text:
Talk:Nonblocking minimal spanning switch

Early comments
See Clos network (preceding unsigned comment by 06:09, 19 May 2004)

If any of this makes sense to anybody, please try to explain what is going on. My first question would be: what is an nxm multiplexer? AxelBoldt 21:06 Feb 27, 2003 (UTC)

Surely this should be...
Surely this should be titled "Space Division Switching" or something along those lines!

Cleaning up
I've added the reference (paper by Clos) to the article. (Found without too much difficulty using Google Scholar.)

Finding the reference also allowed additional research into just what this article might be about (since the current article content is not very helpful in that respect). Much of the article is really background for the title subject of the article, not the subject itself. Even so, it does not have enough background (or links to same) to lay an adequate basis for the title subject. For instance, it appears that there should be a description of Clos networks, and definitions given for "strictly nonblocking" and "rearrangeably nonblocking" networks. For useful background in this area see Hunter, Switching Systems and Jajszczyk, 50 Years of Clos Networks, which GScholar turned up.

The actual subject of the article is finally reached in the current so-called "Methods" section, but unfortunately that section is so vague and incomplete as to make, I think, improvement all but impossible for anyone but a subject-area expert (who might be able to find the underlying thought behind the present words, or at least on his own find something useful to say). But that Methods section is central to the article even while being confusingly vague and next to useless. Maybe this article should be tagged with "expert help needed".

-R. S. Shaw 04:11, 9 October 2005 (UTC)

As a semi-expert, I'd appreciate some input on what people would like to know about this subject. When I last worked this article over, I mostly changed the individual sentences into English. There are many Clos network concepts without Wikipedia articles. The distinction between strictly nonblocking (where arbitrary new calls are switchable) and rearrangeably nonblocking (where some new calls require reassignment of nodes currently in use) are much more obscure topics than this algorithm. I'd appreciate it if you changed the section labels to what you think the sections should be. --RSaunders 20:45, 18 October 2005 (UTC)

 * Okay, but I just stumbled on this article because it was in the cleanup category, and looked it over since it seemed the subject could be interesting. Who knows what the audience in general would want. Anyway, some comments:

 * Thanks for doing some translation into English. I did notice the current article was much improved from the original. I do realize we can't write all the Clos-net-related articles one might want before we fix up this article. We don't need to go into "rearrangeably nonblocking" -- it just occurred to me we were rearranging connections, which meant something to me after I had learned a bit about Clos nets.  I changed the "methods" section name to "algorithm", which seemed better to me.
 * Practical implementations of switches
 * I find this section interesting, and should be somewhere in Wikipedia, but maybe not in this article. (It maybe should be a core for a Telephone switch article; that article name now is just a redirect to Telephone exchange.  The latter article seems like it should be left less technical, since its subject is broader, and so not get this stuff.)
 * For this NonblockingMSS article, the problem with this section is that it is too detailed and confusing as background to the article subject. In particular, how the time-division multiplexing works and is equivalent to a crossbar is hard to comprehend, and so distracts in a major way from simply setting necessary background.
 * Problem Space
 * Although there is some switch background, a main difficulty when we get to the "Algorithm" section is that the article assumes the reader knows the nature of the realm in which the algorithm is going to be working, but that really hasn't been described. Now that I've  personally studied Clos networks in external sources, I think I have the general idea, but on my first reading I sure didn't.  From my new perspective I think the single earlier sentence "a switch can be created using a series of layers called subswitches" is pretty much the entire "description" of the problem/solution space, and it surely seems inadequate.  There seems to be a lack of words, whether or not there could be a diagram to help.
 * I guess the "layers of switches" is the problem space and needs expansion. Isn't it something like a bunch of subswitches which can be thought of as crossbars, organized somehow? Maybe one "layer" of subswitches is connected to inputs and to another layer, then layer to layer, then to a layer that connects to outputs?  Are there 3 layers like I guess a Clos network has, or arbitrary number, or what?  Is the connection topology between the layers random or structured somehow, or ?
 * The first step in the Nonblocking Minimal Spanning Switch algorithm is to search for a subswitch that contains the needed in and out connections.
 * Confusion point: if there are layers like I just wrote of, then a particular subswitch could not be directly ("directly" is implied and meant, isn't it?) connected to both input and output since connections to those would exist only in completely distinct layers of subswitches.
 * More confusion: why is this mentioned? It seems like saying "the problem is solved if the problem is solved." Isn't this "search" trivial? Wouldn't one start with the originating (in) line, and immediately have the subswitch it connects to, and then see if that subswitch has a direct (I assume it's direct, but the article doesn't say) connection to the destination (out) line? It's not really a "search" to simply determine if the subswitch in hand has the desired output - it's not looking at any other subswitch or part of the network.
 * If a subswitch with the needed pair of connections can't be found, a pair of subswitches and a connection between them will be needed. One subswitch has a needed input connection. The other has a needed output connection.
 * This seems like a somewhat confused statement of the real starting conditions for the algorthm. My guess is that one is given the in and out lines, and trivially from that one has the in and out subswitches (so these are essentially givens for the problem), and the point of the entire algorithm is to find the needed connection between them.  Is that right? Seems like the article should actually state what is given and what is being sought through the action of the algorithm.
 * The pair of subswitches' connections can be reorganized with a topological sort.
 * Uh, OK, it's introducing a new concept (although if I check the referenced topological sorting article, I don't immediately get help, since it says things like a graph can have "many valid topological sorts" which sounds distressingly vague for current purposes.) Well, I'm ready for further expansion and explanation...
 * The connections of both subswitches are placed in a list that also includes the desired new connection.
 * Whew. I wonder exactly what this means. I have a vague idea what placing [some kind (what kind?) of a representation of] a connection into a list might mean, but I'd like some help from the article on this point. Are these list entries including all the intermediate subswitches for each established connection?  In one entry big entry per overall connection or a bunch of entries?    But then the text gets really murky with "includes the desired new connection" - if you have the connection already, wouldn't you be done?  The point the algoritm is to find the/a connection, and we haven't yet, so you can't yet list it!
 * topological sort??
 * And now we've reached the end of the paragraph on the topological sort, and still have close to zero idea what it is or how it forms a part of the NMSS algorithm. The next and final paragraph has very, very unclear relationship to this one - is that paragraph's actions somehow part of the topological sort, or is still preparing more data for the topological sort, or is it done after the topological sort?  Are these two paragraphs even related to one another or are they talking about completely different subjects?
 * Connections are traced through the list by the Nonblocking Minimal Spanning Switch algorithm.
 * Since we don't know the nature of the contents of this list, figuring out how tracing could be done using it is mysterious.
 * Starting from some input or output, ...
 * Why is this so vague? Is it trying to say we're avoiding starting with some other kind of entry that might be in the list?  (Again, what kind of entries might there be?)  Why can't it just start with, say, the input we were given to establish a new connection for? Just more puzzlement.
 * ''...the software traces a connection to an output, ...
 * Uh, even if we started with an output, as was a stated possibility??? puzzle, puzzle
 * ...then traces the other connection at that output to an input, and so forth, until it comes to an end.
 * Huh? What does "comes to an end" mean?  Apparently an output or an input isn't an end (we kept tracing), so what is an end?
 * Each time it traces from input to output, the connection is placed in one subswitch, and removed from the list.
 * Uh, what is this operation "connection is placed in one subswitch"? How is it "placed"? Is this placing into another list of some kind?  And in what subswitch??  An arbitrary one?  Doesn't the subswitch have to be carefully chosen in some manner?  I suppose the phrase "removed from the list" would make sense at this point, if we only had previously gotten a good idea of what this list looks like.
 * ''When it traces from output to input, the connection is placed in the other subswitch"
 * Umm, what "other" subswitch? Aren't there many subswitches under consideration?
 * To complete correctly, tracing must begin with single connection inputs and outputs, and only then trace double-ended inputs and outputs, which might form loops.
 * This has a plausible ring to it, but I'm so lost by this time I can only ignore it. Also, I don't remember the article even hinting at what "double-ended inputs and outputs" might be.
 * Telephone exchange
 * This other article has a section which seems to almost verbatim copy the original version of the Algorithm section of this NMSS article. If this article gets improved, maybe something (like replacement with a link) should be done with that section.

 * Yours for article improvement -R. S. Shaw 23:25, 19 October 2005 (UTC)

As a semi-expert :-), I could clean this up, but I think most of the article needs to be cut, because it does not directly relate to the type of switch in the title. The article reads more like an overview of switching technology. I agree that much of what is here could be useful in other articles, but I'm not in a position to create several articles out of one. Elasticsoul 22:24, 26 October 2005 (UTC)

I'm a domain expert. I actually have written the software to control a small three-layer switch using the Clos algorithm, and read some of the research on it, and its application. Further, the text resembles some text that I originally wrote for telephone switch several years ago. I noticed when it migrated to telephone exchange, but haven't followed it much since. I'm terribly sorry that it seemed unclear. I just now tried to add lots and lots of context to the algorithm and explain it better. And maybe it does belong in telephone switch, but if you put it there, the British are quite likely to move it to telephone exchange, which some of them believe is an exact cognate (they do not seem to distinguish the machine from the area code). Best wishes User:Ray Van De Walker

Diagram of rerouting
I added a diagram of rerouting a NMSS so additional signals can be routed. (Actually, 2 diagrams). Let me know if there are any comments on how they came out. RJFJR 21:48, 13 May 2007 (UTC)

Geometric series?
''However, the crossbar switch does so at the expense of using N2 (N squared) simple SPST switches. For large N (and the practical requirements of a phone switch are considered large) this geometric growth was prohibitively expensive.''

What has the geometric series to do with this? Thanks, --Abdull (talk) 09:03, 4 August 2008 (UTC)
 * Nothing. It should be geometric growth, I think. Ray Van De Walker 23:14, 13 December 2009 (UTC)


text:
The Moonies

The Moonies were an alternative rock band from Liverpool, England that existed from 2001 to 2005. The band consisted of singer/guitarist Steve Banks, singer/bassist Ryan Clarke, guitarist Steve Charmley, and drummer Mike Berry.

History
Banks, Clarke, and Berry were members of the Moggs, who released an album titled Sweet Corn & Soft Porn for Foulplay Records in 2000. The Moggs lost their recording contract, and in 2001 recruited Charmley and adopted the new name the Moonies as a tribute to Keith Moon. Under this name they signed with EMI. The band was known for their emo-oriented sound mixed with lyrics and fashion styles reminiscent of 1960s British rock.

The band released several singles in the UK starting in 2002, and released the album It's Amazing! in Japan in 2003. They toured the UK several times with the Buzzcocks and recorded an unreleased song featuring Buzzcocks singer Pete Shelley. Charmley left the Moonies in 2004, followed by Banks in 2005, after which the group disbanded.

Members

 * Steve Banks (vocals/guitar)
 * Ryan Clarke (vocals/bass)
 * Mike Berry (drums)
 * Steve Charmley (guitar)

Discography
Albums

 * The Rock and Roll EP (EP, 2003)
 * It's Amazing! (2003)

Singles

 * "Blue' (2002)
 * "Cool" (2002)
 * "I Would Give It All Up for Your Love" (2003)

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

title: The Moonies


text:
Mickey Deans

Michael DeVinko, Jr. (September 24, 1934 – July 11, 2003), known as Mickey Deans, was an American musician and entrepreneur. He is best known as the fifth husband and widower of actress and singer Judy Garland.

Career
He was a disco owner, jazz pianist, and drug dealer. During the 1950s and 1960s, he appeared at the popular New York City nightspot Jilly's on West 52nd Street. He also worked in Los Angeles, Reno, Miami Beach and the Virgin Islands. By the time Deans met singer and actress Judy Garland in 1967, he had switched his occupation from musician to working as the manager of the Manhattan discothèque Arthur, owned by Sybil Burton, on East 54th Street.

Life with Judy Garland
Deans met Garland at her hotel in New York City on March 10, 1967. A mutual friend of theirs asked Deans to deliver a package of amphetamines to Garland's room in the St. Regis. He was dressed as a doctor, and he "delivered the medication she needed to get herself together to fly to work on Valley of the Dolls."

After two years of intermittent dating, they were married on March 15, 1969, in London. Although hundreds of guests were invited to the reception at Quaglino's, only 50 people attended. In particular, Liza Minnelli, Garland's eldest child, did not, saying "I can't make it Mamma but I promise I'll come to your next one!".

In her book Me and My Shadows: Living With the Legacy of Judy Garland, Garland's daughter Lorna Luft writes that when her mother married Deans, she was in the final stages of prescription drug addiction and “was dying in front of his eyes.”

Rosalyn Wilder, who worked as a production assistant at Talk of the Town from 1959 to 1979, and who was present at Garland's wedding to Deans, describes Deans as the “dreadful man who became her husband. … I mean if she put an advert in a newspaper for the most unsuitable person to take care of her, she wouldn’t have had a better response. … I don’t know what possessed… well, I know what possessed her because he gave in to her and he fed her all the things she wanted.”

After the wedding, Deans tried to turn Garland's finances around. He envisioned a documentary and a chain of Judy Garland movie theaters. Neither materialized.

Life after Judy Garland's death
Garland's daughter, Lorna Luft, recalled sharing a limousine with Deans after her mother's funeral in 1969. He insisted on stopping at a Manhattan office and it became clear to Lorna that he was striking his book deal only hours after her mother's funeral service.

"In a move that takes my breath away to this very day when I think of it, Mickey had scheduled a meeting and wanted me to go along," Lorna wrote, adding that Deans and another man "discussed some sort of business deal" in her presence. "Months later, someone told me the other man was a publisher, and that Mickey had arranged to stop by on the way back from my mother’s funeral to cut a deal on a Judy Garland biography. I don't know if it was true, but his book did come out a couple of years later under the title, ‘Weep No More, My Lady.’ Needless to say, I didn't buy a copy. Mickey Deans. What a putz."

After Garland's death, Deans had a four-year relationship with Rose Driscoll, and they adopted a son, Richard.

He was later suspected in the 1983 murder of his boss, Roy Radin.

In 1985, he bought Franklin Castle, a historic four-story stone mansion on Franklin Boulevard in Cleveland, Ohio. for $93,000 ($ today) and remodeled the home for $2.1 million.

Death
Deans died in Northfield Center, Ohio, on July 11, 2003, after a long illness. He was 68.

Books

 * Weep No More, My Lady ISBN 0-515-02989-0 Pyramid Books Edition, & ISBN 0-8161-6039-2 G. K. Hall (1972), a.k.a. Judy's Story (1974)
 * Luft, Lorna. Me and My Shadows: A Family Memoir ISBN 0671019007 Gallery Books (1999)

title: Mickey Deans


text:
Mettmann (district)

Mettmann is a Kreis (district) in the middle of North Rhine-Westphalia, Germany. Neighboring are the Ennepe-Ruhr, Rhein-Kreis Neuss, Rheinisch-Bergischer Kreis and the district-free cities Cologne, Leverkusen, Wuppertal, Solingen, Düsseldorf, Duisburg, Mülheim, Essen. It is the most densely populated rural district in Germany; it borders Düsseldorf Airport in the northwestern district borders, on the city limits of Ratingen, and is also near Cologne Bonn Airport. It was named after Mettmann, its district seat.

History
In the 12th century the area of today's district became the property of the earldom Berg. When Napoleon occupied the area in 1792 it became a canton in the arrondissement Düsseldorf.

After the Congress of Vienna the area went to Prussia in 1816 as part of the province Westphalia, and the first district Mettmann was created. This district was merged from 1820 till 1861 with the district Elberfeld, which then became a city in its own right, now it is part of Wuppertal. In 1877 the administrative seat was moved to Vohwinkel. In 1929 more of the area was given over to Wuppertal, and the district was merged with the district Düsseldorf (Landkreis, excluding the city itself), to form the district Düsseldorf-Mettmann, with administrative seat in Düsseldorf. In 1952 the seat went back to Mettmann, and after the last reorganization of 1975/76 the district got back the old name. The cities Langenfeld and Monheim were included into the district then, while Kettwig became part of the city Essen.

Geography
The district Mettmann is located just south of the Ruhr area, next to the big cities of Düsseldorf.

Most famous is the valley of the small river Düssel, the Neanderthal, where the fossils of the human species Neanderthals were first found.

Coat of arms
The red lion with the blue tongue is the sign of the earldom Berg. In the black border it has the symbols for the two most important industries of the district - the production of locks, and agriculture. The coat of arms was designed in 1936 by the painter and herald Pagenstecher from Düsseldorf.

title: Mettmann (district)


text:
Online analytical processing

In computing, online analytical processing, or OLAP, is an approach to quickly answer multi-dimensional analytical (MDA) queries. OLAP is part of the broader category of business intelligence, which also encompasses relational databases, report writing and data mining. Typical applications of OLAP include business reporting for sales, marketing, management reporting, business process management (BPM), budgeting and forecasting, financial reporting and similar areas, with new applications emerging, such as agriculture.

The term OLAP was created as a slight modification of the traditional database term online transaction processing (OLTP).

OLAP tools enable users to analyse multidimensional data interactively from multiple perspectives. OLAP consists of three basic analytical operations: consolidation (roll-up), drill-down, and slicing and dicing. Consolidation involves the aggregation of data that can be accumulated and computed in one or more dimensions. For example, all sales offices are rolled up to the sales department or sales division to anticipate sales trends. By contrast, the drill-down is a technique that allows users to navigate through the details. For instance, users can view the sales by individual products that make up a region's sales. Slicing and dicing is a feature whereby users can take out (slicing) a specific set of data of the OLAP cube and view (dicing) the slices from different viewpoints. These viewpoints are sometimes called dimensions (such as looking at the same sales by salesperson, or by date, or by customer, or by product, or by region, etc.).

Databases configured for OLAP use a multidimensional data model, allowing for complex analytical and ad hoc queries with a rapid execution time. They borrow aspects of navigational databases, hierarchical databases and relational databases.

OLAP is typically contrasted to OLTP (online transaction processing), which is generally characterized by much less complex queries, in a larger volume, to process transactions rather than for the purpose of business intelligence or reporting. Whereas OLAP systems are mostly optimized for read, OLTP has to process all kinds of queries (read, insert, update and delete).

Overview of OLAP systems
At the core of any OLAP system is an OLAP cube (also called a 'multidimensional cube' or a hypercube). It consists of numeric facts called measures that are categorized by dimensions. The measures are placed at the intersections of the hypercube, which is spanned by the dimensions as a vector space. The usual interface to manipulate an OLAP cube is a matrix interface, like Pivot tables in a spreadsheet program, which performs projection operations along the dimensions, such as aggregation or averaging.

The cube metadata is typically created from a star schema or snowflake schema or fact constellation of tables in a relational database. Measures are derived from the records in the fact table and dimensions are derived from the dimension tables.

Each measure can be thought of as having a set of labels, or meta-data associated with it. A dimension is what describes these labels; it provides information about the measure.

A simple example would be a cube that contains a store's sales as a measure, and Date/Time as a dimension. Each Sale has a Date/Time label that describes more about that sale.

For example: Sales Fact Table +-+--+ | sale_amount | time_id | +-+--+           Time Dimension |     2008.10|     1234 |+     +-+---+ +-+--+    |     | time_id | timestamp         | |    +-+---+                               +>|   1234  | 20080902 12:35:43 |                                     +-+---+

Multidimensional databases
Multidimensional structure is defined as "a variation of the relational model that uses multidimensional structures to organize data and express the relationships between data". The structure is broken into cubes and the cubes are able to store and access data within the confines of each cube. "Each cell within a multidimensional structure contains aggregated data related to elements along each of its dimensions". Even when data is manipulated it remains easy to access and continues to constitute a compact database format. The data still remains interrelated. Multidimensional structure is quite popular for analytical databases that use online analytical processing (OLAP) applications. Analytical databases use these databases because of their ability to deliver answers to complex business queries swiftly. Data can be viewed from different angles, which gives a broader perspective of a problem unlike other models.

Aggregations
It has been claimed that for complex queries OLAP cubes can produce an answer in around 0.1% of the time required for the same query on OLTP relational data. The most important mechanism in OLAP which allows it to achieve such performance is the use of aggregations. Aggregations are built from the fact table by changing the granularity on specific dimensions and aggregating up data along these dimensions, using an aggregate function (or aggregation function). The number of possible aggregations is determined by every possible combination of dimension granularities.

The combination of all possible aggregations and the base data contains the answers to every query which can be answered from the data.

Because usually there are many aggregations that can be calculated, often only a predetermined number are fully calculated; the remainder are solved on demand. The problem of deciding which aggregations (views) to calculate is known as the view selection problem. View selection can be constrained by the total size of the selected set of aggregations, the time to update them from changes in the base data, or both. The objective of view selection is typically to minimize the average time to answer OLAP queries, although some studies also minimize the update time. View selection is NP-Complete. Many approaches to the problem have been explored, including greedy algorithms, randomized search, genetic algorithms and A* search algorithm.

Some aggregation functions can be computed for the entire OLAP cube by precomputing values for each cell, and then computing the aggregation for a roll-up of cells by aggregating these aggregates, applying a divide and conquer algorithm to the multidimensional problem to compute them efficiently. For example, the overall sum of a roll-up is just the sum of the sub-sums in each cell. Functions that can be decomposed in this way are called decomposable aggregation functions, and include  and , which can be computed for each cell and then directly aggregated; these are known as self-decomposable aggregation functions.

In other cases the aggregate function can be computed by computing auxiliary numbers for cells, aggregating these auxiliary numbers, and finally computing the overall number at the end; examples include  (tracking sum and count, dividing at the end) and   (tracking max and min, subtracting at the end). In other cases the aggregate function cannot be computed without analyzing the entire set at once, though in some cases approximations can be computed; examples include  and  ; for example, the median of a set is not the median of medians of subsets. These latter are difficult to implement efficiently in OLAP, as they require computing the aggregate function on the base data, either computing them online (slow) or precomputing them for possible rollouts (large space).

Types
OLAP systems have been traditionally categorized using the following taxonomy.

Multidimensional OLAP (MOLAP)
MOLAP (multi-dimensional online analytical processing) is the classic form of OLAP and is sometimes referred to as just OLAP. MOLAP stores this data in an optimized multi-dimensional array storage, rather than in a relational database.

Some MOLAP tools require the pre-computation and storage of derived data, such as consolidations – the operation known as processing. Such MOLAP tools generally utilize a pre-calculated data set referred to as a data cube. The data cube contains all the possible answers to a given range of questions. As a result, they have a very fast response to queries. On the other hand, updating can take a long time depending on the degree of pre-computation. Pre-computation can also lead to what is known as data explosion.

Other MOLAP tools, particularly those that implement the functional database model do not pre-compute derived data but make all calculations on demand other than those that were previously requested and stored in a cache.

Advantages of MOLAP
 * Fast query performance due to optimized storage, multidimensional indexing and caching.
 * Smaller on-disk size of data compared to data stored in relational database due to compression techniques.
 * Automated computation of higher level aggregates of the data.
 * It is very compact for low dimension data sets.
 * Array models provide natural indexing.
 * Effective data extraction achieved through the pre-structuring of aggregated data.

Disadvantages of MOLAP
 * Within some MOLAP systems the processing step (data load) can be quite lengthy, especially on large data volumes. This is usually remedied by doing only incremental processing, i.e., processing only the data which have changed (usually new data) instead of reprocessing the entire data set.
 * Some MOLAP methodologies introduce data redundancy.

Products
Examples of commercial products that use MOLAP are Cognos Powerplay, Oracle Database OLAP Option, MicroStrategy, Microsoft Analysis Services, Essbase, TM1, Jedox, and icCube.

Relational OLAP (ROLAP)
ROLAP works directly with relational databases and does not require pre-computation. The base data and the dimension tables are stored as relational tables and new tables are created to hold the aggregated information. It depends on a specialized schema design. This methodology relies on manipulating the data stored in the relational database to give the appearance of traditional OLAP's slicing and dicing functionality. In essence, each action of slicing and dicing is equivalent to adding a "WHERE" clause in the SQL statement. ROLAP tools do not use pre-calculated data cubes but instead pose the query to the standard relational database and its tables in order to bring back the data required to answer the question. ROLAP tools feature the ability to ask any question because the methodology is not limited to the contents of a cube. ROLAP also has the ability to drill down to the lowest level of detail in the database.

While ROLAP uses a relational database source, generally the database must be carefully designed for ROLAP use. A database which was designed for OLTP will not function well as a ROLAP database. Therefore, ROLAP still involves creating an additional copy of the data. However, since it is a database, a variety of technologies can be used to populate the database.

Advantages of ROLAP

 * ROLAP is considered to be more scalable in handling large data volumes, especially models with dimensions with very high cardinality (i.e., millions of members).
 * With a variety of data loading tools available, and the ability to fine-tune the extract, transform, load (ETL) code to the particular data model, load times are generally much shorter than with the automated MOLAP loads.
 * The data are stored in a standard relational database and can be accessed by any SQL reporting tool (the tool does not have to be an OLAP tool).
 * ROLAP tools are better at handling non-aggregatable facts (e.g., textual descriptions). MOLAP tools tend to suffer from slow performance when querying these elements.
 * By decoupling the data storage from the multi-dimensional model, it is possible to successfully model data that would not otherwise fit into a strict dimensional model.
 * The ROLAP approach can leverage database authorization controls such as row-level security, whereby the query results are filtered depending on preset criteria applied, for example, to a given user or group of users (SQL WHERE clause).

Disadvantages of ROLAP

 * There is a consensus in the industry that ROLAP tools have slower performance than MOLAP tools. However, see the discussion below about ROLAP performance.
 * The loading of aggregate tables must be managed by custom ETL code. The ROLAP tools do not help with this task. This means additional development time and more code to support.
 * When the step of creating aggregate tables is skipped, the query performance then suffers because the larger detailed tables must be queried. This can be partially remedied by adding additional aggregate tables, however it is still not practical to create aggregate tables for all combinations of dimensions/attributes.
 * ROLAP relies on the general purpose database for querying and caching, and therefore several special techniques employed by MOLAP tools are not available (such as special hierarchical indexing). However, modern ROLAP tools take advantage of latest improvements in SQL language such as CUBE and ROLLUP operators, DB2 Cube Views, as well as other SQL OLAP extensions. These SQL improvements can mitigate the benefits of the MOLAP tools.
 * Since ROLAP tools rely on SQL for all of the computations, they are not suitable when the model is heavy on calculations which don't translate well into SQL. Examples of such models include budgeting, allocations, financial reporting and other scenarios.

Performance of ROLAP
In the OLAP industry ROLAP is usually perceived as being able to scale for large data volumes, but suffering from slower query performance as opposed to MOLAP. The OLAP Survey, the largest independent survey across all major OLAP products, being conducted for 6 years (2001 to 2006) have consistently found that companies using ROLAP report slower performance than those using MOLAP even when data volumes were taken into consideration.

However, as with any survey there are a number of subtle issues that must be taken into account when interpreting the results.
 * The survey shows that ROLAP tools have 7 times more users than MOLAP tools within each company. Systems with more users will tend to suffer more performance problems at peak usage times.
 * There is also a question about complexity of the model, measured both in number of dimensions and richness of calculations. The survey does not offer a good way to control for these variations in the data being analyzed.

Downside of flexibility
Some companies select ROLAP because they intend to re-use existing relational database tables—these tables will frequently not be optimally designed for OLAP use. The superior flexibility of ROLAP tools allows this less than optimal design to work, but performance suffers. MOLAP tools in contrast would force the data to be re-loaded into an optimal OLAP design.

Hybrid OLAP (HOLAP)
The undesirable trade-off between additional ETL cost and slow query performance has ensured that most commercial OLAP tools now use a "Hybrid OLAP" (HOLAP) approach, which allows the model designer to decide which portion of the data will be stored in MOLAP and which portion in ROLAP.

There is no clear agreement across the industry as to what constitutes "Hybrid OLAP", except that a database will divide data between relational and specialized storage. For example, for some vendors, a HOLAP database will use relational tables to hold the larger quantities of detailed data, and use specialized storage for at least some aspects of the smaller quantities of more-aggregate or less-detailed data. HOLAP addresses the shortcomings of MOLAP and ROLAP by combining the capabilities of both approaches. HOLAP tools can utilize both pre-calculated cubes and relational data sources.

Vertical partitioning
In this mode HOLAP stores aggregations in MOLAP for fast query performance, and detailed data in ROLAP to optimize time of cube processing.

Horizontal partitioning
In this mode HOLAP stores some slice of data, usually the more recent one (i.e. sliced by Time dimension) in MOLAP for fast query performance, and older data in ROLAP. Moreover, we can store some dices in MOLAP and others in ROLAP, leveraging the fact that in a large cuboid, there will be dense and sparse subregions.

Products
The first product to provide HOLAP storage was Holos, but the technology also became available in other commercial products such as Microsoft Analysis Services, Oracle Database OLAP Option, MicroStrategy and SAP AG BI Accelerator. The hybrid OLAP approach combines ROLAP and MOLAP technology, benefiting from the greater scalability of ROLAP and the faster computation of MOLAP. For example, a HOLAP server may store large volumes of detailed data in a relational database, while aggregations are kept in a separate MOLAP store. The Microsoft SQL Server 7.0 OLAP Services supports a hybrid OLAP server

Comparison
Each type has certain benefits, although there is disagreement about the specifics of the benefits between providers.

 * Some MOLAP implementations are prone to database explosion, a phenomenon causing vast amounts of storage space to be used by MOLAP databases when certain common conditions are met: high number of dimensions, pre-calculated results and sparse multidimensional data.
 * MOLAP generally delivers better performance due to specialized indexing and storage optimizations. MOLAP also needs less storage space compared to ROLAP because the specialized storage typically includes compression techniques.
 * ROLAP is generally more scalable. However, large volume pre-processing is difficult to implement efficiently so it is frequently skipped. ROLAP query performance can therefore suffer tremendously.
 * Since ROLAP relies more on the database to perform calculations, it has more limitations in the specialized functions it can use.
 * HOLAP attempts to mix the best of ROLAP and MOLAP. It can generally pre-process swiftly, scale well, and offer good function support.

Other types
The following acronyms are also sometimes used, although they are not as widespread as the ones above:

 * WOLAP – Web-based OLAP
 * DOLAP – Desktop OLAP
 * RTOLAP – Real-time OLAP
 * GOLAP – Graph OLAP
 * CaseOLAP – Context-aware Semantic OLAP, developed for biomedical applications. The CaseOLAP platform includes data preprocessing (e.g., downloading, extraction, and parsing text documents), indexing and searching with Elasticsearch, creating a functional document structure called Text-Cube,    and quantifying user-defined phrase-category relationships using the core CaseOLAP algorithm.

APIs and query languages
Unlike relational databases, which had SQL as the standard query language, and widespread APIs such as ODBC, JDBC and OLEDB, there was no such unification in the OLAP world for a long time. The first real standard API was OLE DB for OLAP specification from Microsoft which appeared in 1997 and introduced the MDX query language. Several OLAP vendors – both server and client – adopted it. In 2001 Microsoft and Hyperion announced the XML for Analysis specification, which was endorsed by most of the OLAP vendors. Since this also used MDX as a query language, MDX became the de facto standard. Since September-2011 LINQ can be used to query SSAS OLAP cubes from Microsoft .NET.

History
The first product that performed OLAP queries was Express, which was released in 1970 (and acquired by Oracle in 1995 from Information Resources). However, the term did not appear until 1993 when it was coined by Edgar F. Codd, who has been described as "the father of the relational database". Codd's paper resulted from a short consulting assignment which Codd undertook for former Arbor Software (later Hyperion Solutions, and in 2007 acquired by Oracle), as a sort of marketing coup.

The company had released its own OLAP product, Essbase, a year earlier. As a result, Codd's "twelve laws of online analytical processing" were explicit in their reference to Essbase. There was some ensuing controversy and when Computerworld learned that Codd was paid by Arbor, it retracted the article. The OLAP market experienced strong growth in the late 1990s with dozens of commercial products going into market. In 1998, Microsoft released its first OLAP Server – Microsoft Analysis Services, which drove wide adoption of OLAP technology and moved it into the mainstream.

OLAP clients
OLAP clients include many spreadsheet programs like Excel, web application, SQL, dashboard tools, etc. Many clients support interactive data exploration where users select dimensions and measures of interest. Some dimensions are used as filters (for slicing and dicing the data) while others are selected as the axes of a pivot table or pivot chart. Users can also vary aggregation level (for drilling-down or rolling-up) the displayed view. Clients can also offer a variety of graphical widgets such as sliders, geographic maps, heat maps and more which can be grouped and coordinated as dashboards. An extensive list of clients appears in the visualization column of the comparison of OLAP servers table.

Market structure
Below is a list of top OLAP vendors in 2006, with figures in millions of US Dollars.

Open-source

 * Apache Pinot is used at LinkedIn, Cisco, Uber, Slack, Stripe, DoorDash, Target, Walmart, Amazon, and Microsoft to deliver scalable real time analytics with low latency. It can ingest data from offline data sources (such as Hadoop and flat files) as well as online sources (such as Kafka). Pinot is designed to scale horizontally.
 * Mondrian OLAP server is an open-source OLAP server written in Java. It supports the MDX query language, the XML for Analysis and the olap4j interface specifications.
 * Apache Doris is an open source real-time analytical database based on MPP architecture. It can support both high-concurrency point query scenarios and high-throughput complex analysis.
 * Apache Druid is a popular open-source distributed data store for OLAP queries that is used at scale in production by various organizations.
 * Apache Kylin is a distributed data store for OLAP queries originally developed by eBay.
 * Cubes (OLAP server) is another light-weight open-source toolkit implementation of OLAP functionality in the Python programming language with built-in ROLAP.
 * ClickHouse is a fairly new column orientated DBMS focusing on fast processing and response times.
 * DuckDB is an in-process SQL OLAP database management system.
 * MonetDB is a mature open-source column-oriented SQL RDBMS designed for OLAP queries.

title: Online analytical processing


text:
Mitcham Junction station

Mitcham Junction is a National Rail station served by Southern and Thameslink trains, and a Tramlink stop. It is in the London Borough of Merton and is in Travelcard Zone 4.

The station opened on 1 October 1868 specifically to provide an interchange between the new "South London & Sutton Junction Railway", later re-branded as part of the Portsmouth Line, and the existing "Wimbledon & Croydon Railway".

Despite its name, Mitcham Junction is no longer a railway junction; one of the lines that crossed here (the W&CR) has become a grade-separated tramline, the Croydon Tramlink. Only the Portsmouth Line remains, used by services from Sutton and beyond to London Victoria, and from Sutton to London Blackfriars and beyond. The line still has sharp curves at either end of the station where the junctions were located and speed is limited to 30 mph.

The platforms can accommodate 7 coaches. For longer trains selective door opening is used.

Location
Mitcham Junction is not near the centre of Mitcham but on Mitcham Common next to Mitcham Golf Club, and not far from the historic Cricket Green Conservation Area. The nearest railway station to the commercial centre of Mitcham is Mitcham Eastfields, between Mitcham Junction and Streatham, which opened in June 2008.

National Rail
National Rail services at Mitcham Junction are operated by Southern and Thameslink using and  EMUs.

The typical off-peak service in trains per hour is:

 * 2 tph to London Victoria
 * 2 tph to St Albans City via Herne Hill
 * 2 tph to Sutton
 * 2 tph to Dorking of which 1 continues to Horsham

During the peak hours, additional services between London Victoria and Epsom also call at the station.

On Saturday evenings (after approximately 18:45) and on Sundays, there is no service south of Dorking to Horsham.

London Trams
London Trams operate services at Mitcham Junction using Bombardier CR4000 and Stadler Variobahn Trams.

The typical off-peak service in trams per hour is:
 * 6 tph to Beckenham Junction via West Croydon
 * 6 tph to Elmers End via West Croydon
 * 12 tph to Wimbledon

The stop is also served by a number of early morning and late evening services to and from New Addington.

Connections
London Buses routes 127 and S1 serve the station and tram stop.

title: Mitcham Junction station


text:
ATC

ATC may refer to:

Medicine

 * Acute traumatic coagulopathy
 * Anaplastic thyroid cancer, a form of thyroid cancer
 * Anatomical Therapeutic Chemical Classification System, a WHO drug classification system
 * Certified athletic trainer, post-nominal letters

Military

 * Air Training Command, the predecessor to Air Education and Training Command in the United States Air Force
 * Air Training Corps, United Kingdom, for cadets
 * Air Transport Command, a United States Army Air Force command during World War II
 * New Zealand Air Training Corps, for cadets

Education

 * Academy for Technology and the Classics, a charter school in Santa Fe, New Mexico, US
 * Advanced Technology College, a community college in Daytona Beach, Florida, US
 * Applied Technology College, one of several campuses in the Utah College of Applied Technology system
 * Ambrose Treacy College, high school in Brisbane, Australia

Other organizations

 * Alabang Town Center, a shopping mall in Muntinlupa, Philippines
 * Altice (company), Luxembourg-based multinational telecoms company by Euronext stock symbol
 * American Technology Corporation, former name of LRAD Corporation, a sound technology company
 * American Thermoplastic Company, a United States plastics manufacturer
 * American Tobacco Campus, a former property of the American Tobacco Company in Durham, North Carolina, US
 * American Tractor Corporation, a 1950 manufacturer of crawler tractors which merged into Case
 * American Truck Company
 * American-Turkish Council, a business association promoting U.S.-Turkish relations
 * Appalachian Trail Conservancy, a United States nonprofit organization
 * Applied Technology Council, a nonprofit founded in 1973 which studies the effects of hazards on the built environment
 * Aquarian Tabernacle Church, a Wiccan church founded in 1979
 * Argentina Televisora Color, former name of the television network Televisión Pública (Canal 7)
 * Atlanta Track Club, a non-profit running organization based in Atlanta, Georgia
 * Atlanta Transit Company, former rail operator in Georgia, United States
 * Auckland Theatre Company, professional theatre company, New Zealand
 * Australian Telecommunications Corporation, formerly Australian Telecommunications Commission
 * Automobile and Touring Club for United Arab Emirates, a member of the Fédération Internationale de l'Automobile

Science and technology

 * Ancillary Terrestrial Component, a U.S. Federal Communications Commission-approved technique for using a network of cell-phone towers to supplement a Mobile Satellite Service
 * UK Astronomy Technology Centre, based at the Royal Observatory in Edinburgh, Scotland, part of the Science and Technology Facilities Council
 * Athletic Trainer Certified, a healthcare professional board certified in athletic training
 * Automatic taxonomy construction, a branch of natural language processing, which applies software programs to generate taxonomical classifications from a body of texts.
 * Average total cost, in economics
 * Automatic tool changer, a mechanism allowing CNC machines to switch cutting tools without operator intervention.
 * Alternate technical concept, in engineering and construction.
 * USENIX Annual Technical Conference, an annual peer-reviewed conference
 * ATC, a codon for the amino acid isoleucine

Transport

 * Advanced transportation controller, a standardization effort as part of the United States Department of Transportation's Intelligent transportation system
 * All-terrain cycle, a type of all-terrain vehicle
 * Alvarado Transportation Center, an intermodal transit center in New Mexico
 * American Truck Corporation, a USA importer and/or assembler of Tatra and Kraz trucks.
 * Armored Troop Carrier (LCM), used in the Vietnam War
 * Automatic traffic counter, see traffic count

Aviation

 * Air traffic control, a service provided to aircraft by ground-based controllers
 * Air traffic controller, people who expedite and maintain a safe and orderly flow of air traffic in the global air traffic control system
 * Air Tanzania Corporation, the former state-owned airline of Tanzania

Rail

 * Automatic train control, a safety system for railways
 * Automatic and Track Control, a tokenless form of safeworking on single rail lines used in Victoria, Australia

Games and hobbies

 * Armacham Technology Corporation, a fictional technology developer in the video game F.E.A.R.
 * Artist trading cards, individual fine art miniatures
 * 'Any to Come', a type of conditional bet; See Glossary of bets offered by UK bookmakers

Sports

 * SV Atlétiko Tera Corá, a Bonaire football club

Other uses

 * All Things Comedy, a comedy company and podcast aggregate
 * All Things Considered, a news show on National Public Radio in the United States
 * Against the Current (band), an American pop rock band
 * Argentina Televisora Color, former name for Televisión Pública in Argentina
 * A Touch of Class (band), a pop music group from Germany and the UK (1999–2003)
 * Agreement on Textiles and Clothing, a World Trade Organization regulation for trade in textiles and garments
 * Audio Tele-Conference
 * ATC (belay device), a type of belay device for rock climbing
 * Authorization to Carry, a permit issued in Canada to carry Restricted firearms and Prohibited handguns

title: ATC


text:
Matrix representation of conic sections

In mathematics, the matrix representation of conic sections permits the tools of linear algebra to be used in the study of conic sections. It provides easy ways to calculate a conic section's axis, vertices, tangents and the pole and polar relationship between points and lines of the plane determined by the conic. The technique does not require putting the equation of a conic section into a standard form, thus making it easier to investigate those conic sections whose axes are not parallel to the coordinate system.

Conic sections (including degenerate ones) are the sets of points whose coordinates satisfy a second-degree polynomial equation in two variables, $$Q(x,y) = Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.$$ By an abuse of notation, this conic section will also be called $Q$ when no confusion can arise.

This equation can be written in matrix notation, in terms of a symmetric matrix to simplify some subsequent formulae, as

$$\begin{pmatrix} x & y \end{pmatrix} \begin{pmatrix} A & B/2 \\ B/2 & C \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} D & E \end{pmatrix} \begin{pmatrix} x\\y \end{pmatrix} + F = 0.$$

The sum of the first three terms of this equation, namely $$Ax^2 + Bxy + Cy^2 = \begin{pmatrix} x & y \end{pmatrix} \begin{pmatrix} A & B/2 \\ B/2 & C \end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix},$$ is the quadratic form associated with the equation, and the matrix $$A_{33} = \begin{pmatrix} A & B/2 \\ B/2 & C\end{pmatrix}$$ is called the matrix of the quadratic form. The trace and determinant of $$A_{33} $$ are both invariant with respect to rotation of axes and translation of the plane (movement of the origin).

The quadratic equation can also be written as

$$\mathbf{x}^\mathsf{T} A_Q\mathbf{x} = 0,$$

where $$\mathbf{x}$$ is the homogeneous coordinate vector in three variables restricted so that the last variable is 1, i.e.,

$$\begin{pmatrix} x \\ y \\ 1 \end{pmatrix}$$

and where $$A_Q$$ is the matrix

$$A_Q = \begin{pmatrix} A & B/2 & D/2 \\ B/2 & C & E/2 \\ D/2 & E/2 & F \end{pmatrix}.$$ The matrix $$A_Q$$ is called the matrix of the quadratic equation. Like that of $$A_{33}$$, its determinant is invariant with respect to both rotation and translation.

The 2 × 2 upper left submatrix (a matrix of order 2) of $A_{Q}$, obtained by removing the third (last) row and third (last) column from $A_{Q}$ is the matrix of the quadratic form. The above notation $A_{33}$ is used in this article to emphasize this relationship.

Classification
Proper (non-degenerate) and degenerate conic sections can be distinguished based on the determinant of $A_{Q}$:

If $$\det A_Q = 0$$, the conic is degenerate.

If $$\det A_Q \neq 0$$ so that $Q$ is not degenerate, we can see what type of conic section it is by computing the minor, $$\det A_{33}$$:

 * $Q$ is a hyperbola if and only if $$ \det A_{33} < 0 $$,
 * $Q$ is a parabola if and only if $$ \det A_{33} = 0 $$, and
 * $Q$ is an ellipse if and only if $$ \det A_{33} > 0 $$.

In the case of an ellipse, we can distinguish the special case of a circle by comparing the last two diagonal elements corresponding to the coefficients of $x^{2}$ and $y^{2}$:

 * If $A = C$ and $B = 0$, then $Q$ is a circle.

Moreover, in the case of a non-degenerate ellipse (with $$\det A_{33} > 0 $$ and $$\det A_Q \ne 0$$), we have a real ellipse if $$(A + C)\det A_Q < 0$$ but an imaginary ellipse if $$(A + C)\det A_Q > 0$$. An example of the latter is $$x^2 + y^2 + 10 = 0 $$, which has no real-valued solutions.

If the conic section is degenerate ($$\det A_Q = 0$$), $$\det A_{33}$$ still allows us to distinguish its form:

 * Two intersecting lines (a hyperbola degenerated to its two asymptotes) if and only if $$\det A_{33} < 0$$.
 * Two parallel straight lines (a degenerate parabola) if and only if $$\det A_{33} = 0$$. These lines are distinct and real if $$D^2+E^2 > 4(A+C)F$$, coincident if $$D^2+E^2 = 4(A+C)F$$, and non-existent in the real plane if $$D^2+E^2 < 4(A+C)F$$.
 * A single point (a degenerate ellipse) if and only if $$\det A_{33} > 0$$.

The case of coincident lines occurs if and only if the rank of the 3 × 3 matrix $$A_Q$$ is 1; in all other degenerate cases its rank is 2.

Central conics
When $$ \det A_{33} \neq 0 $$ a geometric center of the conic section exists and such conic sections (ellipses and hyperbolas) are called central conics.

Center
The center of a conic, if it exists, is a point that bisects all the chords of the conic that pass through it. This property can be used to calculate the coordinates of the center, which can be shown to be the point where the gradient of the quadratic function $Q$ vanishes—that is, $$ \nabla Q = \left[ \frac{\partial Q}{\partial x}, \frac{\partial Q}{\partial y} \right] = [0,0]. $$ This yields the center as given below.

An alternative approach that uses the matrix form of the quadratic equation is based on the fact that when the center is the origin of the coordinate system, there are no linear terms in the equation. Any translation to a coordinate origin $(x_{0}, y_{0})$, using $x* = x – x_{0}$, $y* = y − y_{0}$ gives rise to

$$\begin{pmatrix} x^* + x_0 & y ^* + y_0 \end{pmatrix} \begin{pmatrix}A & B/2\\B/2 & C \end{pmatrix} \begin{pmatrix} x^* + x_0\\y^* + y_0 \end{pmatrix} + \left(\begin{matrix}D & E \end{matrix}\right) \left(\begin{matrix}x^* + x_0 \\ y^* + y_0\end{matrix}\right) +F= 0. $$

The condition for $(x_{0}, y_{0})$ to be the conic's center $(x_{c}, y_{c})$ is that the coefficients of the linear $x*$ and $y*$ terms, when this equation is multiplied out, are zero. This condition produces the coordinates of the center: $$ \begin{pmatrix} x_c \\ y_c \end{pmatrix} = \begin{pmatrix} A & B/2 \\ B/2 & C \end{pmatrix}^{\!-1} \begin{pmatrix} -D/2 \\ -E/2 \end{pmatrix} = \begin{pmatrix} (BE-2CD)/(4AC-B^2) \\ (DB-2AE)/(4AC-B^2) \end{pmatrix}.$$

This calculation can also be accomplished by taking the first two rows of the associated matrix $A_{Q}$, multiplying each by $(x, y, 1)^{⊤}$ and setting both inner products equal to 0, obtaining the following system:

$$\begin{align} Ax + (B/2)y + D/2 &= 0, \\ (B/2)x + Cy + E/2 &= 0. \end{align}$$

This yields the above center point.

In the case of a parabola, that is, when $4AC − B^{2} = 0$, there is no center since the above denominators become zero (or, interpreted projectively, the center is on the line at infinity.)

Centered matrix equation
A central (non-parabola) conic $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$ can be rewritten in centered matrix form as $$\begin{pmatrix} x-x_c & y-y_c \end{pmatrix} \begin{pmatrix} A & B/2 \\ B/2 & C \end{pmatrix} \begin{pmatrix} x-x_c \\ y-y_c \end{pmatrix} = K,$$ where $$K = -\frac{\det (A_Q)}{AC-(B/2)^2} = -\frac{\det(A_Q)}{\det(A_{33})}.$$

Then for the ellipse case of $AC > (B/2)^{2}$, the ellipse is real if the sign of $K$ equals the sign of $(A + C)$ (that is, the sign of each of $A$ and $C$), imaginary if they have opposite signs, and a degenerate point ellipse if $K = 0$. In the hyperbola case of $AC < (B/2)^{2}$, the hyperbola is degenerate if and only if $K = 0$.

Standard form of a central conic
The standard form of the equation of a central conic section is obtained when the conic section is translated and rotated so that its center lies at the center of the coordinate system and its axes coincide with the coordinate axes. This is equivalent to saying that the coordinate system's center is moved and the coordinate axes are rotated to satisfy these properties. In the diagram, the original $xy$-coordinate system with origin $O$ is moved to the $x'y'$-coordinate system with origin $O'$.

The translation is by the vector $$\mathbf{t} = \begin{pmatrix} x_c \\ y_c \end{pmatrix}.$$

The rotation by angle $α$ can be carried out by diagonalizing the matrix $A_{33}$. Thus, if $$\lambda_1$$ and $$\lambda_2$$ are the eigenvalues of the matrix A33, the centered equation can be rewritten in new variables $x'$ and $y'$ as

$$\lambda_1 x'^2 + \lambda_2 y'^2 = - \frac{\det A_Q}{\det A_{33}}.$$

Dividing by $$K = -\frac{\det A_Q}{\det A_{33}}$$ we obtain a standard canonical form.

For example, for an ellipse this form is $$\frac{{x'}^2}{a^2} + \frac{{y'}^2}{b^2} = 1.$$ From here we get $a$ and $b$, the lengths of the semi-major and semi-minor axes in conventional notation.

For central conics, both eigenvalues are non-zero and the classification of the conic sections can be obtained by examining them.
 * If $λ_{1}$ and $λ_{2}$ have the same algebraic sign, then $Q$ is a real ellipse, imaginary ellipse or real point if $K$ has the same sign, has the opposite sign or is zero, respectively.
 * If $λ_{1}$ and $λ_{2}$ have opposite algebraic signs, then $Q$ is a hyperbola or two intersecting lines depending on whether $K$ is nonzero or zero, respectively.

Axes
By the principal axis theorem, the two eigenvectors of the matrix of the quadratic form of a central conic section (ellipse or hyperbola) are perpendicular (orthogonal to each other) and each is parallel to (in the same direction as) either the major or minor axis of the conic. The eigenvector having the smallest eigenvalue (in absolute value) corresponds to the major axis.

Specifically, if a central conic section has center $(x_{c}, y_{c})$ and an eigenvector of $A_{33}$ is given by $v(v_{1}, v_{2})$ then the principal axis (major or minor) corresponding to that eigenvector has equation, $$ \frac{x-x_c}{v_1} = \frac{y-y_c}{v_2}. $$

Vertices
The vertices of a central conic can be determined by calculating the intersections of the conic and its axes — in other words, by solving the system consisting of the quadratic conic equation and the linear equation for alternately one or the other of the axes. Two or no vertices are obtained for each axis, since, in the case of the hyperbola, the minor axis does not intersect the hyperbola at a point with real coordinates. However, from the broader view of the complex plane, the minor axis of an hyperbola does intersect the hyperbola, but at points with complex coordinates.

Poles and polars
Using homogeneous coordinates, the points $$\mathbf{p} = \begin{pmatrix} p_0 \\ p_1 \\ p_2 \end{pmatrix} $$ and $$\mathbf{r} = \begin{pmatrix} r_0 \\ r_1 \\ r_2 \end{pmatrix} $$ are conjugate with respect to the conic $Q$ provided $$ \mathbf{p}^\mathsf{T} A_Q \mathbf{r} = 0.$$

The conjugates of a fixed point $p$ either form a line or consist of all the points in the plane of the conic. When the conjugates of $p$ form a line, the line is called the polar of $p$ and the point $p$ is called the pole of the line, with respect to the conic. This relationship between points and lines is called a polarity.

If the conic is non-degenerate, the conjugates of a point always form a line and the polarity defined by the conic is a bijection between the points and lines of the extended plane containing the conic (that is, the plane together with the points and line at infinity).

If the point $p$ lies on the conic $Q$, the polar line of $p$ is the tangent line to $Q$ at $p$.

The equation, in homogeneous coordinates, of the polar line of the point $p$ with respect to the non-degenerate conic $Q$ is given by $$ \mathbf{p}^T A_Q \begin{pmatrix} x \\ y \\ z \end{pmatrix} = 0.$$

Just as $p$ uniquely determines its polar line (with respect to a given conic), so each line determines a unique pole $p$. Furthermore, a point $p$ is on a line $L$ which is the polar of a point $r$, if and only if the polar of $p$ passes through the point $r$ (La Hire's theorem). Thus, this relationship is an expression of geometric duality between points and lines in the plane.

Several familiar concepts concerning conic sections are directly related to this polarity. The center of a non-degenerate conic can be identified as the pole of the line at infinity. A parabola, being tangent to the line at infinity, would have its center being a point on the line at infinity. Hyperbolas intersect the line at infinity in two distinct points and the polar lines of these points are the asymptotes of the hyperbola and are the tangent lines to the hyperbola at these points of infinity. Also, the polar line of a focus of the conic is its corresponding directrix.

Tangents
Let line $L$ be the polar line of point $p$ with respect to the non-degenerate conic $Q$. By La Hire's theorem, every line passing through $p$ has its pole on $L$. If $L$ intersects $Q$ in two points (the maximum possible) then the polars of those points are tangent lines that pass through $p$ and such a point is called an exterior or outer point of $Q$. If $L$ intersects $Q$ in only one point, then it is a tangent line and $p$ is the point of tangency. Finally, if $L$ does not intersect $Q$ then $p$ has no tangent lines passing through it and it is called an interior or inner point.

The equation of the tangent line (in homogeneous coordinates) at a point $Q$ on the non-degenerate conic $p$ is given by,

$$ \mathbf{p}^\mathsf{T} A_Q \begin{pmatrix} x \\ y \\ z \end{pmatrix} = 0. $$

If $p$ is an exterior point, first find the equation of its polar (the above equation) and then the intersections of that line with the conic, say at points $s$ and $t$. The polars of $s$ and $t$ will be the tangents through $p$.

Using the theory of poles and polars, the problem of finding the four mutual tangents of two conics reduces to finding the intersection of two conics.

title: Matrix representation of conic sections


text:
Brocken

The Brocken, also sometimes referred to as the Blocksberg, is the highest peak in the Harz mountain range and also the highest peak in Northern Germany; it is near Schierke in the German state of Saxony-Anhalt between the rivers Weser and Elbe. Although its elevation of 1141 m is below alpine dimensions, its microclimate resembles that of mountains of about 2000 m. The peak above the tree line tends to have a snow cover from September to May, and mists and fogs shroud it up to 300 days of the year. The mean annual temperature is only 2.9 °C. It is the easternmost mountain in northern Germany; travelling east in a straight line, the next prominent elevation would be in the Ural Mountains in Russia.

The Brocken has always played a role in legends and has been connected with witches and devils; Johann Wolfgang von Goethe took up the legends in his play Faust. The Brocken spectre is a common phenomenon on this misty mountain, where a climber's shadow cast upon fog creates eerie optical effects.

Today the Brocken is part of the Harz National Park and hosts a historic botanical garden of about 1,600 alpine mountain plants. A narrow gauge steam railway, the Brocken Railway, takes visitors to the railway station at the top on 1125 m.

FM-radio and television broadcasting make major use of the Brocken. The old television tower, the Sender Brocken, is now used as hotel and restaurant. It also has an observation deck, open to tourists.

Location
The Brocken rises over the Harz National Park in the district of Harz, whose main town of Wernigerode lies about 12 km east-northeast of the mountain. The state boundary with Lower Saxony runs past the Brocken some 2 km to the west. At the southeastern foot of the Brocken lies the spa resort of Schierke.

Somewhat to the north below the summit of the Brocken is a reservoir, the Brockenteich, constructed in 1744. On or near the mountain are the source areas of the rivers Bode, Ecker, Ilse and Oder. The rounded summit of the Brocken is treeless, but vegetated with dwarf shrubs.

Summit and subpeaks
The highest point on the Brocken reaches an elevation of (1141.1 m). Its subpeaks include the Heinrichshöhe (1040 m), Königsberg (1034 m) and Kleiner Brocken ("Little Brocken") (1018 m).

Before 1989 the height of the Brocken was recorded in almost all the relevant maps and books as (1142 m). A survey of the summit at the beginning of the 1990s based on the current reference system, however, gave the height as just 1141.1 m. In order to provide a reference point for the old data, in the mid-1990s granite boulders were set on the highest point of the Brocken, which not only matched the old given height, but exceeded it by about a metre. A bench mark of "1142 m" was recorded on the summit stone. This height on the upper plate refers to the line on the lower plate.

Geology
From a geological point of view the Brocken and its surrounding terrain, the Brocken massif, consists mainly of granite (called Brocken granite), an igneous rock. The granitic plutons of the Harz – the Brocken, Ramberg and Oker plutons – emerged towards the end of the Harz mountain-building phase of the Upper Carboniferous, about 300 million years ago. First, alkaline magma intruded into the overlying sediments, crystallized out and formed gabbro and diorite massifs, such as the Harzburg gabbro. A little later, silica-rich granitic magma rose, some intruding into voids and gaps in the older rocks, but most being created by the melting of existing sediments. On the boundary between granite and host rock, the so-called contact zone, a great variety of transitions may be seen. For example, the summit of the Achtermannshöhe consists of contact-metamorphosed hornfels of the contact zone that, here, lies over the Brocken granite. The subsequent erosion of the Harz mountains that followed the uplifting of the Harz during the Upper Cretaceous saw the disappearance of the protective hornfels summit, thus exposing the granite that had crystallized underground during the Upper Carboniferous. The alleged hardness of Brocken granite is not the reason for the height of the mountain, but the geological fact that it was well protected by its weather-resistant hornfels crest for a long time before erosion set in.

Only in recent geological times, since the tertiary period, did the typical, rounded, spheroidal weathering of granite outcrops and granite boulders of the Brocken take place. Such blockfields are very rare in Central Europe outside the Alps and are subject to conservation measures. They originated mainly under periglacial conditions, i.e. during the course of the ice ages, and their retreat. Today's blockfields of Brocken granite, as well as other rocks in the Harz National Park, particularly in the Oker valley, are therefore at least 10,000 years old. Physical weathering, such as frost shattering, has played a key role in their formation, resulting in giant piles of loosely stacked rocks. In 2006, the granite blockfields of the Brocken, together with 76 other interesting geotopes, were designated as a "National Geotope".

Climate
The Brocken is a place of extreme weather conditions. Due to its exposed location in the north of Germany its peak lies above the natural tree line. The climate on the Brocken is like that of the alpine 1600 - 2200 m zone or even that of Iceland. This is due to its short summers and very long winters, with many months of continuous snow cover, strong storms and low temperatures even in summer. The summit, however, does not have an alpine climate, as the average summer temperature is above 10 C.

Due to its significant height difference compared with the surrounding terrain the Brocken has the highest precipitation of any point in northern central Europe, with an average annual precipitation (1961–1990) of 1814 mm. Its average annual temperature is 2.9 C.

The Brocken weather station has recorded the following extreme values: The Brocken also holds the record for the greatest number of days of mist and fog in a single calendar year in Germany, 330 days in 1958, and has an average of 120 days of snowfall per year.
 * Its highest temperature was 29.7 C on 25 July 2019.
 * Its lowest temperature was -28.4 C on 1 February 1956.
 * In 1973 it had 205 days of snow cover.
 * Its greatest depth of snow was 380 cm on 14 and 15 April 1970.
 * Its highest measured wind speed was 263 km/h on 24 November 1984.
 * Its greatest annual precipitation was 2335 mm in 1981.
 * Its least annual precipitation was 984 mm in 1953.
 * The longest annual sunshine was 2004.5 hours in 1921.
 * The shortest annual sunshine was 972.2 hours in 1912.

Flora
The harsh climate of the Brocken makes it a habitat for rare species. The mountain's summit is a subalpine zone with flora and fauna almost comparable to those of north Scandinavia and the Alps. The Brocken is the only mountain in Germany's Central Uplands whose summit lies above the treeline, so that only very small spruce grow there and much of it is covered by a dwarf shrub heathland. In the Brocken Garden, established in 1890, flora are nurtured by national park employees; visitors are allowed to view it as part of regular guided tours. The garden does not just display plants from the Brocken, but also high mountain flora from other regions and countries.

Amongst the typical species of the Brocken that are rarely if ever found elsewhere in North Germany and which occur above about (1050 m) are the variant of the alpine pasqueflower known as the Brocken flower or Brocken anemone (Pulsatilla alpina subsp. alba), hawkweeds like the Brocken hawkweed (Hieracium negrescens) and the alpine hawkweed (Hieracium alpinum), vernal grasses (Anthoxanthum), the lady's mantle (Alchemilla), the tormentil (Potentilla tormentilla), the alpine clubmoss (Diphasiastrum alpinum), the lichens, Iceland moss (Cetraria islandica) and reindeer lichen (Cladonia rangiferina). The crowberry is also referred to here as the Brocken myrtle (Brockenmyrte).

On the raised bogs around the summit of the Brocken there are e.g. cottongrasses, sundews and the dwarf birch (Betula nana).

Fauna
Several animal species have adapted to the conditions of life on the Brocken. For example, the water pipit (Anthus aquaticus) and the ring ouzel both breed in the area around the summit.

The viviparous lizard occurs on the Brocken in a unique, dark-colored variant, ''Lacerta vivipara aberr. negra''. The common frog (Rana temporaria) can also be found here. Insects are very numerous. There are many beetles including ground beetles such as Amara erratica, and hundreds of species of butterfly. The cabbage white here produces only one generation per year compared with two in the lowlands.

Some mammal and bird species that occur here are relics of the ice age, including the northern bat (Eptesicus nils soni), the alpine shrew (Sorex alpinus) and the ring ouzel.

Ascent, construction and use
The first documented ascent of the Brocken was in 1572 by the physician and botanist, Johannes Thal from Stolberg, who in his book Sylva Hercynia described the flora of the mountain area. In 1736 Count Christian Ernst of Stolberg-Wernigerode had the Wolkenhäuschen ("Clouds Cabin") erected at the summit, a small refuge that is still preserved. He also had a mountain lodge built on the southern slope, named Heinrichshöhe after his son Henry (Heinrich) Ernest. The first inn on the Brocken summit was built around 1800.

Between 1821 and 1825 Carl Friedrich Gauss used the line of sight to the Großer Inselsberg in the Thuringian Forest and the Hoher Hagen mountain near Göttingen for triangulation in the course of the geodesic survey of the Kingdom of Hanover. A measurement carried out by the military staff of Prussia in 1850 found the Brocken's height to be at its present level of 1141.1 m. After the first Brocken lodge had been destroyed by a fire, a new hotel opened in 1862. The Brocken Garden, a botanical garden, was laid out in 1890 by Professor Albert Peter of Göttingen University on an area of 4600 m2 granted by Count Otto of Stolberg-Wernigerode. It was Germany's first Alpine garden.

The narrow gauge Brocken Railway was opened on 27 March 1899. Brocken station is one of the highest railway stations in Germany lying at a height of (1125 m). Its gauge is. In 1935 the Deutsche Reichspost made the first television broadcast from the Brocken using a mobile transmitter and, in the following year, the first television tower in the world was built on the mountain; carrying the first live television broadcast of the Summer Olympics in Berlin. The tower continued functioning until September 1939, when the authorities suspended broadcasting on the outbreak of World War II.

In 1937 the Brocken, together with the Wurmberg, Achtermann and Acker-Bruchberg were designated as the Upper Harz (Oberharz) nature reserve.

During an air attack by the United States Army Air Forces on 17 April 1945 the Brocken Hotel and the weather station were destroyed by bombing. The television tower, however, survived. From 1945 until April 1947, the Brocken was occupied by US troops. As part of the exchange of territory (specified at the Yalta Conference) the mountain was transferred to the Soviet occupation zone. Before the Americans left the Brocken in 1947, however, they disabled the rebuilt weather station and the television tower.

The ruins of the Brocken Hotel were blown up in 1949. From 1948 to 1959 part of the Brocken was reopened to tourists. Although a pass was required, these were freely issued. From August 1961 the Brocken, which lay in East Germany's border zone, immediately adjacent to West Germany, was declared a military exclusion zone and was therefore no longer open to public access. Extensive military installations were built on and around the summit. The security of the area was the responsibility of the border guards of the 7th Schierke Border Company, which was stationed in platoon strength on the summit. For accommodation, they used the Brocken railway station. The Soviet Red Army also used a large portion of territory. In 1987, the goods traffic on the Brocken Railway ceased due to poor track conditions.

The Brocken was extensively used for surveillance and espionage purposes. On the summit were two large and powerful listening stations, which could capture radio traffic in almost all of Western Europe. One belonged to Soviet military intelligence, the GRU, and was also the westernmost outpost of the Soviets in Germany; the other was Department III of the Ministry for State Security in the GDR. The listening posts were codenamed "Yenisei" and "Urian". Between 1973 and 1976 a new modern television tower was built for the second channel of the GDR's television service, the Deutscher Fernsehfunk. Today it is used by the public Zweites Deutsches Fernsehen (ZDF) television network. The Stasi (East German secret police) used the old tower until 1985, when they moved to a new building – now a museum. To seal the area, the entire Brocken plateau was then surrounded by a concrete wall, built from 2,318 sections, each one 2.4 tonne in weight and 3.60 m high. The whole area was not publicly accessible until 3 December 1989. The wall has since been dismantled, as have the Russian barracks and the domes of their listening posts. Today the old tower beside the lodge again is home to a weather station of the Deutscher Wetterdienst.

Following the fall of the Berlin Wall, beginning on 3 December 1989 the Brocken was again open to the public during a demonstration walk. With German reunification there was a gradual reduction in border security facilities and military installations from 1990. The last Russian soldier left the Brocken on 30 March 1994. The Brocken summit was renaturalised at a cost of millions of euros. It is now a popular tourist destination for visitors to the Harz.

As a protected area since 1939 and due to the decades of restricted access the unique climate of the Brocken provided outstanding conditions. The massif is partly still covered with primary forest extremely rare in Germany. It provides perfect conditions for endangered and nearly extinct species like the Eurasian lynx, wildcats and capercaillies. The Brocken was therefore declared part of a national park in 1990.

Name and significance
The widespread use of the name "Brocken" did not occur until towards the end of the Middle Ages. Hitherto the region had just been described as the Harz. This was primarily because, until then, the focus had been mining. The first record of a placename that resembles the present name of the mountain goes back, however, to the year 1176 when it is referred to as broke in the Saxon World Chronicle (Sächsische Weltchronik). Another early written reference to the mountain, this time as the Brackenberg, appears in 1490 in a letter from Count Henry of Stolberg. Other early documented names of the Brocken were, in 1401, the Brockenberg, in 1424 the Brocberg, in 1495 mons ruptus (Latin), in 1511 the Brogken and Brockin, in 1531 the Brogken, in 1540 the Brokenberg and, in 1589, the Brackenberg. In Old Saxon-Germanic times, a large portrait of Wodin is supposed to have been found on the Brocken. In addition, animal and human sacrifices were offered by the Saxons to their supreme god, Odin, on the blockfields of the summit until they renounced them as part of their baptismal vows when Christianity spread to the region under Charles the Great.

As far as the origin of the name is concerned, there are several interpretations: In the town records (Stadtbuch) of Osterwieck an entry for the Brocken was found in the year 1495 under the Latin name of mons ruptus, which means "broken hill". Its Low German name, broken, as the mountain had become named in 1176 in the Saxon World Chronicle and also in English, means "broken". On the one hand, this explanation of its meaning can be attributed to the fact that the two mountains, "Kleiner Brocken" and "Großer Brocken", were formed by the breakup of a single massif. On the other hand, its meaning may refer to the serious erosion of the mountain. In other words, it refers to the fact that the Brocken was eroded or "broken down" to its present size.

But the most likely derivation of the name comes from the shape of the mountain as a whole. A brocken in German is a large, shapeless mass. The size of the Brocken may thus have given it its name. Since the term "block" has a similar meaning, this could also be the derivation of its alternative name, the Blocksberg. The true origin of the name Blocksberg, however, should not be seen as "block" in the sense of "mass", but rather the German word block (as in block of wood) in witchcraft.

Another theory holds that the name "Brocken" is derived from bruch, a word used in northern Germany for bog or moor, which commonly used to be spelt as bruoch or brok. It is however doubtful that this fact was primarily responsible for its name. Another possibility is that its name is derived from the fields of boulders strewn over the summit and the slopes of the mountain. This derivation for the name "Brocken" is, however, unlikely because such blockfields are also found on other mountains in the Harz. Moreover, the regions concerned were hardly known at the time when the term was used. Another presumption is based on the reference in a letter written in 1490 by Count Henry of Stolberg-Wernigerode, where he uses the term Brackenberg. However the suggestion that this referred to old, unusable timber, which was called bracken, is disputed.

Tourism
Today a narrow gauge railway, the Brocken Railway, once more shuttles between Wernigerode, Drei Annen Hohne, Schierke and the Brocken. The trains are regularly hauled by steam locomotives.

At the summit is the Brockenhaus with a museum on the history of the mountain and the Brocken Garden (a botanical garden), which is managed by the Harz National Park. In addition there are restaurants and the Brocken Hotel, which is run by the Brocken publican (Brockenwirt), Hans Steinhoff. Important publicans in the past included Johann Friedrich Gerlach from 1801 to 1834, Carl Eduard Nehse between 1834 and 1850, who brought out a map of the Brocken in 1849 and the Brocken Register (Brockenstammbuch) in 1850, as well as Rudolf Schade from 1908 to 1927, who considerably increased the repute and the size of guest facilities on the Brocken. The area around the Brocken is especially popular with hikers. The Goethe Way (Goetheweg) is a well known trail that leads to the summit of the Brocken. It is named after Johann Wolfgang von Goethe, who more-or-less followed this route in 1777. Many paths lead to the local towns of Schierke, Braunlage and Sankt Andreasberg. The 100 km Harz Witches' Path also runs from the Brocken eastwards to Thale and westwards via Torfhaus and Altenau to Osterode. The "Bad Harzburg Devil's Path" runs from the Brocken to Bad Harzburg. Mountain bikers also use the trails.

From Schierke a metalled road leads to the summit, which is used by horse-drawn wagons, as well as touring and racing cyclists. Because of the situation in the national park, vehicles with internal combustion engines are only allowed with special permission.

There are also hiking paths to Brocken from Schierke, Wernigerode and Ilsenburg.

Worthy of special mention is the bearer of the Badge of Honour of Saxony-Anhalt, Benno Schmidt (born 1932) – also known as Brocken Benno – of Wernigerode, who has climbed the mountain since 1989, almost daily, with more than 8,888 ascents (as of September 2020) and whose feat has been registered in the Guinness Book of World Records.

Sports
Two well-known running events pass over the Brocken: the Ilsenburg Brocken Run (beginning of September, 26 km, of which 12 km uphill, has taken place since the 1920s) and the Brocken Marathon which is part of the Harz Mountain Run with its start and finish south of Wernigerode. Both start in the valley, climb the Brocken and return. The most challenging part in each case is the last four kilometres to the Brocken summit, for which in both races, a separate mountains classification is given. This section is a concrete slab track with a steady incline of about 20% and the runners are exposed above the tree line, often to a sharp, icy wind. Of the just under 1,000 people who usually achieve it, only 50 negotiate this section without stopping to walk.

Since 2004, the Brocken Challenge, an ultra marathon 84 km from Göttingen to the Brocken summit, has been staged in February each year. The proceeds from this event go to charity. The runs are conducted in accordance with the rules of the national park.

The 87 km "Brocken Climb" from Göttingen to the Brocken has taken place annually since 2003. More than 300 people take part in these two-day hikes in June.

In early May each year the Braunschweig-Brocken Ultra Run takes place with 2 × 75 km legs spread over two days. The participants run from Braunschweig to Schierke, cross the Brocken, overnight in Schierke and run back again the next day. Overall, it is therefore a 150 km race.

Transmission site
Since the 1930s various radio and television transmitters have been erected on the Brocken, see Brocken Transmitter.

Brocken House
Brocken House (Brockenhaus), the modern information centre for the Harz National Park, is located in the converted "Stasi Mosque" (Stasi-Moschee), a former surveillance installation for the Ministry for State Security. The historic antenna equipment in the dome may be visited. Behind the building is checkpoint No. 9 on the Harzer Wandernadel hiking trail network.

Weather station
The extreme weather conditions of the Brocken are of special meteorological interest. From 1836 the Brockenwirt, who also ran the guest house and restaurant, kept meteorological records. The first weather station on the Brocken was built in 1895. Technically poor and too small, it was partially demolished in 1912 and replaced with a large stone construction, the Hellman Observatory, that was not completed until the First World War. In 1917 the academic and nature lover George Grobe took over running the observation post, his daughter supporting him until his death in 1935. Today's weather station started life in 1939. Measurements were interrupted at the end of the Second World War as a result of military bombardment, but began again in 1947. On 16 March 2010 the Brocken Weather Station became a climate reference station to provide uninterrupted, long-term climatic observations.

Literary mentions

 * Goethe described the Brocken in his Faust, first published in 1808, as the center of revelry for witches on Walpurgisnacht (30 April; the eve of St Walpurga's Day).
 * Now, to the Brocken, the witches ride;
 * The stubble is gold and the corn is green;
 * There is the carnival crew to be seen,
 * And Squire Urianus will come to preside.
 * So over the valleys, our company floats,
 * With witches a-farting on stinking old goats.
 * Goethe may have gained inspiration from two rock formations on the mountain's summit, the Teufelskanzel (Devil's Pulpit) and the Hexenaltar (Witches' Altar).

 * The Brocken is similarly mentioned in many other literary and musical works based on Faust
 * Another famous visitor on the Brocken, author Heinrich Heine, wrote his book Die Harzreise ("The Harz Journey") published in 1826. He says:
 * The mountain somehow appears so Germanically stoical, so understanding, so tolerant, just because it affords a view so high and wide and clear. And should such mountain open its giant eyes, it may well see more than we, who like dwarfs just trample on it, staring from stupid eyes.

 * The summit register entry Many stones, tired bones, views: none, Heinrich Heine ("Viele Steine, müde Beine, Aussicht keine, Heinrich Heine") is a popular, though unsourced phrase related to the weary ascent and the mostly foggy conditions.
 * The teacher Heinrich Pröhle collected the Brockensagen tales and legends as well as the etymology of the geographic names in the Harz. He carefully examined the Teufelskanzel and the Hexenaltar, mentioned above.
 * Henry James has his character Basil Ransom ask: "What kind of meetings do you refer to? You speak as if it were a rendezvous of witches on the Brocken" in the first chapter of his novel The Bostonians (1886).
 * Slothrop and Geli Tripping experience the famous Brocken Spectre in Thomas Pynchon's novel Gravity's Rainbow, as the Mittelbau-Dora labour camp in the Harz mountains north of Nordhausen from 1943 was the home of the V-2 rocket production. In David Foster Wallace's Pynchon-influenced Infinite Jest the characters Remy Marathe and Hugh Steeply also experience the Brocken spectre on a ridge in the desert outside Tucson.

In popular culture

 * "Black Sabbath", the first track of the debut album of the early occult rock band Coven, starts with the line "They journeyed far to Brocken Mountain pinnacle".
 * The progressive metal band Fates Warning titled their debut album Night on Bröcken (note the "Heavy metal umlaut"). The title track refers to Witches Sabbath on Walpurgis Night.
 * The song "Born in a Burial Gown" by Cradle of Filth (from the album Bitter Suites to Succubi) contains an allusion to the Brocken's history as a witches' gathering-place.
 * The indie rock band Liars' album They Were Wrong, So We Drowned is a concept album loosely based on tales of the gatherings of witches on the Brocken as well as witch trials.
 * The novel Cloud Castles by Michael Scott Rohan features the Brocken as the home and body of Chernobog
 * Bibi Blocksberg, a German audio drama for children about a witch, refers to an alternate name for the Brocken (Blocksberg).
 * The Brocken is mentioned in the novel Bald Mountain by Sergej Golovachov.
 * The Brocken is mentioned in episode 546 of the TV anime series Detective Conan.
 * There are two German fictional characters in the anime/manga Kinnikuman who are called Brockenman and Brocken Jr.
 * There is a German black metal band named Brocken Moon.
 * Brocken spectres is the topic of Polish poetic folk band "Na Bani" titled "Brocken" from the album "20 lat z górą".
 * German heavy metal band Edguy mentions Brocken and Walpurgisnacht in their song "Angel Rebellion" from the album Kingdom of Madness.
 * Caleb Carr published "The Legend of Broken" in 2012. The fictional kingdom of Broken occupies the Harz Mountains, and in particular Brocken, which for centuries had been considered the seat of supernatural doings, because, Carr demonstrates, of the ignorance and superstition of humans.

title: Brocken


text:
Mitcham railway station (England)

Mitcham railway station was a railway station in Mitcham in the London Borough of Merton, England. It closed after the last train ran on 31 May 1997. Mitcham tram stop replaced the station, and is located just east of the original site.

History
After the Surrey Iron Railway (SIR) went out of business in 1846, the Wimbledon and Croydon Railway (W&CR) took over the route and opened the station on 22 October 1855. The route was then operated as a conventional railway, until it was closed by Railtrack after the last train on 31 May 1997, for conversion to tram operation. Station Court, on the north of the tram line and east of London Road A217 was originally used as a merchant's home.

title: Mitcham railway station (England)


text:
Talk:The Moonies

Untitled
Who the f keeps changing this article?Tom Breen 00:45, 29 April 2006 (UTC)

I can hardly wait for Google to index this page... --Uncle Ed 21:47 Feb 27, 2003 (UTC)

Fair use rationale for Image:Moonies3.jpg
Image:Moonies3.jpg is being used on this article. I notice the image page specifies that the image is being used under fair use but there is no explanation or rationale as to why its use in this Wikipedia article constitutes fair use. In addition to the boilerplate fair use template, you must also write out on the image description page a specific explanation or rationale for why using this image in each article is consistent with fair use.

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BetacommandBot (talk) 23:30, 2 January 2008 (UTC)

Fair use rationale for Image:Moonieslogo1.jpg
Image:Moonieslogo1.jpg is being used on this article. I notice the image page specifies that the image is being used under fair use but there is no explanation or rationale as to why its use in this Wikipedia article constitutes fair use. In addition to the boilerplate fair use template, you must also write out on the image description page a specific explanation or rationale for why using this image in each article is consistent with fair use.

Please go to the image description page and edit it to include a fair use rationale. Using one of the templates at Fair use rationale guideline is an easy way to insure that your image is in compliance with Wikipedia policy, but remember that you must complete the template. Do not simply insert a blank template on an image page.

If there is other fair use media, consider checking that you have specified the fair use rationale on the other images used on this page. Note that any fair use images lacking such an explanation can be deleted one week after being tagged, as described on criteria for speedy deletion. If you have any questions please ask them at the Media copyright questions page. Thank you.

BetacommandBot (talk) 23:31, 2 January 2008 (UTC)

title: Talk:The Moonies


text:
Talk:Nuclear

Untitled
I don't believe this is a long enough definition for the word Nuclear. It doesn't have enough detail either.

 * Then add the detail instead of complaining. --snoyes 22:03 Feb 27, 2003 (UTC)

I can't believe it. I enter a good site on your article, and I'm accused of vandalism! Thanks a lot Snoyes! — Preceding unsigned comment added by 205.213.111.51 (talk) 15:18, 30 September 2004 (UTC)

 * You can believe it because nuclear is very fantastic 🤞 41.114.142.96 (talk) 16:50, 15 August 2023 (UTC)

 * Well, there were several problems with your addition. Firstly, this is a disambiguation page, not an article page. If you think your link is relevant to one of the topics listed on this page, you should go to that topic, and add the link there. But also, the link you added does not resolve to a real web site. Maybe you mistyped it? BTW, it wasn't snoyes who reverted your edit. Securiger 15:55, 30 Sep 2004 (UTC)

OK.

Pronunciation?
Is the pronunciation really under dispute? Are there any sources for pronouncing it as "new-kyoo-ler"?

It has to be "new-clee-ar", phonetically. "new-kyoo-lar" is just a sort of slang.
 * "Nucular" is how George Bush pronounces it. It's immature and childish. Shandristhe azylean 22:46, 15 August 2006 (UTC)

Granted that the accepted standard pronunciation is "new-clee-ar", that is not what the IPA phonetic transcription in the article now indicates. It needs to be corrected. I am not familiar with the technicalities of using the IPA font. Is there someone out there who could fix it? --A R King 19:33, 18 August 2007 (UTC)

Removing page.
Nuclear is a form of the word nucleus. It only belongs in a dictionary, not in an encyclopedia.David R. Ingham 00:29, 20 January 2006 (UTC)

 * Nuclear is some of a electromagnets 41.114.142.96 (talk) 16:52, 15 August 2023 (UTC)
 * Nuclear power in south Africa Research please 41.114.142.96 (talk) 16:53, 15 August 2023 (UTC)

Pronunc, dispute
Was going to suggest that the now WIDESPREAD ("wrong") nucular pronunciation means that this article needs to mention this debate, but I seem to have been beaten to it. It is elitist vs the common man, it is partisan and political. It is also a bone of contention in UK or USA use of English. The shortened form "nuke" has led to this situation. What a surprise that Bush's enemies are already in evidence here. Some would say that if his pronunciation of certain words is all you can find to complain about then he must be an superb President. We need to mention that the different pronunciations are a cause of substantial argument and mutual antagonism. There aren't many words you can say that about. --81.105.251.160 15:41, 29 September 2006 (UTC)

title: Talk:Nuclear