|
|
| Line 1: |
Line 1: |
| {{no footnotes|date=March 2013}}
| | There are some things in life you will surely enjoy doing yourself: building a, making a sand castle, perhaps even something as ambitious as buying a home. If you do not know what you"re getting yourself in-to but self-filing for divorce really can be a headache. It may even be an painful experience, therefore make certain you have your ducks in a row before you do your research. <br><br>Listed below are a number of tips on just how to start the crucial job of filing for divorce so it is hassle-free. <br><br>The main reason you could file for divorce without the help of a lawyer or legal service is as it is merely a good [http://Answers.Yahoo.com/search/search_result?p=deal+cheaper&submit-go=Search+Y!+Answers deal cheaper]. The filing fee, to begin with, will likely be over $100 and if your answer is recorded, that number will only rise. If you want to prevent any additional charges, particularly thinking about the emotional and financial toll a divorce takes on everybody else concerned It"s understandable. <br><br>But unless you have a solid grasp on which you"re doing, then that lawyer or legal support may be a resource you regret perhaps not purchasing. None the less, here"s how to file for divorce by yourself. <br><br>There are certainly a couple of needs that you"ve to meet up before you should also consider filing for divorce. In certain states, you should have to meet a lot more than the following requirements, but here is a basic idea: <br><br>" Whatever state you"re filing for divorce in, you will need to have lived there for a minimum of six months. In terms of district, your residency requirement is 90 days. (These state and county residency requirements can vary, depending on the place of the United States.) <br><br>" Have you got legal grounds to divorce? "Irreconcilable difficulties" may be the most commonly cited explanation for a divorce and it"s an extremely wide definition, meaning it"s rare that it"s illegal for anyone to file for divorce. To get a second interpretation, consider checking out: [http://xtlm.info/2014/08/20/appropriate-problem-expert-attorneys/ patent pending]. Then you"ve legal grounds, if you"ve experienced marital issues that have hurt the union and are irreconcilable. Get further on [http://journals.fotki.com/tinyglass828/Cannon-Meat/ visit site] by visiting our impressive article. There is also the choice of "incurable sanity" that is only utilized in extreme circumstances. <br><br>" You will file for divorce in your county; the action for divorce should be shown in the court of one"s authority. It might hard or a easy to track down the correct court. <br><br>If your divorce is uncontested, it is fairly uncomplicated to file for divorce. It"s as soon as your spouse reacts with a counteraction of some kind that it begins to get difficult and legal counsel is not really expendable. You"ll find also online resources at this time that enable you to do your entire processing on the Internet; these often cost money to work with however. <br><br>In conclusion, to self file for divorce, you"ll need to fulfill the above demands and file a petition for divorce using the proper court in your state and county. (More used counties may have multiple areas that you"ll have to review before filing.) Remember to look at your neighborhood needs, for legal grounds and jurisdictions, so you are not struck with any unpleasant surprises and hangups.. To read more, please check out: [http://hesey.info/news/2014/08/legal-problem-expert-solicitors/ purchase here].<br><br>In case you have virtually any inquiries relating to exactly where along with tips on how to utilize [http://storify.com/imperfecti113 free health clinics], it is possible to email us in our web-site. |
| In [[mathematics]], the '''Cayley–Dickson construction''', named after [[Arthur Cayley]] and [[Leonard Eugene Dickson]], produces a sequence of [[algebra over a field|algebras]] over the [[field (mathematics)|field]] of [[real number]]s, each with twice the [[dimension of a vector space|dimension]] of the previous one. The algebras produced by this process are known as '''Cayley–Dickson algebras'''. They are useful [[composition algebra]]s frequently applied in [[mathematical physics]].
| |
| | |
| The Cayley–Dickson construction defines a new algebra based on the [[direct sum]] of an algebra with itself, with multiplication defined in a specific way and an [[involution (mathematics)|involution]] known as [[Complex conjugate|conjugation]]. The product of an element and its conjugate (or sometimes the square root of this) called the [[norm (mathematics)|norm]].
| |
| | |
| The symmetries of the real field disappear as the Cayley–Dickson construction is repeatedly applied: first losing order, then commutativity of multiplication, and next associativity of multiplication.
| |
| | |
| More generally, the Cayley–Dickson construction takes any algebra with involution to another algebra with involution of twice the dimension.<ref name=Sch45>Schafer (1995) p.45</ref>
| |
| | |
| == Complex numbers as ordered pairs ==
| |
| | |
| The [[complex numbers]] can be written as [[ordered pair]]s (''a'', ''b'') of [[real number]]s ''a'' and ''b'', with the addition operator being component-by-component and with multiplication defined by
| |
| | |
| : <math>(a, b) (c, d) = (a c - b d, a d + b c).\,</math>
| |
| | |
| A complex number whose second component is zero is associated with a real number: the complex number (''a'', 0) is the real number ''a''.
| |
| | |
| Another important operation on complex numbers is conjugation. The conjugate (''a'', ''b'')<sup>*</sup> of (''a'', ''b'') is given by
| |
| | |
| : <math>(a, b)^* = (a, -b).\,</math> | |
| | |
| The conjugate has the property that
| |
| | |
| : <math>(a, b)^* (a, b)
| |
| = (a a + b b, a b - b a) = (a^2 + b^2, 0),\,</math>
| |
| | |
| which is a non-negative real number. In this way, conjugation defines a ''[[norm (mathematics)|norm]]'', making the complex numbers a [[normed vector space]] over the real numbers: the norm of a complex number ''z'' is
| |
| | |
| : <math>|z| = (z^* z)^{1/2}.\,</math>
| |
| | |
| Furthermore, for any nonzero complex number ''z'', conjugation gives a [[inverse element|multiplicative inverse]],
| |
| | |
| : <math>z^{-1} = {z^* / |z|^2}.\,</math>
| |
| | |
| In as much as complex numbers consist of two independent real numbers, they form a 2-dimensional [[vector space]] over the real numbers.
| |
| | |
| Besides being of higher dimension, the complex numbers can be said to lack one algebraic property of the real numbers: a real number is its own conjugate.
| |
| | |
| == Quaternions ==
| |
| | |
| The next step in the construction is to generalize the multiplication and conjugation operations.
| |
| | |
| Form ordered pairs <math>(a, b)</math> of complex numbers <math>a</math> and <math>b</math>, with multiplication defined by
| |
| | |
| : <math>(a, b) (c, d) | |
| = (a c - d^* b, d a + b c^*).\,</math>
| |
| | |
| Slight variations on this formula are possible; the resulting constructions will yield structures identical up to the signs of bases.
| |
| | |
| The order of the factors seems odd now, but will be important in the next step. Define the conjugate <math>(a, b)^*\,</math> of <math>(a, b)</math> by
| |
| | |
| : <math>(a, b)^* = (a^*, -b).\,</math>
| |
| | |
| These operators are direct extensions of their complex analogs: if <math>a</math> and <math>b</math> are taken from the real subset of complex numbers, the appearance of the conjugate in the formulas has no effect, so the operators are the same as those for the complex numbers. | |
| | |
| The product of an element with its conjugate is a non-negative real number:
| |
| | |
| : <math>(a, b)^* (a, b)
| |
| = (a^*, -b) (a, b)
| |
| = (a^* a + b^* b, b a^* - b a^*)
| |
| = (|a|^2 + |b|^2, 0 ).\,</math>
| |
| | |
| As before, the conjugate thus yields a norm and an inverse for any such ordered pair. So in the sense we explained above, these pairs constitute an algebra something like the real numbers. They are the [[quaternions]], named by [[William Rowan Hamilton|Hamilton]] in 1843.
| |
| | |
| Inasmuch as quaternions consist of two independent complex numbers, they form a 4-dimensional vector space over the real numbers.
| |
| | |
| The multiplication of quaternions is not quite like the multiplication of real numbers, though. It is not [[commutative]], that is, if <math>p</math> and <math>q</math> are quaternions, it is not generally true that <math>p q = q p</math>.
| |
| | |
| == Octonions ==
| |
| {{main|Octonion}}
| |
| From now on, all the steps will look the same.
| |
| | |
| This time, form ordered pairs <math>(p, q)</math> of
| |
| quaternions <math>p</math> and <math>q</math>, with multiplication and conjugation defined exactly as for the quaternions:
| |
| : <math>(p, q) (r, s)
| |
| = (p r - s^* q, s p + q r^*).\,</math>
| |
| | |
| Note, however, that because the quaternions are not commutative, the order of the factors in the multiplication formula becomes important—if the last factor in the multiplication formula were <math>r^*q</math> rather than
| |
| <math>qr^*</math>, the formula for multiplication of an element by its conjugate would not yield a real number.
| |
| | |
| For exactly the same reasons as before, the conjugation operator yields a norm and a multiplicative inverse of any nonzero element.
| |
| | |
| This algebra was discovered by [[John T. Graves]] in 1843, and is called the [[octonions]] or the "[[Arthur Cayley|Cayley]] numbers".
| |
| | |
| Inasmuch as octonions consist of two quaternions, the octonions form an 8-dimensional vector space over the real numbers.
| |
| | |
| The multiplication of octonions is even stranger than that of quaternions. Besides being non-commutative, it is not [[associative]]: that is, if <math>p</math>, <math>q</math>, and <math>r</math> are octonions, it is generally not true that
| |
| :<math>(p q) r = p (q r).\ </math>
| |
| | |
| For the reason of this non-associativity, octonions have [[Octonion#Properties|no matrix representation]].
| |
| | |
| == Further algebras ==
| |
| | |
| The algebra immediately following the octonions is called the [[sedenions]]. It retains an algebraic property called [[power associativity]], meaning that if <math>s</math> is a sedenion, <math>s^n s^m = s^{n + m}</math>, but loses the property of being an [[alternative algebra]] and hence cannot be a [[composition algebra]].
| |
| | |
| The Cayley–Dickson construction can be carried on ''[[ad infinitum]]'', at each step producing a power-associative algebra whose dimension is double that of the algebra of the preceding step. All the algebras generated in this way over a field are ''quadratic'': that is, each element satisfies a quadratic equation with coefficients from the field.<ref name=Sch50>Schafer (1995) p.50</ref>
| |
| | |
| == General Cayley–Dickson construction ==
| |
| {{harvtxt|Albert|1942|p= 171}} gave a slight generalization, defining the product and involution on ''B''=''A''⊕''A'' for ''A'' an [[*-algebra|algebra with involution]] (with (''xy'')<sup>*</sup> = ''y''<sup>*</sup>''x''<sup>*</sup>) to be
| |
| : <math>(p, q) (r, s)
| |
| = (p r - \gamma s^* q, s p + q r^*)\,</math>
| |
| :<math>(p, q)^* = (p^*, -q)\ </math>
| |
| for γ an additive map that commutes with * and left and right multiplication by any element. (Over the reals all choices of γ are equivalent to −1, 0 or 1.) In this construction, ''A'' is an algebra with involution, meaning: | |
| *''A'' is an abelian group under +
| |
| *''A'' has a product that is left and right distributive over +
| |
| *''A'' has an involution *, with ''x''** = ''x'', (''x'' + ''y'')* = ''x''* + ''y''*, (''xy'')* = ''y''*''x''*.
| |
| The algebra ''B''=''A''⊕''A'' produced by the Cayley–Dickson construction is also an algebra with involution.
| |
| | |
| ''B'' inherits properties from ''A'' unchanged as follows.
| |
| *If ''A'' has an identity 1<sub>''A''</sub>, then ''B'' has an identity (1<sub>''A''</sub>, 0).
| |
| *If ''A'' has the property that ''x'' + ''x''<sup>*</sup>, ''xx''<sup>*</sup> associate and commute with all elements, then so does ''B''. This property implies that any element generates a commutative associative *-algebra, so in particular the algebra is power associative.
| |
|
| |
| Other properties of ''A'' only induce weaker properties of ''B'':
| |
| *If ''A'' is commutative and has trivial involution, then ''B'' is commutative.
| |
| *If ''A'' is commutative and associative then ''B'' is associative.
| |
| *If ''A'' is associative and ''x'' + ''x''<sup>*</sup>, ''xx''<sup>*</sup> associate and commute with everything, then ''B'' is an [[alternative algebra]].
| |
| | |
| == References ==
| |
| {{reflist}}
| |
| *{{Citation | last1=Albert | first1=A. A. | author1-link=Abraham Adrian Albert | title=Quadratic forms permitting composition | jstor=1968887 | mr=0006140 | year=1942 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=43 | pages=161–177 | doi=10.2307/1968887 | issue=1}} (see p. 171)
| |
| * {{Citation | last1=Baez | first1=John | author1-link=John Baez | title=The Octonions | url=http://math.ucr.edu/home/baez/octonions/octonions.html | year=2002 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=39 | pages=145–205 | doi=10.1090/S0273-0979-01-00934-X | issue=2}}. ''(See "[http://math.ucr.edu/home/baez/octonions/node5.html Section 2.2, The Cayley-Dickson Construction]")''
| |
| *{{Citation | last1=Dickson | first1=L. E. | author1-link=Leonard Dickson | title=On Quaternions and Their Generalization and the History of the Eight Square Theorem | jstor=1967865 | publisher=Annals of Mathematics | series=Second Series | year=1919 | journal=[[Annals of Mathematics]] | issn=0003-486X | volume=20 | issue=3 | pages=155–171 | doi=10.2307/1967865}}
| |
| * {{Citation | last1=Kantor | first1=I. L. | last2=Solodownikow | first2=A. S. | title=Hyperkomplexe Zahlen | publisher=B.G. Teubner | location=Leipzig | year=1978}}
| |
| * {{Citation | last1=Hamilton | first1=William Rowan | author1-link=William Rowan Hamilton | title=On Quaternions | url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quatern2/Quatern2.html | year=1847 | journal=Proceedings of the Royal Irish Academy | issn=1393-7197 | volume=3 | pages=1–16}}
| |
| * Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in ''Symmetries in Complex Analysis'' by Bruce Gilligan & Guy Roos, volume 468 of ''Contemporary Mathematics'', [[American Mathematical Society]], ISBN 978-0-8218-4459-5 .
| |
| * {{citation | first=Richard D. | last=Schafer | year=1995 | origyear=1966 | zbl=0145.25601 | title=An introduction to non-associative algebras | publisher=[[Dover Publications]] | isbn=0-486-68813-5 }}
| |
| | |
| ==External links==
| |
| * Hyperjeff, ''[http://history.hyperjeff.net/hypercomplex.html Sketching the History of Hypercomplex Numbers]'' (1996–2006).
| |
| | |
| {{Number Systems}}
| |
| | |
| {{DEFAULTSORT:Cayley-Dickson construction}}
| |
| [[Category:Hypercomplex numbers]]
| |
There are some things in life you will surely enjoy doing yourself: building a, making a sand castle, perhaps even something as ambitious as buying a home. If you do not know what you"re getting yourself in-to but self-filing for divorce really can be a headache. It may even be an painful experience, therefore make certain you have your ducks in a row before you do your research.
Listed below are a number of tips on just how to start the crucial job of filing for divorce so it is hassle-free.
The main reason you could file for divorce without the help of a lawyer or legal service is as it is merely a good deal cheaper. The filing fee, to begin with, will likely be over $100 and if your answer is recorded, that number will only rise. If you want to prevent any additional charges, particularly thinking about the emotional and financial toll a divorce takes on everybody else concerned It"s understandable.
But unless you have a solid grasp on which you"re doing, then that lawyer or legal support may be a resource you regret perhaps not purchasing. None the less, here"s how to file for divorce by yourself.
There are certainly a couple of needs that you"ve to meet up before you should also consider filing for divorce. In certain states, you should have to meet a lot more than the following requirements, but here is a basic idea:
" Whatever state you"re filing for divorce in, you will need to have lived there for a minimum of six months. In terms of district, your residency requirement is 90 days. (These state and county residency requirements can vary, depending on the place of the United States.)
" Have you got legal grounds to divorce? "Irreconcilable difficulties" may be the most commonly cited explanation for a divorce and it"s an extremely wide definition, meaning it"s rare that it"s illegal for anyone to file for divorce. To get a second interpretation, consider checking out: patent pending. Then you"ve legal grounds, if you"ve experienced marital issues that have hurt the union and are irreconcilable. Get further on visit site by visiting our impressive article. There is also the choice of "incurable sanity" that is only utilized in extreme circumstances.
" You will file for divorce in your county; the action for divorce should be shown in the court of one"s authority. It might hard or a easy to track down the correct court.
If your divorce is uncontested, it is fairly uncomplicated to file for divorce. It"s as soon as your spouse reacts with a counteraction of some kind that it begins to get difficult and legal counsel is not really expendable. You"ll find also online resources at this time that enable you to do your entire processing on the Internet; these often cost money to work with however.
In conclusion, to self file for divorce, you"ll need to fulfill the above demands and file a petition for divorce using the proper court in your state and county. (More used counties may have multiple areas that you"ll have to review before filing.) Remember to look at your neighborhood needs, for legal grounds and jurisdictions, so you are not struck with any unpleasant surprises and hangups.. To read more, please check out: purchase here.
In case you have virtually any inquiries relating to exactly where along with tips on how to utilize free health clinics, it is possible to email us in our web-site.