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## Is the "-ney" in "hackney" a diminutive suffix?

1

Is the "-ney" in "hackney" a diminutive suffix?

What are some other words with the same suffix? Honey, Whitney,...?

5

Hackney is a placename, once a parish northwest of London in the County of Middlesex; the name derives from OE Hacan ieg, Haca's isle, designating the meadows rising out of the marshes along the Thames.

The superior grazing afforded by these meadows made the area famous for its horses, which became known as hackneys. The term designated a light horse: not a small horse but an ordinary riding horse, as opposed to the heavy horse which was used for drawing heavy loads or ridden by a fully armored knight in combat.

That name subsequently passed from the horses to a form of light carriage which was the ancestor of the modern taxicab.

The shortened version, hack, came to designate horses offered for hire; this use was eventually extended, figuratively and derogatorily, to hired workers, particularly prostitutes and writers.

But what I have learned is that hack means horse, and hackney means small horse. – Tim – 2013-12-21T03:57:20.410

1@Tim I'm afraid you are misinformed. – StoneyB on hiatus – 2013-12-21T04:11:16.490

I searched "define Hackney" in Google. – Tim – 2013-12-21T04:31:01.493

1@Tim Well, the first half-dozen sources I get on that search have no such distinction; but Google respondes differently in different locations. Point me to a source and I'll address it. – StoneyB on hiatus – 2013-12-21T13:53:06.707

Followed the search, the top result returned by Google says "a horse or pony of a light breed with a high-stepping trot, used in harness." – Tim – 2013-12-21T14:44:54.787

@Tim I've added some clarification. But hack derives from hackney, not the other way round. A hackney is not a small hack. – StoneyB on hiatus – 2013-12-21T15:25:12.560

Wow, in Hackney, Islington, there is a pub called "Nags Head"! nag is slang for horse, well I never! – Mari-Lou A – 2013-12-22T01:38:54.670

3

No, this ending is not dimiutive. A partial list of words (including names) can be obtained on an Ubuntu Linux system easily:

$grep 'ney$' /usr/share/dict/words
Barney
Britney
Brittney
Cagney
Carney
Chaney
Cheney
Cockney
Courtney
Delaney
Disney
Haney
Hockney
Kinney
McCartney
McKinney
Mooney
Mulroney
Orkney
Penney
Rodney
Romney
Rooney
Sidney
Sweeney
Sydney
Taney
Tawney
Tunney
Whitney
attorney
baloney
blarney
boloney
boney
chimney
chutney
cockney
gurney
hackney
honey
jitney
journey
kidney
looney
money
phoney
stoney
tourney

No obvious pattern emerges among these words which would suggest that "ney" functions as some kind of independent unit.

Nouns that end in the [n] sound can often form adjectives ending in [ni]. This is usually written "ny". Note that in the above list, "boney" and "stoney" are just misspellings. The correct spellings are "bony" and "stony".

"Baloney" was derived from "bologna", probably by way of "bologna sausage" which is just called "bologna", a confusing spelling to semi-literates who turned it into "baloney".

There is no obvious connection among nouns like "money", "honey", or "attorney". Moreover, there are nouns which end with the same syllables, but are spelled just with a "y", rather than "ey", like "pony" or the flower "peony". "Pony" could just as easily be spelled "poney". There is no rule by which we can deduce that "honey" and "money" do not sound like "pony" or "phony". The "ey" spelling is not what encodes the difference. They just have to be memorized.

Let's focus on nouns ended with "-ney". Does it have some meaning? – Tim – 2013-12-21T03:08:44.320

0

It's a guess. A shot in the dark, so to speak.

If I had access to the Oxford English Dictionary I'd check there first, but I don't. I'm left with etymonline dictionary. It could be that -ney is a deviation or reflects the morphological change of the word, new.

Old English neowe, niowe, earlier niwe "new, fresh, recent, novel, unheard-of, different from the old; untried, inexperienced," from Proto-Germanic newjaz (cf. Old Saxon niuwi, Old Frisian nie, Middle Dutch nieuwe, [...] Danish and Swedish ny, Gothic niujis "new")

Otherwise it could derive from the Dutch ny

Adjective ny (neuter nyt, definite and plural ny or nye, comparative nyere, superlative nyest)

• new
• fresh
• recent
• novel
• other
• different

Hypothesis (I'm guessing here) It could be that "ney" was tagged onto the names of British towns that were growing and developing from the the middle ages and onwards. But as I'm not a historian, I am not going to spend the next two or four hours researching into the dates of towns, to see whether a smaller or older town/village existed earlier. I trust someone more expert than I will provide an answer.
Definitely, food for thought.


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# Homotopies of edge paths in simplicial complexes

Suppose $K$ is a simplicial complex with base vertex $v_0$ and $p$ is an edge path in $K$ based at $v_0$ (so a sequence of vertices connected by edges starting and ending at $v_0$). In other words, $p$ is a simplicial map $p:M\to K$ where $M$ is some subdivision of $\partial\Delta_{1}$. If the loop $|p|:|\partial\Delta_{2}|\to |K|$ on realizations is null-homotopic is seems, according to this page, that there is a sequence of edge paths (where each step moves edge(s) across a 2-simplex) from $p$ to the constant edge path.

I would like to know if $|p|$ being null-homotopic also means there is a subdivision $N$ of the standard 2-simplex $\Delta_2$ so the subcomplex of $N$ corresponding to the boundary $\partial\Delta_{2}$ is precisely $M$ and a simplicial map $h:N\to K$ and $h$ restricted to $M$ is $p$? Perhaps this subdivision can be "built" out of the moves?

Of course, you can take a simplicial approximation to a homotopy $|\Delta_2|\to|K|$ that extends $|p|$, but what I am asking seems more combinatorial. If anyone can suggest texts/papers for me to look into that treat this question that would also be very nice!

-

Essentially, what you are asking is the following: let $K$ be a simplicial set, and let $p: \partial \Delta[2] \to K$ be a "loop" (simplicially) such that the geometric realization of $p$ is trivial. Then, after subdividing $K$ appropriately to some $K'$, you are asking whether one can "fill in" $p$ to get a map $\Delta[2] \to K$. The answer is yes, at least with Kan's notion of subdivision (see this article). The point is that, for a fibrant simplicial set (i.e. a Kan complex), a map $p: \partial \Delta[2] \to K$ is trivial on the geometric realizations precisely if it can be filled in to a map $\Delta[2] \to K$; that's essentially because of the (rather deep) equivalence between the homotopy categories of Kan complexes and CW complexes, which implies that a "simplicial" definition of the homotopy groups is the same as the homotopy groups of the realization.
In this case, the point becomes that subdividing infinitely gives a "fibrant replacement" for any simplicial set $K$, so one way to compute the homotopy groups of a realization of a simplicial set is to take the (combinatorial) homotopy groups of the infinite subdivision of $K$.

format: plain


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# Weka for text processing

## Overview

As discussed in the text processing notes, text files typically need to be converted into “feature vectors” before machine learning algorithms can be applied. The most common feature vector is a vector where each dimension is a different word, and the value in that dimension is either 0/1 binary value (yes or no the word is in that document), an integer count (the frequency of the word in that document), or a real value (often the tf-idf score of that word in that document).

The ARFF files we will be working with will always look like this:

@relation some-description-of-the-data
@attribute contents string
@attribute class {ham,spam}

@data
'Go until jurong point, crazy.. Available only in bugis n great world la e buffet... Cine there got amore wat...',ham
'Ok lar... Joking wif u oni...',ham
'Free entry in 2 a wkly comp to win FA Cup final tkts 21st May 2005. Text FA to 87121 to receive entry question(std txt rate)T&Cs apply 08452810075over18s',spam
'U dun say so early hor... U c already then say...',ham
'Nah I dont think he goes to usf, he lives around here though',ham
...

Of course, the classes (ham/spam) may change. These files are very easy to generate, should you actually wish to use text classification for your own purposes.

We’ll practice with the sms-spam.arff file, which has examples of SMS text msg ham and spam. This collection was produced by Tiago A. Almeida from the Federal University of Sao Carlos and José María Gómez Hidalgo from the R&D Department of Optenet. You may find their paper interesting; it examines the performance of various machine learning algorithms on this data set.

## Steps

### Strings to binary feature vectors

Load the ARFF file and choose a filter.

Choose the StringToWordVector filter, in the Unsupervised > Attribute section.

Click the filter name to get more options.

Change “attributeIndices” to “1” to just select the first attribute (the contents of the SMS message). Change “attributeNamePrefix” to “word-“ so that all the new features look like “word-foo” and “word-bar”, etc. Change “wordsToKeep” to “100000” to keep them all.

Click “Apply”

Now you’ll have lots more attributes. You can click “Undo” if you don’t like it.

Everything works better if the class (ham/spam) is last. After our filter, it’s not anymore. Let’s fix that. Choose a new filter, “Reorder,” under Unsupervised > Attribute. Click the filter name to change its properties. Type “2-last,1” to make the new order the second-to-last attributes followed by the first attribute. This puts the first attribute at the end.

Click “Apply.” Now the class (ham/spam) should be the last attribute again.

### Strings to frequency vectors

Follow the steps above (“Strings to binary feature vectors”) except use the following changes in the StringToWordVector filter:

Change “outputWordCounts” to “True.”

### Strings to tf-idf vectors

Follow the steps above (“Strings to binary feature vectors”) except use the following changes in the StringToWordVector filter:

• Change “IDFTransform” to “True”
• Change “TFTransform” to “True”
• Change “normalizeDocLength” to “Normalize all data”

## Applying the same transformations for prediction

Words in a new set of documents may not exactly match those in the training set. But the training set only contains attributes for words found in the training set (and possibly not all such words if the “wordsToKeep” parameter was small). However, in order to use a trained model for prediction, all of the same attributes must be present in the test ARFF file.

To ensure the training ARFF and testing ARFF have the same attributes, we need to apply the same transformations to each dataset. Thus, we need to write the transformations in a script, to be run from the command line. Furthermore, when applying StringToWordVector on the test dataset, we need to give the StringToWordVector filter the training ARFF file.

Here are all the steps shown above for TF-IDF feature vectors as a series of command-line statements, in two scripts. First, train.sh:

#!/bin/sh

java -cp /opt/weka/weka-3.8.0.jar weka.filters.unsupervised.attribute.StringToWordVector \
-R 1 -P word- -W 100000 -prune-rate -1.0 -T -N 1 \
-stemmer weka.core.stemmers.NullStemmer -stopwords-handler weka.core.stopwords.Null \
-M 1 -tokenizer "weka.core.tokenizers.WordTokenizer -delimiters \" \\\\r\\\\n\\\\t.,;:\\\\\'()\!?\\\\\\\"\"" \
-i sms-spam.arff -o sms-spam-1.arff

java -cp /opt/weka/weka-3.8.0.jar weka.filters.unsupervised.attribute.Reorder \
-R 2-last,first -i sms-spam-1.arff -o sms-spam-2.arff

java -cp /opt/weka/weka-3.8.0.jar weka.classifiers.bayes.NaiveBayes \
-t sms-spam-2.arff -d mymodel

Next, make a test ARFF file:

@relation sms-spam-test
@attribute contents string
@attribute class {ham,spam}

@data
'This is a test of the spam detector. Is this message spam?',?
'Buy now! Earn money! Cash prize!',?
'Hey honey, can you pick up peanut butter',?

And run test.sh:

#!/bin/sh

# notice use of batch option: -b
# notice input of training data: -i sms-spam.arff, plus testing input/output: -r and -s
# processed training data must be saved again, for some reason: -o sms-spam-1.arff
# (otherwise it dumps the result to the console)

java -cp /opt/weka/weka-3.8.0.jar weka.filters.unsupervised.attribute.StringToWordVector \
-b -R 1 -P word- -W 100000 -prune-rate -1.0 -T -N 1 \
-stemmer weka.core.stemmers.NullStemmer -stopwords-handler weka.core.stopwords.Null \
-M 1 -tokenizer "weka.core.tokenizers.WordTokenizer -delimiters \" \\\\r\\\\n\\\\t.,;:\\\\\'()\!?\\\\\\\"\"" \
-i sms-spam.arff -o sms-spam-1.arff -r sms-spam-test.arff -s sms-spam-test-1.arff

java -cp /opt/weka/weka-3.8.0.jar weka.filters.unsupervised.attribute.Reorder \
-R 2-last,first -i sms-spam-test-1.arff -o sms-spam-test-2.arff

java -cp /opt/weka/weka-3.8.0.jar weka.classifiers.bayes.NaiveBayes \
-T sms-spam-test-2.arff -l mymodel -classifications weka.classifiers.evaluation.output.prediction.CSV

Result:

=== Predictions on test data ===

inst#,actual,predicted,error,prediction
1,1:?,2:spam,,1
2,1:?,2:spam,,1
3,1:?,1:ham,,1

CSCI 431 material by Joshua Eckroth is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License. Source code for this website available at GitHub.


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# What's The Discrete Math Derivative Equivalent?

• I

## Main Question or Discussion Point

$$ƒ = b^n$$
$$b,n,I ∈ ℤ$$

Condition: Upon choosing a base value $b$..

$$n | b^n ≤ I$$
(n is determined based off the value of $b$ to yield the highest ƒ without going over $I$)
$$1<b<L , L<<I$$
where $I$ is some large number, and $L$ is also sufficiently large such that we want to avoid going through each base integer via trial and error....

How might I determine the base value that yields a value $ƒ$ that is closest to $I$?

## Answers and Replies

Related Set Theory, Logic, Probability, Statistics News on Phys.org
andrewkirk
Science Advisor
Homework Helper
Gold Member
I'm not sure exactly what you are asking.
Are you asking how to, given ##L,I\in\mathbb{Z}## with ##0<<L<<I##, find the combination ##b,n\in\mathbb{Z}## that maximizes ##b^n## subject to the constraints
1. ##1<b<L##; and
2. ##b^n<I##

If so then you could set ##Hi=\lfloor\log_2 I\rfloor## and ##Lo=\lceil \log_L I\rceil##, then find
$$b^n=max_{Lo\leq m\leq Hi}\bigg( \lfloor I^\tfrac{1}{m}\rfloor\bigg)^m$$

where ##n## is the value of ##m## that delivers that maximum.
That will give a much smaller search space than ##1,...,L##.

Last edited:


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# Shape, transverse size, and charged hadron multiplicity of jets in pp collisions at \sqrt{s} = 7 TeV

CMS Collaboration; Amsler, C; Chiochia, V; et al (2012). Shape, transverse size, and charged hadron multiplicity of jets in pp collisions at \sqrt{s} = 7 TeV. Journal of High Energy Physics, 6:160.

## Statistics

### Citations

3 citations in Web of Science®
19 citations in Scopus®


text:
show · ec.weierstrass_isomorphism all knowls · up · search:

Two Weierstrass models $E$, $E'$ over a field $K$ with Weierstrass coefficients $[a_1,a_2,a_3,a_4,a_6]$ and $[a'_1,a'_2,a'_3,a'_4,a'_6]$ are isomorphic over $K$ if there exist $u\in K^*$ and $r,s,t\in K$ such that \begin{aligned} u a_1' &= a_1+2s, \\ u^2a_2' &= a_2-sa_1+3r-s^2, \\ u^3a_3' &= a_3+ra_1+2t, \\ u^4a_4' &= a_4-sa_3+2ra_2-(t+rs)a_1+3r^2-2st, \\ u^6a_6' &= a_6+ra_4+r^2a_2+r^3-ta_3-t^2-rta_1. \\ \end{aligned} The set of transformations with parameters $[u,r,s,t]\in K^*\times K^3$ form the group of Weierstrass isomorphisms, which acts on both the set of all Weierstrass models over $K$ and also on the subset of smooth models, preserving the point at infinity. The discriminants $\Delta$, $\Delta'$ of the two models are related by $$u^{12}\Delta' = \Delta.$$

In the smooth case such a Weierstrass isomorphism $[u,r,s,t]$ induces an isomorphism between the two elliptic curves $E$, $E'$ they define. In terms of affine coordinates this is given by $$(x,y) \mapsto (x',y')$$ where \begin{aligned} x &= u^2x' + r \\ y &= u^3y' + su^2x' + t \\ \end{aligned}

Authors:
Knowl status:
• Review status: reviewed
• Last edited by David Roe on 2018-12-13 14:17:44
Referred to by:
History:


text:
History

Help on query formulation
Estimation of the value of $π$ using Monte Carlo method and related study of errors. (English)
Math. Sch. (Leicester) 42, No. 5, 21-23 (2013).
From the text: The number it holds a special interest in the history as well as in the current use of mathematics and science. This constant, is a number whose value is close to 3.14. It is defined mathematically as “the ratio of the circumference of a circle to the diameter of a circle". $π$ has numerous applications in mathematics as well as in science and technology. However, it is an irrational number, i.e. it cannot be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are co-prime integers and $q\ne 0$. Because it is irrational, no exact value of $n$ can been found. Only estimates of $n$ have been put up. There are several methods to get an approximate value of $π$. One of them, discussed in this article, is by making use of the Monte Carlo method. This article describes a simulation method using the probability that a point chosen at random inside a square lies inside the inscribed circle.
Classification: K50 N50 K90 F50


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# networkx.algorithms.bridges.bridges¶

bridges(G, root=None)[source]

Generate all bridges in a graph.

A bridge in a graph is an edge whose removal causes the number of connected components of the graph to increase. Equivalently, a bridge is an edge that does not belong to any cycle.

Parameters
Gundirected graph
rootnode (optional)

A node in the graph G. If specified, only the bridges in the connected component containing this node will be returned.

Yields
eedge

An edge in the graph whose removal disconnects the graph (or causes the number of connected components to increase).

Raises
NodeNotFound

If root is not in the graph G.

Notes

This is an implementation of the algorithm described in _[1]. An edge is a bridge if and only if it is not contained in any chain. Chains are found using the networkx.chain_decomposition() function.

Ignoring polylogarithmic factors, the worst-case time complexity is the same as the networkx.chain_decomposition() function, $$O(m + n)$$, where $$n$$ is the number of nodes in the graph and $$m$$ is the number of edges.

References

1

https://en.wikipedia.org/wiki/Bridge_%28graph_theory%29#Bridge-Finding_with_Chain_Decompositions

Examples

The barbell graph with parameter zero has a single bridge:

>>> G = nx.barbell_graph(10, 0)
>>> list(nx.bridges(G))
[(9, 10)]


text:
SKY-MAP.ORG
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# HD 118216

Contents

### Images

DSS Images   Other Images

### Related articles

 Mass loss and orbital period decrease in detached chromospherically active binariesThe secular evolution of the orbital angular momentum (OAM), thesystemic mass (M=M1+M2) and the orbital period of114 chromospherically active binaries (CABs) were investigated afterdetermining the kinematical ages of the subsamples which were setaccording to OAM bins. OAMs, systemic masses and orbital periods wereshown to be decreasing by the kinematical ages. The first-orderdecreasing rates of OAM, systemic mass and orbital period have beendetermined as per systemic OAM, per systemic mass and per orbitalperiod, respectively, from the kinematical ages. The ratio of d logJ/dlogM= 2.68, which were derived from the kinematics of the presentsample, implies that there must be a mechanism which amplifies theangular momentum loss (AML) times in comparison to isotropic AML ofhypothetical isotropic wind from the components. It has been shown thatsimple isotropic mass loss from the surface of a component or bothcomponents would increase the orbital period. A Comparative Study of Flaring Loops in Active StarsDynamo activity in stars of different types is expected to generatemagnetic fields with different characteristics. As a result, adifferential study of the characteristics of magnetic loops in a broadsample of stars may yield information about dynamo systematics. In theabsence of direct imaging, certain physical parameters of a stellarmagnetic loop can be extracted if a flare occurs in that loop. In thispaper we employ a simple nonhydrodynamic approach introduced by Haisch,to analyze a homogeneous sample of all of the flares we could identifyin the EUVE DS database: a total of 134 flares that occurred on 44 starsranging in spectral type from F to M and in luminosity class from V toIII. All of the flare light curves that have been used in the presentstudy were obtained by a single instrument (EUVE DS). For each flare, wehave applied Haisch's simplified approach (HSA) in order to determineloop length, temperature, electron density, and magnetic field. For eachof our target stars, a literature survey has been performed to determinequantitatively the extent to which our results are consistent withindependent studies. The results obtained by HSA are found to be wellsupported by results obtained by other methods. Our survey suggeststhat, on the main sequence, short loops (with lengths<=0.5R*) may be found in stars of all classes, while thelargest loops (with lengths up to 2R*) appear to be confinedto M dwarfs. Based on EUVE data, the transition from small to largeloops on the main sequence appears to occur between spectral types K2and M0. We discuss the implications of this result for dynamo theories. On the Temperature-Emission Measure Distribution in Stellar CoronaeStrong peaks in the emission measure-temperature (EM-T ) distributionsin the coronae of some binary stars are associated with the presence ofhot (107 K), dense (up to 1013 cm -3)plasma. These peaks are very reminiscent of those predicted to arise inan impulsively heated solar corona. A coronal model comprised of manyimpulsively heated strands is adapted to stellar parameters. It is shownthat the properties of the EM-T distribution can be accounted for ingeneral terms provided the emission comes from many very small loops(length under 103 km) with intense magnetic fields (1 kG)distributed across part of the surface of the star. The heating requiresevents that generally dissipate between 1026 and 1028 ergs, which is in the range of solar microflares. This impliesthat such stars must be capable of generating regions of localizedintense magnetic fields. A Large Millimeter Flare on the RS CVn Binary σ GeminorumThe flaring behavior of active RS CVn binaries has been studiedextensively at centimeter wavelengths, but very little has beenpublished concerning RS CVn flares in the millimeter region, where thenonthermal gyrosynchrotron emission is much more likely to be opticallythin. We present 3 mm (99 GHz) observations of a large flare on theactive binary σ Gem using the Owens Valley Millimeter Array. Two 3hr observations in 2004 February showed only 3 σ upper limits of 3mJy, while the third observation on 2004 April 22 showed a large flareoutburst that filled the 6 hr observation and reached a peak fluxdensity of 90 mJy. The strongest brightening was preceded by at least a4 hr period of significantly elevated emission at 20-40 mJy. Theseobservations demonstrate the feasibility of detailed multifrequencystudies of nonthermal electron distributions in active star coronae withthe next generation of millimeter and centimeter radio arrays. MERLIN Astrometry of 11 Radio StarsWe report accurate positions in the International Celestial ReferenceFrame (ICRF) for 11 radio stars. Observations were made using theMulti-Element Radio Linked Interferometer Network at a radio frequencyof 5 GHz. The positions are estimated to be accurate at the 5 mas level.Positions were obtained directly in the ICRF by phase referencing theradio stars to ICRF quasars whose positions are estimated to be accurateat the 0.25 mas level. We use our results together with results ofprevious observations to obtain proper-motion estimates for these stars.The average proper-motion uncertainties are 1.1 mas yr-1 inμαcosδ and 1.2 mas yr-1 inμδ, comparable to the Hipparcos values. X-ray luminosity function of faint point sources in the Milky WayWe assessed the contribution to the X-ray (above 2 keV) luminosity ofthe Milky Way by different classes of low-mass binary systems and singlestars. We began by using the RXTE Slew Survey of the sky at|b|>10° to construct an X-ray luminosity function (XLF) of nearbyX-ray sources in the range 1030 ergs-1 Extragalactic Source Counts in the 20-50 keV Energy Band from the Deep Observation of the Coma Region by INTEGRAL IBISWe present the analysis of serendipitous sources in a deep, 500 ks, hardX-ray observation of the Coma Cluster region with the IBIS instrument onboard INTEGRAL. In addition to the Coma Cluster, the final 20-50 keVimage contains 12 serendipitous sources with statistical significance>4 σ. We use these data (after correcting for expected numberof false detections) to extend the extragalactic source counts in the20-50 keV energy band down to a limiting flux of1.0×10-11 ergs s-1 cm-2 (~=1mcrab). This is a more than a factor of 10 improvement in sensitivitycompared to the previous results in this energy band obtained with theHEAO 1 A-4 instrument. The derived source counts are consistent with theEuclidean relation N(>f)~f-3/2. A large fraction ofidentified serendipitous sources are low-redshift, z<0.02, activegalactic nuclei (AGNs), mostly of Seyfert 1 type. The surface density ofhard X-ray sources is (1.4+/-0.5)×10-2 deg-2above a flux threshold of 10-11 ergs s-1cm-2. These sources directly account for ~3% of the cosmicX-ray background in the 20-50 keV energy band. Given the low redshiftdepth of our sample, we expect that similar sources at higher redshiftsaccount for a significant fraction of the hard X-ray background. Ourfield covers only 3% of the sky; a systematic analysis of otherextragalactic INTEGRAL observations can produce much larger sourcesamples and is, therefore, critically important. Kinematics of chromospherically active binaries and evidence of an orbital period decrease in binary evolution The kinematics of 237 chromospherically active binaries (CABs) werestudied. The sample is heterogeneous with different orbits andphysically different components from F to M spectral-type main-sequencestars to G and K giants and supergiants. The computed U, V, W spacevelocities indicate that the sample is also heterogeneous in velocityspace. That is, both kinematically younger and older systems exist amongthe non-evolved main sequence and the evolved binaries containing giantsand subgiants. The kinematically young (0.95 Gyr) subsample (N= 95),which is formed according to the kinematical criteria of moving groups,was compared with the rest (N= 142) of the sample (3.86 Gyr) toinvestigate any observational clues of binary evolution. Comparing theorbital period histograms between the younger and older subsamples,evidence was found supporting the finding of Demircan that the CABs losemass (and angular momentum) and evolve towards shorter orbital periods.The evidence of mass loss is noticeable on the histograms of the totalmass (Mh+Mc), which is compared between theyounger (only N= 53 systems available) and older subsamples (only N= 66systems available). The orbital period decrease during binary evolutionis found to be clearly indicated by the kinematical ages of 6.69, 5.19and 3.02 Gyr which were found in the subsamples according to the periodranges of logP<= 0.8, 0.8 < logP<= 1.7 and 1.7 < logP<=3, respectively, among the binaries in the older subsample. X-ray astronomy of stellar coronaeX-ray emission from stars in the cool half of the Hertzsprung-Russelldiagram is generally attributed to the presence of a magnetic coronathat contains plasma at temperatures exceeding 1 million K. Coronae areubiquitous among these stars, yet many fundamental mechanisms operatingin their magnetic fields still elude an interpretation through adetailed physical description. Stellar X-ray astronomy is thereforecontributing toward a deeper understanding of the generation of magneticfields in magnetohydrodynamic dynamos, the release of energy in tenuousastrophysical plasmas through various plasma-physical processes, and theinteractions of high-energy radiation with the stellar environment.Stellar X-ray emission also provides important diagnostics to study thestructure and evolution of stellar magnetic fields from the first daysof a protostellar life to the latest stages of stellar evolution amonggiants and supergiants. The discipline of stellar coronal X-rayastronomy has now reached a level of sophistication that makes tests ofadvanced theories in stellar physics possible. This development is basedon the rapidly advancing instrumental possibilities that today allow usto obtain images with sub-arcsecond resolution and spectra withresolving powers exceeding 1000. High-resolution X-ray spectroscopy has,in fact, opened new windows into astrophysical sources, and has played afundamental role in coronal research. The Geneva-Copenhagen survey of the Solar neighbourhood. Ages, metallicities, and kinematic properties of 14 000 F and G dwarfsWe present and discuss new determinations of metallicity, rotation, age,kinematics, and Galactic orbits for a complete, magnitude-limited, andkinematically unbiased sample of 16 682 nearby F and G dwarf stars. Our63 000 new, accurate radial-velocity observations for nearly 13 500stars allow identification of most of the binary stars in the sampleand, together with published uvbyβ photometry, Hipparcosparallaxes, Tycho-2 proper motions, and a few earlier radial velocities,complete the kinematic information for 14 139 stars. These high-qualityvelocity data are supplemented by effective temperatures andmetallicities newly derived from recent and/or revised calibrations. Theremaining stars either lack Hipparcos data or have fast rotation. Amajor effort has been devoted to the determination of new isochrone agesfor all stars for which this is possible. Particular attention has beengiven to a realistic treatment of statistical biases and errorestimates, as standard techniques tend to underestimate these effectsand introduce spurious features in the age distributions. Our ages agreewell with those by Edvardsson et al. (\cite{edv93}), despite severalastrophysical and computational improvements since then. We demonstrate,however, how strong observational and theoretical biases cause thedistribution of the observed ages to be very different from that of thetrue age distribution of the sample. Among the many basic relations ofthe Galactic disk that can be reinvestigated from the data presentedhere, we revisit the metallicity distribution of the G dwarfs and theage-metallicity, age-velocity, and metallicity-velocity relations of theSolar neighbourhood. Our first results confirm the lack of metal-poor Gdwarfs relative to closed-box model predictions (the G dwarfproblem''), the existence of radial metallicity gradients in the disk,the small change in mean metallicity of the thin disk since itsformation and the substantial scatter in metallicity at all ages, andthe continuing kinematic heating of the thin disk with an efficiencyconsistent with that expected for a combination of spiral arms and giantmolecular clouds. Distinct features in the distribution of the Vcomponent of the space motion are extended in age and metallicity,corresponding to the effects of stochastic spiral waves rather thanclassical moving groups, and may complicate the identification ofthick-disk stars from kinematic criteria. More advanced analyses of thisrich material will require careful simulations of the selection criteriafor the sample and the distribution of observational errors.Based on observations made with the Danish 1.5-m telescope at ESO, LaSilla, Chile, and with the Swiss 1-m telescope at Observatoire deHaute-Provence, France.Complete Tables 1 and 2 are only available in electronic form at the CDSvia anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/418/989 RXTE all-sky slew survey. Catalog of X-ray sources at |b|>10oWe report results of a serendipitous hard X-ray (3-20 keV), nearlyall-sky (|b|>10o) survey based on RXTE/PCA observationsperformed during satellite reorientations in 1996-2002. The survey is80% (90%) complete to a 4σ limiting flux of ≈ 1.8 (2.5) ×10-11 erg s-1 cm-2 in the 3-20 keVband. The achieved sensitivity in the 3-8 keV and 8-20 keV subbands issimilar to and an order of magnitude higher than that of the previouslyrecord HEAO-1 A1 and HEAO-1 A4 all-sky surveys, respectively. A combined7× 103 sq deg area of the sky is sampled to flux levelsbelow 10-11 erg s-1 cm-2 (3-20 keV). Intotal 294 sources are detected and localized to better than 1 deg. 236(80%) of these can be confidently associated with a known astrophysicalobject; another 22 likely result from the superposition of 2 or 3closely located known sources. 35 detected sources remain unidentified,although for 12 of these we report a likely soft X-ray counterpart fromthe ROSAT all-sky survey bright source catalog. Of the reliablyidentified sources, 63 have local origin (Milky Way, LMC or SMC), 64 areclusters of galaxies and 100 are active galactic nuclei (AGN). The factthat the unidentified X-ray sources have hard spectra suggests that themajority of them are AGN, including highly obscured ones(NH>1023 cm-2). For the first timewe present a log N-log S diagram for extragalactic sources above4× 10-12 erg s-1 cm-2 at 8-20keV.Table 2 is only available in electronic form at the CDS via anonymousftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/418/927 Stellar Coronal AstronomyCoronal astronomy is by now a fairly mature discipline, with a quartercentury having gone by since the detection of the first stellar X-raycoronal source (Capella), and having benefitted from a series of majororbiting observing facilities. Serveral observational characteristics ofcoronal X-ray and EUV emission have been solidly established throughextensive observations, and are by now common, almost text-book,knowledge. At the same time the implications of coronal astronomy forbroader astrophysical questions (e.g.Galactic structure, stellarformation, stellar structure, etc.) have become appreciated. Theinterpretation of stellar coronal properties is however still often opento debate, and will need qualitatively new observational data to bookfurther progress. In the present review we try to recapitulate our viewon the status of the field at the beginning of a new era, in which thehigh sensitivity and the high spectral resolution provided by Chandraand SMM-Newton will address new questions which were not accessiblebefore. The Structure of Stellar Coronae in Active Binary SystemsA survey of 28 stars (22 active binary systems, plus six single stars orwide binaries for comparison) using extreme ultraviolet spectra has beenconducted to establish the structure of stellar coronae in active binarysystems from the emission measure distribution (EMD), electrondensities, and scale sizes. Observations obtained by the ExtremeUltraviolet Explorer satellite (EUVE) during 9 years of operation areincluded for the stars in the sample. EUVE data allow a continuous EMDto be constructed in the range logTe(K)~5.6-7.4, using ironemission lines. These data are complemented with IUE observations tomodel the lower temperature range [logTe(K)~4.0-5.6].Inspection of the EMD shows an outstanding narrow enhancement, orbump'' peaking around logTe(K)~6.9 in 25 of the stars,defining a fundamental coronal structure. The emission measure per unitstellar area decreases with increasing orbital (or photometric) periodsof the target stars; stars in binaries generally have more material atcoronal temperatures than slowly rotating single stars. High electrondensities (Ne>~1012 cm-3) arederived at ~logTe(K)~7.0 for some targets, implying smallemitting volumes.The observations suggest the magnetic stellar coronae of these stars areconsistent with two basic classes of magnetic loops: solar-like loopswith maximum temperature around logTe(K)~6.3 and lowerelectron densities (Ne>~109-1010.5cm-3), and hotter loops peaking aroundlogTe(K)~6.9 with higher electron densities(Ne>~1012 cm-3). For the most activestars, material exists at much higher temperatures[logTe(K)>=6.9] as well. However, current ab initiostellar loop models cannot reproduce such a configuration. Analysis ofthe light curves of these systems reveals signatures of rotation ofcoronal material, as well as apparent seasonal (i.e., year-to-year)changes in the activity levels. Very Long Baseline Interferometry Imaging of the RS Canum Venaticorum Binary Star System HR 5110We present VLBI images of the RS CVn binary star HR 5110 (=BH CVn; HD118216), obtained from observations made at 8.4 GHz on 1994 May 29/30 insupport of the NASA/Stanford Gravity Probe B project. Our images show anemission region with a core-halo morphology. The core was 0.39+/-0.09mas (FWHM) in size, or 66%+/-20% of the 0.6+/-0.1 mas diameter of thechromospherically active K subgiant star in the binary system. The halowas 1.95+/-0.22 mas (FWHM) in size, or 1.8+/-0.2 times the 1.1+/-0.1 masseparation of the centers of the K and F stars. (The uncertainties givenfor the diameter of the K star and its separation from the F star eachreflect the level of agreement of the two most recent publisheddeterminations.) The core increased significantly in brightness over thecourse of the observations and seems to have been the site of flareactivity that generated an increase in the total flux density of ~200%in 12 hr. The fractional circular polarization simultaneously decreasedfrom ~10% to 2.5%. The 100 Brightest X-Ray Stars within 50 Parsecs of the SunBased on the Hipparcos and Tycho-2 astrometric catalogs and the ROSATsurveys, a sample of 100 stars most luminous in X-rays within or arounda distance of 50 pc is culled. The smallest X-ray luminosity in thesample, in units of 1029 ergs s-1, isLX=9.8 the strongest source in the solar neighborhood is IIPeg, a RS CVn star, at LX=175.8. With respect to the originof X-ray emission, the sample is divided into partly overlapping classesof pre-main-sequence, post-T Tauri, and very young ZAMS objects (typeXY), RS CVn-type binary stars (type RS), other active short-periodbinaries, including binary BY Dra-type objects (type XO), apparentlysingle or long-period binary active evolved stars (type XG), contactbinaries of WU UMa kind (type WU), apparently single or long-periodbinary variable stars of BY Dra kind (type BY), and objects of unknownnature (type X?). Chromospherically active, short-period binaries (RSand XO) make up 40% of the brightest X-ray emitters, followed by youngstars (XY) at 30% and unknown sources (X?) at 15%. The fraction ofspectroscopically single evolved X-ray emitters of spectral classes IVand III is quite large (10%). The sources identified as RS CVn-typestars (RS, 23 objects) are considerably stronger in X-ray than theXY-objects and the other active binaries (XO and WU, 20 objects). Sevenobjects have LX>100, all RS except one XY, viz., BO Mic. Onlyfive (22%) RS objects have LX<25, while only three (10%)XY stars have LX>25. Formally, the limit of LX=25could serve as a statistical criterion to differentiate RS and XY stars.However, the other short-period binaries (including eclipsing stars ofAlgol and β Lyr type) have a distribution of LX verysimilar to the XY objects. The contact binaries (WU) appear to be muchweaker in X-rays than their detached counterparts of RS type, but thesample of the former is too small (three objects) to reach a firmconclusion. Sources matched with giants (either single or in binaries)are found to be significantly harder, with only 7% of hardness ratiosbelow 0, than subgiants (66% of HR1<0) and dwarfs (59% of HR1<0).Almost all objects in the sample are binary or multiple stars; thefraction of components (FC), defined as the total number of componentsin all binary and multiple systems divided by the sum of the totalnumber of components and single stars, is at least 0.90. The FC for theXY objects reaches 0.81, and for the unknown type 0.89. About 70% of RSobjects have also visual or astrometric companions, which makes themhierarchical multiple systems. The RS objects (mostly old, evolvedstars) and the XY stars have quite different kinematics. While the RSobjects move at considerable velocities in apparently random directionswith respect to the local standard of rest, the young stars have smallerand orderly velocities and tend to comprise expanding mini-associationssuch as the β Pic and the Tucana groups. The majority of the youngX-ray active stars belong to the Pleiades stream with the meanheliocentric velocity (U,V,W)=(-9.6,-21.8,-7.7) km s-1. Astrometric Positions and Proper Motions of 19 Radio StarsWe have used the Very Large Array, linked with the Pie Town Very LongBaseline Array antenna, to determine the astrometric positions of 19radio stars in the International Celestial Reference Frame (ICRF). Thepositions of these stars were directly linked to the positions ofdistant quasars through phase-referencing observations. The positions ofthe ICRF quasars are known to 0.25 mas, thus providing an absolutereference at the angular resolution of our radio observations. Averagevalues for the errors in our derived positions for all sources were 13and 16 mas in αcosδ and δ, respectively, withaccuracies approaching 1-2 mas for some of the stars observed.Differences between the ICRF positions of the 38 quasars and thosemeasured from our observations showed no systematic offsets, with meanvalues of -0.3 mas in αcosδ and -1.0 mas in δ.Standard deviations of the quasar position differences of 17 and 11 masin αcosδ and δ, respectively, are consistent with themean position errors determined for the stars. Our measured positionswere combined with previous Very Large Array measurements taken from1978 to 1995 to determine the proper motions of 15 of the stars in ourlist. With mean errors of ~1.6 mas yr-1, the accuracies ofour proper motions approach those derived from Hipparcos and, for a fewof the stars in our program, are better than the Hipparcos values.Comparing the positions of our radio stars with the Hipparcos Catalogue,we find that at the epoch of our observations, the two frames arealigned to within formal errors of approximately 3 mas. This resultconfirms that the Hipparcos frame is inertial at the expected level. VLA Radio Positions of Stars: 1978-1995VLA astrometric positions of the radio emission from 52 stars arereported, from observations obtained between 1978 and 1995. Thepositions of these stars have been obtained and reduced in a uniformmanner. Based on our measurements, the offset of the optical (Hipparcos)frame from the radio reference frame is in agreement with the Hipparcosextragalactic link results, within their mean errors. Comparison of theVLA measurements with the Hipparcos optical positions confirms earlierestimates of the accuracy of these positions as 30 mas. Long-termmeasurements of UX Ari have improved its proper motion. Lithium and rotation in F and G dwarfs and subgiantsLithium abundances have been determined in 127 F and G Pop I stars basedon new measurements of the equivalent width of the lambda 6707 ÅLi I line from their high resolution CCD spectra. Distances and absolutemagnitudes of these stars have been obtained from the HipparcosCatalogue and their masses and ages derived, enabling us to investigatethe behaviour of lithium as a function of these parameters. Based ontheir location on the HR diagram superposed on theoretical evolutionarytracks, the sample of the stars has been chosen to ensure that they havemore or less completed their Li depletion on the main sequence. A largespread in the Li abundances is found at any given effective temperatureespecially in the already spun down late F and early G stars. Thisspread persists even if the Li-dip'' stars that have evolved from themain sequence temperature interval 6500-6800 K are excluded. Stars inthe mass range up to 2 M/Msun when divided into threemetallicity groups show a linear correlation between Li abundance andmass, albeit with a large dispersion around it which is not fullyaccounted for by age either. The large depletions and the observedspread in Li are in contrast to the predictions of the standard stellarmodel calculations and suggest that they are aided by non-standardprocesses depending upon variables besides mass, age and metallicity.The present study was undertaken to examine, in particular, the effectsof rotation on the depletion of Li. No one-to-one correlation is foundbetween the Li abundance and the present projected rotational velocity.Instead the observed abundances seem to be dictated by the rotationalhistory of the star. However, it is noted that even this interpretationis subject to the inherent limitation in the measurement of the observedLi EQW for large rotational velocities.Table 1 is only available in electronic form at the CDS via anonymousftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/409/251 Catalogue of averaged stellar effective magnetic fields. I. Chemically peculiar A and B type starsThis paper presents the catalogue and the method of determination ofaveraged quadratic effective magnetic fields < B_e > for 596 mainsequence and giant stars. The catalogue is based on measurements of thestellar effective (or mean longitudinal) magnetic field strengths B_e,which were compiled from the existing literature.We analysed the properties of 352 chemically peculiar A and B stars inthe catalogue, including Am, ApSi, He-weak, He-rich, HgMn, ApSrCrEu, andall ApSr type stars. We have found that the number distribution of allchemically peculiar (CP) stars vs. averaged magnetic field strength isdescribed by a decreasing exponential function. Relations of this typehold also for stars of all the analysed subclasses of chemicalpeculiarity. The exponential form of the above distribution function canbreak down below about 100 G, the latter value representingapproximately the resolution of our analysis for A type stars.Table A.1 and its references are only available in electronic form atthe CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/407/631 and Tables 3 to 9are only available in electronic form at http://www.edpsciences.org The Hamburg/RASS Catalogue of optical identifications. Northern high-galactic latitude ROSAT Bright Source Catalogue X-ray sourcesWe present the Hamburg/RASS Catalogue (HRC) of optical identificationsof X-ray sources at high-galactic latitude. The HRC includes all X-raysources from the ROSAT Bright Source Catalogue (RASS-BSC) with galacticlatitude |b| >=30degr and declination delta >=0degr . In thispart of the sky covering ~ 10 000 deg2 the RASS-BSC contains5341 X-ray sources. For the optical identification we used blue Schmidtprism and direct plates taken for the northern hemisphere Hamburg QuasarSurvey (HQS) which are now available in digitized form. The limitingmagnitudes are 18.5 and 20, respectively. For 82% of the selectedRASS-BSC an identification could be given. For the rest either nocounterpart was visible in the error circle or a plausibleidentification was not possible. With ~ 42% AGN represent the largestgroup of X-ray emitters, ~ 31% have a stellar counterpart, whereasgalaxies and cluster of galaxies comprise only ~ 4% and ~ 5%,respectively. In ~ 3% of the RASS-BSC sources no object was visible onour blue direct plates within 40\arcsec around the X-ray sourceposition. The catalogue is used as a source for the selection of(nearly) complete samples of the various classes of X-ray emitters. The HRX-BL Lac sample - Evolution of BL Lac objectsThe unification of X-ray and radio selected BL Lacs has been anoutstanding problem in the blazar research in the past years. Recentinvestigations have shown that the gap between the two classes can befilled with intermediate objects and that apparently all differences canbe explained by mutual shifts of the peak frequencies of the synchrotronand inverse Compton component of the emission. We study the consequencesof this scheme using a new sample of X-ray selected BL Lac objectscomprising 104 objects with z<0.9 and a mean redshift bar {z} = 0.34.77 BL Lacs, of which the redshift could be determined for 64 (83%)objects, form a complete sample. The new data could not confirm ourearlier result, drawn from a subsample, that the negative evolutionvanishes below a synchrotron peak frequency log nupeak =16.5. The complete sample shows negative evolution at the 2sigma level(< Ve/Va > = 0.42 +/- 0.04). We concludethat the observed properties of the HRX BL Lac sample show typicalbehaviour for X-ray selected BL Lacs. They support an evolutionarymodel, in which flat-spectrum radio quasars (FSRQ) with high energeticjets evolve towards low frequency peaked (mostly radio-selected) BL Lacobjects and later on to high frequency peaked (mostly X-ray selected) BLLacs.Appendix (Tables 8 and 9, Fig. 8) is only available in electronic format http://www.edpsciences.org Automated spectroscopic abundances of A and F-type stars using echelle spectrographs. II. Abundances of 140 A-F stars from ELODIEUsing the method presented in Erspamer & North (\cite{erspamer},hereafter Paper I), detailed abundances of 140 stars are presented. Theuncertainties characteristic of this method are presented and discussed.In particular, we show that for a S/N ratio higher than 200, the methodis applicable to stars with a rotational velocity as high as 200 kms-1. There is no correlation between abundances and Vsin i,except a spurious one for Sr, Sc and Na which we explain by the smallnumber of lines of these elements combined with a locally biasedcontinuum. Metallic giants (Hauck \cite{hauck}) show larger abundancesthan normal giants for at least 8 elements: Al, Ca, Ti, Cr, Mn, Fe, Niand Ba. The anticorrelation for Na, Mg, Si, Ca, Fe and Ni with Vsin isuggested by Varenne & Monier (\cite{varenne99}) is not confirmed.The predictions of the Montréal models (e.g. Richard et al.\cite{richard01}) are not fulfilled in general. However, a correlationbetween left [(Fe)/(H)right ] and log g is found for stars of 1.8 to 2.0M_sun. Various possible causes are discussed, but the physical realityof this correlation seems inescapable.Based on observations collected at the 1.93 m telescope at theObservatoire de Haute-Provence (St-Michel l'Observatoire, France) andCORALIE.Based on observations collected at the Swiss 1.2 m Leonard Eulertelescopes at the European Southern Observatory (La Silla, Chile).Tables 5 and 6 are only available in electronic form at the CDS viaanonymous ftp to cdsarc.u.strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/398/1121 Origin of Prolonged X-Ray Flares on Active Late-Type StarsSoft X-ray data for prolonged flares in subgiants in RS CVn binarysystems and some other active late-type stars (AB Dor, Algol) areanalyzed. During these nonstationary events, a large amount of hotplasma with temperatures exceeding 108 K exists for many hours.Numerical simulations of gas-dynamical processes in the X-raysourcegiant loopscan yield reliable estimates of theplasma parameters and flare-source size. This confirms that suchphenomena exist while considerable energy is supplied to the top part ofa giant loop or system of loops. Refined estimates of the flare energy(up to 1037 erg) and scales contradict the widely accepted idea thatprolonged X-ray flares are associated with the evolution of localmagnetic fields. The energy of the current component of the large-scalemagnetic field arising during the ejection of magnetic field by plasmajets or stellar wind is estimated. Two cases are considered: a globalstellar field and fields connecting regions with oppositely directedunipolar magnetic fields. The inferred energy of the current componentof the magnetic field associated with distortion of the initial MHDconfiguration is close to the total flare energy, suggesting thatlarge-scale magnetic fields play an important role in prolonged flares.The flare process encompasses some portion of a streamer belt and maypropagate along the entire magnetic equator of the star during the mostpowerful prolonged events. HIPPARCOS age-metallicity relation of the solar neighbourhood disc starsWe derive age-metallicity relations (AMRs) and orbital parameters forthe 1658 solar neighbourhood stars to which accurate distances aremeasured by the HIPPARCOS satellite. The sample stars comprise 1382 thindisc stars, 229 thick disc stars, and 47 halo stars according to theirorbital parameters. We find a considerable scatter for thin disc AMRalong the one-zone Galactic chemical evolution (GCE) model. Orbits andmetallicities of thin disc stars show now clear relation each other. Thescatter along the AMR exists even if the stars with the same orbits areselected. We examine simple extension of one-zone GCE models whichaccount for inhomogeneity in the effective yield and inhomogeneous starformation rate in the Galaxy. Both extensions of the one-zone GCE modelcannot account for the scatter in age - [Fe/H] - [Ca/Fe] relationsimultaneously. We conclude, therefore, that the scatter along the thindisc AMR is an essential feature in the formation and evolution of theGalaxy. The AMR for thick disc stars shows that the star formationterminated 8 Gyr ago in the thick disc. As already reported by Grattonet al. (\cite{Gratton_et.al.2000}) and Prochaska et al.(\cite{Prochaska_et.al.2000}), thick disc stars are more Ca-rich thanthin disc stars with the same [Fe/H]. We find that thick disc stars showa vertical abundance gradient. These three facts, the AMR, verticalgradient, and [Ca/Fe]-[Fe/H] relation, support monolithic collapseand/or accretion of satellite dwarf galaxies as likely thick discformation scenarios. Tables 2 and 3 are only available in electronicform at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)or via http:/ /cdsweb.u-strasbg.fr/ cgi-bin/qcat?J/ A+A/394/927 Rotational velocities of A-type stars in the northern hemisphere. II. Measurement of v sin iThis work is the second part of the set of measurements of v sin i forA-type stars, begun by Royer et al. (\cite{Ror_02a}). Spectra of 249 B8to F2-type stars brighter than V=7 have been collected at Observatoirede Haute-Provence (OHP). Fourier transforms of several line profiles inthe range 4200-4600 Å are used to derive v sin i from thefrequency of the first zero. Statistical analysis of the sampleindicates that measurement error mainly depends on v sin i and thisrelative error of the rotational velocity is found to be about 5% onaverage. The systematic shift with respect to standard values fromSlettebak et al. (\cite{Slk_75}), previously found in the first paper,is here confirmed. Comparisons with data from the literature agree withour findings: v sin i values from Slettebak et al. are underestimatedand the relation between both scales follows a linear law ensuremath vsin inew = 1.03 v sin iold+7.7. Finally, thesedata are combined with those from the previous paper (Royer et al.\cite{Ror_02a}), together with the catalogue of Abt & Morrell(\cite{AbtMol95}). The resulting sample includes some 2150 stars withhomogenized rotational velocities. Based on observations made atObservatoire de Haute Provence (CNRS), France. Tables \ref{results} and\ref{merging} are only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.125.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/393/897 A highly sensitive search for magnetic fields in B, A and F starsCircular spectropolarimetric observations of 74 stars were obtained inan attempt to detect magnetic fields via the longitudinal Zeeman effectin their spectral lines. The sample observed includes 22 normal B, A andF stars, four emission-line B and A stars, 25 Am stars, 10 HgMn stars,two lambda Boo stars and 11 magnetic Ap stars. Using the Least-SquaresDeconvolution multi-line analysis approach (Donati et al.\cite{donati97etal}), high precision Stokes I and V mean signatures wereextracted from each spectrum. We find absolutely no evidence formagnetic fields in the normal, Am and HgMn stars, with upper limits onlongitudinal field measurements usually considerably smaller than anypreviously obtained for these objects. We conclude that if any magneticfields exist in the photospheres of these stars, these fields are notordered as in the magnetic Ap stars, nor do they resemble the fields ofactive late-type stars. We also detect for the first time a field in theA2pSr star HD 108945 and make new precise measurements of longitudinalfields in five previously known magnetic Ap stars, but do not detectfields in five other stars classified as Ap SrCrEu. We also report newresults for several binary systems, including a new vsin i for therapidly rotating secondary of the Am-delta Del SB2 HD 110951. Based onobservations obtained using the MuSiCoS spectropolarimeter on theBernard Lyot telescope, l'Observatoire du Pic du Midi, France. Search for point sources of gamma radiation above 15 TeV with the HEGRA AIROBICC arrayA search for potential point sources of very high energy gamma rays hasbeen carried out on the data taken simultaneously by the HEGRA AIROBICCand Scintillator arrays from August 1994 to March 2000. The list ofsought sources includes supernova remnants, pulsars, AGNs and binarysystems. The energy threshold is around 15 TeV. For the Crab Nebula, amodest excess of 2.5 standard deviations above the cosmic ray backgroundhas been observed. Flux upper limits (at 90% c.l.) of around 1.3 timesthe flux of the Crab Nebula are obtained, in average, for the candidatesources. A different search procedure has been used for an all-skysearch which yields absolute flux upper limits between 4 and 9 crabsdepending on declination, in the band from delta = 0 to delta =60o. The full versions of Tables 1 and 2, including thecoordinates of the sources, are available in electronic form at the CDSvia anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/390/39 Optical positions of 55 radio stars from astrolabe observations from the Yunnan ObservatoryThe observations by the photoelectric astrolabe at Yunnan Observatoryrelative to the Hipparcos Catalogue and the optical positions of 55radio stars were obtained from observations between 1991 and 2000. Theyall resulted from processing the photon counts obtained by means of theastrolabe after the automation of the instrument. There are 46 stars incommon with the Hipparcos Catalogue. Tables \ref{t1} and \ref{t2} arealso available in electronic form at the CDS via anonymous ftpcdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/383/1062. A complete sample of Seyfert galaxies selected at 0.25 keVWe have used the ROSAT Bright Source Catalogue to extract a completesample of sources selected in the band from 0.1-0.4keV. This0.25keV-selected sample is composed of 54 Seyfert galaxies, 25 BLLacertae objects, 4 clusters and 27 Galactic stars or binaries.Seyfert-type galaxies with ultrasoft' X-ray spectra can very often beclassed optically as narrow-line Seyfert 1s (NLS1s). Such objects arereadily detected in 0.25keV surveys; the sample reported here contains20 NLS1s, corresponding to a 40 per cent fraction of the Seyferts.Optical spectra of the Seyfert galaxies were gathered for correlativeanalysis, which confirmed the well-known relations between X-ray slopeand optical spectral properties (e.g. [Oiii]/Hβ ratio; Feiistrength, Hβ width). The various intercorrelations are most likelydriven, fundamentally, by the shape of the photoionizing continuum inSeyfert nuclei. We argue that a steep X-ray spectrum is a betterindicator of an extreme' set of physical properties in Seyfert galaxiesthan is the narrowness of the optical Hβ line. The correlationstudies were also used to isolate a number of Seyfert galaxies withapparently `anomalous' properties. Of particular interest are the sixobjects with relatively weak permitted line emission (Hβ and Feii)and weak optical continua. Such objects are rare in most surveys, buttwo of these (IC 3599 and NGC 5905) are known to be transient activegalactic nuclei in which the X-ray flux has faded by factors ~100. Ifthe other four objects also turn out to be transient, this woulddemonstrate that 0.25keV surveys provide an efficient way of finding aninteresting class of object. Finally, the luminosity function of the0.25keV-selected Seyfert galaxies was determined and broken down intosubsamples to investigate the relative space densities of Seyferts whenseparated on the basis of either X-ray slope or Hβ linewidth. Catalogue of Apparent Diameters and Absolute Radii of Stars (CADARS) - Third edition - Comments and statisticsThe Catalogue, available at the Centre de Données Stellaires deStrasbourg, consists of 13 573 records concerning the results obtainedfrom different methods for 7778 stars, reported in the literature. Thefollowing data are listed for each star: identifications, apparentmagnitude, spectral type, apparent diameter in arcsec, absolute radiusin solar units, method of determination, reference, remarks. Commentsand statistics obtained from CADARS are given. The Catalogue isavailable in electronic form at the CDS via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcar?J/A+A/367/521
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Quiz 5 / Re: LEC5101 quiz#5 FiveC
« on: November 06, 2020, 01:57:49 AM »
Was $y_2(t)$ given? I had same question but only $y_1(t)$ was given.

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Dynamic Programming

The term dynamic programming has a curious origin. When Richard Bellman late in the 1940s were seeking for a viral definition of his method, his boss was apparently not very inclined on science and in particular on mathematical research, the activities that Bellman was deep into when he formulate his famous equation. Likewise in marketing campaigns, where names and terms are carefully selected for attention teasing, Bellman coined the definition that combines the multi-staging and time-varying of ‘dynamic’ with the optimization archetyping of ‘programming’, coincidentally induced by the fame of Dantzig’s linear programming for mathematical optimization.

DP describes problems that involve dynamic processes for finding the best decision one after another. The backbone idea rolls around the splitting of a problem till its atomic parts are identified, then trimming those parts with a objective function. Basically, the core concept of recursion. Considering for example the Fibonacci algorithm:

The node fib(2) appear twice in the tree, therefore Fibonacci unveils an overlapping structure. It is eligible to be an DP algorithm. Not all recursive algorithms are inherently overlapping, look at binary tree search, merge and quick sort, for example. They don’t manifest the property of traversing smaller chunk of data more then once, hence they cannot join Dynamic Programming’s family. When the sub-problems are repeated in the problem as whole, and their values are evaluated all over again unless a caching mechanism (memoization) could be displaced for preventing such inefficiency.

Bellman also conceived the principle of optimality according to which an optimal policy should always hands out the optimal decision from any state or action previously done. However the chessboard pieces’ are laid, or however pedestrian are crossing the road or waiting on the platform, the agent will move pieces properly in case of chess and it will skirt traversing people in case of self-driving car. The automatic actor will always follow the best strategy from the first step and thereafter. This is for introducing another property of DP methods, which is the optimal structure.

When a courier has to deliver a packet from $$A$$ to the destination $$F$$, he will catch the shortest way highlighted in blue. The optimal decision for completing the journey $$A \rightarrow F$$, in the shortest path problem, includes necessarily the optimal solution $$C \rightarrow F$$. As you may have noticed, The recursion sweats also out of this property where nested sub-problems are always optimal on their way to the final target. The shortest path problem present the optimal sub-structure, which claims that an optimal solution of a problem includes necessarily the optimal solutions of its sub-problems. Not all recursive problems are optimal even in their encapsulated structure, and the counterfactual is given by slightly different dilemma, the longest path problem. The longest way $$A \rightarrow F$$ includes the node $$C$$, but if we choose to start from $$C$$, the path is not included in the way starting from $$A$$.

Dynamic programming in reinforcement learning

The first time DP and RL were mentioned together was by Minsky in the 1961 and it took form of the Bellman equation, because RL problems have expose usually an overlapping and optimal structure, ideal for being solved by DP.

Markov decision process

Reinforcement learning is a class of methods for determining the optimal policy an agent should apply for maximize its return in a given environment. The entities with a role in RL are the state $$S$$, the mutable conditions an agent experience in an environment, the action $$A$$ the agent execute in a particular state, and the reward $$R$$, if any, which score the goodness of action taken in a state. These are the pillars of the markov decision process, the framework that formalizes the policy $$π$$ as a sequence of steps $$(S_t, A_t, R_t)$$, aggregated in episodes.

The policy is a mapping from states and related actions, RL tells us how the agent’s policy changes as result of the experience. For finding good policies, we need to estimate how good it is, in terms of future rewards, to be in a particular state. Value functions $$v_\pi(s)$$ define the expected return when starting from a given state $$s$$ and following $$\pi$$ thereafter. their fundamental property is that they satisfy recursive relationships similar to what we already have seen for dynamic programming. Hence, the idea of DP in RL is the use of value functions to organize the search for good policies.

The Gridworld example

Problem: Find all trajectories towards the flags from any cell

Among many examples I may show for explaining what does DP stands for, this very simple GridWorld conveys the idea of an agent displaced in an 2D environment. The reward drives the agent to move toward the flagged corners regardless the position where the agent is located. How the agent could learn the optimal ways? I give you an hint, let’s start from the end, the flagged boxes. In that position you don’t have anymore rewards to gather, the task is brilliantly completed, so we assume the terminal’s state value is is equal to zero.

Iteration 1. Terminal state value is zero

From end position, step back and look around. Among the actions you can do, the move that catapult you into the terminal state is more rewarding than others (zero instead of -1). Congratulations! You already have solved a piece of the puzzle. It is just the last action, you still have to solve the rest of the grid. In that position, as in any other cell of the grid, you need to assign a value to the cell where you are in that particular moment, in order to pave the way to the final goal. That value $$Q$$ indicates how good it is to be there. Not all cells carry the same value, some are more valuable than others. The agent, when choosing the next move, will move to the cell with highest value, among the surroundings. The value would not be just the reward you get in there (-1), but the reward plus the average of the values of the cells proximate to you. $$Q_t = r_t + \sum_{n=t+1}{Q_n}$$

Iteration 2: set the terminal state value to _zero_

Going backward till the initial position, you see all rewards on the way to the terminal state waiting for you to being picked up. This is the value of your state. You are in the middle of the trajectory towards the end, and the value of your state is equal of the reward in that state plus the sum of the discounted rewards thereafter. Clearly, the agent moves toward the most promising among the surrounding cells, once it has realized their value. Let’s keep going. The grid starts to unveil the optimal trajectories by just going backward and evaluating what could be the best move, taking the cell with highest value. A pattern is identifiable, something that programmers knows well, the recursion, and simplified as:

$Q_t=r_t+argmax(Q_{t+1})$

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### Biscuit Decorations

Andrew decorated 20 biscuits to take to a party. He lined them up and put icing on every second biscuit and different decorations on other biscuits. How many biscuits weren't decorated?

### Constant Counting

You can make a calculator count for you by any number you choose. You can count by ones to reach 24. You can count by twos to reach 24. What else can you count by to reach 24?

### Skip Counting

Find the squares that Froggie skips onto to get to the pumpkin patch. She starts on 3 and finishes on 30, but she lands only on a square that has a number 3 more than the square she skips from.

# Lots of Biscuits!

##### Stage: 1 Challenge Level:

Finlay from Stalyhill Junior School answered the final part of this problem, which asked how many more biscuits the children would need to bake so that they could share them equally with Miss King too. Finlay said:

They would need to make $4$ more biscuits. They would then get $4$ biscuits each as $24$ divided by $6$ is $4$.

Well done, Finlay.  Emerson and Hamish from St Peter's added a bit more detail:

If you add all the biscuits up you get $20$. There are six people so you need to find the multiple of $6$ that is closest to $20$ but that is also above it. The answer is therefore $24$.  $24-20=4$ Therefore they need to bake $4$ more biscuits.

Ed from St Peter's sent  in answers to all parts of the problem.  He said:

Ali and Danny baked $12$ biscuits. If the biscuits were shared between Ali and Danny, they would receive $6$ each.
Will, Jess and Karni baked $8$ biscuits. If the biscuits were shared between Will, Jess and Karni they would receive $2 \frac{2}{3}$ each.
There were $20$ biscuits baked all together. If they shared all the biscuits between the $5$ children , each child would receive $4$ bisuits.
Yes, they can share all the biscuits between all the children and Miss King. Each person would receive $3\frac{1}{3}$ biscuits. The children would have to bake $4$ more biscuits to make $24$ in total and therefore each person receiving $4$ biscuits each.

Thank you, Ed.  It's a shame you didn't tell us how you worked out this solution.


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### Sidebar

Reference

analysis:amplipex:artifacts

## Artifact Removal

A nice artifact detection and removal walkthrough can be found at FieldTrip Artifact Documentation

### Visual Artifact Removal

This section addresses visual artifact removal using the FieldTrip toolbox FieldTrip Visual Artifact walkthrough

Load the data and define the trials as per Loading Amplipex data into fieldtrip. Once this has been done we can begin by setting up our cfg (configuration) structure and using the ft_databrowser function

cfg = data_trl.cfg;
cfg.channel     = 'all';
cfg.trials      = 'all';
cfg.latency     = 'maxperlength';
cfg.method      = 'Summary';
cfg.keepchannel = 'yes';
cfg.metric      = 'var';
cfg.alim        = [];
cfg.coninuous   = 'yes';
cfg.viewmode    = 'vertical'; % this shows all of the channels vertically
cfg = ft_databrowser(cfg,data_trl);
data_trl.cfg = cfg; % rewrite the cfg back into the data set

The browser will let you scroll through the trials and use the cursor to highlight a section of the data containing an artifact.

The selected artifact regions are not altered within the data itself, instead the start and end of each artifact are saved as part of the cfg structure: data_trl.cfg.artfctdef.visual.artifact . So now can change these regions to any value of our choice.

%% Mannual Artifact removal
art_value = 0; %Value to replace the data within artifact regions
artifacts = data_trl.cfg.artfctdef.visual.artifact;
trials = data_trl.cfg.trl;
for iArt = 1:length(artifacts)
for iTrial = 1:length(trials)
if trials(iTrial,1)<= artifacts(iArt,1) && trials(iTrial,2)>= artifacts(iArt,2)
if artifacts(iArt,1)-trials(iTrial,1) ==0
data_trl.trial{1,iTrial}(1:16,(1:artifacts(iArt,2)-trials(iTrial,1)))=art_value;
else
data_trl.trial{1,iTrial}(1:16,(artifacts(iArt,1)-trials(iTrial,1):artifacts(iArt,2)-trials(iTrial,1)))=art_value;
end
end
end
end

Now we can visualize the data regions that have been replaced to ensure they have been correctly converted.

cfg.viewmode    = 'vertical'; % this shows all of the channels vertically
ft_databrowser(cfg,data_trl);

From here we can run any further ft analysis.


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# Pricing of Swaption by Proxy and Monte Carlo

here's the problem. Suppose you want to compute the price of a Call option on a Swap contract. Let $T$ and $T+S$ the times (in year fraction) where the Swap lives and suppose that the fluxes of the swap are every $6$ months in the times: $$T=T_0< T_1< \dots < T_N = T+S$$ Let $S_{fwd}$ the swap forward rate: $$S_{fwd} = S(t; T, T+S) = \frac{B(t, T) - B(t, T+S)}{\sum_{i=1}^M (T_i - T_{i-1})B(t, T_i)} = 2\frac{B(t, T) - B(t, T+S)}{\sum_{i=1}^M B(t, T_i)}$$ while the spot Swap rate is: $$S_t(s) = \frac{1-B(t, t+s)}{\sum_{i=1}^M (T_i - T_{i-1})B(t, T_i)}$$ where $B(0, T)$ is the usual risk-free discount. The above formula is just the rate which let the swap be fair. ($S_t(s)$ is just $S_{fwd}$ with $t=T$)

Our Call option will use this $S_{fwd}$ as strike and it will exercise $2$ times a year so that the payoff will be:

$$\sum_i^{2M}\mathbb{E}^{T_i} \left[ \frac{1}{2}B(T, T_i)(S_T(s) - S_{fwd})^+ | \mathcal{F}_T\right] = A(T)(S_T(s) - S{fwd})^+$$

Now we can compute the expected value. In order to have a simpler term we can (I have to...for the exercise spirit this passage is mandatory) change of measure:

$$B(0, T)\mathbb{E}^T \left[ A(T)(S_T(s) - S{fwd})^+ \right] = A(0) \mathbb{E}^A \left[(S_T(s) - S_{fwd} )^+ \right].$$

Finally I come to the main question: At this point the exercise ask me to do some approximations and to estimate the errors. One of this approximations is to compute the integral in the forward measure $T$ (the measure we are using before the change of measure) even if we've changed of numeraire. It seems like a good idea since all the trajectories are available in that measure and, in this way, no Girsanov or Brownian motion corrections is needed.

So that we have the following approximation: $$A(0) \mathbb{E}^A \left[(S_T(s) - S_{fwd} )^+ \right] = A(0) \mathbb{E}^T \left[(S_T(s) - S_{fwd} )^+ \right] + \epsilon$$

The problem is that I don't understand how to compute the error $\epsilon$ generated by a change of measure: in fact it is a "theoretical" technique and I cannot see how to "write down" the error (the estimate can be done both in the analytic way and in a numerical way: Montecarlo, finite differences...).

Thank you in advice. I know it could seem like confusing...and in fact it is form me a little bit.

Ciao! AM

• When you use the Girsanov theorem, what are the parameters you use to adjust the drift? Is there an error on these? – will Oct 30 '17 at 17:17
• I'm not using Girsanov. After the change of measure I just use the forward measure again. Of course is not correct but I would like to understand how far I am from the real value if I "ignore" the change of measure. – clarkmaio Oct 30 '17 at 17:19
• Ignoring the change of measure is the same as saying the correlation is zero... – will Oct 30 '17 at 17:20
• Could you explain this? I'm not following – clarkmaio Oct 30 '17 at 17:22
• Have a look in this question, where the effective forward, $F' = F \exp (\hat{c}) = F \exp \left(\hat{ \sigma}_{DOM} \cdot \rho \cdot \hat{\sigma}_{FOR} \cdot T\right)$. This is a change of measure from the domestic to foreign measure, and is very similar to what you're asking. If the correlation is zero though, there is no adjustment. – will Oct 30 '17 at 17:25


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# Sum of Roots

Algebra Level 3

$\large 2x^{2} - 4x + 1 = 0$

If $$\alpha$$ and $$\beta$$ are the roots of the equation above, then what is the value of $$\frac{1}{ \alpha + 2\beta } + \frac{1}{ \beta + 2\alpha }$$?

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# Multiplying Vectors II: The Vector Product

Last time, we looked at the scalar, or dot, product of vectors, focusing on proving the equivalence of two ways to define it. This time, we’ll look at the vector, or cross, product in the same way. The distinction between dot and cross product reflects the symbol used, u · v vs. u × v, which is a creative use of the two different symbols that evolved for multiplication of numbers. The real distinction, though, is between the product being a scalar (number) and a vector.

## What is the cross product?

As we saw last time, the dot product is hard to explain in simple geometric terms, so that it seems unlikely that anyone would invent it in that form. Rather, it is a simple way to combine two vectors algebraically, that has properties suitable for multiplication, and that happens to incorporate the angle between the two vectors, specifically its cosine, making it very useful. The same is true of the cross product, which is complicated to describe geometrically, but is (in some sense) very neat algebraically, and turns out to involve the sine of the angle between them.

I’ll start, as I did last time, with a question about the essential meaning of the operation; this time I have to use a question from 2016 that was not published in the Ask Dr. Math archive. It starts with the geometric description, with a mention of the algebraic form using determinants, which we’ll talk more about below.

What does the cross product of two vectors actually represent?

Ex. a X b = |a||b|sin(angle between the vectors).u, where u is a unit vector in the direction of a X b

My Physics professor once stated that there is a physical definition of cross product of vectors. Though he didn't elaborate, I am still thinking what it might be.

I tried breaking the vectors and using the determinant technique to evaluate the result. I carefully examined the determinant and the definition to understand. Upon finding that the determinant represents a parallelogram's area, I concluded that it might represent vector area of the parallelogram. However, I still don't understand what the physical meaning is.

Here is an example showing a cross product:

Vectors u and v form a parallelogram with diagonal u + v; the cross product u×v is perpendicular to that plane, and its magnitude is $$|\mathbf{u}| |\mathbf{v}|\sin(\theta)$$, which is the area of that parallelogram.

I replied, first giving references to the three pages we’ll be looking at below and next time, then continuing:

These give successively more detailed answers to what the cross product is. I think your description as "vector area of a parallelogram" (or "directed area") is good. I don't know just what your professor might have had in mind as a "physical definition"; it is *defined* mathematically, not by physics, but it can be *applied* in physics in various ways that justify its use. It appears in several rather distinct fields of physics, so no one of those can be called the meaning of the cross product in general.

Avinash’s term “vector area”, which I suggested calling a “directed area”, refers to the fact that the magnitude of the cross product is the area of the parallelogram defined by the vectors, while its direction is perpendicular to the plane of the two vectors.

For a particularly simple example, consider the cross product of the unit vectors i and j, which is a vector perpendicular to both, with magnitude 1, the area of a 1 by 1 square:

Avinash replied,

When replying to my question, you mentioned "directed area". I am a bit confused, as to how area being a scalar can be called a vector.

I know, I talked about vector area and so you might assume that I know about it.But the truth is that I had to accept the fact as is.

I hope that you can give me a convincing explanation of vector area.

Well, that's why we don't just call it area! Area itself is a scalar; but we can associate it with a direction, and it happens that the cross product naturally combines them.

I think you made up the term "vector area"; when we talk about that or "directed area", we simply mean attaching a direction to an area. That is mentioned here:

https://en.wikipedia.org/wiki/Cross_product

In mathematics and vector calculus, the *cross product*
or *vector product* (occasionally *directed area product*
to emphasize the geometric significance) is a binary
operation on two vectors in three-dimensional space (R^3)
and is denoted by the symbol x.

This directed area is just the area times a unit vector perpendicular to the plane (which defines the direction of the plane).

The term “vector area” is in fact used, though not usually in this context.

We can also talk about "signed area", which is closely related; this commonly arises in connection with a formula for the area of the region enclosed by a polygon, which gives a positive area if you go around counterclockwise, but a negative area in the other direction. This formula is closely related to the cross product. Here are two discussions of this:

http://mathforum.org/library/drmath/view/60583.html

Geometric Proof of Area of Triangle Formula
http://mathforum.org/library/drmath/view/72141.html

This last link mentions the cross product, and provides an introductory proof of what we’ll be seeing later; I previously discussed it in Polygon Coordinates and Areas.

Avinash pursued the question:

Thank you for your response.You helped me a lot, but could you please explain why do we associate direction with an area? It is kind of hard to come up with an idea of combining area with direction.
I mean, how could someone think of using directions in area, a scalar quantity? Is it because of applications in finding flux or something alike?

This is the core of the question: Why do all this at all?

I said,

The first reason is probably that when we look for a way to make a vector perpendicular to the plane of two given vectors, it just happens that the magnitude of the cross product, which does that job, is equal to the area. So we accept that as a gift! It's not that we were seeking that goal; it was just handed to us as a nice result.

It also turns out the this idea is useful, as you mention, for concepts like flux. Someone might have thought of that first; I don't know. But however it came about, it's a good idea. Everything works together very neatly.

As we pursue a proof below, we’ll see that what I said is exactly right. Avinash concluded:

Thank you, thank you, thank you, thanks a million for answering all my questions without grumbling. If I had asked all these questions to my math professor, he would have regarded it as 'stupid' and 'irrelevant'. Glad that the members of Dr.Math are always ready to help.

To which I replied,

You're welcome.

I actually like the odd questions that don't really have answers, but that make you look deeper into things! I don't get those enough in my classes.

Now let’s look at the first of the archived answers I referred Avinash to.

## How (and why) to invent two products

This question is from 2012:

Crossed up by Scalars and Vectors

I have never really understood why dot products and cross products have been defined the way they are. Why is the result of a dot product a scalar, and that of a vector product a vector?

I have consulted many books, and I have searched the net, but nobody seems to give any plausible explanation for this.

Doctor Jerry replied, starting with the dot product, much as we saw last time:

Hello Karthik,

Thanks for writing to Dr. Math.

Given the idea of a vector, it is natural to look for a way to calculate the angle t between vectors (a1, a2, a3) and (b1, b2, b3). Letting ||a|| be the length of the vector a, if you apply the Law of Cosines to the triangle with sides a, b, and a - b, you will see

(a1 - b1)^2 + (a2 - b2)^2 + (a3 - b3)^2 =
a1^2 + a2^2 + a3^2 + b1^2 + b2^2 + b3^2 - 2||a||*||b||* cos[t]

Simplify and you will find this, where "x y" means "x times y":

a1 b1 + a2 b2 + a3 b3 = ||a||*||b||*cos[t]

This makes it clear that the left side is a useful combination of a and b.

This is about as concise an answer as one could give. We can imagine starting with the Law of Cosines in the hope of finding the angle. Having found this simple formula, we just define the left-hand side as $$\mathbf{a}\cdot\mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$$ and we have proved that $$\mathbf{a}\cdot\mathbf{b} = ||\mathbf{a}|| ||\mathbf{b}||\cos(\theta)$$

(By the way, notice that for ease of typing, we have often used $$|\mathbf{u}|$$ for magnitude, i.e. length, but the proper notation is $$||\mathbf{u}||$$.)

Now we can try something similar to invent the cross product:

For the cross product, if you have vectors a = (a1, a2, a3) and b = (b1, b2, b3), it seems clear that a vector that is perpendicular to both of a and b would be useful. So, we seek a vector (c1, c2, c3) such that

a.c = 0
b.c = 0

If you solve this system of two equations for c1 and c2, you will find

c1 = (a3 b2 - a2 b3)/(a2 b1 - a1 b2) * c3
c2 = (-a3 b1 + a1 b3)/(a2 b1 - a1 b2) * c3

If you choose c3 as a2 b1 - a1 b2, this will give a solution and clean up the mess a bit.

Having defined the dot product, we use it to find whether two vectors are perpendicular. The resulting system of equations looks like this: $$\left\{\begin{matrix}a_1c_1+a_2c_2+a_3c_3=0\\ b_1c_1+b_2c_2+b_3c_3=0\end{matrix}\right.$$ There are only two equations for three variables, so the solution is not unique; as it turns out, it only determines the proportions (i.e., the direction of the resulting vector).

He treats $$c_3$$ as a constant in order to solve (a quick way would be to use Cramer’s Rule), observing that the solution really expresses $$c_1$$ and $$c_2$$ as multiples of $$c_3$$, so that taking $$c_3$$ as the denominator of both fractions gives a convenient result: $$\mathbf{a}\times\mathbf{b} = \mathbf{c} = (a_3b_2-a_2b_3, a_1b_3-a_3b_1, a_2b_1-a_1b_2)$$ This has a nice pattern to it, which can be restated in various ways, including using determinants, which are just a shorthand way to write exactly these kinds of expressions.

Notice that this approach, as I suggested earlier, focuses on making a perpendicular vector; we don’t even know yet the significance of its magnitude. We’ll get to that in a moment.

Note, again, that the original goal was not to find a scalar product and a vector product! It was to find an angle, which led to a scalar, and a perpendicular vector. When these were discovered, it was natural to name them the scalar product and the vector product, and eventually to use the dot and cross multiplication symbols to distinguish them. The actual development of these ideas started from a different perspective, as vectors themselves were only gradually being developed; the dot and cross representations came around 1880, after 40 years or so of development.

## Can we prove that |u×v| = |u||v|sin θ ?

We’ve derived the definition of the cross product as a vector perpendicular to the given vectors, but its magnitude was just whatever arose from the easiest choice. Can we prove that the magnitude is what we say it is, the area of the parallelogram? I don’t find that in any published answers, so I’ll use two that were not archived. First, consider this one from 2003:

How can one prove that for 2 vectors a and b that the vector product (a cross b)= mod(a).mod(b).sin(theta).n

Where does the sine come in to the equation - is it through the determinant definition of the cross product?

Hi Peter,

I assume that n is a unit vector in the direction of (a x b), so your question is essentially why

|a x b| = |a||b|sin(θ) ,

where θ is the angle between a and b.  Here, |a| is the modulus of a vector a.

This quantity is also the area of the parallelogram with sides a and b.

- - - - -
a /:
/ :h      /
/  :
/θ  :     /
----------
b

The altitude h is |a|sin(θ), so the area is |a||b|sin(θ).

The fact that the magnitude is the area is not part of the proof, but a useful fact. Now, where can we can get a sine from? You might think of the Law of Sines, as we used the Law of Cosines for the dot product, but I don’t think anything would come of that. But since we already know that the dot product is related to the cosine, we might try to use that:

sin^2(θ) = 1 - cos^2(θ)

[   a.b  ]2
= 1 - [ ------ ]
[ |a||b| ]

[|a|^2|b|^2 - (a.b)^2 ]
= -----------------------
|a|^2 |b|^2

so we want

[|a||b|sin(θ)]^2 = |a|^2|b|^2 - (a.b)^2 .

Just expand the right side, simplify, and you will have |a x b|^2 .

We now have to show that the right side agrees with our algebraic definition of the cross product. Doctor Fenton left that to Peter; let’s give it a try:

$$|a|^2|b|^2 – (a\cdot b)^2 \\= (a_1^2+a_2^2+a_3^2)(b_1^2+b_2^2+b_3^2) – (a_1b_1+a_2b_2+a_3b_3)^2\\ =(a_1^2b_1^2+a_1^2b_2^2+a_1^2b_3^2+a_2^2b_1^2+a_2^2b_2^2+a_2^2b_3^2+a_3^2b_1^2+a_3^2b_2^2+a_3^2b_3^2) \\- (a_1^2b_1^2+a_2^2b_2^2+a_3^2b_3^2+2a_1a_2b_1b_2+2a_2a_3b_2b_3+2a_1a_3b_1b_3)\\ =a_1^2b_1^2+a_1^2b_2^2+a_1^2b_3^2+a_2^2b_1^2+a_2^2b_2^2+a_2^2b_3^2+a_3^2b_1^2+a_3^2b_2^2+a_3^2b_3^2\\- a_1^2b_1^2-a_2^2b_2^2-a_3^2b_3^2-2a_1a_2b_1b_2-2a_2a_3b_2b_3-2a_1a_3b_1b_3\\ =a_1^2b_2^2+a_1^2b_3^2+a_2^2b_1^2+a_2^2b_3^2+a_3^2b_1^2+a_3^2b_2^2-2a_1a_2b_1b_2-2a_2a_3b_2b_3-2a_1a_3b_1b_3$$

But what is $$|a \times b|^2$$?

$$|a \times b|^2 \\= |(a_3b_2-a_2b_3, a_1b_3-a_3b_1, a_2b_1-a_1b_2)|^2 \\= (a_3b_2-a_2b_3)^2+(a_1b_3-a_3b_1)^2+(a_2b_1-a_1b_2)^2 \\= (a_3^2b_2^2-2a_2a_3b_2b_3+a_2^2b_3^2)+(a_1^2b_3^2-2a_1a_3b_1b_3+a_3^2b_1^2)+(a_2^2b_1^2-2a_1a_2b_1b_2+a_1^2b_2^2) \\= a_1^2b_2^2+a_1^2b_3^2+a_2^2b_1^2+a_2^2b_3^2+a_3^2b_1^2+a_3^2b_2^2-2a_1a_2b_1b_2-2a_2a_3b_2b_3-2a_1a_3b_1b_3$$

They are equal!

## Using vector algebra

The hard part there was the algebra at the end. With more expansive understanding of the algebra of vectors (provable from the purely algebraic definition we are using), it is possible to make the work look a little neater, at least. Here is an unarchived question from 2006; I’m going to abridge this discussion heavily, removing a number of wrong paths along the way:

Prove that ||A x B|| = ||A||.||B||.sin θ using the properties of cross product.

Properties are:

1. A x B = -B x A
2. A x (B + C) = A x B + A x C
3. A x A = 0
4. r(A x B) = r(A) x B = A x (rB)
5. A x (B x C) = (C.A)B - (A.B)C
6. A.(B x C) = (A x B).C
7. B.(B x C) = (B x C)C = 0

I have no idea where to start! I have been given the hint of "Use the properties, especially 5, 6, and 7", but to me the properties seem useless because the question I'm trying to prove doesn't seem to be related to the properties.  I'm getting really, really confused by the fact that I'm trying to do something with magnitude, but I'm supposed to be manipulating properties that don't involve the magnitude symbols.

Hi Lucy,

||v||^2 = v.v    for any vector v.

Applying that to this problem gives you a  dot product of two vectors which are themselves cross-products.  Property 6 will let you rewrite that, and then you need to use property 5 to further rewrite the result.  You will also need to use the property of dot products that

A.B = ||A|| ||B|| cos(θ)

where θ is the angle between A and B.

Lucy responded (omitting some side trails):

Hi Dr. Fenton,

I've looked at your suggestions and I've gotten a lot farther, but I'm still not quite there yet.  So far I have:

Since ||v||^2 = v.v, we can square both sides and get

(||A x B||)^2 = (A x B).(A x B)

Hi Lucy,

For convenience, I rewrite the properties:

Properties are:

1. A x B = -B x A
2. A x (B + C) = A x B + A x C
3. A x A = 0
4. r(A x B) = r(A) x B = A x (rB)
5. A x (B x C) = (C.A)B - (A.B)C
6. A.(B x C) = (A x B).C
7. B.(B x C) = (B x C)C = 0

First, apply property 6.  It's somewhat confusing to have A,B, and C represent both specific vectors (in your problem) and generic vectors (in the formulas).  Let me rewrite property 6 in terms of generic vectors U, V, and W:

U.(V x W) = (U x V).W .

Now apply it to (A x B).(A x B), with U = A x B, V = A, and W = B:

(A x B).(A x B) = ((A x B) x A).B .

Next, you want to apply property 5 to simplify the triple vector product on the right side, but that requires the first cross-product to be in the second factor, so you must first apply property 1, and then you can apply property 5, to get

-((A.B)A - (A.A)B).B       .

Now, you have a vector dotted with the difference of two vectors.  The dot products A.B and A.A are just constants (scalars): it's like

-(sA - tB).B  .

Compute that dot product, using the distributive property, before applying the dot product identity.  Then, put back in the values of s and t.

Lucy did that:

After applying the distributive property, I get

-((A.B)A).B - ((A.A)B).B)

Doctor Fenton continued:

This expression is equal to

-(A.B)(A.B) + (A.A)(B.B) = ||A||^2 ||B||^2 - (A.B)^2

Now apply the identity

A.B = ||A|| ||B|| cos(θ) .

Lucy was able to finish:

I got it!! Now I have

||A||^2 ||B||^2 - (||A|| ||B|| cos(θ))^2
||A||^2 ||B||^2 - ||A||^2 ||B||^2 cos^2(θ)
||A||^2 ||B||^2 (1 - cos^2(θ))
||A||^2 ||B||^2 (sin^2(θ))
||A|| ||B|| sin(θ)

Which is the end of the proof! Thank you so much for helping me, I really appreciate this wonderful favour you have done for me and the time you took to type out your thorough answers.

Putting all this together, here is the work, which parallels what we did with components before:

$$(||A \times B||)^2 = (A \times B)\cdot(A \times B) \\= ((A \times B) \times A)\cdot B \\= -((A\cdot B)A – (A\cdot A)B)\cdot B \\= -((A\cdot B)A)\cdot B – ((A\cdot A)B)\cdot B) \\= -(A\cdot B)(A\cdot B) + (A\cdot A)(B\cdot B) \\= ||A||^2 ||B||^2 – (A\cdot B)^2 \\= ||A||^2 ||B||^2 – (||A|| ||B|| cos(\theta))^2 \\= ||A||^2 ||B||^2 – ||A||^2 ||B||^2 \cos^2(\theta) \\= ||A||^2 ||B||^2 (1 – \cos^2(\theta)) \\= ||A||^2 ||B||^2 (\sin^2(\theta))$$

$$||A \times B|| = ||A|| ||B|| \sin (\theta)$$

For a very different perspective, you may want to read this, from 2001, which was the second link I gave to Avinash:

Cross Products; Rotating in Three Dimensions

Next time we’ll look at the third link, a long discussion of dot and cross products and a lot more.

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text:
## Hokkaido Mathematical Journal

### Maximization of quadratic forms expressed by distance matrices

#### Abstract

If $x,y,z$ are real numbers satisfying $x+y+z=1$, then the maximum of the quadratic form $axy+bxz+cyz$ with positive constants $a,b,c$ is $$\displaystyle{\frac{abc}{2ab+2ac+2bc-a^2-b^2-c^2}}$$ under the assumption $\sqrt{a}<\sqrt{b}+\sqrt{c}$. Extending this fact, we give the maximum of the quadratic form $\displaystyle\sum_{1 \le i< j \le n} a_{ij}x_i x_j$ in $n$-variables $x_1,\ldots,x_n$ satisfying $\displaystyle\sum_{i=1}^{n} x_i = 1$ with constants $a_{ij} \ge 0$ under certain assumptions.

#### Article information

Source
Hokkaido Math. J., Volume 35, Number 3 (2006), 641-658.

Dates
First available in Project Euclid: 29 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1285766422

Digital Object Identifier
doi:10.14492/hokmj/1285766422

Mathematical Reviews number (MathSciNet)
MR2275987

Zentralblatt MATH identifier
1124.15022

Subjects
Primary: 15A63: Quadratic and bilinear forms, inner products [See mainly 11Exx]
Secondary: 15A48

#### Citation

IZUMINO, Saichi; NAKAMURA, Noboru. Maximization of quadratic forms expressed by distance matrices. Hokkaido Math. J. 35 (2006), no. 3, 641--658. doi:10.14492/hokmj/1285766422. https://projecteuclid.org/euclid.hokmj/1285766422


text:
# Updating The Build Plan

If you want to update the build plan for yourself. You can use these commands.

To update the nixpkgs base:

nix-prefetch-git https://github.com/NixOS/nixpkgs | tee nix/nixpkgs-src.json

To update the haskell.nix package set

nix-prefetch-git https://github.com/input-output-hk/haskell.nix | tee nix/haskell-nix-src.json

If you make modifications to the package.yaml or to the stack.yaml, you will need to run stack-to-nix using the haskell.nix tools.

stack-to-nix -o nix/ --stack-yaml stack.yaml

Remember, any modifications you make to the build plan will bring you out of lockstep with Shakebook’s official cache, at which point you may want to set up your own.


text:
# Texas Go Math Grade 7 Module 2 Answer Key Rates and Proportionality

Refer to our Texas Go Math Grade 7 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 7 Module 2 Answer Key Rates and Proportionality.

## Texas Go Math Grade 7 Module 2 Answer Key Rates and Proportionality

Texas Go Math Grade 7 Module 2 Are You Ready? Answer Key

Divide.

Question 1.
$$\frac{3}{4}$$ ÷ $$\frac{4}{5}$$ _____________
Answer:
Multiply by the reciprocal of the divisor:
= $$\frac{3}{4}$$ × $$\frac{5}{4}$$
= $$\frac{15}{16}$$

Question 2.
$$\frac{5}{9}$$ ÷ $$\frac{10}{11}$$ _____________
Answer:
Multiply by the reciprocal of the divisor:
= $$\frac{5}{9}$$ × $$\frac{11}{10}$$
= $$\frac{11}{18}$$

Question 3.
$$\frac{3}{8}$$ ÷ $$\frac{1}{2}$$ _____________
Answer:
Multiply by the reciprocal of the divisor:
= $$\frac{3}{8}$$ × $$\frac{2}{1}$$
= $$\frac{3}{4}$$

Question 4.
$$\frac{16}{21}$$ ÷ $$\frac{8}{9}$$ _____________
Answer:
Multiply by the reciprocal of the divisor:
= $$\frac{16}{21}$$ × $$\frac{9}{8}$$
= $$\frac{6}{7}$$

Write the ordered pair for each point.

Question 5.
B ____________
Answer:
B(-4, 1)

Question 6.
C ____________
Answer:
C(3, 0)

Question 7.
D _____________
Answer:
D(5, 4)

Question 8.
E _____________
Answer:
E(-2, -2)

Question 9.
F _____________
Answer:
F(0, 0)

Question 10.
G _____________
Answer:
G(-4, 0)

Texas Go Math Grade 7 Module 2 Reading Start – Up Answer Key

Visualize Vocabulary

Use the ✓ words to complete the graphic. You can put more than one word in each bubble.

Understand Vocabulary

Match the term on the left to the definition on the right.

Answer:
1. rate of change ………. B. A rate that describes how one quantity changes in relation to another quantity.
2. proportion ………… A. Statement that two rates or ratios are equivalent
3. proportion ……….. C. Rate in which the second quantity is one unit

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