The Sorbian languages are two related, but only mutually intelligible, West Slavic languages spoken by the Sorbs, a West Slavic minority in the Lusatia region of eastern Germany. They are classified under the West Slavic branch of the Indo-European languages and are therefore related to the other two West Slavic subgroups: Lechitic and Czech–Slovak; the languages have been known as Wendish or Lusatian. Their collective ISO 639-2 code is wen; the two Sorbian languages and literary standards are Upper Sorbian, spoken by about 40,000 people in Saxony, Lower Sorbian spoken by about 10,000 people in Brandenburg. The area where the two languages are spoken is known as Lusatia. After the settlement of the Germanic territories by the Sorbs' Slavic ancestors in the fifth and sixth centuries, the Sorbian language had been in use in much of what was the southern half of East Germany for several centuries, still had its stronghold in Lusatia, where it enjoys national protection and fostering to the present day.

Outside Lusatia, it has been superseded by German. From the 13th century on, the language suffered official discrimination. Bible translations into Sorbian provided the foundations for its writing system. In Germany and Lower Sorbian are recognized and protected as minority languages. In the home areas of the Sorbs, both languages are recognized as second official languages next to German; the city of Bautzen in Upper Lusatia is the centre of Upper Sorbian culture. Bilingual signs can be seen around the city, including the name of the city, "Bautzen/Budyšin"; the city of Cottbus is considered the cultural centre of Lower Sorbian. Sorbian has been spoken in the small Sorbian settlement of Serbin in Lee County, a few speakers still remain there; until 1949, newspapers were published in Sorbian there. The local dialect has been influenced by surrounding speakers of German and English; the German terms "Wends" and "Wendish" once denoted "Slav" generally. Both Upper and Lower Sorbian have the dual for nouns, pronouns and verbs.

For example, the word ruka is used for one hand, ruce for two hands, ruki for more than two hands. As with most of the Slavic languages, Sorbian uses no articles; the Sorbian languages are declined in six or seven cases: Nominative Accusative Dative Genitive Instrumental Locative Vocative The following is selected vocabulary from the two Sorbian languages compared with other Slavic languages. Sorbian alphabet Wends List of Sorbian-language writers Low Lusatian German White Serbia Euromosaic information page Kurs serskeje rěce / Bluń, introductory texts of the lessons included in the Sorbian language textbook Curs practic de limba sorabă

In algebraic geometry, the Iitaka dimension of a line bundle L on an algebraic variety X is the dimension of the image of the rational map to projective space determined by L. This is 1 less than the dimension of the section ring of L R = ⨁ d = 0 ∞ H 0; the Iitaka dimension of L is always less than or equal to the dimension of X. If L is not effective its Iitaka dimension is defined to be − ∞ or said to be negative; the Iitaka dimension of L is sometimes called L-dimension, while the dimension of a divisor D is called D-dimension. The Iitaka dimension was introduced by Shigeru Iitaka. A line bundle is big if it is of maximal Iitaka dimension, that is, if its Iitaka dimension is equal to the dimension of the underlying variety. Bigness is a birational invariant: If f: Y → X is a birational morphism of varieties, if L is a big line bundle on X f*L is a big line bundle on Y. All ample line bundles are big. Big line bundles need not determine birational isomorphisms of X with its image. For example, if C is a hyperelliptic curve its canonical bundle is big, but the rational map it determines is not a birational isomorphism.

Instead, it is a two-to-one cover of the canonical curve of C, a rational normal curve. The Iitaka dimension of the canonical bundle of a smooth variety is called its Kodaira dimension. Consider on complex algebraic varieties in the following. Let K be the canonical bundle on M; the dimension of H0, holomorphic sections of Km, is denoted by called m-genus. Let N = N becomes to be all of the positive integer with non-zero m-genus; when N is not empty, for m ∈ N m-pluricanonical map Φ m K is defined as the map Φ m K: M ⟶ P N z ↦ where φ i are the bases of H0. The image of Φ m K, Φ m K is defined as the submanifold of P N. For certain m let Φ m k: M → W = Φ m K ⊂ P N be the m-pluricanonical map where W is the complex manifold embedded into projective space PN. In the case of surfaces with κ=1 the above W is replaced by a curve C, an elliptic curve. We want to extend this fact to the general dimension and obtain the analytic fiber structure depicted in the upper right figure. Given a birational map φ: M ⟶ W, m-pluricanonical map brings the commutative diagram depicted in the left figure, which means that Φ m K = Φ m K, i.e. m-pluricanonical genus is birationally invariant.

It is shown by Iitaka that given n-dimensional compact complex manifold M with its Kodaira dimension κ satisfying 1 ≤ κ ≤ n-1, there are enough large m1,m2 such that Φ m 1 K: M ⟶ W m 1 and Φ m 2 K: M ⟶ W m 2 are birationally equivalent, which means there are the birational map φ: W m 1 ⟶ W m 2. Namely, the diagram depicted in the right figure is commutative. Furthermore, one can select M ∗ {\displaystyle

Signalling block systems enable the safe and efficient operation of railways by preventing collisions between trains. The basic principle is that a route is broken up into a series of blocks, only one train may occupy a block at a time, that the blocks are sized to allow a train to stop within them; this ensures. A block system is referred to as the method of working in the UK, method of operation in the US and safeworking in Australia. In most examples, a system of signals is used to control flow between the blocks; when a train enters a block, signals at both ends change to indicate that the block is occupied using red lamps or indicator flags. When a train first enters a block, the rear of the same train has not yet left the previous block, so both blocks are marked as occupied; this ensures there is less than one block length on either end of the train, marked as occupied, so any other train approaching this section will have enough room to stop in time if the first train stops dead on the tracks.

The occupied block will only be marked unoccupied when the end of the train has left it, leaving the entire block clear. Block systems have the disadvantage that they limit the number of trains on a particular route to something smaller than the number of blocks. Since the route has a fixed length, increasing the number of trains requires more blocks, which means the blocks are shorter, which means the trains have to operate at lower speeds in order to safely stop; as a result, the number and size of blocks are related to the route's overall route capacity and cannot be changed without changes to the signals all along the line. Block systems are used to control trains between stations and yards, not within the yards, where some other method will be used. Any block system is defined by its associated physical equipment and by the application of a relevant set of rules; some systems involve the use of signals. Some systems are designed for single track railways for which a danger exists of both head-on and rear-end collision, as opposed to double track, whose main danger is a rear-end collision.

The basic problem for train control is the long stopping distances of a loaded train. This is far longer than the operator's eyesight at night or in bad weather; the distances are great enough that local terrain may block sighting of trains ahead, the routing of the rails, around bends and such, may make it difficult to know where to look for another train. This leads to the possibility that a train may break down on the tracks, the following train comes upon it when rounding a bend, or sees its rear signal lamp. In these situations there will not be enough room for the train to stop; this is known as the "brick wall criterion". In the case of two operational trains, differences in speed may be great enough that a faster train may not have time to slow down to match the speed of the one in front before it overtakes it. Blocks avoid these problems by ensuring there is a certain minimum distance between trains, a distance, set to ensure any train operating within the speed and load limits will have time to stop before reaching a train ahead of it.

There are many potential solutions to implementing such a system. Most rail routes have a sort of natural block layout inherent in the layout of the railway stations; this provides the ability to implement a set of blocks using manual signalling based at these locations. In this case, the station operator places a flag indicating a train has just left the station, removes it only after a fixed time. Trains operate according to a strict timetable, as such, cannot leave a station until an appointed time, until any other trains they were to meet at that station have arrived. If one train is delayed, all trains it is scheduled to meet are delayed; this can lead to all trains on the railway being affected. This method is not authorised for use in many high-traffic railway systems Popular on single track lines in North America up until the 1980s, Train Order operation was less a block system and more of a system of determining which trains would have the right of way when train movements would come into conflict.

Trains would make use of a predetermined operating plan known as the timetable which made use of fixed passing locations referred to as stations. Amendments to the operating plan would come from a train dispatcher in the form of train orders, transmitted to the trains via intermediaries known as agents or operators at train order stations; this method is not authorised for use in the UK. A similar system, known as Telegraph and Crossing Order, was used in the 19th century, but after three serious head-on collisions in the 1870s its use was condemned. In North American train order system was implemented on top of other block systems when those block systems needed to be superseded. For example, where manual or automatic block was implemented, train orders would be used to authorize movements into occupied blocks, against the current of traffic or where no current of traffic was established. If a single track branch line is a dead end with a simple shuttle train service a single token is sufficient.

The driver of any train entering the branch line must be in possession of the token, no collision with another train is possible. For convenience in passing it from hand to hand, the token was in the form of a staff 800 mm long and 40 mm diameter, i