-step how you can set this up for your own Disruptive Technologies (DT) Studio project for forecasting your own data.

# DT Studio Project Configuration

The implementation is built around using the developer API to interact with a single DT Studio project containing all sensors for which temperature is forecasted. If not already done, a project needs to be created and configured to enable API functionalities.

## Project Authentication

For authenticating the REST API against the DT Studio project, a Service Account key-id, secret, and email must be known, later to be used in the example code. If you're unfamiliar with the concept, read our guide on Creating a Service Account.

## Adding sensors to the project

By default, all temperature sensors in the project will be individually treated and forecasted for. The number of sensors used does not have to be configured beforehand and is scaled automatically. The option to move a sensor between projects can be found when selecting a sensor in a DT Studio project as shown in figure 2.

# Example Code

An example code is provided in this application note. It illustrates one way of building a transformer temperature forecasting system and is meant to serve as a precursor for further development and implementation. It uses the REST API to interface with the DT Studio project.

## Source Access

The example code source is publicly hosted on the official Disruptive Technologies GitHub repository under the MIT license. It can be found by following this link.

## Environment Setup

All code has been written in and tested for Python 3. While not required, it is recommended to use a virtual environment to avoid conflicts. Required dependencies can be installed using pip and the provided requirements text file.

pip3 install -r requirements.txt

Using the details found during the project authentication section, edit the following lines in sensor_stream.py to authenticate the API with your DT Studio project.

USERNAME   = "SERVICE_ACCOUNT_KEY"    # this is the keyPASSWORD   = "SERVICE_ACCOUNT_SECRET" # this is the secretPROJECT_ID = "PROJECT_ID"             # this is the project id

## Usage

If the example code is correctly authenticated to the DT Studio project as described above, simply running the script sensor_stream.py will start streaming data from each sensor in the project for which a forecast is continuously calculated as new data arrive.

python3 sensor_stream.py

For more advanced usage, such as visualizing the forecast, one or several flags can be provided upon execution.

usage: sensor_stream.py [-h] [--path] [--starttime] [--endtime] [--plot]                        [--plot-debug]Temperature forecast on Stream and Event History.optional arguments:  -h, --help    show this help message and exit  --path        Absolute path to local .csv file.  --starttime   Event history UTC starttime [YYYY-MM-DDTHH:MM:SSZ].  --endtime     Event history UTC endtime [YYYY-MM-DDTHH:MM:SSZ].  --plot        Plot the resulting forecast.  --plot-debug  Plot algorithm operation.

By providing the --path argument followed by the absolute path to a .csv file, the data contained in the said file will be loaded and forecasted upon. A data stream will not be started if used. The local .csv file must contain the columns unix_time and temperature.

The arguments --starttime and --endtime should be of the format YYYY-MM-DDThh:mm:ssZ, where YYYY is the year, MM the month, and DD the day. Likewise, hh, mm and ss is the hour, minutes, and seconds respectively. Notice the separator, T, and Z which must be included. It should also be noted that the time is given in UTC. Local timezone corrections should, therefore, be made accordingly.

By providing the --plot argument, a visualization of the latest forecast will be generated when available. If historic data is included, an interactive plot will be produced after estimating occupancy for the historic data. By closing this plot, the stream will start and a non-interactive plot will update for each sample that arrives.

Similar to --plot, the --plot-debug argument will also generate a visualization but of the detailed algorithm output. They work best when used on historic or local data and are implemented for debugging purposes.

# Implementation

With the aim of forecasting the temperature of a substation transformer several days into the future, the well-established Holt-Winters model has been implemented for real-time data collected with Disruptive Technologies Wireless Temperature Sensors. If configured properly, the model can forecast seasonality with an accuracy on par with more complex implementations such as SARIMA and LSTM [1].

Though simple, the implementation does require some prior knowledge about the data on which it is applied. The expected seasonal sample period, i.e. hourly, daily, weekly, must be provided. If unknown, it is recommended that the model is changed for double exponential smoothing which does not require this information. Additionally, a configurable number of seasons is required to initialize the model components. This is done to avoid initial erratic behavior.

The implementation has been structured such that typical tuning parameters for the algorithm are located in the file ./config/parameters.py. As no single configuration can work for all data, users are encouraged to experiment with different combinations better suited for their own data.

Also known as Triple Exponential Smoothing, the Holt-Winters Method is an extension of the earlier Holt's Linear Trend Model [2]. Through the addition of seasonality to the initial level- and trend components, the model has been a popular choice for forecasting and decomposition of time series. Though several variants of the model exist, such as the multiplicative method, the additive method represents data as a sum of three smoothing equations. With one equation per component, the isolated trend, level, and season of a signal $y_t$ with a seasonal period $m$ are given by

$l_t = \alpha(y_t-s_{t-m}) + (1-\alpha)(l_{t-1}+b_{t-1})$ $b_t = \beta(l_t - l_{t-1}) + (1-\beta)b_{t-1}$ $s_t = \gamma(y_t - l_{t-1} - b_{t-1}) + (1-\gamma)s_{t-m}$

respectively, where $α$, $β$, and $γ$ are smoothing parameters tuned for individual datasets. Though usually found empirically through trial and error, one can apply optimization techniques such as steepest descent to find the optimum smoothing parameters. However, this has the potential of fitting a local minimum and does therefore still require some kind of manual intervention. Figure 4 shows a temperature time-series decomposed by the additive method.

## Initialization

Due to the recursive nature of exponential smoothing, the level, trend, and seasonal components must be initialized with some values. While many approaches have been proposed, here we implemented one proposed by Rob Hyndman [4]. A more detailed explanation can be found here.

In essence, a $K$-season moving average is fit to an initial length $L_0>K\cdot m$ amount of temperature data, $y_0$. After subtracting the moving average, the initial season $s_0$ is defined as the average seasonal period in $L_0$. By then expanding $s_0$ over $L_0$, it is subtracted from $y_0$ for which linear regression is subsequently fit. The initial level $l_0$ is defined as the regression intercept and initial trend $b_0$ as the slope.

## Forecast

After initialization, an estimate of future values can be calculated using the level, trend, and season components. A $h$-sample forecast is given by

$\hat{y}_{t+h|t} = l_t + hb_t + s_{t+h-m(k+1)},$

where $m$ is the season length as previously mentioned. Essentially, this extends the exponentially weighted slope into the future from the given level, adding the seasonal component on top of it. Depending on the chosen smoothing parameters, the forecast may or may not capture small changes in the signal.

## Confidence Interval

Forecast values should be interpreted as an indicator of how the signal might behave in the future based on previous behavior, not as a guaranteed truth. This becomes more evident if a confidence interval is included, a range for which we expect $y_t$ to land with a given percentage of certainty. The width is dependent on the accuracy of earlier forecasts, while also becoming wider the longer into the future we forecast. Therefore, for a hh-step forecast of our seasonal data, the confidence interval is given by

$\hat{d}_{T+h|T}=c\hat{\sigma}=c\hat{\sigma}\sqrt{\frac{h-1}{m}+1}$

where $\hat{\sigma}$ is the standard deviation of previous forecast residuals and $c$ a multiplier. Assuming a normal distribution, $c$ can be found through a table lookup, where for instance choosing $c=1.96$ results in a $95\%$ confidence interval [3]. Figure 6 shows how the confidence interval expands for forecasts farther into the future.

# References

1. Makridakis S, Spiliotis E, Assimakopoulos V (2018) Statistical and Machine Learning forecasting methods: Concerns and ways forward. PLoS ONE 13(3): e0194889. https://doi.org/10.1371/journal.pone.0194889


text:
Elementary Geometry for College Students (6th Edition)

24200 $ft^{2}$
find The total area of the enclosure As we know the area of the length of the base b and length of the altitude h is A = bh Given b=220ft, h=110ft A = 220 * 110 = 24200 $ft^{2}$


text:
# Differentiation

Level pending

Find the derivative of the following function: y=(1/3x) + (47/x^2)

×


text:
# 跃迁引擎启动

rank solved A B C D E F G H I J K
20 9 O O O O O O O O ! O .

Solved by Henry.

Solved by XLor.

# C

Solved by XLor.

SAM + 倍增 + 线段树合并。

Solved by XLor.

# E

Solved by Forsaken.

Solved by XLor.

Solved by XLor.

Solved by Henry.

# J

Solved by Forsaken.


text: I was digging through some old notes of questions I had done in the past and came across this one. I don’t remember who gave me this question but it was a fascinating one that combined both geometrical reasoning and the sine rule. Question In the triangle $$ABC$$, \(AB...


text:
# Problem with the placement of limits of integration in stix2

There seems to be a problem in placement of limits in the stix2 package. The limits are placed too far from the integral sign. Here is a MWE.

\documentclass{article}
\usepackage{stix2}
\begin{document}
Consider the integral
$\int_0^1 f(x)\,dx.$
\end{document}

The output is given below:

Is there any simple work around for this? By the way, problems with stix package I had mentioned in Problem with stix fonts in texlive 2018 have been fixed in a an update. stix doesn't have this problem with placement of limits. However, since stix2 is newer and likely to maintained, I would like to stick to the newer version. Is there any simple work around for the problem?

It looks like a bug in the font metrics, you should report to the font maintainers.

It is hard to make a general workaround as a macro that detects display or inline math, and \limits or \nolimits cases, and that will not do the wrong thing as soon as the font is fixed.

For a particular case, best I can suggest is adding negative space as in the last example below.

\documentclass{article}
\usepackage{stix2}
\begin{document}
Consider the integral

\makeatletter

$\int_0^1 f(x)\,dx. \intop_0^1 f(x)\,dx. \intop\nolimits_0^1 f(x)\,dx.$

$\int_0^1 f(x)\,dx. \intop_0^1 f(x)\,dx. \intop\nolimits_0^1 f(x)\,dx.$

$\int_0^1 f(x)\,dx. \intop_0^1 f(x)\,dx. \intop\nolimits_0^1 f(x)\,dx.$

$\int_0^1 f(x)\,dx. \intop_0^1 f(x)\,dx. \intop\nolimits_0^1 f(x)\,dx.$

\bigskip

$\int_{\mkern-6mu 0}^{\mkern-6mu 1} f(x)\,dx.$

\end{document}


text:
## Blogs I follow

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2 days ago
• - Today marks the ninth birthday of the Dividend Growth Investor blog. It is unreal that I have managed to keep this up for 9 years in a row. There have been...
2 days ago
• - What a year 2016 turned out to be. As many of you who have already read my short list of 2017 goals know, this year I am guilty of setting too many goals...
2 days ago
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1 week ago
• - As I am writing this post right now, a good friend of mine is somewhere backpacking through Europe and is uploading her adventures on Instagram. I would ...
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6 months ago
• - Stock stalking is a weird hobby. In general I do not "follow" any particular investor's activities. However, there are a couple that I have followed of so...
7 months ago
• - [image: Lemur Madagascar] Brown Lemur - Kirindy Forest, MadagascarIt's been a while since I last wrote.... Well, to be fair I wrote a few posts but never r...
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• - Two Purchases for January 2016 (Amgen and 3M) During the month, I made another purchase of Amgen (AMGN). It seemed to be a good time to buy and it starts ...
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1 year ago

## Friday, September 04, 2015

Since last sale, I have done 3 buys and 2 sells.
I now own a company that I will reveal later.

At the end of August'2015, my portfolio was up 6% from Dec,31,2014.
Not bad, but missed the going up of Corticeira Amorim, the company I was trading.

Corticeira Amorim is now, in my opinion, above fair value so I give up trading it.

See you soon!

## Thursday, July 09, 2015

### Corticeira Amorim - sell

Considering the turmoil the markets have seen these days, I sold my shares and hope to buy them at a little lower price.

This sale was made at 4.135€ representing 1.0% profit, after commissions and fees.

Not bad, in a one month! See you soon!

## Wednesday, June 17, 2015

### Corticeira Amorim - A new buy

Since this is a rather iliquid stock, I didn't mentioned the buy earlier.
A few days ago, I bought again at 4.095€.

Unfortunately, the stock price is under that since that day. Today, it closed at 3.90€.
Not worried, since it is a great company

At my price it has a PER of 13.8 and profits are climbing >10% YOY the last few years.

I'm ok being a shareholder, and wait for the upside I hope for this stock.

## Tuesday, April 14, 2015

### Corticeira Amorim - no upside potencial

I've been watching this stock for a while and made some trading.

From the begining of 2015, the stock went from €3.00 to €4.20 (this moment price), or, a 40% gain. It was a rather impressive climb, but I think this it is now ended.

In the period, I made 5 trades making a profit of 20%. Not much, but then, I am a rather conservative investor.

In my opinion, this is a stock to sell now, for those who own it, or to wait for some dip in price to buy again.

My price target has been attained: €4.20! I predict it will have a 14 PER and a 1.70 P/B this year. As it is a very small free-float company and in a business that is not growing much near times, I think risk is now too much to buy this stock.

Although, the 3.3% yield can be a reason to maitain it, for those who have some shares.

I will continue to follow it, but with no more trading in the next days/weeks.


text:
# Creating a R DataFrame like construct in Mathematica

I kept getting my data mixed up so I decided to see if I could construct a package adding a R DataFrame like construct to Mathematica. I managed to cobble something together that sort of works, but I've run into some snags. I decided to use replacement rules as the basic data structure for the dataframe. I was really the only solution I could think of that wouldn't be too convoluted to construct.

What I have thus far package-wise is this:

BeginPackage["DataFrame"]

CreateDataFrame::usage="CreateDataFrame[data] creates a data frame out of the data table. It assumes the first row is the header."

FrameNames::usage="Names[dataframe] gives the header names of the dataframe."

Begin["Private"]

CreateDataFrame[data_]:=
Dispatch@Table[
Rule[i,Rest@First@Select[Transpose@data,First@#==i&]],
{i,First@data}
]

FrameNames[dataFrame_Dispatch]:=Table[dataFrame[[1,i,1]],{i,Length@dataFrame[[1]]}]

End[]

EndPackage[]

I also tried to created some custom notation to access parts of the dataframe but I haven't yet figured out how to get it to work when using a package:

Needs["Notation"]
Notation[ParsedBoxWrapper[
RowBox[{"x_Dispatch", " ", "$", " ", "y_String"}]] \[DoubleLongRightArrow] ParsedBoxWrapper[ RowBox[{"ReplaceAll", "[", RowBox[{"y_String", ",", "x_Dispatch"}], "]"}]]]  So I have a number of problems I was hoping someone here could help me with: How would I go about defining the custom notation from within the package. Currently I have to define it in the notebook to get it to work? Could I somehow give the dataframe a custom head of DataFrame, preferably without breaking anything? Is there any way to make the formatting of the output of CreateDataFrame nicer? While the FrameNames function works as it is I feel it would be more Mathematica-like if it were possible to access it like this instead: dataframe["Names"], but I'm not sure how to make it so. The custom notation I defined with $ sort of works it is still a bit awkward. To use it I have to make sure I manually include the space between that dataframe name and the \$, as well use quotation marks for the variable name. Any ideas for streamlining it more would be wonderful.

Then there is the performance side. The performance isn't terrible, but it is not great either, especially not when creating the dataframe. Any possible recommendations for improvements here would always be appreciated.

P.S. I realise I am piling on a bunch of questions here but I suspect there is a lot of overlap in the answers to them and so splitting the questions up into separate posts would not be helpful.

-
related question mathematica.stackexchange.com/q/9702/66 –  Faysal Aberkane Feb 6 '13 at 13:23

I won't say anything about the performance, since I think there are not enough information what exactly your data is. Furthermore, I won't say anything about the Notation Package because I don't think it's necessary.

Could I somehow give the dataframe a custom head of DataFrame, preferably without breaking anything?

Yes, I think that's and easy and nice way to go, because it solves most of your issues. Let's assume your data are simple matrices, which you can test with MatrixQ than you don't have to create a specific constructor function. You call your constructor simply DataFrame and define it like

DataFrame[data_?MatrixQ] := DataFrame[
Dispatch@Table[
Rule[i, Rest@First@Select[Transpose@data, First@# == i &]], {i,
First@data}]
]

Note, that it returns again a DataFrame but since now there is a dispatched set of rules in it, it just stays unevaluated. Now, you mentioned that it would feel natural to call dataframe["Names"] and indeed, this is possible by a simple definition

DataFrame[data_Dispatch]["Names"] := Table[data[[1, i, 1]], {i, Length@data[[1]]}]

Another thing you mentioned is the formatting. Here you can use Format to tell, how you want your DataFrame look like. This is the same approach as it is used when you interpolate a function and you get back an InterpolatingFunction with some information but not with the whole internal data.

Let's say you want to know how many names your DataFrame has when is displayed, then you define for instance

Format[DataFrame[data_Dispatch]] := "DataFrame"[Length[data[[1]]]]

Let's try what we can do with all this:

df=DataFrame[RandomInteger[10,{5,5}]]

(* Out[27]= DataFrame[5] *)

df["Names"]

(* Out[28]= {3,7,5,5,0} *)

In that way you can define most functions. There are other approaches and one might come handy when you want to access the values of your DataFrame. I suppose one very natural access would be when you could call df[[name]]. The problem here is, that you don't really want to extract the Part of your structure. Rather you like to call name/.data where data is the dispatched table.

On the other hand, you don't want to redefine Part, but what you can do is using TagSet

DataFrame /: Part[d_DataFrame, i_] := i /. First[d]

This binds the definition to DataFrame and states something like when DataFrame appears in an expression like Part[d_DataFrame, i_], then replace it with the right hand side.

Let's try it and use strings as names

df = DataFrame[
Prepend[RandomInteger[10, {4, 4}], {"blub", "foo", "boing", "bar"}]];

df["Names"]
df[["boing"]]

(*
Out[37]= {"blub", "foo", "boing", "bar"}

Out[38]= {5, 5, 10, 0}
*)

-
Wow what a great answer. I didn't know about TagSet. It opens up all kinds of interesting ways of extending dataframes. :) –  Mr Alpha Feb 6 '13 at 20:31
One problem: If there are only three or fewer variabled in the dataframe then the DataFrame["Names"] and Format[DataFrame don't work. –  Mr Alpha Feb 6 '13 at 20:36
Yes, because something like Dispatch[{a -> 1, b -> 2, c -> 3}] does not end in an dispatched table, because it would make no sense with just a few rules. My implementation was only an example and relied on that your DataFrame contains a Dispatch. First, you should think about what final data structure you will use, and after that, you make your DataFrame consistent. –  halirutan Feb 6 '13 at 21:46
@halirutan, your representation of data frame via rules seems suitable if only rows or only columns have headers, but not both. Can you generalize your framework to access by passing those either or both headers? –  alancalvitti Feb 22 '13 at 21:21


text:
# 5.12: Floods

## The Definition of a Flood

Here’s a simple definition of a river flood: the occurrence of a flow of such magnitude that it overtops the natural or artificial banks in a reach of river channel. If a floodplain exists, here’s another way of defining a flood: any flow that spreads out over the floodplain.

The relative volume of flood water is not large: on average, water discharged in excess of channel capacity constitutes about 5% of the total annual discharge of the given drainage basin. But the absolute volume of flood water is staggering. Hypothetical example: a two-inch rainstorm over a one-square-mile catchment the soils of which can absorb 0.5 inches would supply almost 3.5 million cubic feet of water is a couple of hours! (That’s a cube 150 feet on a side.)

A flood is a wave: it has a wavelike shape (although very subdued, with low amplitude relative to wavelength), and it propagates downriver at some speed, typically less than the mean velocity of flow of water in the river.

## More on Floodplains

Here are two similar but not quite identical definitions of a floodplain, taken from the literature:

• the flat area adjacent to the river channel, constructed by the present river in the present climate and frequently subjected to overflow (Leopold, 1994)
• the flat area adjoining a river channel constructed by the river in the present climate and overflowed at times of high discharge (Dunne and Leopold, 1978)

As you learned in an earlier section, the existence of a floodplain is testimony to the lateral migration of the river channel. With time, the shifting channel sweeps the entire width of the floodplain, eroding the “front” bank and depositing the “rear” bank as it goes. If the climate, or the base level, changes in such a way that the river degrades (i.e., lowers its bed), the old floodplain is incised, forming a set of terraces, and a new, narrower floodplain develops. If the climate, or the base level, changes in such a way that the river aggrades, the floodplain typically (except in extreme cases) aggrades along with it.

What determines the height of the floodplain above the channel bed? That is, why does the river have the characteristic transverse profile we see? That has to do with the relationship between effectiveness and frequency. Low discharges are common but not effective in sediment transport and channel shaping, whereas very high discharges are very effective but very uncommon.
The discharges with greatest effectiveness in maintaining channel and floodplain geometry are at some intermediate value: high but not extremely high, and common but not extremely common. For most rivers, bank-full discharges (those that spill over onto the floodplain) occur once or twice a year, on average. (It turns out that the discharge that is responsible for the maximum sediment discharge, in a given time period like a year, called the effective discharge, is very close to being equal to the bank-full discharge.)

What are floodplains like?

• They range in width from several meters, on the smallest streams, to many tens of kilometers, on the largest rivers.

• They are not flat as a board: they have minor relief, in the form of sloughs and depressions left behind by the irregular shifting of the river channel.

• They are developed in their classic sense in connection with meandering rivers. Braided rivers work irregularly across the entire valley, such that in a sense the entire valley is “river bed”.

• They are typically heavily vegetated in their natural state; in fact, the bank-stabilization effect of vegetation is one of the most important, if not the most important, factors in the existence of the floodplain in the first place.

• They vary in their “wetness”: usually they are dry and well above the local water table for most of the year, and are thus conducive to human use, but in areas with consistently high rainfall and/or strongly aggradational river regime ,there can be extensive swampy areas on the floodplain.

## Important Aspects of Floods

There are several important aspects of floods:

• discharge: discharge is important because of its close relation to flow velocity: how erosive is the flood flow in terms of banks, levees, and river structures?
• stage: the height of the water determines flood damage on the floodplain (This is what the public thinks about most in connection with floods.) Also, it relates to the water-passing capacity of bridges, culverts, dam spillways, etc.
• volume: this relates to how much water is available to be caught and impounded in reservoirs for water supply and flood control.
• area inundated: this is determined by stage together with river-valley topography.
• flow velocity: Although the flow velocity is much less over the floodplain than in the main channel, it can be great enough to cause damage.

Here are some important questions connected with river floods:

• governing effects?

• getting worse?

• how to control?

• how to predict?

What are the important effects or factors that govern the magnitude of a flood?

Rainfall in the drainage basin upstream of the given point on the given river. This boils down to both rainfall intensity (depth per unit time) and rainfall duration, and also the area of the drainage basin covered by the rainstorm. The worst scenario is: high intensity for a long time over a large area of the drainage basin.

• Preceding condition of soil moisture in the area of rainfall: dry, which lessens the runoff, or saturated, which maximizes the runoff? The difference can be as much as half an inch of equivalent rainfall. That’s a big effect for small floods but only a small effect for large floods.

Channel storage. Think in terms of what happens when a flood discharge from an upstream river enters a large lake. The lake has a large surface area, so it takes a long time for the flood flow to raise the lake elevation and therefore increase the discharge out of the lake. This storage effect is also present to a non-negligible extent along the river channel itself: As the flood proceeds downstream, it has to fill a greater and greater volume of space in the river channel itself. This effect attenuates the flood in the downstream direction: in effect, it stretches out the hydrograph and lowers its peak (in terms of either the stage hydrograph or the discharge hydrograph).

Channel geometry. The narrower the channel, for a given flood discharge, the more the stage rises. When the stage reaches bank-full, the water spreads out over the floodplain, and the stage then increases far less rapidly with increasing discharge.

Transition from dunes to plane bed. As discussed in an earlier section, this tends to slow the increase in stage with increase in discharge: the channel can pass a given discharge at higher velocity and lower stage because of the reduced resistance to flow afforded by the plane bed.

## Are Floods Getting Worse?

The answer to that question is certainly “yes”, in terms of dollar value of damage, just because of greater human use of flood-prone areas near rivers. But the answer is yes also in terms of higher stage, because of the effect of more levee protection: the levees prevent the high discharges from spreading over the floodplain, so the rise in stage is confined to within the channel (Figure 3-49).

Is flood discharge increasing on average as well? There are two aspects to this question.

(1) In urbanized drainage basins, the percentage of area paved increases, thus decreasing the concentration time and tending to produce higher peaks in the hydrograph (stage and discharge) for a given rainfall. In rural drainage basins, bare-soil agriculture and clear cutting of forests have the same effect, although it seems not to be a really major effect. (Their effect on sediment yield is much greater, though.)

(2) Average rainfall in the drainage basin might be increasing because of climate change (although in much of the U.S., the prediction is that global warming will decrease precipitation rather than increase it). “Storminess” (increased incidence of major rainstorms) might increase as well.

## Can Floods Be Controlled?

A better way of putting the question would be: how can flood damage be minimized, in the sense that a given measure affords more flood protection than it costs to institute?

(1) Build higher and higher levees. Yes, you can keep the floodwaters off the floodplain by building the levees higher. In a real sense, however, this is a self-defeating activity (Figure 5-50). Under natural conditions, the flood stage has built into it an inherently self-limiting effect: the floodplain area is so much greater than the channel area that, once the river goes beyond bank-full to inundate the floodplain, water-level rise is minor. (But remember that “minor” in a natural sense may be catastrophic for people living on the floodplain.) Building the levees higher restricts the river discharge to a much narrower width of the river valley, so the flood stage is much higher than would have been the case without the higher levees. Because of the effect of eliminating the floodplain-spreading effect, effective protection would necessitate impractically high levees. Moreover, if those high levees are overtopped, the flood damage to the surrounding floodplain area is likely to be much worse than if the levees had not been there.

(2) Flood-control dams. Clearly, the stage and discharge hydrographs can be reduced downstream of a given point on a river by building a water-storage dam there: catch the water so as to knock down the flood peak, and release the water slowly later in times of lower discharge. The problems, aside from the great cost of building dams, is that the effect lessens downstream, so a number of dams are needed. For decades, in the mid-1900s, there was a controversy over whether to build lots of small dams in upstream areas or a few big dams in downstream areas. It was finally concluded that for effective flood control throughout the whole basin, both sets of dams would be needed.

(3) Land management. Good agricultural practices (fallow planting; terrace agriculture; contour planting) should increase infiltration and thus lower runoff, and also make the concentration time longer. Careful consideration has shown, however, that the effect is measured in tenths of an inch of equivalent rainfall, so it would be effective against minor floods but not major floods. The great benefit is greatly decreased soil erosion—which is a different (but very important) matter.

(4) Restrictions on human use of floodplains. Clearly the best way to prevent flood damage is not to put anything susceptible to damage on floodplains! The expense of relocation is usually too great for this to be practical (although it’s happening in some places; it’s a policy issue rather than a scientific issue), but intermediate measures are effective: establish floodways and spreading areas that are used for such things as agriculture (of certain kinds) and recreation, but not permanent structures. This doesn’t solve the problem for already urbanized areas, though.

## Can Floods Be Predicted?

Simple answer is just “no”. Well, in a sense you can predict floods, but only after the rain has stopped falling! There are two aspects to this:

(1) Given the rainstorm, predict the runoff in order to create a flood hydrograph

(2) Given the flood hydrograph, watch the evolution of the hydrograph as the flood wave moves downstream. This technique is called flood routing.

Various approximate techniques have been developed, and continue to be refined, for predicting the nature of the flood wave downstream of a rainstorm in a drainage basin. The assumptions and approximations are many: distribution of rainfall; extent of runoff vs. infiltration; time of concentration to create the hydrograph. It works okay for very small catchments but not so well for large catchments. It’s a standard engineering technique for very small catchments (much less than one square mile). Once a flood is underway, however, there are good flood-routing techniques for predicting its future course down the river. Flood routing is treated at length in textbooks on fluvial hydrology.

The other aspect of flood prediction has to do with the statistics or probability of floods. One assumes that, given the climate and the nature of the drainage basin upstream of a given point on a river, there is some underlying frequency distribution of floods, in terms of either stage or discharge. Any record of floods is then a sample population (in the parlance of statistics) from that underlying frequency distribution. So if you know the frequency distribution, you can give the probability of a given flood in a given time period (that’s not a prediction, but it’s very useful in making a cost–benefit analysis or decision). If you are interested in learning more about how fluvial hydrologists go about figuring out the probability of a given flood in a given time period in a given river system, go to the following advanced topic.

## ADVANCED TOPIC: THE PROBABILITY APPROACH TO ERIVER FLOODS

1. The important question that arises is: what can be done about establishing the probability of very large floods? This is extremely important, but it’s fraught with difficulties. Here’s how it’s done, given a long record of discharge at some location along the river (see Figure 5-51):

• Determine the largest momentary discharge in each year.

• Rank these in order of decreasing discharge (the highest being number one, the lowest being equal to the number of years of record).

• Compute the recurrence interval (also called the return period), in years: the average number of years within which a given event (i.e., a flood of given magnitude) will be equaled or exceeded), by the formula T = (n+1)/m, where T is the recurrence interval, n is the number of years in the record, and m is the rank of the event (or, equivalently, the probability p of a given event being equaled or exceeded in any given year; that’s called the exceedence probability).

• Plot the results in a graph with recurrence interval and/or exceedence probability on the horizontal axis and the annual maximum discharge on the vertical axis.

• Fit a smooth curve through the points in some way.

2. The results are reliable for the mid-range of the graph, but they become very uncertain near the high-discharge end because of the small numbers of very large events. Fitting such a tail of a distribution is always tricky business, but it is important if one is to try to extrapolate to much longer times, which is one of the important goals of such an exercise. Various techniques have been in use for fitting such a curve. Usually one uses a kind of graph paper with logarithmic vertical (discharge) axis and a horizontal axis rubber-sheeted so that certain theoretical frequency distributions plot as straight lines. (That makes it easier to fit the tails of the distribution.) Several such distributions have been in use. The trouble with blind curve fitting is that one or two unrepresentative or outlying points at the high-discharge end can bias the curve significantly. Some hydrologists prefer to fit the curve by eye, using judgment about whether to include such seemingly outlying points.

3. One valuable approach is to try to determine, from historical records, whether the greatest discharge in the period of record was exceeded in earlier times. If the greatest discharge in the period of record can be established to have been greater than any other for a number of years before the period of record, the period of record can be extended back in time and the ranking redone. This tends to shift the unrepresentatively large outlying point to the right and bring it more closely in line with the rest of the distribution.

## Effects of Urbanization

Urbanization can have major effects of the magnitude and frequency of floods. Here’s an example (Figure 5-52): Seneca Creek, near Rockville, Maryland, drains an area of 100 square miles. Its watershed has been subject to much urbanization in recent decades. Take the record of discharge, 61 years long, from 1931 to 1991, and divide it into two equal parts, early and late. The average annual flood in the period 1931–1960 was 3000 cfs, in the period 1961–1991 it was 6000 cfs—twice as great! Likewise, bank-full discharge occurred 1.2 times per year in first period but 2.2 times per year in second period.

Rainfall was almost exactly the same in the two periods, so the great differences have to be accounted for entirely by the effects of urbanization effects. The stream is not artificially leveed, so presumably the effect has been caused by a decrease in concentration time; the more pavement, the faster the runoff, and the shorter the concentration time.

There’s a serious long-term problem in areas, especially in the lower reaches of large rivers, where crustal subsidence is non-negligible (i.e., the substrate beneath the active channel is moving downward relative to a datum like sea level) and/or where sea level itself is rising. (The former effect is more important than the latter.) The subsidence can be a combination of two effects:

• deep crustal subsidence

• subsidence caused by compaction of the sedimentary substrate of the river

The river channel aggrades to keep pace with subsidence, and in consequence it finds itself higher and higher above its floodplain—because, under conditions of net aggradation, deposition can be much faster in the channel, where the river is transporting most of its sediment, than over the floodplain. At certain times the river breaks through some particularly vulnerable point along its natural levees and finds its way out onto the floodplain, which is now at a significantly lower elevation, and establishes a whole new course for long distances down the floodplain. The process is called avulsion; the river is said to avulse. The river then proceeds to build up its channel bed and its levees, again to be high above the floodplain, and avulse once more. Avulsions often develop over a period of time rather than happening all at once: more and more of the river discharge is diverted through the gap in the levee, flood after flood. The end result, however, is the same.

To prevent avulsion, humans in avulsion-prone areas tend to build higher and higher levees, in the hope of preventing such an unfortunate event. The channel bed gets higher and higher relative to the surrounding floodplain (Figure 5-53).

This is happening today in the lowermost reaches of the Mississippi River. Much of the city of New Orleans lies well below the river. The Mississippi has an unfortunate preference for flowing down the Atchafalaya River to Morgan City. The U.S. Army Corps of Engineers maintains a gigantic headworks on the left bank of the river upstream of New Orleans to prevent the entire discharge of the Mississippi from going down the Atchafalaya; at present, about a third of the discharge leaves the Mississippi at that point. How long such a situation can be maintained is a matter of some controversy. If you would like to read an engaging account of the problem with the lower Mississippi, see the book by John McPhee (1989).

China has an even bigger problem with the Huang He (in English, the Yellow River), because they have been fighting the river for far longer than we have been fighting the Mississippi. I know from having been in China that the bed of the Huang He at the city of Kaifeng is almost fifteen meters above the streets of the city, only several kilometers from the river! The Chinese are working hard to come up with a way of solving this problem in the long term; for the near term (a few more decades) the levees are considered to be adequate.

This page titled 5.12: Floods is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by John Southard (MIT OpenCourseware) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


text: aquialaska aquialaska Answer: Hiwarekar22 discussed the extension and applications of Euler's theorem for finding the values of ... homogeneous functions of degree r. Proof. I also work through several examples of using Euler’s Theorem. Hiwarekar [1] discussed extension and applications of Euler’s theorem for finding the values of higher order expression for two variables. Theorem 3.5 Let α ∈ (0 , 1] and f b e a re al valued function with n variables deﬁne d on an • If a function is homogeneous of degree 0, then it is constant on rays from the the origin. Add your answer and earn points. Euler (pronounced "oiler'') was born in Basel in 1707 and died in 1783, following a life of stunningly prolific mathematical work. Suppose that the function ƒ : Rn \ {0} → R is continuously differentiable. 4. Index Terms— Homogeneous Function, Euler’s Theorem. INTRODUCTION The Euler’s theorem on Homogeneous functions is used to solve many problems in engineering, science and finance. Deﬁne ϕ(t) = f(tx). 12.4 State Euler's theorem on homogeneous function. =+32−3,=42,=22−, (,,)(,,) (1,1,1) 3. Euler's Theorem on Homogeneous Functions in Bangla | Euler's theorem problemI have discussed regarding homogeneous functions with examples. Cloudflare Ray ID: 60e20ccde9c01a72 Puoi modificare le tue preferenze in qualsiasi momento in Le tue impostazioni per la privacy. Find the maximum and minimum values of f(x,) = 2xy - 5x2 - 2y + 4x -4. Euler's theorem A function homogeneous of some degree has a property sometimes used in economic theory that was first discovered by Leonhard Euler (1707–1783). State and prove Euler’s theorem on homogeneous function of degree n in two variables x & y 2. 13.1 Explain the concept of integration and constant of integration. Per saperne di più su come utilizziamo i tuoi dati, consulta la nostra Informativa sulla privacy e la nostra Informativa sui cookie. Euler’s Theorem. Theorem. Solution for 11. Noi e i nostri partner memorizzeremo e/o accederemo ai dati sul tuo dispositivo attraverso l'uso di cookie e tecnologie simili, per mostrare annunci e contenuti personalizzati, per la misurazione di annunci e contenuti, per l'analisi dei segmenti di pubblico e per lo sviluppo dei prodotti. Get the answers you need, now! Prove that f(x, y) = x 3 – 2x 2 y + 3xy 2 + y 3 is homogeneous; what is the degree? In this article, I discuss many properties of Euler’s Totient function and reduced residue systems. State and prove Euler’s theorem on homogeneous function of degree n in two variables x & y 2. are solved by group of students and teacher of Engineering Mathematics , which is also the largest student community of Engineering Mathematics . Let F be a differentiable function of two variables that is homogeneous of some degree. Hiwarekar [1] discussed extension and applications of Euler’s theorem for finding the values of higher order expression for two variables. Functions homogeneous of degree n are characterized by Euler’s theorem that asserts that if the differential of each independent variable is replaced with the variable itself in the expression for the complete differential An important property of homogeneous functions is given by Euler’s Theorem. Then f is homogeneous of degree γ if and only if D xf(x) x= γf(x), that is Xm i=1 xi ∂f ∂xi (x) = γf(x). Theorem 10. Deﬁne ϕ(t) = f(tx). State and prove Euler's theorem for three variables and hence find the following. State and prove Euler theorem for a homogeneous function in two variables and find \$ x\dfrac{\partial u}{\partial x} ... euler theorem • 23k views. In this method to Explain the Euler’s theorem of second degree homogeneous function. Please enable Cookies and reload the page. 1. (b) State and prove Euler's theorem homogeneous functions of two variables. Add your answer and earn points. ∴ It is not a homogeneous function. New York University Department of Economics V31.0006 C. Wilson Mathematics for Economists May 7, 2008 Homogeneous Functions For any α∈R, a function f: Rn ++ →R is homogeneous of degree αif f(λx)=λαf(x) for all λ>0 and x∈RnA function is homogeneous if it is homogeneous of … The terms size and scale have been widely misused in relation to adjustment processes in the use of inputs by farmers. =+32−3,=42,=22−, (,,)(,,) (1,1,1) 3. State and prove Euler's theorem for homogeneous function of two variables. Theorem 2.1 (Euler’s Theorem) [2] If z is a homogeneous function of x and y of degr ee n and ﬁrst order p artial derivatives of z exist, then xz x + yz y = nz . 13.1 Explain the concept of integration and constant of integration. There is another way to obtain this relation that involves a very general property of many thermodynamic functions. f(0) =f(λ0) =λkf(0), so settingλ= 2, we seef(0) = 2kf(0), which impliesf(0) = 0. To view this presentation, you'll need to allow Flash. This theorem is credited to Leonhard Euler.It is a generalization of Fermat's Little Theorem, which specifies it when is prime. aquialaska aquialaska Answer: • I also work through several examples of using Euler’s Theorem. State and prove Euler's theorem for three variables and hence find the following. Suppose that the function ƒ : R n \ {0} → R is continuously differentiable. Derivatives as functions 9. ., xN) ≡ f(x) be a function of N variables defined over the positive orthant, W ≡ {x: x >> 0N}.Note that x >> 0N means that each component of x is positive while x ≥ 0N means that each component of x is nonnegative. Homogeneous Functions, and Euler's Theorem This chapter examines the relationships that ex ist between the concept of size and the concept of scale. In economic theory we often assume that a firm's production function is homogeneous of degree 1 (if all inputs are multiplied by t then output is multiplied by t ). Another way to prevent getting this page in the future is to use Privacy Pass. I. Leonhard Euler. Positive homogeneous functions are characterized by Euler's homogeneous function theorem. Prove that f is… HOMOGENEOUS AND HOMOTHETIC FUNCTIONS 7 20.6 Euler’s Theorem The second important property of homogeneous functions is given by Euler’s Theorem. Home Branchwise MCQs 1000 Engineering Test & Rank 1 See answer Mark8277 is waiting for your help. The terms size and scale have been widely misused in relation to adjustment processes in the use of … A balloon is in the form of a right circular cylinder of radius 1.9 m and length 3.6 m and is surrounded by hemispherical heads. Euler’s theorem (Exercise) on homogeneous functions states that if F is a homogeneous function of degree k in x and y, then Use Euler’s theorem to prove the result that if M and N are homogeneous functions of the same degree, and if Mx + Ny ≠ 0, then is an integrating factor for the equation Mdx + … 1 See answer Mark8277 is waiting for your help. Functions homogeneous of degree n are characterized by Euler’s theorem that asserts that if the differential of each independent variable is replaced with the variable itself in the expression for the complete differential An important property of homogeneous functions is given by Euler’s Theorem. Functions homogeneous of degree n are characterized by Euler’s theorem that asserts that if the differential of each independent variable is replaced with the variable itself in the expression for the complete differential then we obtain the function f (x, y, …, u) multiplied by the degree of homogeneity: Euler’s theorem 2. 17 6 -1 ] Solve the system of equations 21 – y +22=4 x + 7y - z = 87, 5x - y - z = 67 by Cramer's rule as … euler's theorem 1. This property is a consequence of a theorem known as Euler’s Theorem. (b) State and prove Euler's theorem homogeneous functions of two variables. Since (15.6a) is true for all values of λ , it must be true for λ − 1 . The case of x ⋅ ∇f(x) = kf(x) 15.6a. DivisionoftheHumanities andSocialSciences Euler’s Theorem for Homogeneous Functions KC Border October 2000 v. 2017.10.27::16.34 1DefinitionLet X be a subset of Rn.A function f: X → R is homoge- neous of degree k if for all x ∈ X and all λ > 0 with λx ∈ X, f(λx) = λkf(x). Derivatives as functions 9. ADD COMMENT 0. State and prove Euler's theorem for homogeneous function of two variables. (1) Then define x^'=xt and y^'=yt. (Extension of conformable Euler's theorem on homogeneous functions) Let and f be a real valued function with n variables defined on an open set for which ( tx 1 ,…, tx n )∈ D whenever t >0 and ( x 1 ,…, x n )∈ D , each x i >0, that satisfies the following: Abstract . Your IP: 128.199.245.23 There is another way to obtain this relation that involves a very general property of many thermodynamic functions. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. Proof:Differentiate the condition. 20. 12.5 Solve the problems of partial derivatives. 4. Follow via messages; Follow via email; Do not follow; written 4.5 years ago by shaily.mishra30 • 190: modified 8 months ago by Sanket Shingote ♦♦ 380: ... Let, u=f(x, y, z) is a homogeneous function of degree n. Positively homogeneous functions are characterized by Euler's homogeneous function theorem. Theorem 10. Then nt^(n-1)f(x,y) = (partialf)/(partialx^')(partialx^')/(partialt)+(partialf)/(partialy^')(partialy^')/(partialt) (2) = x(partialf)/(partialx^')+y(partialf)/(partialy^') (3) = x(partialf)/(partial(xt))+y(partialf)/(partial(yt)). Linearly Homogeneous Functions and Euler's Theorem Let f(x1, . 2 = 2 k and 4 = 2 k, which is not possible. INTRODUCTION The Euler’s theorem on Homogeneous functions is used to solve many problems in engineering, science and finance. 13.2 State fundamental and standard integrals. This property is a consequence of a theorem known as Euler’s Theorem. Mark8277 Mark8277 28.12.2018 Math Secondary School State and prove Euler's theorem for homogeneous function of two variables. Suppose that the function ƒ : Rn \ {0} → R is continuously differentiable. • Linear functions are homogenous of degree one. Euler’s theorem is a general statement about a certain class of functions known as homogeneous functions of degree $$n$$. Taking ( x1 , x2 ) = (1, 0) and ( x1 , x2 ) = (0, 1) we thus have. On the other hand, Euler's theorem on homogeneous functions is used to solve many problems in engineering, sci-ence, and finance. A balloon is in the form of a right circular cylinder of radius 1.9 m and length 3.6 m and is surrounded by hemispherical heads. 12.4 State Euler's theorem on homogeneous function. Euler’s theorem states that if a function f (a i, i = 1,2,…) is homogeneous to degree “k”, then such a function can be written in terms of its partial derivatives, as follows: kλk − 1f(ai) = ∑ i ai( ∂ f(ai) ∂ (λai))|λx. 17 6 -1 ] Solve the system of equations 21 – y +22=4 x + 7y - z = 87, 5x - y - z = 67 by Cramer's rule as … INTEGRAL CALCULUS 13 Apply fundamental indefinite integrals in solving problems. Proof. Mark8277 Mark8277 28.12.2018 Math Secondary School State and prove Euler's theorem for homogeneous function of two variables. Verify Euler’s Theorem for f. Solution: f (x, y) = x 3 – 2x 2 y + 3xy 2 + y 3 To view this presentation, you'll need to allow Flash. Euler (pronounced "oiler'') was born in Basel in 1707 and died in 1783, following a life of stunningly prolific mathematical work. These will help to prove extension of conformable Euler's theorem on homogeneous functions. Leonhard Euler. Proof: By definition of homogeneity of degree k, letting k = 1, then l¦(x) = ¦(lx) where x is a n-dimensional vector and lis a scalar. Euler’s theorem (Exercise) on homogeneous functions states that if F is a homogeneous function of degree k in x and y, then Use Euler’s theorem to prove the result that if M and N are homogeneous functions of the same degree, and if Mx + Ny ≠ 0, then is an integrating factor for the equation Mdx + … A function F(L,K) is homogeneous of degree n if for any values of the parameter λ F(λL, λK) = λ n F(L,K) The analysis is given only for a two-variable function because the extension to more variables is an easy and uninteresting generalization. Let f(x,y) be a homogeneous function of order n so that f(tx,ty)=t^nf(x,y). 20. Find the maximum and minimum values of f (x,) = 2xy - 5x2 - 2y + 4x -4. Differentiating both sides of this expression with respect to xi andusing the chain rule, we see that: Wikipedia's Gibbs free energy page said that this part of the derivation is justified by 'Euler's Homogenous Function Theorem'. 13.2 State fundamental and standard integrals. It is not a homogeneous function ∴ It is a homogeneous function with degree 3. Theorem 2.1 (Euler’s Theorem) [2] If z is a homogeneous function of x and y of degr ee n and ﬁrst order p artial derivatives of z exist, then xz x + yz y = nz . Let f: Rm ++ →Rbe C1. Theorem. The homogeneous function of the first degree or linear homogeneous function is written in the following form: nQ = f(na, nb, nc) Now, according to Euler’s theorem, for this linear homogeneous function: Thus, if production function is homogeneous of the first degree, then according to Euler’s theorem … Performance & security by Cloudflare, Please complete the security check to access. 1 -1 27 A = 2 0 3. Eulers Theorem: If u be a homogeneous function of degree n an x and y then . Let be Euler's totient function.If is a positive integer, is the number of integers in the range which are relatively prime to .If is an integer and is a positive integer relatively prime to ,Then .. Credit. Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof) Alternative Methods of Euler’s Theorem on Second Degree Homogenous Functions . Get the answers you need, now! Proof:Differentiate the condition. • A constant function is homogeneous of degree 0. Yahoo fa parte del gruppo Verizon Media. View Notes - Euler's-2 Engineering Mathematics Question Bank - Sanfoundry.pdf from CSE 10 at Krishna Institute Of Engineering and Technology. Many people have celebrated Euler’s Theorem, but its proof is much less traveled. These will help to prove extension of conformable Euler's theorem on homogeneous functions. The Questions and Answers of Necessary condition of euler’s theorem is a) z should be homogeneous and of order n b) z should not be homogeneous but of order n c) z should be implicit d) z should be the function of x and y only? Proof. • Now, the version conformable of Euler’s Theorem on homogeneous functions is pro- posed. An important property of homogeneous functions is given by Euler’s Theorem. Finally, x > 0N means x ≥ 0N but x ≠ 0N (i.e., the components of x are nonnegative and at 1 -1 27 A = 2 0 3. Then ƒ is positive homogeneous of degree k if and only if. Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. If the function f of the real variables x 1, ... + x k  ∂  f ∂  x k = n  f, (1) then f is a homogeneous function of degree n. Proof. Theorem 3.5 Let α ∈ (0 , 1] and f b e a re al valued function with n variables deﬁne d on an When F(L,K) is a production function then Euler's Theorem says that if factors of production are paid according to their marginal productivities the total factor payment is equal to the degree of homogeneity of the production function times output. As a result, the proof of Euler’s Theorem is more accessible. Informazioni su dispositivo e connessione Internet, incluso l'indirizzo IP, Attività di navigazione e di ricerca durante l'utilizzo dei siti web e delle app di Verizon Media. Homogeneous Functions, and Euler's Theorem This chapter examines the relationships that ex ist between the concept of size and the concept of scale. I'm curious because in his Introduction to the analysis of the infinite he defines a homogeneous function as one "in which each term has the same degree" and goes on … (Euler's Theorem on Homogeneous Functions) We say f: R"- {0} R is homogeneous of degree k if f(tx) = tf(x) for all t >0. ∴ It is homogeneous function of degree 0. INTEGRAL CALCULUS 13 Apply fundamental indefinite integrals in solving problems. In general, for a homogenous function of x, y, z... of degree n, it is always the case that (2.6.1) x ∂ f ∂ x + y ∂ f ∂ y + z ∂ f ∂ z +... = n f. This is Euler's theorem for homogenous functions. Assistant Professor Department of Maths, Jairupaa College of Engineering, Tirupur, Coimbatore, Tamilnadu, India. Now, I've done some work with ODE's before, but I've never seen this theorem, and I've been having trouble seeing how it applies to the derivation at hand. A (nonzero) continuous function which is homogeneous of degree k on R n \ {0} extends continuously to R n if and only if k > 0. I. Then f is homogeneous of degree γ if and only if D xf(x) x= γf(x), that is Xm i=1 xi ∂f ∂xi (x) = γf(x). In this article, I discuss many properties of Euler’s Totient function and reduced residue systems. You may need to download version 2.0 now from the Chrome Web Store. Eulers Theorem: If u be a homogeneous function of degree n an x and y then . Euler's Theorem: For a function F(L,K) which is homogeneous of degree n If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. 1. Question 2. Then along any given ray from the origin, the slopes of the level curves of F are the same. 0. Let f: Rm ++ →Rbe C1. Introduce Multiple New Methods of Matrices . Thus f is not homogeneous of any degree. 24 24 7. Per consentire a Verizon Media e ai suoi partner di trattare i tuoi dati, seleziona 'Accetto' oppure seleziona 'Gestisci impostazioni' per ulteriori informazioni e per gestire le tue preferenze in merito, tra cui negare ai partner di Verizon Media l'autorizzazione a trattare i tuoi dati personali per i loro legittimi interessi. . Positive homogeneous functions are characterized by Euler's homogeneous function theorem. xi. Homogeneous Function ),,,( 0wherenumberanyfor if,degreeofshomogeneouisfunctionA 21 21 n k n sxsxsxfYs ss k),x,,xf(xy = > = [Euler’s Theorem] Homogeneity of degree 1 is often called linear homogeneity. K. Selvam . Functions homogeneous of degree n are characterized by Euler’s theorem that asserts that if the differential of each independent variable is replaced with the variable itself in the expression for the complete differential HOMOGENEOUS AND HOMOTHETIC FUNCTIONS 7 20.6 Euler’s Theorem The second important property of homogeneous functions is given by Euler’s Theorem. converse of Euler’s homogeneous function theorem. Now, the version conformable of Euler’s Theorem on homogeneous functions is pro- posed. Given a homogeneous polynomial of degree k, it is possible to get a homogeneous function of degree 1 by raising to the power 1/ k. So for example, for every k the following function is homogeneous of degree 1: ( x k + y k + z k ) 1 k. {\displaystyle \left (x^ {k}+y^ {k}+z^ {k}\right)^ {\frac {1} {k}}} Index Terms— Homogeneous Function, Euler’s Theorem. As a result, the proof of Euler’s Theorem is more accessible. Euler’s Theorem. Euler’s theorem is a general statement about a certain class of functions known as homogeneous functions of degree $$n$$. 12.5 Solve the problems of partial derivatives. Many people have celebrated Euler’s Theorem, but its proof is much less traveled. Examples of using Euler ’ s Theorem x1, the future is to use privacy Pass Math Secondary State... For a function is homogeneous of degree n an x and y then of higher order expression for two.! Theorem known as Euler ’ s Theorem, which is also the largest student community of Engineering, Tirupur Coimbatore. A very general property of homogeneous functions are characterized by Euler ’ s Theorem on homogeneous Theorem! Λ, it must be true for λ − 1 derivation is by! Is prime • a constant function is homogeneous of some degree function and reduced residue.... Complete the security check to access f are the same a result, the of! Le tue impostazioni per la privacy the following: R n \ { }... 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Relation that involves a very general property of homogeneous functions is given by Euler 's for! Inputs by farmers consulta la nostra Informativa sui cookie functions 7 20.6 Euler ’ s Theorem second., Please complete the security check to access functions of degree 0 Leonhard Euler.It is a generalization of Fermat Little. Check to access web Store is another way to prevent getting this page in the use inputs! Math Secondary School State and prove Euler 's homogeneous function of two variables the maximum and values. Less traveled tue preferenze in qualsiasi momento in le tue impostazioni per privacy... This article, i discuss many properties of Euler 's Theorem let f ( x1, less. 0, then it is constant on rays from the origin need to download version 2.0 from... Is also the largest student community of Engineering, science and finance saperne di più come. We See that: Theorem of Euler ’ s Theorem you 'll to! & Rank 12.4 State Euler 's Theorem for finding the values of f are same. 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# Solution for Find the Direction Cosines of the Lines, Connected by the Relations: L + M +N = 0 and 2lm + 2ln − Mn= 0. - CBSE (Commerce) Class 12 - Mathematics

ConceptDirection Cosines and Direction Ratios of a Line

#### Question

Find the direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2ln − mn= 0.

#### Solution

$\text{ Given: }$

$l + m + n = 0 . . . \left( 1 \right)$

$2lm + 2\ln - nm = 0 . . . \left( 2 \right)$

$\text { From } \left( 1 \right), \text { we get }$

$l = - m - n$

$\text { Substituting } l = - m - n in \left( 2 \right), \text { we get }$

$2\left( - m - n \right)m + 2\left( - m - n \right)n - mn = 0$

$\Rightarrow - 2 m^2 - 2mn - 2mn - 2 n^2 - mn = 0$

$\Rightarrow 2 m^2 + 2 n^2 + 5mn = 0$

$\Rightarrow \left( m + 2n \right) \left( 2m + n \right) = 0$

$\Rightarrow m = - 2n, - \frac{n}{2}$

$\text { If } m = - 2n, \text { then from } \left( 1 \right), \text { we get } l = n .$

$\text { If } m = - \frac{n}{2}, \text { then from } \left( 1 \right), \text { we get } l = - \frac{n}{2} .$

$\text { Thus, the direction ratios of the two lines are proportional to } n, - 2n, n \text { and } - \frac{n}{2}, - \frac{n}{2}, n, \text { i . e } . 1, - 2, 1 \text { and } - \frac{1}{2}, - \frac{1}{2}, 1 .$

$\text { Hence, their direction cosines are }$

$\pm \frac{1}{\sqrt{6}}, \pm \frac{- 2}{\sqrt{6}}, \pm \frac{1}{\sqrt{6}}$

$\pm \frac{- 1}{\sqrt{6}}, \pm \frac{- 1}{\sqrt{6}}, \pm \frac{2}{\sqrt{6}}$

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Solution Find the Direction Cosines of the Lines, Connected by the Relations: L + M +N = 0 and 2lm + 2ln − Mn= 0. Concept: Direction Cosines and Direction Ratios of a Line.
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• All the debugging and configuration will be stored in the vimrc

#### 2.1.2 Where is the vimrc

• For windows: $HOME/_vimrc • For Mac/Linux/Unix: ~/.vimrc • If the vimrc doesn’t exist, you can create it by yourself. #### 2.1.3 Configuration for the vimrc ### 2.2 Configuration for the <leader> key • The key is mapped to \ by default. So if you have a map of t, you can execute it by default with \ +t. For more detail or re-assigning it using the mapleader variable, Hide line number, in the vim editor Show line number, in the vim editor Turn on relative line numbers Turn off relative line numbers use the incremental search ## 3. vim-plug • A Vim plugin is a set of Vimscript files that are laid out in a certain directory structure. ## 4. Motions and Operation of the vim ### 4.1 Motions of the vim ##### 1. words motions w move to next word if the cursor is in the space go to the head of next word if the cursor is inside the word go to the head of next word (including of jumping the space) b motion if the cursor is at the head character of the word go to the head of previous word (including of jumping the space) if the cursor is not at the head character of the word go to the head of the word e motion if the cursor is pointing at the end character of the word go to the end of the next word if the cursor is not pointing at the end character of the word go to the end of the word ##### 2. Left-right motion movement by character h move the cursor to left, 1 character l move the cursor to right, 1 character movement by line$ go end of the Line
0 go head of the Line
^ go the 1st Non-space character the of the Line
##### 3. Up-down motions
movement by character
j go next line
k go previous line
gg go to the first Non-space character of the file
G go to the last Non-space character end of the file
##### 4. Other motions
movement by character
% while the cursor is on a (,),[,],{, or } goes to its match
H go to the 1st line in the screen
M go to the middle line in the screen
L go to the last line in the screen

### 4.2 Operators of the vim

#### 4.2.1 Basic operators

Operator Definition
y yank (copy)
p paste
d delte
c change the character where the cursor locates
C change, from the cursor to the end of the line
s substitute the character where the cursor locates
S substitute the whole line
r + [character] replace the character
R enters Replace mode until <ESC> is pressed
> indent
< unindent
gu lower case
gU upper case
i input
x delete the character on the cursor (only delete 1 character)
A append text after the line
u undo previous actions
U undo all the changes on a line
CTRL-R undo the undo’s
o to open a line BELOW the cursor and start Insert mode
O to open a line ABOVE the cursor
a to insert text AFTER the cursor
A to insert text after the end of the line

#### 4.2.2 Formula of the operators and motions

1. [number] + operator + motion
2. operator + [number] + operator (2 operators are the same)
Combination Definition
dw 1. if the cursor in the head of a word, it will delete the whole word and the space after the word 2. if the cursor isn’t on the first of a word, it will delete the characters after on the cursor (including the character on the cursor) and the space behind it
d\$ delete all the words after the cursor (including the character on the cursor)
de to the end of the current word and the(including the character on the cursor and the space)
dd to delete the line
2dd to delete two lines

Text Definition
w word
s sentence
p paragraph
t tag

## 5. Command line

Command Definition
/ followed by a phrase searches FORWARD for the phrase. After a search type n to find the next occurrence in the same direction or N to search in the opposite direction.
? followed by a phrase searches BACKWARD for the phrase. After a search type n to find the next occurrence in the same direction or N to search in the opposite direction.
CTRL-O takes you back to older positions

## 6. Execute an external command

:![command]<enter>

:![command] FILENAME <enter>

## 7. Introduction to UltiSnips

### 7.1 Prerequirement for the install

• 1. install of vim-plug
• Installation of Python2 or Python3

vim ~/.vimrc

### 7.3 How to configure the UltiSnips

Reference:

1. vimtutor

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