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The **chain** had to be paid for by the **chain** shipping companies themselves and was made of seamless steel links. The individual links were from good, weldable bars with a low carbon content. Depending on the section of the river the bars had a typical thickness of 18 to 27 mm. Despite that, there were frequent breakages. The **chain** had shackles at an interval of several 100 m that could be opened when two **chain** boats met. Most of these high quality chains were made in England or France.

A longer **chain** of 100 ft, with a hundred 1 ft links, was devised in the UK in the late 18th century by Jesse Ramsden, though it never supplanted Gunter's chain. Surveyors also sometimes used such a device, and called it the engineer's chain.

Saw **chain** - **Chain** arrangements

Skip **chain**

Saw **chain** - Joining **chain**

Large scale operators and retail shops may buy bulk **chain** on reels. This must then be cut and joined to length which is done by inserting rivet pins. These non-reusable pins are usually supplied already installed into a half-link and must be peened over against a half-link on the other side. As this peening is done with a bench-mounted rotary tool, rather than hammering, it is referred to as "rivet spinning". The tool is usually hand-cranked, or may be electrically powered for mass production.

Saw **chain** - **Chain** arrangements

Semi-Skip **chain**

The map hd A + d B h is easily verified to induce the zero map on homology, for any h. It immediately follows that f and g induce the same map on homology. One says f and g are **chain** homotopic (or simply homotopic), and this property defines an equivalence relation between **chain** maps.

Uses for **chain** include:

Roller **chain** - **Chain** standards

Typically chains with parallel shaped links have an even number of links, with each narrow link followed by a broad one. Chains built up with a uniform type of link, narrow at one and broad at the other end, can be made with an odd number of links, which can be an advantage to adapt to a special chainwheel-distance, on the other side such a **chain** tends to be not so strong.

Saw **chain** - Joining **chain**

Chains are usually bought ready-joined to length, to suit a particular bar. All chainsaws have adjustable bar mounts to allow their **chain** tightness to be adjusted, allowing for any wear in the **chain** linkages. There is no requirement to remove links to shorten worn chains, chains will wear out on their cutting teeth before wear in their pivots becomes a problem. The adjustment also permits enough slack to allow a **chain** to be installed, so there is no need for a "split link" when fitting, as for bicycles.

A continuous map f between topological spaces X and Y induces a **chain** map between the singular **chain** complexes of X and Y, and hence induces a map f * between the singular homology of X and Y as well. When X and Y are both equal to the n-sphere, the map induced on homology defines the degree of the map f.

Roller **chain** - **Chain** standards

A typical bicycle **chain** (for derailleur gears) uses narrow 1/2" pitch chain. The width of the **chain** is variable, and does not affect the load capacity. The more sprockets at the rear wheel (historically 3-6, nowadays 7-12 sprockets), the narrower the chain. Chains are sold according to the number of speeds they are designed to work with, for example, "10 speed chain". Hub gear or single speed bicycles use 1/2" x 1/8" chains, where 1/8" refers to the maximum thickness of a sprocket that can be used with the chain.

Addition **chain** - Brauer **chain**

A Brauer **chain** or star addition **chain** is an addition **chain** in which one of the summands is always the previous chain: that is,

Saw **chain** - **Chain** arrangements

The terms used to describe **chain** arrangements can be confusing. Most modern chains do not have only cutter teeth and drive links. There are tie straps which separate the cutters from each other.

Roller **chain** - **Chain** strength

The standard minimum ultimate strength of the ANSI 29.1 steel **chain** is 12,500 x (pitch, in inches) 2. X-ring and O-Ring chains greatly decrease wear by means of internal lubricants, increasing **chain** life. The internal lubrication is inserted by means of a vacuum when riveting the **chain** together.

A **chain** map sends cycles to cycles and boundaries to boundaries, and thus induces a map on homology.

A **chain** homotopy offers a way to relate two **chain** maps that induce the same map on homology groups, even though the maps may be different. Given two **chain** complexes A and B, and two **chain** maps f,g : A → B, a **chain** homotopy is a sequence of homomorphisms h n : A n → B n+1 such that hd A + d B h = f − g. The maps may be written out in a diagram as follows, but this diagram is not commutative.

Roller **chain** - **Chain** strength

The most common measure of roller chain's strength is tensile strength. Tensile strength represents how much load a **chain** can withstand under a one-time load before breaking. Just as important as tensile strength is a chain's fatigue strength. The critical factors in a chain's fatigue strength is the quality of steel used to manufacture the chain, the heat treatment of the **chain** components, the quality of the pitch hole fabrication of the linkplates, and the type of shot plus the intensity of shot peen coverage on the linkplates. Other factors can include the thickness of the linkplates and the design (contour) of the linkplates. The rule of thumb for roller **chain** operating on a continuous drive is for the **chain** load to not exceed a mere 1/6 or 1/9 of the chain's tensile strength, depending on the type of master links used (press-fit vs. slip-fit). Roller chains operating on a continuous drive beyond these thresholds can and typically do fail prematurely via linkplate fatigue failure.

A **chain** map f between two **chain** complexes and is a sequence f_\bullet of homomorphisms for each n that commutes with the boundary operators on the two **chain** complexes, so. This is written out in the following commutative diagram. :

In India, a revenue **chain** with 16 links and of length is used in cadastral surveys.

Addition **chain** - **Chain** length

Let l(n) denote the smallest s so that there exists an addition **chain** of length s which computes n. It is known that :,where \nu(n) is Hamming weight (the number of ones) of the binary expansion of n.

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