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s article is about mathematics. For other uses, see Isomorphism (disambiguation). The group of fifth roots of unity under multiplication is isomorphic to the group of rotations of the regular pentagon under composition. <span>In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that admits an inverse. [note 1] Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences. For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if and only if it is bijective. In topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or bicontinuous functions. In mathematical analysis, where the morphisms are differentiable functions, isomorphisms are also called diffeomorphisms. A canonical isomorphism is a canonical map that is an isomorphism. Two objects are said to be canonically isomorphic if there is a canonical isomorphism between them. For example, the canonical map from a finite-dimensional vector space V to its second dual space is a canonical isomorphism; on the other hand, V is isomorphic to its dual space but not canonically in general. Isomorphisms are formalized using category theory. A morphism f : X → Y in a category is an isomorphism if it admits a two-sided inverse, meaning that there is another morphism g : Y → X in that category such that gf = 1 X and fg = 1 Y , where 1 X and 1 Y are the identity morphisms of X and Y, respectively. [1] Contents 1 Examples 1.1 Logarithm and exponential 1.2 Integers modulo 6 1.3 Relation-preserving isomorphism 2 Isomorphism vs. bijective morphism 3 Applications 4 Relation with equ

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and exponential 1.2 Integers modulo 6 1.3 Relation-preserving isomorphism 2 Isomorphism vs. bijective morphism 3 Applications 4 Relation with equality 5 See also 6 Notes 7 References 8 Further reading 9 External links Examples[edit] <span>Logarithm and exponential[edit] Let be the multiplicative group of positive real numbers, and let be the additive group of real numbers. The logarithm function satisfies for all , so it is a group homomorphism. The exponential function satisfies for all , so it too is a homomorphism. The identities and show that and are inverses of each other. Since is a homomorphism that has an inverse that is also a homomorphism, is an isomorphism of groups. Because is an isomorphism, it translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to multiply real numbers using a ruler and a table of logarithms, or using a slide rule with a logarithmic scale. Integers modulo 6[edit] Consider the group , the integers from 0 to 5 with addition modulo 6. Also consider the group , the ordered pairs where the x coordinates can be 0 or 1, and the

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The endothelium is: 1. Anti-inflammatory 2. Anti‐thrombotic 3. Anti‐proliferative 4. Anti‐oxidative 5. Vasodilatory 6. Selectively permeable

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The endothelium is: 1. Anti-inflammatory 2. Anti‐thrombotic 3. Anti‐proliferative 4. Anti‐oxidative 5. Vasodilatory 6. Selectively permeable

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The endothelium is: 1. Anti-inflammatory 2. Anti‐thrombotic 3. Anti‐proliferative 4. Anti‐oxidative 5. Vasodilatory 6. Selectively permeable

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The endothelium is: 1. Anti-inflammatory 2. Anti‐thrombotic 3. Anti‐proliferative 4. Anti‐oxidative 5. Vasodilatory 6. Selectively permeable

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The endothelium is: 1. Anti-inflammatory 2. Anti‐thrombotic 3. Anti‐proliferative 4. Anti‐oxidative 5. Vasodilatory 6. Selectively permeable

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The endothelium is: 1. Anti-inflammatory 2. Anti‐thrombotic 3. Anti‐proliferative 4. Anti‐oxidative 5. Vasodilatory 6. Selectively permeable

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Widowmaker From Overwatch Wiki Jump to: navigation, search Widowmaker "One shot, one kill." Real Name <span>Amélie Lacroix Age 33 Nationality French Occupation Assassin Base of Operations Annecy, France Affiliation Talon Relations Gérard Lacroix (husband, deceased) Quotes

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Widowmaker From Overwatch Wiki Jump to: navigation, search Widowmaker "One shot, one kill." Real Name <span>Amélie Lacroix Age 33 Nationality French Occupation Assassin Base of Operations Annecy, France Affiliation Talon Relations Gérard Lacroix (husband, deceased) Quotes

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of her targets through walls and objects for a moderate amount of time. This enhanced vision is shared with her allies. Ability Breakdown[edit | edit source] Ability Ammo Fire Rate Damage Headshot Duration Reload/Cooldown <span>Widow's Kiss: Assault 30 10 rps 13 ✓ - 1.5s Widow's Kiss: Sniper 30 (3 per shot) 1 rps 17 - 150 ✓ - 1.5s Grappling Hook - - - - - 12s Venom Mine - - 15 dps(75 total)

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of her targets through walls and objects for a moderate amount of time. This enhanced vision is shared with her allies. Ability Breakdown[edit | edit source] Ability Ammo Fire Rate Damage Headshot Duration Reload/Cooldown <span>Widow's Kiss: Assault 30 10 rps 13 ✓ - 1.5s Widow's Kiss: Sniper 30 (3 per shot) 1 rps 17 - 150 ✓ - 1.5s Grappling Hook - - - - - 12s Venom Mine - - 15 dps(75 total) - 5s

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cts for a moderate amount of time. This enhanced vision is shared with her allies. Ability Breakdown[edit | edit source] Ability Ammo Fire Rate Damage Headshot Duration Reload/Cooldown Widow's Kiss: Assault 30 10 rps <span>13 ✓ - 1.5s Widow's Kiss: Sniper 30 (3 per shot) 1 rps 17 - 150 ✓ - 1.5s Grappling Hook - - - - - 12s Venom Mine - - 15 dps(75 total) - 5s 15s

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erate amount of time. This enhanced vision is shared with her allies. Ability Breakdown[edit | edit source] Ability Ammo Fire Rate Damage Headshot Duration Reload/Cooldown Widow's Kiss: Assault 30 10 rps 13 ✓ - <span>1.5s Widow's Kiss: Sniper 30 (3 per shot) 1 rps 17 - 150 ✓ - 1.5s Grappling Hook - - - - - 12s Venom Mine - - 15 dps(75 total) - 5s 15s Infra-Sight