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pieces and decomposition
It is equivalent to saying that the prime decomposition of the manifold has no acyclic components, and turns out to be equivalent to the condition that all geometric pieces of the manifold have geometries based on the two Thurston geometries S < sup > 2 </ sup >× R and S < sup > 3 </ sup >.
Laczkovich's decomposition uses about 10 < sup > 50 </ sup > different pieces.
The Bolyai-Gerwien theorem is a related but much simpler result: it states that one can accomplish such a decomposition of a simple polygon with finitely many polygonal pieces if both translations and rotations are allowed for the reassembly.
There is a unique minimal way of cutting an irreducible oriented 3-manifold along tori into pieces that are Seifert manifolds or atoroidal called the JSJ decomposition, which is not quite the same as the decomposition in the geometrization conjecture, because some of the pieces in the JSJ decomposition might not have finite volume geometric structures.
It is possible to choose a " canonical " decomposition into pieces with geometric structure, for example by first cutting the manifold into prime pieces in a minimal way, then cutting these up using the smallest possible number of tori.
However this minimal decomposition is not necessarily the one produced by Ricci flow ; if fact, the Ricci flow can cut up a manifold into geometric pieces in many inequivalent ways, depending on the choice of initial metric.
Roughly speaking, the Ricci flow contracts positive curvature regions and expands negative curvature regions, so it should kill off the pieces of the manifold with the " positive curvature " geometries S < sup > 3 </ sup > and S < sup > 2 </ sup > × R, while what is left at large times should have a thick-thin decomposition into a " thick " piece with hyperbolic geometry and a " thin " graph manifold.
Unlike the generalized solution to Tarski's circle-squaring problem, the axiom of choice is not required for the proof, and the decomposition and reassembly can actually be carried out " physically "; the pieces can, in theory, be cut with scissors from paper and reassembled by hand.
Warning: the JSJ decomposition is not quite the same as the decomposition in the geometrization conjecture, because some of the pieces in the JSJ decomposition might not have finite volume geometric structures.
Early in 1981, Yau suggested to Richard Hamilton that he use the Ricci flow to realize naturally the canonical decomposition of a three-dimensional manifold into pieces, each of which has a geometric structure, in the Thurston program.
That is, it is an algebra over a commutative ring or field with a decomposition into " even " and " odd " pieces and a multiplication operator that respects the grading.
An abstract definition of ( real ) Hodge structure is now given: for a real vector space W, a Hodge structure of integer weight k on W is a direct sum decomposition of W < sup > C </ sup > = W ⊗ C, the complexification of W, into graded pieces W < sup > p, q </ sup > where k = p + q, and the complex conjugation of W < sup > C </ sup > interchanges this subspace with W < sup > q, p </ sup >.
This is essentially the content of Hilbert's third problem – more precisely, not all polyhedral pyramids are scissors congruent ( can be cut apart into finite pieces and rearranged into the other ), and thus volume cannot be computed purely by using a decomposition argument.
In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces.
The decomposition of a ( meromorphic ) elliptic function into pieces of ' three kinds ' parallels the representation as ( i ) a constant, plus ( ii ) a linear combination of translates of the Weierstrass zeta function, plus ( iii ) a function with arbitrary poles but no residues at them.
Manifold decomposition works in two directions: one can start with the smaller pieces and build up a manifold, or start with a large manifold and decompose it.

pieces and constructed
Roofs are constructed from Alpine rocks such as pieces of schist, gneiss or slate.
Items such as billiard balls, dominoes and pieces for games like chess, checkers, and backgammon are constructed of Bakelite for its look, durability, fine polish, weight, and sound.
It was built in the Ionic order and consists of seven fluted columns, unusually carved from single pieces of stone ( most columns were constructed from a series of discs joined together ).
Viols were first constructed much like the vihuela de mano, with all surfaces, top, back, and sides made from flat slabs or pieces of joined wood, bent or curved as required.
It was constructed of four pieces ( the dome and three lateral bands ) and held together by 1, 800 buttons.
Several large pieces of sculpture found on Fermilab and designed by Wilson include Tractricious, a free-standing arrangement of steel tubes near the Industrial Complex constructed from parts and materials recycled from the Tevatron collider, and the soaring Broken Symmetry, which greets those entering the campus via the Pine Street entrance.
The larvae feed for several years on rotting deciduous wood, growing through three larval stages until eventually pupating inside a pupal cell constructed from surrounding wood pieces and soil particles.
For more modern pieces the pattern was constructed carefully on paper, in much the same way as cross-stitch patterns are created.
It was hoped that a grand epic could be constructed using pieces preserved in folklore.
When visited by Oglethorpe in February the settlers had already constructed " a battery of four pieces of cannon, built a guardhouse, a storehouse, a chapel, and several huts for particular people ".
In the 1860s during the American Civil War, housing for a battery of four field artillery pieces was constructed at the tip of the peninsula.
The Scottish heraldic term for an X-shaped cross is a ' saltire ', from the old French word saultoir or salteur ( itself derived from the Latin saltatorium ), a word for both a type of stile constructed from two cross pieces and a type of cross-shaped stirrup-cord.
One of the most effective types of RAM comprises arrays of pyramid shaped pieces, each of which is constructed from a suitably lossy material.
The, or neck of the shamisen is usually constructed such that it is divided into three or four pieces that fit and lock together.
In 1912, the Danish government constructed the " Flak Fort " on the Salthom Flak sands just north of the island proper, stationing a number of artillery pieces ranging in calibre from 47 mm to 290 mm.
Since Gallatin had based his principal proposals on the known advantageous natural geographic features of the country, many of his proposals were the locations of future navigation improvements surveyed, authorized and constructed starting with the 1824 General Survey Act and the first of many pieces of rivers and harbors legislation, as well by individual state-built improvements.
The nest of the Long-tailed Tit is constructed from four materials-lichen, feathers, spider egg cocoons and moss, over 6000 pieces in all for a typical nest.
The following year the college was granted pieces of land upon which the college farm and the Lake Kariba Research Station were constructed.
The rooms of the chancery often had walls full of pigeonholes, constructed to hold rolled up pieces of parchment for safekeeping or ready reference, a precursor to the book shelf.
It was constructed at the price of 2000 pieces of gold ( zlatých ) at the expense of cardinal Péter Pázmány
Aluminum is less costly than tin-plated steel but offers the same resistance to corrosion in addition to greater malleability, resulting in ease of manufacture ; this gave rise to the two-piece can, where all but the top of the can is simply stamped out of a single piece of aluminium, rather than laboriously constructed from two pieces of steel.
The wall lattice of a ger is constructed of straight pieces as opposed to the yurt's curved lattice.
* The narthex contains two stained glass windows constructed from over 8, 000 pieces of glass collected from England, France and Germany.
Although the chambers are paved with slabs of masonry the walls are constructed of pieces of rock roughly-formed into bricks.

pieces and using
One example is the Banach – Tarski paradox which says that it is possible to decompose (" carve up ") the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original.
The latter two completed his design for an altarpiece of the Vision of Saint Nicholas ( San Nicola da Tolentino, Rome ) using two separate marble pieces linked together in one event and place, yet successfully separating the divine and earthly spheres.
The plant is normally grown in well-drained loamy soil, using pieces of mature root planted at the start of the rainy season.
After the Second World War, factories were retrofitted to produce Bakelite using a more efficient extrusion process which increased production and enabled the uses of Bakelite to extend into other genres: jewelry boxes, desk sets, clocks, radios, game pieces like chessmen, poker chips, billiard balls and Mah Jong sets.
The objective is to be first to race one's pieces across the hexagram-shaped gameboard into " home "— the corner of the star opposite one's starting corner — using single-step moves or moves which jump over other pieces.
Computer controlled music is also found in the performance pieces by the Canadian composer Udo Kasemets such as the Marce ( ntennia ) l Circus C ( ag ) elebrating Duchamp ( 1987 ), a realization of the Marcel Duchamp process piece Erratum Musical using an electric model train to collect a hopper-car of stones to be deposited on a drum wired to an Analog: Digital converter, mapping the stone impacts to a score display ( performed in Toronto by pianist Gordon Monahan during the 1987 Duchamp Centennial ), or his installations and performance works ( e. g. Spectrascapes ) based on his Geo ( sono ) scope ( 1986 ) 15x4-channel computer-controlled audio mixer.
; Virtual Execution System ( VES ): The VES loads and executes CLI-compatible programs, using the metadata to combine separately generated pieces of code at runtime.
Most crochet uses far less than 1 / 3 more yarn than knitting for comparable pieces, and a crocheter can get similar feel and drape to knitting by using a larger hook or thinner yarn.
These shows have featured large collections of his gridfonts, his ambigrams ( pieces of calligraphy created with two readings, either of which is usually obtained from the other by rotating or reflecting the ambigram, but sometimes simply by " oscillation ", like the Necker Cube or the rabbit / duck figure of Joseph Jastrow ), and his " Whirly Art " ( music-inspired visual patterns realized using shapes based on various alphabets from India ).
Ethiopians eat with their right hands, using pieces of injera to pick up bites of entrées and side dishes.
These are made using assortments of different pieces of quest material found in the game.
Before the widespread use of non-linear editing systems, the initial editing of all films was done with a positive copy of the film negative called a film workprint ( cutting copy in UK ) by physically cutting and pasting together pieces of film, using a splicer and threading the film on a machine with a viewer such as a Moviola, or " flatbed " machine such as a K .- E .- M. or Steenbeck.
FIG has a policy of only using four of the five pieces of apparatus and changes them for different Olympic cycles.
The Art Nouveau movement in particular made great use of glass, with René Lalique, Émile Gallé, and Daum of Nancy important names in the first French wave of the movement, producing colored vases and similar pieces, often in cameo glass, and also using lustre techniques.
The idea of using computers to search for relevant pieces of information was popularized in the article As We May Think by Vannevar Bush in 1945.
Medea, using her sorcery, claimed to Pelias ' daughters that she could make their father younger by chopping him up into pieces and boiling the pieces in a cauldron of water and magical herbs.
Ornamental pieces may also be knitted separately and then attached using applique.
Mies designed modern furniture pieces using new industrial technologies that have become popular classics, such as the Barcelona chair and table, the Brno chair, and the Tugendhat chair.
It was advantageous for NASA – even before the Shuttle was scheduled to be retired in 2010 – to keep using many pieces of 1970s technology rather than to upgrade those systems.
A " smith " of any type is one who shapes metal pieces, often using a forge or mould, into useful objects or to be part of a more complex structure.
A program interpreting such structural markup may apply its own rules or styles for presenting the various pieces of text, using different typefaces, boldness, font size, indentation, colour, or other styles, as desired.

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