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| | The writer is called Irwin. Managing individuals has been his day job for a while. To perform baseball is the pastime he will never stop doing. South Dakota is where I've usually been living.<br><br>My web page ... at home std testing ([http://www.streaming.iwarrior.net/user/WMcneil please click the up coming document]) |
| |+Tessarine multiplication
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| In [[mathematics]], a '''tessarine''' is a [[hypercomplex number]] of the form
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| :<math>t = w + x i + y j + z k, \quad w, x, y, z \in \mathbb{R}</math>
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| where <math> i j = j i = k, \quad i^2 = -1, \quad j^2 = +1 .</math>
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| The tessarines are best known for their subalgebra of '''real tessarines''' <math> t = w + y j \ </math>,
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| also called [[split-complex number]]s, which express the parametrization of the [[unit hyperbola]].
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| [[James Cockle (lawyer)|James Cockle]] introduced the tessarines in 1848 in a series of articles in [[Philosophical Magazine]]. Cockle used tessarines to isolate the hyperbolic cosine series and the hyperbolic sine series in the exponential series. He also showed how [[zero divisor]]s arise in tessarines, inspiring him to use the term "impossibles."
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| In 1892 [[Corrado Segre]] introduced '''bicomplex numbers''' in [[Mathematische Annalen]], which form an algebra equivalent to the tessarines (see section below). As '''commutative hypercomplex numbers''', the tessarine algebra has been advocated by Clyde M. Davenport (1991, 2008) (exchange ''j'' and −''k'' in his multiplication table). Davenport has noted the isomorphism with the direct sum of the complex number plane with itself. Tessarines have also been applied in [[digital signal processing]] (see Pei (2004) and Alfsmann (2006,7). In 2009 mathematicians proved a [[motor variable#Polynomial factorization|fundamental theorem of tessarine algebra]]: a polynomial of degree ''n'' with tessarine coefficients has ''n''<sup>2</sup> roots, counting multiplicity.<ref>Robert D. Poodiack & Kevin J. LeClair (2009) "Fundamental theorems of algebra for the perplexes", [[The College Mathematics Journal]] 40(5):322–35</ref>
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| ==Linear representation==
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| For tessarine <math> t = w + xi + yj + zk, \ </math> note that <math>t = (w + xi) + (y + zi) j \ </math> since {{nowrap|1=''ij'' = ''k''}}.
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| The mapping
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| :<math>t \mapsto \begin{pmatrix} p & q \\ q & p \end{pmatrix}, \quad p = w + xi, \quad q = y + zi </math>
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| is a linear representation of the algebra of tessarines as a subalgebra of {{nowrap|2 × 2}} complex matrices.
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| For instance, {{nowrap|1=''ik'' = ''i''(''ij'') = (''ii'')''j'' = −''j''}} in the linear representation is
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| :<math>\begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix} \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} .</math>
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| Note that unlike most matrix algebras, this is a [[commutative]] algebra.
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| ==Isomorphisms to other number systems==
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| In general the tessarines form an [[Algebra over a field|algebra]] of dimension two over the [[complex number]]s with [[basis (linear algebra)|basis]] {{nowrap|{ 1, ''j'' }<nowiki/>}}.
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| ===Bicomplex number===
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| [[Corrado Segre]] read [[W. R. Hamilton]]'s ''Lectures on Quaternions'' (1853) and the works of [[William Kingdon Clifford]]. Segre used some of Hamilton's notation to develop his system of '''bicomplex numbers''': Let ''h'' and ''i'' be square roots of −1 that commute with each other. Then, presuming [[associativity]] of multiplication, the product ''hi'' must have +1 for its square. The algebra constructed on the basis {{nowrap|{ 1, ''h'', ''i'', ''hi'' }<nowiki/>}} is then nearly the same as James Cockle's tessarines. Segre noted that elements
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| :<math> g = (1 - hi)/2, \quad g' = (1 + hi)/2 </math> are [[idempotent]]s.
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| When bicomplex numbers are considered to have basis {{nowrap|{ 1, ''h'', ''i'', −''hi'' }<nowiki/>}} then there is no difference between them and tessarines. Looking at the linear representation of these [[ring isomorphism|isomorphic]] algebras shows agreement in the fourth dimension when the negative sign is used; just consider the sample product given above under linear representation.
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| The [[University of Kansas]] has contributed to the development of bicomplex analysis. In 1953, a Ph.D. student James D. Riley had his thesis "Contributions to the theory of functions of a bicomplex variable" published in the [[Tohoku Mathematical Journal]] (2nd Ser., 5:132–165). Then, in 1991, [[emeritus]] professor G. Baley Price published his book on bicomplex numbers, [[multicomplex number]]s, and their function theory. Professor Price also gives some history of the subject in the preface to his book. Another book developing bicomplex numbers and their applications is by Catoni, Bocaletti, Cannata, Nichelatti & Zampetti (2008).
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| === Direct sum C + C ===
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| The [[direct sum]] of the complex field with itself is denoted {{nowrap|'''C''' ⊕ '''C'''}}. The product of two elements <math>(a \oplus b)</math> and <math> (c \oplus d)</math> is <math> a c \oplus b d </math> in this [[direct sum of modules#Direct sum of algebras|direct sum algebra]].
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| '''Proposition:''' The algebra of tessarines is isomorphic to {{nowrap|'''C''' ⊕ '''C'''}}.
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| proof: Every tessarine has an expression <math>t = u + v j \ </math> where ''u'' and ''v'' are complex numbers. Now if <math>s = w + z j \ </math> is another tessarine, their product is
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| :<math> t s = (u w + v z) + (u z + v w) j .</math>
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| The isomorphism mapping from tessarines to {{nowrap|'''C''' ⊕ '''C'''}} is given by
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| :<math>t \mapsto (u+v) \oplus (u - v) , \quad s \mapsto (w + z) \oplus (w - z).</math>
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| In {{nowrap|'''C''' ⊕ '''C'''}}, the product of these images, according to the algebra-product of {{nowrap|'''C''' ⊕ '''C'''}} indicated above, is
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| :<math>(u + v)(w + z) \oplus (u - v)(w - z).</math>
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| This element is also the image of ''ts'' under the mapping into {{nowrap|'''C''' ⊕ '''C'''}}. Thus the products agree, the mapping is a homomorphism; and since it is [[bijective]], it is an isomorphism.
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| ===Conic quaternion / octonion / sedenion, bicomplex number===
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| When ''w'' and ''z'' are both [[complex number]]s
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| : <math>w :=~a + ib</math>
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| : <math>z :=~c + id</math>
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| (with ''a'', ''b'', ''c'', ''d'' real) then ''t'' algebra is isomorphic to [[conic quaternion]]s <math>a + bi + c \varepsilon + d i_0</math>, to bases <math>\{ 1,~i,~\varepsilon ,~i_0 \}</math>, in the following identification:
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| : <math>1 \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} \qquad i \equiv \begin{pmatrix} i & 0 \\ 0 & i\end{pmatrix} \qquad \varepsilon \equiv \begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix} \qquad i_0 \equiv \begin{pmatrix} 0 & i \\ i & 0\end{pmatrix}.</math>
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| They are also isomorphic to "bicomplex numbers" (from [[multicomplex number]]s) to bases <math>\{ 1,~i_1, i_2, j \}</math> if one identifies:
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| :<math>1 \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} \qquad i_1 \equiv \begin{pmatrix} i & 0 \\ 0 & i\end{pmatrix} \qquad i_2 \equiv \begin{pmatrix} 0 & i \\ i & 0\end{pmatrix} \qquad j \equiv \begin{pmatrix} 0 & -1 \\ -1 & 0\end{pmatrix}.</math>
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| Note that ''j'' in bicomplex numbers is identified with the opposite sign as ''j'' from above.
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| When ''w'' and ''z'' are both [[quaternion]]s (to bases <math>\{ 1,~i_1,~i_2,~i_3 \}</math>), then ''t'' algebra is isomorphic to [[conic octonion]]s; allowing [[octonion]]s for ''w'' and ''z'' (to bases <math>\{ 1,~i_1, \dots, ~i_7 \}</math>) the resulting algebra is identical to [[conic sedenion]]s.
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| ===Quotient rings of polynomials===
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| A modern approach to tessarines uses the [[polynomial ring]] {{nowrap|'''R'''[''X'',''Y'']}} in two indeterminates ''X'' and ''Y''. Consider these three second degree [[polynomial]]s <math>X^2 + 1,\ Y^2 - 1,\ XY - YX</math>. Let ''A'' be the [[ideal (ring theory)|ideal]] generated by them. Then the [[quotient ring]] {{nowrap|'''R'''[''X'',''Y'']/''A''}} is isomorphic to the ring of tessarines. In this quotient ring approach, individual tessarines correspond to [[coset]]s with respect to the ideal ''A''. Note that <math>(XY)^2 + 1 \in A</math> can be proven using computations with cosets.
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| Now consider the alternative ideal ''B'' generated by <math>X^2 + 1,\ Y^2 + 1,\ XY - YX</math>.
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| In this case one can prove <math>(XY)^2 - 1 \in B</math>. The [[ring isomorphism]] {{nowrap|'''R'''[''X'',''Y'']/''A'' ≅ '''R'''[''X'',''Y'']/''B''}} involves a [[change of basis]] exchanging <math>Y \leftrightarrow XY</math>.
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| The approach to tessarines by James Cockle resembles the use of ideal ''A'', while Corrado Segre's bicomplex numbers correspond to the use of ideal ''B''.
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| Alternatively, suppose the field '''C''' of ordinary complex numbers is presumed given, and '''C'''[''X''] is the ring of polynomials in ''X'' with complex coefficients. Then the quotient {{nowrap|'''C'''[''X'']/(''X''<sup>2</sup> − 1)}} is another presentation of bicomplex numbers.
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| ==Algebraic properties==
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| Tessarines with ''w'' and ''z'' complex numbers form a [[commutative]] and [[associative]] quaternionic [[ring (mathematics)|ring]] (whereas [[quaternion]]s are not commutative). They allow for powers, roots, and logarithms of <math>j \equiv \varepsilon</math>, which is a non-real root of 1 (see [[conic quaternion]]s for examples and references). They do not form a [[field (mathematics)|field]] because the [[idempotent]]s
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| : <math>\begin{pmatrix} z & \pm z \\ \pm z & z \end{pmatrix} \equiv z (1 \pm j) \equiv z (1 \pm \varepsilon)</math>
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| have [[determinant]] / [[absolute value|modulus]] 0 and therefore cannot be inverted multiplicatively. In addition, the arithmetic contains [[zero divisor]]s
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| : <math>\begin{pmatrix} z & z \\ z & z \end{pmatrix} \begin{pmatrix} z & -z \\ -z & z \end{pmatrix}
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| \equiv z^2 (1 + j )(1 - j)
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| \equiv z^2 (1 + \varepsilon )(1 - \varepsilon) = 0.</math>
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| In contrast, the [[quaternion]]s form a [[skew field]] without zero-divisors, and can also be represented in [[Quaternion#Representing_quaternions_by_matrices|2×2 matrix form]].
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| ==Polynomial roots==
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| Write {{nowrap|1=<sup>2</sup>'''C''' = '''C''' ⊕ '''C'''}} and represent elements of it by ordered pairs (''u'',''v'') of complex numbers. Since the algebra of tessarines '''T''' is isomorphic to <sup>2</sup>'''C''', the [[ring of polynomials|rings of polynomials]] '''T'''[X] and <sup>2</sup>'''C'''[''X''] are also isomorphic, however polynomials in the latter algebra split:
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| :<math>\sum_{k=1}^n (a_k , b_k ) (u , v)^k \quad = \quad \left({\sum_{k=1}^n a_i u^k} ,\quad \sum_{k=1}^n b_k v^k \right).</math>
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| In consequence, when a polynomial equation <math>f(u,v) = (0,0)</math> in this algebra is set, it reduces to two polynomial equations on '''C'''. If the degree is ''n'', then there are ''n'' [[root of a function|roots]] for each equation: <math>u_1, u_2, \dots, u_n,\ v_1, v_2, \dots, v_n .</math>
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| Any ordered pair <math>( u_i , v_j ) \!</math> from this set of roots will satisfy the original equation in <sup>2</sup>'''C'''[''X''], so it has ''n''<sup>2</sup> roots.
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| Due to the isomorphism with '''T'''[''X''], there is a correspondence of polynomials and a correspondence of their roots. Hence the tessarine polynomials of degree ''n'' also have ''n''<sup>2</sup> roots, counting [[multiplicity (mathematics)|multiplicity of roots]].
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| ==Notes and references==
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| {{reflist}}
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| * Daniel Alfsmann (2006) [http://www.eurasip.org/proceedings/eusipco/eusipco2006/papers/1568981962.pdf On families of 2^N dimensional hypercomplex algebras suitable for digital signal processing], 14th European Signal Processing Conference, Florence, Italy.
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| * Daniel Alfsmann & Heinz G Göckler (2007) [http://www.dsv.rub.de/imperia/md/content/public/eusipco2007_hyperbolic.pdf On Hyperbolic Complex LTI Digital Systems]
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| * F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti. (2008) ''The Mathematics of Minkowski Space-Time with an Introduction to Commutative Hypercomplex Numbers'', [[Birkhäuser Verlag]], Basel ISBN 978-3-7643-8613-9.
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| * James Cockle in London-Dublin-Edinburgh [[Philosophical Magazine]], series 3
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| ** 1848 On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra, 33:435–9.
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| ** 1849 On a New Imaginary in Algebra 34:37–47.
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| ** 1849 On the Symbols of Algebra and on the Theory of Tessarines 34:406–10.
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| ** 1850 On Impossible Equations, on Impossible Quantities and on Tessarines 37:281–3.
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| * Clyde Davenport (1991) ''A Hypercomplex Calculus with Applications to Special Relativity'' ISBN 0-9623837-0-8 .
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| * Clyde Davenport (2008) [http://home.comcast.net/~cmdaven/hyprcplx.htm Commutative Hypercomplex Mathematics].
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| * Soo-Chang Pei, Ja-Han Chang & Jian-Jiun Ding (2004) "Commutative reduced biquaternions and their Fourier transform for signal and image processing", ''IEEE Transactions on Signal Processing'' 52:2012–31.
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| * G. Baley Price (1991) ''An introduction to multicomplex spaces and functions'', [[Marcel Dekker]] ISBN 0-8247-8345-X .
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| *{{Citation
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| | authorlink = Corrado Segre
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| | last = Segre
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| | first = Corrado
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| | title = Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici (The real representation of complex elements and hyperalgebraic entities)
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| | journal = [[Mathematische Annalen]]
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| | volume = 40
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| | pages = 413–467
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| | year = 1892
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| | url = http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0040&DMDID=DMDLOG_0034&L=1
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| }}. (see especially pages 455–67)
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| {{Number Systems}}
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| [[Category:Hypercomplex numbers]]
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| [[Category:Quaternions]]
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| [[Category:Matrices]]
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