manifold

man·i·fold

M0081000 (măn′ə-fōld′)adj.1. Many and varied; of many kinds; multiple: our manifold failings.2. Having many features or forms: manifold intelligence.3. Being such for a variety of reasons: a manifold traitor.4. Consisting of or operating several devices of one kind at the same time.n.1. A whole composed of diverse elements.2. One of several copies.3. A pipe or chamber having multiple apertures for making connections.4. Mathematics A topological space in which each point has a neighborhood that is equivalent to a neighborhood in Euclidean space. The surface of a sphere is a two-dimensional manifold because the neighborhood of each point is equivalent to a part of the plane.tr.v. man·i·fold·ed, man·i·fold·ing, man·i·folds 1. To make several copies of, as with carbon paper.2. To make manifold; multiply.

[Middle English, from Old English manigfeald : manig, many; see many + -feald, -fald, -fold.]

man′i·fold′ly adv.man′i·fold′ness n.

manifold

(ˈmænɪˌfəʊld) adj1. of several different kinds; multiple: manifold reasons. 2. having many different forms, features, or elements: manifold breeds of dog. n3. something having many varied parts, forms, or features4. (Printing, Lithography & Bookbinding) a copy of a page, book, etc5. (General Engineering) a chamber or pipe with a number of inlets or outlets used to collect or distribute a fluid. In an internal-combustion engine the inlet manifold carries the vaporized fuel from the carburettor to the inlet ports and the exhaust manifold carries the exhaust gases away6. (Mathematics) maths a. a collection of objects or a setb. a topological space having specific properties7. (Philosophy) (in the philosophy of Kant) the totality of the separate elements of sensation which are then organized by the active mind and conceptualized as a perception of an external objectvb8. (Printing, Lithography & Bookbinding) (tr) to duplicate (a page, book, etc)9. to make manifold; multiply[Old English manigfeald. See many, -fold] ˈmaniˌfolder n ˈmaniˌfoldly adv ˈmaniˌfoldness n

man•i•fold

(ˈmæn əˌfoʊld)

adj. 1. of many kinds; numerous and varied: manifold duties. 2. having numerous different parts, features, or forms: a manifold social program. 3. using or operating similar or identical devices at the same time. 4. being such for many reasons: a manifold enemy. n. 5. something having many different parts or features. 6. a carbon copy; facsimile. 7. a pipe or fitting with several openings for funneling the flow of liquids or gases, as in the exhaust system of an automobile engine. 8. a set of elements having in common a number of topologic properties. adv. 9. very much; in great measure: to multiply burdens manifold. v.t. 10. to make copies of, as with carbon paper. [before 1000; Middle English; Old English manigf(e)ald] man′i•fold`ly, adv. man′i•fold`ness, n. syn: See many.

manifold

Past participle: manifolded
Gerund: manifolding

Imperative
manifold
manifold

Present
I manifold
you manifold
he/she/it manifolds
we manifold
you manifold
they manifold

Preterite
I manifolded
you manifolded
he/she/it manifolded
we manifolded
you manifolded
they manifolded

Present Continuous
I am manifolding
you are manifolding
he/she/it is manifolding
we are manifolding
you are manifolding
they are manifolding

Present Perfect
I have manifolded
you have manifolded
he/she/it has manifolded
we have manifolded
you have manifolded
they have manifolded

Past Continuous
I was manifolding
you were manifolding
he/she/it was manifolding
we were manifolding
you were manifolding
they were manifolding

Past Perfect
I had manifolded
you had manifolded
he/she/it had manifolded
we had manifolded
you had manifolded
they had manifolded

Future
I will manifold
you will manifold
he/she/it will manifold
we will manifold
you will manifold
they will manifold

Future Perfect
I will have manifolded
you will have manifolded
he/she/it will have manifolded
we will have manifolded
you will have manifolded
they will have manifolded

Future Continuous
I will be manifolding
you will be manifolding
he/she/it will be manifolding
we will be manifolding
you will be manifolding
they will be manifolding

Present Perfect Continuous
I have been manifolding
you have been manifolding
he/she/it has been manifolding
we have been manifolding
you have been manifolding
they have been manifolding

Future Perfect Continuous
I will have been manifolding
you will have been manifolding
he/she/it will have been manifolding
we will have been manifolding
you will have been manifolding
they will have been manifolding

Past Perfect Continuous
I had been manifolding
you had been manifolding
he/she/it had been manifolding
we had been manifolding
you had been manifolding
they had been manifolding

Conditional
I would manifold
you would manifold
he/she/it would manifold
we would manifold
you would manifold
they would manifold

Past Conditional
I would have manifolded
you would have manifolded
he/she/it would have manifolded
we would have manifolded
you would have manifolded
they would have manifolded

Thesaurus

Noun	1.	manifold - a pipe that has several lateral outlets to or from other pipesexhaust manifold - a manifold that receives exhaust gases from the cylinders and conducts them to the exhaust pipeinlet manifold - manifold that carries vaporized fuel from the carburetor to the inlet valves of the cylindersintake manifold - a manifold consisting of a pipe to carry fuel to each cylinder in an internal-combustion enginepipage, pipe, piping - a long tube made of metal or plastic that is used to carry water or oil or gas etc.
	2.	manifold - a lightweight paper used with carbon paper to make multiple copies; "an original and two manifolds"manifold paperpaper - a material made of cellulose pulp derived mainly from wood or rags or certain grasses
	3.	manifold - a set of points such as those of a closed surface or an analogue in three or more dimensionsmathematical space, topological space - (mathematics) any set of points that satisfy a set of postulates of some kind; "assume that the topological space is finite dimensional"
Verb	1.	manifold - make multiple copies of; "multiply a letter"re-create, copy - make a replica of; "copy that drawing"; "re-create a picture by Rembrandt"
	2.	manifold - combine or increase by multiplication; "He managed to multiply his profits"multiplyincrease - make bigger or more; "The boss finally increased her salary"; "The university increased the number of students it admitted"double, duplicate - increase twofold; "The population doubled within 50 years"triple, treble - increase threefold; "Triple your income!"quadruple - increase fourfold; "His stock earning quadrupled"quintuple - increase fivefold; "The population of China quintupled"proliferate - cause to grow or increase rapidly; "We must not proliferate nuclear arms"
Adj.	1.	manifold - many and varied; having many features or forms; "manifold reasons"; "our manifold failings"; "manifold intelligence"; "the multiplex opportunities in high technology"multiplexmultiple - having or involving or consisting of more than one part or entity or individual; "multiple birth"; "multiple ownership"; "made multiple copies of the speech"; "his multiple achievements in public life"; "her multiple personalities"; "a pineapple is a multiple fruit"

manifold

adjective (Formal) numerous, many, various, varied, multiple, diverse, multiplied, diversified, abundant, assorted, copious, multifarious, multitudinous, multifold The difficulties are manifold.

Translations

manifold

Formal1. a chamber or pipe with a number of inlets or outlets used to collect or distribute a fluid. In an internal-combustion engine the inlet manifold carries the vaporized fuel from the carburettor to the inlet ports and the exhaust manifold carries the exhaust gases away 2. Mathsa. a collection of objects or a set b. a topological space having specific properties 3. (in the philosophy of Kant) the totality of the separate elements of sensation which are then organized by the active mind and conceptualized as a perception of an external object

Manifold

Component that distributes the water in a home-run plumbing system. It has one inlet and many outlets, each of which feeds one fixture or appliance.

Manifold

a system of devices and equipment for the startup and continuous trouble-free operation of oil and gas wells. It consists of piping and connections, gates, valves, tee joints, fourway pieces, standpipes (risers), shock absorbers, small cocks, compensators, and bypasses. In flowing-well and compressor-lift oil production, the manifold is connected to the Christmas tree mainly by means of flange couplings and ends with the line of pipes supplying the output to the gauging devices.

Manifold

a mathematical concept that refines and generalizes to any number of dimensions the concept of a curve or surface without singular points (that is, curves without points of self-intersection, end points, and the like and surfaces without self-intersections, boundaries, and so forth).

Examples of one-dimensional manifolds are a line, a parabola, a circle, an ellipse, and, in general, any curve at each point of which there exists a neighborhood that is a one-to-one bicontinuous (or, to use topological terminology, homeomorphic) image of an interval (the interior of a line segment). An interval is itself a one-dimensional manifold, although a line segment is not a manifold, since its end points lack such neighborhoods.

Two-dimensional manifolds include any region in the plane (for example, the interior of the circle x² + y² < r²), the plane itself, a paraboloid, a sphere, an ellipsoid, a torus, and others. Each of their points has a neighborhood homeomorphic to the interior of a circle. This requirement eliminates, for example, a conical surface (its vertex, which is common to its two nappes, lacks the type of neighborhood required). However, we distinguish a particular class of objects that do not satisfy this requirement—the manifolds with a boundary (for example, the closed circle x² + y² ≤r²).

Three-dimensional manifolds include ordinary Euclidean space as well as any open set in Euclidean space. A characteristic feature of three-dimensional manifolds is that each point of such a manifold has a neighborhood homeomorphic to the interior of a sphere.

Figure 1. One-dimensional manifolds

Manifolds are either closed or open (see below for definition). In the case of one dimension, every closed manifold is homeomorphic to a circle, and every open manifold, to a line (Figure 1 depicts one-dimensional manifolds and neighborhoods of a point P in each of them). The closed manifolds in the case of two dimensions are already quite varied. They belong to infinitely many topological types: the sphere, which is a surface of genus 0 (Figure 2,a); the torus, which is a surface of genus 1 (Figure 2,b); the two-holed doughnut, which is a surface of genus 2 (Figure 2,c); and, in general, the “sphere with n handles,” which is a surface of genus n (Figure 2,d depicts such a surface for n = 3). All topological types of closed two-dimensional orientable manifolds are exhausted by these examples. There exists an infinite number of closed two-dimensional nonorientable manifolds, namely, the one-sided surfaces, for example, the projective plane and the one-sided torus (Klein bottle).

We also have a classification of open two-dimensional mani-folds. At this time (1974), we have no complete classification of three-dimensional manifolds, or even of closed three-dimensional manifolds.

Figure 2. Examples of closed two-dimensional manifolds

A manifold of n-dimensions (or n-dimensional manifold) is a Hausdorff topological space having the following properties: (1) each point in it has a neighborhood homeomorphic to the interior of an n-dimensional sphere and (2) the entire space can be represented as a sum of a finite or countably infinite set of such neighborhoods. A manifold is said to be closed if it is compact and open if it is not. The requirement that a manifold be connected, that is, that every two points in it can be connected by a continuous arc, is sometimes added to the definition of a manifold.

The concept of a manifold of any (natural) number of dimensions n was introduced into mathematics to meet the highly varied needs of geometry, mathematical analysis, mechanics, and physics. The importance of a sufficiently broad interpretation of a manifold as a topological space is that any kind of object, for example, line, sphere, and matrix, can be points of manifolds thus defined.

A special kind of manifold is a smooth, or differentiable, mani-fold. It is possible to study differentiable functions on a smooth manifold, as well as differentiable mappings of a smooth mani-fold into itself or into other smooth manifolds. Smooth mani-folds are of particular importance in modern mathematics since they are the most widely used in applications and in related fields (for example, configuration spaces and phase spaces in mechanics and physics). It is possible to introduce a metric on a smooth manifold, thus converting it into a Riemannian manifold. On such manifolds it is possible to introduce a differential geometry. For example, by introducing a metric in the configuration space of a mechanical system, it is possible to interpret trajectories of motions as geodetic lines in this space. A manifold for whose elements there is defined a (differentiable) multiplication that makes the manifold into a group is said to be a Lie group.

The concept of manifold plays a major role in the theory of algebraic functions, continuous groups, and other fields. The properties of a manifold that remain invariant under topological transformations (topological properties) are essential in these applications. They include the orientability or nonorientability of a manifold. The study of these properties constitutes one of the most important problems in topology.

REFERENCES

Aleksandrov, P. S., and V. A. Efremovich. Ocherk osnovnykh poniatii topologii. Moscow-Leningrad, 1936.
Aleksandrov, P. S. Kombinatornaia topologiia. Moscow-Leningrad, 1947.
Lang, S. Vvedenie v teoriiu differentiruemykh mnogoobrazii. Moscow, 1967. (Translated from English).

N. V. EFIMOV

manifold

[′man·ə‚fōld] (engineering) The branch pipe arrangement which connects the valve parts of a multicylinder engine to a single carburetor or to a muffler. (mathematics) A topological space which is locally euclidean; there are four types: topological, piecewise linear, differentiable, and complex, depending on whether the local coordinate systems are obtained from continuous, piecewise linear, differentiable, or complex analytic functions of those in euclidean space; intuitively, a surface.

manifold

A section of duct, a fitting, or a pipe with a number of branches which are close together.

manifold

A chamber having a number of outlets for distributing fluids or gases. Examples: fuel manifold, exhaust manifold, hydraulic manifold, inlet manifold, engine manifold, etc. See exhaust manifold and oxygen manifold.

manifold

man·i·fold

manifold

man•i•fold

manifold

manifold

manifold

manifold

Manifold

Manifold

Manifold

REFERENCES

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Synonyms for manifold

adj numerous

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Synonyms for manifold

noun a pipe that has several lateral outlets to or from other pipes

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noun a lightweight paper used with carbon paper to make multiple copies

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noun a set of points such as those of a closed surface or an analogue in three or more dimensions

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verb make multiple copies of

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verb combine or increase by multiplication

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adj many and varied

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