Misplaced Pages

#33966

31-838: Carnival Sensation , initially named Sensation , was a 70,367  GT cruise ship operated by Carnival Cruise Line . She was built in 1993 in Finland and cruised from Florida, USA to ports in the Bahamas and the Caribbean until the COVID-19 pandemic stopped operations in March 2020. The ship was subsequently sold for scrapping. Sensation was the third Fantasy -class cruise ship operated by Carnival. Built by Kværner Masa-Yards at its Helsinki New Shipyard in Helsinki , Finland, she

62-451: A {\displaystyle a} in the domain of f {\displaystyle f} and setting g ( y ) {\displaystyle g(y)} to the unique element of the pre-image f − 1 [ y ] {\displaystyle f^{-1}[y]} (if it is non-empty) or to a {\displaystyle a} (otherwise). The left inverse g {\displaystyle g}

93-434: A = b , {\displaystyle \forall a,b\in X,\;\;f(a)=f(b)\Rightarrow a=b,} which is logically equivalent to the contrapositive , ∀ a , b ∈ X , a ≠ b ⇒ f ( a ) ≠ f ( b ) . {\displaystyle \forall a,b\in X,\;\;a\neq b\Rightarrow f(a)\neq f(b).} An injective function (or, more generally,

124-426: A = b . {\displaystyle a=b.} Equivalently, if a ≠ b , {\displaystyle a\neq b,} then f ( a ) ≠ f ( b ) {\displaystyle f(a)\neq f(b)} in the contrapositive statement. Symbolically, ∀ a , b ∈ X , f ( a ) = f ( b ) ⇒

155-448: A function f {\displaystyle f} is injective depends on how the function is presented and what properties the function holds. For functions that are given by some formula there is a basic idea. We use the definition of injectivity, namely that if f ( x ) = f ( y ) , {\displaystyle f(x)=f(y),} then x = y . {\displaystyle x=y.} Here

186-497: A mathematical formula. Gross tonnage is based on "the moulded volume of all enclosed spaces of the ship" whereas net tonnage is based on "the moulded volume of all cargo spaces of the ship". In addition, a ship's net tonnage is constrained to be no less than 30% of her gross tonnage. The gross tonnage calculation is defined in Regulation 3 of Annex 1 of The International Convention on Tonnage Measurement of Ships, 1969 . It

217-500: A monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see Homomorphism § Monomorphism for more details. A function f {\displaystyle f} that is not injective is sometimes called many-to-one. Let f {\displaystyle f} be a function whose domain is a set X . {\displaystyle X.} The function f {\displaystyle f}

248-548: A monomorphism) is often denoted by using the specialized arrows ↣ or ↪ (for example, f : A ↣ B {\displaystyle f:A\rightarrowtail B} or f : A ↪ B {\displaystyle f:A\hookrightarrow B} ), although some authors specifically reserve ↪ for an inclusion map . For visual examples, readers are directed to the gallery section. More generally, when X {\displaystyle X} and Y {\displaystyle Y} are both

279-565: Is a function f that maps distinct elements of its domain to distinct elements of its codomain; that is, x 1 ≠ x 2 implies f ( x 1 ) ≠ f ( x 2 ) (equivalently by contraposition , f ( x 1 ) = f ( x 2 ) implies x 1 = x 2 ). In other words, every element of the function's codomain is the image of at most one element of its domain . The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions , which are functions such that each element in

310-443: Is a function g : Y → X {\displaystyle g:Y\to X} such that for every x ∈ X {\displaystyle x\in X} , g ( f ( x ) ) = x {\displaystyle g(f(x))=x} , then f {\displaystyle f} is injective. In this case, g {\displaystyle g}

341-454: Is a linear transformation it is sufficient to show that the kernel of f {\displaystyle f} contains only the zero vector. If f {\displaystyle f} is a function with finite domain it is sufficient to look through the list of images of each domain element and check that no image occurs twice on the list. A graphical approach for a real-valued function f {\displaystyle f} of

SECTION 10

#1732772185034

372-605: Is an example: f ( x ) = 2 x + 3 {\displaystyle f(x)=2x+3} Proof: Let f : X → Y . {\displaystyle f:X\to Y.} Suppose f ( x ) = f ( y ) . {\displaystyle f(x)=f(y).} So 2 x + 3 = 2 y + 3 {\displaystyle 2x+3=2y+3} implies 2 x = 2 y , {\displaystyle 2x=2y,} which implies x = y . {\displaystyle x=y.} Therefore, it follows from

403-408: Is based on two variables, and is ultimately an increasing one-to-one function of ship volume: The value of the multiplier K increases logarithmically with the ship's total volume (in cubic metres) and is applied as an amplification factor in determining the gross tonnage value. K is calculated with a formula which uses the common or base-10 logarithm : Once V and K are known, gross tonnage

434-543: Is bijective. Indeed, f {\displaystyle f} can be factored as In J , Y ∘ g , {\displaystyle \operatorname {In} _{J,Y}\circ g,} where In J , Y {\displaystyle \operatorname {In} _{J,Y}} is the inclusion function from J {\displaystyle J} into Y . {\displaystyle Y.} More generally, injective partial functions are called partial bijections . A proof that

465-489: Is calculated using the formula, whereby GT is a function of V: which by substitution is: Thus, gross tonnage exhibits linearithmic growth with volume, increasing faster at larger volumes. The units of gross tonnage, which involve both cubic metres and log-metres, have no physical significance, but were rather chosen for historical convenience. Since gross tonnage is a bijective function of ship volume, it has an inverse function , namely ship volume from gross tonnage, but

496-421: Is called a retraction of f . {\displaystyle f.} Conversely, f {\displaystyle f} is called a section of g . {\displaystyle g.} Conversely, every injection f {\displaystyle f} with a non-empty domain has a left inverse g {\displaystyle g} . It can be defined by choosing an element

527-424: Is not necessarily an inverse of f , {\displaystyle f,} because the composition in the other order, f ∘ g , {\displaystyle f\circ g,} may differ from the identity on Y . {\displaystyle Y.} In other words, an injective function can be "reversed" by a left inverse, but is not necessarily invertible , which requires that

558-476: Is said to be injective provided that for all a {\displaystyle a} and b {\displaystyle b} in X , {\displaystyle X,} if f ( a ) = f ( b ) , {\displaystyle f(a)=f(b),} then a = b {\displaystyle a=b} ; that is, f ( a ) = f ( b ) {\displaystyle f(a)=f(b)} implies

589-620: The Carnival Sensation undertook cruises from Port Canaveral . During 2007, in common with all of her Fantasy -class sisters, she had the prefix Carnival added to her name. In early 2016, Carnival Sensation moved to Miami , replacing the Carnival Victory , doing 4 and 5 night cruises. Carnival Victory took over the 3- and 4-night cruises to Nassau and Freeport, Bahamas , in the Bahamas . In February 2018,

620-532: The real line R , {\displaystyle \mathbb {R} ,} then an injective function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the horizontal line test . Functions with left inverses are always injections. That is, given f : X → Y , {\displaystyle f:X\to Y,} if there

651-406: The codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces , an injective homomorphism is also called a monomorphism . However, in the more general context of category theory , the definition of

SECTION 20

#1732772185034

682-535: The company announced that Carnival Sensation would take 17 sailings to Cuba in 2019. In July 2020, the company announced that Carnival Sensation would move to Mobile, Alabama to replace Carnival Fantasy at the Alabama Cruise Terminal. Carnival Fantasy was retired and sold amid the ongoing COVID-19 pandemic. Carnival Sensation was to take over itineraries already scheduled aboard Carnival Fantasy and Carnival Fascination . However, it

713-452: The definition that f {\displaystyle f} is injective. There are multiple other methods of proving that a function is injective. For example, in calculus if f {\displaystyle f} is a differentiable function defined on some interval, then it is sufficient to show that the derivative is always positive or always negative on that interval. In linear algebra, if f {\displaystyle f}

744-697: The function is bijective. In fact, to turn an injective function f : X → Y {\displaystyle f:X\to Y} into a bijective (hence invertible) function, it suffices to replace its codomain Y {\displaystyle Y} by its actual image J = f ( X ) . {\displaystyle J=f(X).} That is, let g : X → J {\displaystyle g:X\to J} such that g ( x ) = f ( x ) {\displaystyle g(x)=f(x)} for all x ∈ X {\displaystyle x\in X} ; then g {\displaystyle g}

775-592: The inverse cannot be expressed in terms of elementary functions . A root-finding algorithm may be used for obtaining an approximation to a ship's volume given its gross tonnage. The formula for exact conversion of gross tonnage to volume is: where ln {\displaystyle \ln } is the natural logarithm and W {\displaystyle W} is the Lambert W function . One-to-one function In mathematics , an injective function (also known as injection , or one-to-one function )

806-509: The ship" and is used to determine things such as a ship's manning regulations, safety rules, registration fees, and port dues, whereas the older gross register tonnage is a measure of the volume of only certain enclosed spaces. The International Convention on Tonnage Measurement of Ships, 1969 was adopted by IMO in 1969. The Convention mandated a transition from the former measurements of gross register tonnage (grt) and net register tonnage (nrt) to gross tonnage (GT) and net tonnage (NT). It

837-479: Was floated out on November 1, 1993, and christened Sensation by four Carnival Vice Presidents: Vicki L. Freed, Roberta Jacoby, Cherie Weinstein and Geri Donnelly. Sensation was one of several cruise ships chartered by the US Government in 2005 to provide accommodation for residents and relief workers following Hurricane Katrina . After being released from Federal Emergency Management Agency service,

868-457: Was announced in February 2022, that Carnival would retire both Carnival Sensation as well as Carnival Ecstasy from the fleet. The latter would remain in service until October 2022, and Carnival Sensation would not return to service. In February 2022 it was reported that the ship had been sold for US$ 11 million, and on March 18, 2022 sailed from Miami for Aliağa , Turkey, for scrapping. It

899-633: Was beached on April 5, 2022. Gross tonnage Gross tonnage, along with net tonnage , was defined by the International Convention on Tonnage Measurement of Ships, 1969 , adopted by the International Maritime Organization (IMO) in 1969, and came into force on 18 July 1982. These two measurements replaced gross register tonnage (GRT) and net register tonnage (NRT). Gross tonnage is calculated based on "the moulded volume of all enclosed spaces of

930-451: Was provided to allow ships time to adjust economically, since tonnage is the basis for satisfying manning regulations and safety rules. Tonnage is also the basis for calculating registration fees and port dues. One of the convention's goals was to ensure that the new calculated tonnages "did not differ too greatly" from the traditional gross and net register tonnages. Both GT and NT are obtained by measuring ship's volume and then applying

961-607: Was the first successful attempt to introduce a universal tonnage measurement system. Various methods were previously used to calculate merchant ship tonnage, but they differed significantly and one single international system was needed. Previous methods traced back to George Moorsom of Great Britain 's Board of Trade who devised one such method in 1854. The tonnage determination rules apply to all ships built on or after 18 July 1982. Ships built before that date were given 12 years to migrate from their existing gross register tonnage (GRT) to use of GT and NT. The phase-in period

Carnival Sensation - Misplaced Pages Continue

#33966