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ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(4): Pg.336-338
Rigidity Holder for Cubic Matrices M. H. Rezaeigol and M. H. Hosseini Academic Member of School of Mathematics, University of Birjand, IRAN. (Received on: July 4, 2014) ABSTRACT Let ܲ be a field, and â€ŤÜŻâ€ŹáˆşŕŹˇáˆť (ܲ) is ring of cubic matrices on a field ܲ. In this paper is defined (â€ŤÜĽâ€ŹŕŹľ , â€ŤÜĽâ€ŹŕŹś )-Holder valuation on a ring â€ŤÜŻâ€ŹáˆşŕŹˇáˆť (ܲ) and then is proved, that áˆşâ€ŤÜĽâ€ŹŕŹľ , â€ŤÜĽâ€ŹŕŹś áˆť-Holder valuation on a ring â€ŤÜŻâ€ŹáˆşŕŹˇáˆť (ܲ) is (2, ß™)-Holder equivalent to some classical valuation, where ß™ = áˆşlogáˆş2â€ŤÜĽâ€ŹŕŹľ áˆťáˆťŕŹżŕŹľ . It gives expansion of the theorem of Garcia ([4]) on some class of noncommutative rings (class of cubic matrices on a field ܲ). Keywords: cubic matrices; (â€ŤÜĽâ€ŹŕŹľ , â€ŤÜĽâ€ŹŕŹś )-Holder.
1. INTRODUCTION Definition 1.1. Let Γ be a totally ordered abelian group. matrix valuation on skew field (simple valuation on = ௥ ( )) is a function : → Γ âˆŞ {∞} satisfying: 1. = + ( ), for all , ∈ ; 2. ∇ ≼ min{ , ( )}, for all , ∈ such that ∇ is defined; 3. ( ) is unchanged if any row or column of is multiplied by -1; 4. = ∞ for any singular matrix Üş ∈ ‍;ܣ‏ 5. = 0, where is an identity matrix. Remark 1.2. We observe that when = 1, (i.e., is a division ring), then (1)-(5) simply say that is a valuation on .
Proposition 1.3. Let be a simple valuation on = ௥ ( ). Then we have: •
• • •
If ≠( ). Then ∇ = min , , whenever ∇ is defined in . = 0for any elementary matrix in . ( )is unchanged if multiplied on the left (or right) by an elementary matrix. ( )remains unchanged under any permutation of rows (or columns).
2. CUBIC MATRICES AND RIGIDITY HOLDER Now let given the numerical field . Any system from ଷ elements ŕŻœ,ŕŻ?,௞ ( , , =
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M. H. Rezaeigol, et al., J. Comp. & Math. Sci. Vol.5 (4), 336-338 (2014)
1,2, â‹Ż , ) of field that defined as coordinates , , , is called a 3-dimensional (cubic) matrix of order n on and it is denoted in abbreviated from by a symbol ŕŻœŕŻ?௞ ( , , = 1,2, â‹Ż , ) . Definition 2.1. The valuation on set of cubic matrices áˆşŕŹˇáˆť ( ) on a field is called map | |: áˆşŕŹˇáˆť → â„? âˆŞ {∞}, satisfying the following conditions: áˆşŕŹˇáˆť
1. if = ŕŻœŕŻ?௞ ∈ ௥ ( ) and is a singular matrix, then | | = ∞; áˆşŕŹˇáˆť 2. if = ŕŻœŕŻ?௞ , = ŕŻœŕŻ?௞ ∈ ௥ , 1 ≤ ≤ , ŕ°’ŕŻ?௞ = − ŕ°’ŕŻ?௞ and ŕŻœŕŻ?௞ = ŕŻœŕŻ?௞ for ≠, 1 ≤ , ≤ , then | | = | |; áˆşŕŹˇáˆť 3. if = ŕŻœŕŻ?௞ , = ŕŻœŕŻ?௞ ∈ ௥ and the determinant sum ∇ is | ∇ | ≼ determined, then min{| |, | |}; áˆşŕŹˇáˆť 4. for any matrix = ŕŻœŕŻ?௞ ∈ ௥ , áˆşŕŹˇáˆť = ŕŻœŕŻ?௞ ∈ ௠: | ⊕ | = | | + | |; áˆşŕŹˇáˆť 5. | | = 0, where = ௥ ∈ ௥ ( ) is an identity cubic matrix.
Definition 2.2.Let ଵ , ଶ ≼ 1. Then a ( ଵ , ଶ )-Holder valuation on set of cubic matrices áˆşŕŹˇáˆť ( ) on a field is called map : áˆşŕŹˇáˆť → â„? âˆŞ {∞}, satisfying the following conditions: áˆşŕŹˇáˆť
1. If = ŕŻœŕŻ?௞ ∈ ௥ ( ) and is a singular matrix, then | | = ∞; áˆşŕŹˇáˆť 2. if = ŕŻœŕŻ?௞ , = ŕŻœŕŻ?௞ ∈ ௥ , 1 ≤ ≤ , ŕ°’ŕŻ?௞ = − ŕ°’ŕŻ?௞ and ŕŻœŕŻ?௞ = ŕŻœŕŻ?௞ for ≠, 1 ≤ , ≤ , then | | = | |;
áˆşŕŹˇáˆť
3. if = ŕŻœŕŻ?௞ , = ŕŻœŕŻ?௞ ∈ ௥ ( ) and the determinant sum ∇ is determined, then | ∇ | ≼ ଶ min{| |, | |}; áˆşŕŹˇáˆť 4. for any matrix = ŕŻœŕŻ?௞ ∈ ௥ , áˆşŕŹˇáˆť = ŕŻœŕŻ?௞ ∈ ௠: ଵିଵ + ≤ ⊕ ≤ ଵ + ; áˆşŕŹˇáˆť 5. | | = 0, where = ௥ ∈ ௥ ( ) is an identity cubic matrix. Remark 2.3. We shall notice, that a (1,1)Holder valuation on set of cubic matrices áˆşŕŹˇáˆť ( ) on a field is a classical valuation. Let | |, are two valuations on set of cubic matrices áˆşŕŹˇáˆť ( ) on a field . Then we say, that are ଴ , equivalent, if: ଴ିଵ | |ŕ°ˆ ≤ ≤ ଴ | |ŕ°ˆ for all ∈ áˆşŕŹˇáˆť ( ). Theorem 2.4.(Rigidity Holder for set of cubic matrices). Let : áˆşŕŹˇáˆť â&#x;ś â„? âˆŞ {∞} is a ( ଵ , ଶ )Holder valuations from set of cubic matrices áˆşŕŹˇáˆť ( ), where ଵ ≼ 1, ଶ ≼ 1. Then there exist a classical valuations | | on set of cubic matrices áˆşŕŹˇáˆť ( ), which is (2, )equivalent to valuation , where = log ଶ 2 ଵ ିଵ. REFERENCES 1. Leca M. Lecat, Coup d’oeilsur la theorie des determinants superieursdans son etatactuel. Ann. Soc. Scient. Bruxelles 45 (1926), II, fasc. 1/2, 1-98; fasc. 3/4, 141168; 46 (1926), 15-54; 47, serie A, II, fasc. 1, 1-37 (1927).
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M. H. Rezaeigol, et al., J. Comp. & Math. Sci. Vol.5 (4), 336-338 (2014)
2. Rice L. H. Rice, introduction to higher determinants, Journ. Math. Phyz. 9, 4771 (1930). 3. M.H. Hosseini. Holder rigidity for matrices Fundamental and Applied Mathematics., Volume 10, Number 4, 225-233 (2004). 4. Garc E. Munoz Garcia, Hoelder absolute val-ues are equivalent to classical ones. Proceedings of the American Math. Soc., V. 127, N 7, p. 1967-1971 (1999).
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Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)