|
|
| Line 1: |
Line 1: |
| {{hatnote|For similarly named ordering systems outside mathematics, see [[Alphabetical order]] and [[Collation]].}}
| | And how much of those feelings are caused by pressure from the society in general. But what you can do is find an empty apartment and practice. It can also reduce the self esteem and self confidence of women but natural breast enlargement cream not only enlarges the size of breast and prevents sagging. Basically, it is all one big, puzzling mathematical equation that we don't want to get into here for sake of brevity. When it comes to breast enlargement you have a few choices: very expensive, potentially dangerous, ineffective or perhaps the only one out of all methods is the one mentioned above that's a lot cheaper and actually works for breast enlargement. <br><br>The breast milk from mothers who consider fenugreek has reduced extra fat and cholesterol subject material. Yes, they all have well-endowed breasts, may it be open or artificially enhanced. "We try to eat a Mediterranean diet: lots of greens, lots of fish, lots of vegetables," she said. We come in all different shapes and sizes and we can only play the cards that. Breast enhancement can be achieved by taking capsules, pills and the use of creams which increase both the size and firmness of the breast. <br><br>Silicone breast implants require longer incisions than saline-based implants. * Comfort - While many women who go in for breast augmentation surgery have never before needed a bra for the "comfort factor," this can become an issue with larger size and more bulk. ' However, sometimes the pair does not divide and instead sticks together and travels to its place in the dividing cell. Girls don't want Snow White and Cinderella any more. He performs all types of plastic surgery and specializes in face lifts, rhinoplasty and breast augmentation. <br><br>Or you can simply increase your breast size to a level that compliments the rest of your body. However, it is important to note that, you cannot introduce all these foods into your system at a go. For anyone who uses prescribed medication, breast enlargement pills present a danger. Liposuction results in bruising that will generally fade after 7 days. Know the pros and cons or advantages and disadvantages of undergoing cosmetic surgery can make the decision to go through with the surgery easier. <br><br>Using a breast enlargement cream is another way to stimulate breast growth, most breast creams contain an herb called Pueraria Mirifica which mimics natural estrogen production, when applied directly to the breast it not only stimulates breast growth making the breasts larger, it can also help women reduce sagging in their breasts while at the same time making them fuller and perkier. The patients own tissue is used to reconstruct the breast. If you use implants the surgery will be a combination of enlargement and lift. Wild Yam contains compounds similar to progesterone, and the diosgenin, a procurer of progesterone, makes it a popular choice for treating PMS. You can also massage the underneath of the armpit to promote circulation. <br><br>Saw palmetto is an essential male breast enlargement herb. He is a Board Certified and fully trained Cosmetic Plastic Surgeon, with 6 years of formal training in the field of Aesthetic, Reconstructive and Burn Plastic Surgery. Katie Perry is an online content editor in the Tampa Bay area. s breast, it is not considered to be the best breast enlargement choice when it comes to getting good and greater results. Well, these days women surely have got more conscious about their looks, body, skin, face especially the size and shape of their breast.<br><br>Should you cherished this article and you desire to get more information with regards to natural breast augmentation before and after ([http://www.lucky-house.info/sitemap/ Check Out www.lucky-house.info]) generously pay a visit to our own site. |
| [[File:Orderings; 6 choose 3.svg|thumb|340px|Orderings of the 3-[[subset]]s of <math>\scriptstyle \{1,...,6\}</math> (and the corresponding [[Binary numeral system|binary]] [[Vector (mathematics and physics)|vectors]])<br>When the (blue) triples are in ''lex'' order the (red) vectors are in ''revlex'' order, and vice versa. The arrangements on the right side show ''[[Colexicographical order|colex]]'' and ''revcolex'' order.]]
| |
| In [[mathematics]], the '''lexicographic''' or '''lexicographical order''' (also known as '''lexical order''', '''dictionary order''', '''alphabetical order''' or '''lexicographic(al) product''') is a generalization of the way the [[alphabetical order]] of words is based on the alphabetical order of their component letters.
| |
| | |
| ==Definition==
| |
| Given two [[partially ordered set]]s ''A'' and ''B'', the lexicographical [[order theory|order]] on the [[Cartesian product]] ''A'' × ''B'' is defined as
| |
| :(''a'',''b'') ≤ (''a''′,''b''′) if and only if ''a'' < ''a''′ or (''a'' = ''a''′ and ''b'' ≤ ''b''′).
| |
| | |
| The result is a partial order. If ''A'' and ''B'' are [[total order|totally ordered]], then the result is a total order as well.
| |
| | |
| More generally, one can define the lexicographic order on the Cartesian product of ''n'' ordered sets, on the Cartesian product of a countably infinite family of ordered sets, and on the union of such sets.
| |
| | |
| {{Merge from|Colexicographical order|date=May 2011}}
| |
| | |
| ==Motivation and uses==
| |
| | |
| The name of the lexicographic order comes from its generalizing the order given to words in a [[dictionary]]: a sequence of letters (that is, a ''word'')
| |
| | |
| :''a''<sub>1</sub>''a''<sub>2</sub> ... ''a''<sub>''k''</sub>
| |
| | |
| appears in a dictionary before a sequence
| |
| | |
| :''b''<sub>1</sub>''b''<sub>2</sub> ... ''b''<sub>''k''</sub>
| |
| | |
| if and only if at the first ''i'' where ''a<sub>i</sub>'' and ''b<sub>i</sub>'' differ, ''a<sub>i</sub>'' comes before ''b<sub>i</sub>'' in the [[alphabet]].
| |
| | |
| That comparison assumes both sequences are the same length. To ensure they are the same length, the shorter sequence is usually padded at the end with enough "blanks" (a special symbol that is treated as coming before any other symbol). This also allows ordering of phrases. For the purpose of dictionaries, etc., padding with blank spaces is always done. See [[alphabetical order]].
| |
| | |
| For example, the word "Thomas" appears before "Thompson" in dictionaries because the letter 'a' comes before the letter 'p' in the alphabet. The 5th letter is the first that is different in the two words; the first 4 letters are "Thom" in both. Because it is the first difference, the 5th letter is the most significant difference (for an alphabetical ordering).
| |
| | |
| A lexicographical ordering may not coincide with conventional alphabetical ordering. For example, the numerical order of [[Unicode]] codepoints does not always correspond to traditional alphabetic orderings of the characters, which vary from language to language. So the lexicographic ordering induced by codepoint value sorts strings in an unambiguous canonical order, but it does not necessarily "alphabetize" them in the conventional sense.
| |
| | |
| An important property of the lexicographical order is that it preserves [[well-order]]s, that is, if ''A'' and ''B'' are well-ordered sets, then the product set ''A'' × ''B'' with the lexicographical order is also well-ordered.
| |
| | |
| An important exploitation of lexicographical ordering is expressed in the [[ISO 8601]] date formatting scheme, which expresses a date as YYYY-MM-DD. This date ordering lends itself to straightforward [[sorting algorithm|computerized sorting]] of dates such that the sorting algorithm does not need to treat the numeric parts of the date string any differently from a string of non-numeric characters, and the dates will be sorted into [[Chronology|chronological order]]. Note, however, that for this to work, there must always be four digits for the year, two for the month, and two for the day, so for example single-digit days must be padded with a zero yielding '01', '02', ..., '09'.
| |
| | |
| Another generalization of lexical ordering occurs in [[social choice theory]] (the theory of elections). Consider an election in which there are 4 candidates A, B, C and D, each voter expresses a top-to-bottom ordering of the candidates, and the voters' orderings are as follows:
| |
| | |
| {| class="wikitable"
| |
| !18%
| |
| !17%
| |
| !33%
| |
| !32%
| |
| |-
| |
| |A
| |
| |B
| |
| |C
| |
| |D
| |
| |-
| |
| |B
| |
| |A
| |
| |D
| |
| |B
| |
| |-
| |
| |C
| |
| |C
| |
| |A
| |
| |A
| |
| |-
| |
| |D
| |
| |D
| |
| |B
| |
| |C
| |
| |}
| |
| | |
| The [[Minimax Condorcet|MinMax]] voting method is a simple [[Condorcet method]] that counts the votes as in a round-robin tournament (all possible pairings of candidates) and judges each candidate according to its largest "pairwise" defeat. The winner is the candidate whose largest defeat is the smallest. In the example:
| |
| *The largest defeat of A is by D: '''65%''' (33%+32%) rank D over A.
| |
| *The largest defeat of B is by D: '''65%''' (33%+32%) rank D over B.
| |
| *The largest defeat of C is by A (or B): '''67%''' (18%+17%+32%) rank A over C (and B over C).
| |
| *The largest defeat of D is by C: '''68%''' (18%+17%+33%) rank C over D.
| |
| | |
| MinMax declares a tie between A and B since the largest defeats for both are the same size, 65%. This is like saying "Thomas" and "Thompson" should be at the same position because they have the same first letter. However, if the defeats are compared lexically, we have the MinLexMax method. With MinLexMax, because the largest defeats of A and B are the same size, their next largest defeats are then compared:
| |
| *A's next largest defeat is by B: '''49%'''. (17%+32%) rank B over A.
| |
| *B's next largest defeat is by A: '''51%''' (18%+33%) rank A over B.
| |
| Since B's next largest defeat is larger than A's, MinLexMax elects A, which makes more sense than the MinMax tie since a majority rank A over B.
| |
| | |
| Another usage in social choice theory is the [[Ranked Pairs]] voting method. Although usually defined by a procedure that constructs the order of finish, Ranked Pairs is equivalent to finding which of all possible orders of finish is best according to a minlexmax comparison of the majorities they reverse. In the example above, the Ranked Pairs order of finish is ABCD (which elects A). ABCD affirms the majorities who rank A over B, A over C, B over C and C over D, and reverses the majorities who rank D over A and D over B. The largest majority that ABCD reverses is 65%. The only other ordering that wouldn't reverse a larger majority is BACD (which also reverses 65%). ABCD is a better order of finish than BACD because the lexically relevant set of majorities—the majorities on which ABCD and BACD disagree—is {A over B} and BACD reverses the largest majority in this set.
| |
| | |
| ==Case of multiple products==
| |
| | |
| Suppose
| |
| :<math>
| |
| \{ A_1, A_2, \cdots, A_n \}
| |
| </math> | |
| is an n-tuple of sets, with respective total orderings
| |
| :<math>
| |
| \{ <_1, <_2, \cdots, <_n \}
| |
| </math>
| |
| | |
| The dictionary ordering
| |
| :<math>
| |
| \ \ <^{d}
| |
| </math>
| |
| of
| |
| :<math>
| |
| A_1 \times A_2 \times \cdots \times A_n
| |
| </math>
| |
| is then
| |
| :<math>
| |
| (a_1, a_2, \dots, a_n) <^d (b_1,b_2, \dots, b_n) \iff
| |
| (\exists\ m > 0) \ (\forall\ i < m) (a_i = b_i) \land (a_m <_m b_m)
| |
| </math>
| |
| | |
| That is, if one of the terms
| |
| :<math>
| |
| \ \ a_m <_m b_m
| |
| </math>
| |
| and all the preceding terms are equal.
| |
| | |
| Informally,
| |
| :<math>
| |
| \ \ a_1
| |
| </math>
| |
| represents the first letter,
| |
| :<math>
| |
| \ \ a_2
| |
| </math>
| |
| the second and so on when looking up a word in a dictionary, hence the name.
| |
| | |
| This could be more elegantly stated by recursively defining the ordering of any set
| |
| | |
| :<math>
| |
| \ \ C= A_j \times A_{j+1} \times \cdots \times A_k
| |
| </math>
| |
| | |
| represented by
| |
| :<math>
| |
| \ \ <^d (C)
| |
| </math>
| |
| | |
| This will satisfy
| |
| | |
| :<math>
| |
| a <^d (A_i) a' \iff (a <_i a')
| |
| </math>
| |
| | |
| :<math>
| |
| (a,b) <^d (A_i \times B) (a',b') \iff
| |
| a <^d (A_i) a' \lor ( a=a' \ \land \ b <^d (B) b')
| |
| </math>
| |
| | |
| where
| |
| <math>
| |
| B = A_{i+1} \times A_{i+2} \times \cdots \times A_n.
| |
| </math>
| |
| | |
| To put it more simply, compare the first terms. If they are equal, compare the second terms – and so on. The relationship between the first corresponding terms that are not equal determines the relationship between the entire elements.
| |
| | |
| ==Groups and vector spaces==
| |
| If the component sets are [[ordered group]]s then the result is a non-[[Archimedean group]], because e.g. ''n''(0,1) < (1,0) for all ''n''.
| |
| | |
| If the component sets are [[ordered vector space]]s over '''R''' (in particular just '''R'''), then the result is also an ordered vector space.
| |
| | |
| ==Ordering of sequences of various lengths==
| |
| | |
| Given a partially ordered set ''A'', the above considerations allow to define naturally a lexicographical partial order <math><^\mathrm{d}</math> over the [[free monoid]] ''A''* formed by the set of all [[finite sequence]]s of elements in ''A'', with sequence [[concatenation]] as the monoid operation, as follows:
| |
| | |
| :<math>u <^\mathrm{d} v</math> if
| |
| :* <math>u</math> is a [[prefix]] of <math>v</math>, or
| |
| :* <math>u=wau'</math> and <math>v=wbv'</math>, where <math>w</math> is the longest common prefix of <math>u</math> and <math>v</math>, <math>a</math> and <math>b</math> are members of ''A'' such that <math>a<b</math>, and <math>u'</math> and <math>v'</math> are members of ''A''*.
| |
| | |
| If < is a total order on ''A'', then so is the lexicographic order <<sup>d</sup> on ''A''*. If ''A'' is a finite and totally ordered alphabet, ''A''* is the set of all [[String (computer science)#Formal theory|words]] over ''A'', and we retrieve the notion of dictionary ordering used in lexicography that gave its name to the lexicographic orderings.
| |
| However, in general this is not a [[well-order]], even though it is on the alphabet ''A''; for instance, if ''A'' = {''a'', ''b''}, the [[Formal language|language]] {''a''<sup>''n''</sup>''b'' | ''n'' ≥ 0} has no least element: ... <<sup>d</sup> ''aab'' <<sup>d</sup> ''ab'' <<sup>d</sup> ''b''. A well-order for strings, based on the lexicographical order, is the [[shortlex order]]. | |
| | |
| Similarly we can also compare a finite and an infinite string, or two infinite strings.
| |
| | |
| Comparing strings of different lengths can also be modeled as comparing strings of infinite length by right-padding finite strings with a special value that is less than any element of the alphabet.
| |
| | |
| This ordering is the ordering usually used to order [[String (computer science)|character strings]], including in dictionaries and indexes.
| |
| | |
| === Quasi-lexicographic order ===
| |
| The '''[[Shortlex order|quasi-lexicographic order]]''' on the free monoid ''A''<sup>∗</sup> over an ordered alphabet ''A'' orders strings firstly by length, so that the [[empty string]] comes first, and then within strings of fixed length ''n'', by lexicographic order on ''A''<sup>''n''</sup>.<ref>{{cite book | last=Calude | first=Cristian | authorlink=Cristian S. Calude | title=Information and randomness. An algorithmic perspective | series=EATCS Monographs on Theoretical Computer Science | publisher=[[Springer-Verlag]] | year=1994 | isbn=3-540-57456-5 | zbl=0922.68073 | page=1 }}</ref>
| |
| | |
| ==Generalization==
| |
| Consider the set of functions ''f'' from a [[well-ordered set]] ''X'' to a [[totally ordered set]] ''Y''. For two such functions ''f'' and ''g'', the order is determined by the values for the smallest ''x'' such that ''f''(''x'') ≠ ''g''(''x'').
| |
| | |
| If ''Y'' is also well-ordered and ''X'' is finite, then the resulting order is a well-order. As already shown above, if ''X'' is infinite this is in general not the case.
| |
| | |
| If ''X'' is infinite and ''Y'' has more than one element, then the resulting set ''Y''<sup>''X''</sup> is not a [[countable set]], see also [[Cardinal number#Cardinal exponentiation|cardinal exponentiation]].
| |
| | |
| Alternatively, consider the functions ''f'' from an inversely well-ordered ''X'' to a well-ordered ''Y'' with minimum 0, restricted to those that are non-zero at only a finite subset of ''X''. The result is well-ordered. Correspondingly we can also consider a well-ordered ''X'' and apply lexicographical order where a higher ''x'' is a more significant position. This corresponds to [[Ordinal arithmetic#Exponentiation|exponentiation of ordinal numbers]] ''Y''<sup>''X''</sup>. If ''X'' and ''Y'' are countable then the resulting set is also countable.
| |
| | |
| ==Monomials==
| |
| | |
| In algebra it is traditional to order [[term (mathematics)|terms]] in a [[polynomial]], by ordering the [[monomial]]s in the [[indeterminate (variable)|indeterminate]]s. Such matters are typically left implicit in discussion between humans, but must of course be dealt with exactly in [[computer algebra]], for example for testing the equality of polynomials.
| |
| | |
| More specifically, the definition of [[Gröbner basis|Gröbner bases]] and their computation are heavily based on the choice of an ordering of the monomials. To define such an ordering, one identifies every monomial (for example <math>x_1 x_2^3 x_4 x_5^2</math>) with its vector of exponents (here [1,3,0,1,2]), and one chooses an ordering on these vectors of integers. This ordering must satisfy some further conditions to be '''admissible''' for Gröbner bases; see [[monomial order]] for details and the admissibility conditions.
| |
| | |
| One of these admissible orders is the '''lexicographical order'''. Another one is the '''total degree order''', which consists in comparing first the [[total degree]]s, and then resolving the conflicts by using the lexicographical order.
| |
| More generally, every admissible order may be defined as the lexicographical order on the values of a set of ''n'' [[linear form]]s with [[real number|real]] coefficients applied to the vector of exponents (here ''n'' is the number of variables).
| |
| | |
| ==Decimal fractions==
| |
| For [[Decimal#Decimal_fractions|decimal fractions]] from the decimal point, ''a'' < ''b'' applies equivalently for the numerical order and the lexicographic order on the digital representations, provided that [[0.999...|strings with a recurring decimal 9]] like .399999... and strings with trailing zeros are omitted. With these restrictions, there is an order-preserving bijection between the numbers and the strings.
| |
| | |
| == Reverse lexicographic order ==
| |
| | |
| In a common variation of lexicographic order, one compares elements by reading from the right instead of from the left, i.e., the right-most component is the most significant, e.g. applied in a [[rhyming dictionary]].
| |
| | |
| In the case of monomials one may sort the exponents downward, with the exponent of the first base variable as primary sort key, e.g.:
| |
| : <math> x^2 y z^2 < x y^3 z^2 </math>.
| |
| Alternatively, sorting may be done by the sum of the exponents, downward.
| |
| | |
| == See also ==
| |
| | |
| * [[Collation]]
| |
| * [[Colexicographical order]]
| |
| * [[Kleene–Brouwer order]]
| |
| * [[Lexicographic preferences]]
| |
| * [[Total_order#Orders_on_the_Cartesian_product_of_totally_ordered_sets|Orders on the Cartesian product of totally ordered sets]]
| |
| * [[Ordered_vector_space#Examples|Lexicographic order on the '''R'''<sup>''n''</sup>]]
| |
| * [[Lexicographic order topology on the unit square]]
| |
| * [[Long line (topology)]]
| |
| * [[Product order]]
| |
| * [[Lyndon word]]
| |
| * [[Lexicographically minimal string rotation]]
| |
| * [[File:Wikiversity-logo-en.svg|20px]] [[v:Lexicographic and colexicographic order|Lexicographic and colexicographic order]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| [[Category:Order theory]]
| |
| [[Category:Lexicography]]
| |
And how much of those feelings are caused by pressure from the society in general. But what you can do is find an empty apartment and practice. It can also reduce the self esteem and self confidence of women but natural breast enlargement cream not only enlarges the size of breast and prevents sagging. Basically, it is all one big, puzzling mathematical equation that we don't want to get into here for sake of brevity. When it comes to breast enlargement you have a few choices: very expensive, potentially dangerous, ineffective or perhaps the only one out of all methods is the one mentioned above that's a lot cheaper and actually works for breast enlargement.
The breast milk from mothers who consider fenugreek has reduced extra fat and cholesterol subject material. Yes, they all have well-endowed breasts, may it be open or artificially enhanced. "We try to eat a Mediterranean diet: lots of greens, lots of fish, lots of vegetables," she said. We come in all different shapes and sizes and we can only play the cards that. Breast enhancement can be achieved by taking capsules, pills and the use of creams which increase both the size and firmness of the breast.
Silicone breast implants require longer incisions than saline-based implants. * Comfort - While many women who go in for breast augmentation surgery have never before needed a bra for the "comfort factor," this can become an issue with larger size and more bulk. ' However, sometimes the pair does not divide and instead sticks together and travels to its place in the dividing cell. Girls don't want Snow White and Cinderella any more. He performs all types of plastic surgery and specializes in face lifts, rhinoplasty and breast augmentation.
Or you can simply increase your breast size to a level that compliments the rest of your body. However, it is important to note that, you cannot introduce all these foods into your system at a go. For anyone who uses prescribed medication, breast enlargement pills present a danger. Liposuction results in bruising that will generally fade after 7 days. Know the pros and cons or advantages and disadvantages of undergoing cosmetic surgery can make the decision to go through with the surgery easier.
Using a breast enlargement cream is another way to stimulate breast growth, most breast creams contain an herb called Pueraria Mirifica which mimics natural estrogen production, when applied directly to the breast it not only stimulates breast growth making the breasts larger, it can also help women reduce sagging in their breasts while at the same time making them fuller and perkier. The patients own tissue is used to reconstruct the breast. If you use implants the surgery will be a combination of enlargement and lift. Wild Yam contains compounds similar to progesterone, and the diosgenin, a procurer of progesterone, makes it a popular choice for treating PMS. You can also massage the underneath of the armpit to promote circulation.
Saw palmetto is an essential male breast enlargement herb. He is a Board Certified and fully trained Cosmetic Plastic Surgeon, with 6 years of formal training in the field of Aesthetic, Reconstructive and Burn Plastic Surgery. Katie Perry is an online content editor in the Tampa Bay area. s breast, it is not considered to be the best breast enlargement choice when it comes to getting good and greater results. Well, these days women surely have got more conscious about their looks, body, skin, face especially the size and shape of their breast.
Should you cherished this article and you desire to get more information with regards to natural breast augmentation before and after (Check Out www.lucky-house.info) generously pay a visit to our own site.