A topological approach to divisibility of arithmetical functions and GCD matrices

References

• Alexandroff , P . 1937 . Diskrete Räume . Mat. Sb. , 2 : 501 – 518 .
   Google Scholar
• Altinisik , E , Sagan , BE and Tuglu , N . 2005 . GCD matrices, posets, and nonintersecting paths . Linear Multilinear Algebra , 53 : 75 – 84 .
   Web of Science ®Google Scholar
• Altinisik , E , Tuglu , N and Haukkanen , P . 2004 . A note on bounds for norms of the reciprocal LCM matrix . Math. Inequal. Appl. , 7 : 491 – 496 .
   Web of Science ®Google Scholar
• Apostol , TM . 1976 . Introduction to Analytic Number Theory , Berlin : Springer-Verlag .
   Google Scholar
• Arenas , FG . 1999 . Alexandroff spaces . Acta Math. Univ. Comenianae , 68 : 17 – 25 .
   Google Scholar
• Bhatia , R and Dias da Silva , JA . 2008 . Infinite divisibility of GCD matrices . Am. Math. Monthly , 115 : 551 – 553 .
   Web of Science ®Google Scholar
• Bourque , K and Ligh , S . 1992 . On GCD and LCM matrices . Linear Algebra Appl. , 174 : 65 – 74 .
   Web of Science ®Google Scholar
• Bourque , K and Ligh , S . 1993 . Matrices associated with classes of arithmetical functions . J. Number Theory , 45 : 367 – 376 .
   Web of Science ®Google Scholar
• Bourque , K and Ligh , S . 1995 . Matrices associated with multiplicative functions . Linear Algebra Appl. , 216 : 267 – 275 .
   Web of Science ®Google Scholar
• Conover , RA . 1975 . “ A First Course in Topology: ” . In An Introduction to Mathematical Thinking , Baltimore , MD : Williams & Wilkins .
   Google Scholar
• Dugundji , J . 1966 . Topology , Boston : Allyn and Bacon .
   Google Scholar
• Feng , W , Hong , S and Zhao , J . 2009 . Divisibility properties of power LCM matrices by power GCD matrices on gcd-closed sets . Discrete Math. , 309 : 2627 – 2639 .
   Web of Science ®Google Scholar
• Gierz , G , Hofmann , KH , Keimel , K , Lawson , JD , Mislove , M and Scott , DS . 2003 . Continuous Lattices and Domains (Encyclopedia of Mathematics and its Applications 93) , Cambridge : Cambridge University Press .
   Google Scholar
• Haukkanen , P . 1996 . Some characterizations of totients . Int. J. Math. Math. Sci. , 19 : 209 – 217 .
   Google Scholar
• Haukkanen , P . 2004 . On the ℓ p norm of GCD and related matri.e. JIPAM . J. Inequal. Pure Appl. Math. , 5 : 7 (Article 61)
   Google Scholar
• Haukkanen , P . 2006 . An upper bound for the ℓ p norm of a GCD-related matrix . J. Inequal. Appl. , 2006 : 1 – 6 .
   Web of Science ®Google Scholar
• Haukkanen , P . 2007 . On the maximum row and column sum norm of GCD and related matrices, JIPAM . J. Inequal. Pure Appl. Math. , 8 : 7 (Article 97)
   Google Scholar
• Haukkanen , P , Ilmonen , P , Nalli , A and Sillanpää , J . 2010 . On unitary analogs of GCD reciprocal LCM matrices . Linear Multilinear Algebra , 58 : 599 – 616 .
   Web of Science ®Google Scholar
• Haukkanen , P and Korkee , I . 2005 . Notes on the divisibility of GCD and LCM matrices . Int. J. Math. Math. Sci. , 2005 : 925 – 935 .
   Google Scholar
• Haukkanen , P , Wang , J and Sillanpää , J . 1997 . On Smith's determinant . Linear Algebra Appl. , 258 : 251 – 269 .
   Web of Science ®Google Scholar
• He , C and Zhao , J . 2005 . More on divisibility of determinants of LCM matrices on GCD-closed sets . Southeast Asian Bull. Math. , 29 : 887 – 893 .
   Google Scholar
• Hong , S . 1999 . On the Bourque-Ligh conjecture of least common multiple matrices . J. Algebra , 218 : 216 – 228 .
   Web of Science ®Google Scholar
• Hong , S . 2002 . Gcd-closed sets and determinants of matrices associated with arithmetical functions . Acta Arith. , 101 : 321 – 332 .
   Web of Science ®Google Scholar
• Hong , S . 2002 . On the factorization of LCM matrices on gcd-closed sets . Linear Algebra Appl. , 345 : 225 – 233 .
   Web of Science ®Google Scholar
• Hong , S . 2003 . Divisibility of determinants of least common multiple matrices on GCD-closed sets . Southeast Asian Bull. Math. , 27 : 615 – 621 .
   Google Scholar
• Hong , S . 2003 . Factorization of matrices associated with classes of arithmetical functions . Colloq. Math. , 98 : 113 – 123 .
   Google Scholar
• Hong , S . 2004 . Notes on power LCM matrices . Acta Arith. , 111 : 165 – 177 .
   Web of Science ®Google Scholar
• Hong , S . 2006 . Nonsingularity of matrices associated with classes of arithmetical functions on lcm-closed sets . Linear Algebra Appl. , 416 : 124 – 134 .
   Web of Science ®Google Scholar
• Hong , S . 2008 . Infinite divisibility of Smith matrices . Acta Arith. , 134 : 381 – 386 .
   Web of Science ®Google Scholar
• Hong , S . 2008 . Divisibility properties of power GCD matrices and power LCM matrices . Linear Algebra Appl. , 428 : 1001 – 1008 .
   Web of Science ®Google Scholar
• Hong , S , Shum , KP and Sun , Q . 2006 . On nonsingular power LCM matrices . Algebra Colloq. , 13 : 689 – 704 .
   Web of Science ®Google Scholar
• Hong , S , Zhao , J and Yin , Y . 2008 . Divisibility properties of Smith matrices . Acta Arith. , 132 : 161 – 175 .
   Web of Science ®Google Scholar
• Hong , S , Zhou , X and Zhao , J . 2009 . Power GCD matrices for a UFD . Algebra Colloq. , 16 : 71 – 78 .
   Web of Science ®Google Scholar
• Hua , LK . 1982 . Introduction to Number Theory , Berlin : Springer-Verlag .
   Google Scholar
• Ilmonen , P and Haukkanen , P . 2011 . Smith meets Smith: Smith normal form of Smith matrix, Linear Multilinear Algebra , 59 : 557 – 564 .
   Web of Science ®Google Scholar
• Korkee , I and Haukkanen , P . 2003 . On meet and join matrices associated with incidence functions . Linear Algebra Appl. , 372 : 127 – 153 .
   Web of Science ®Google Scholar
• Korkee , I and Haukkanen , P . 2008 . On the divisibility of meet and join matrices . Linear Algebra Appl. , 429 : 1929 – 1943 .
   Web of Science ®Google Scholar
• Kukieła , MJ . 2010 . On homotopy types of Alexandroff spaces . Order , 27 : 9 – 21 .
   Web of Science ®Google Scholar
• MacDuffee , CC . 1946 . The Theory of Matrices , New York : Chelsea Publishing Company .
   Google Scholar
• McCarthy , PJ . 1986 . Introduction to Arithmetical Functions , New York : Springer-Verlag Universitext .
   Google Scholar
• Newman , M . 1972 . Integral Matrices , New York : Academic Press .
   Google Scholar
• Porubský , S . 2000 . “ Topologies induced by systems of divisors ” . In Algebraic Number Theory and Diophantine Analysis (Procedings of International Conference, Graz, 1998) , Edited by: Halter-koch , F and Tichy , RF . 391 – 404 . Berlin : de Gruyter .
   Google Scholar
• Qian , G and Lu , J . 2010 . LU-factorizations of symmetric matrices with applications . Asian-Eur. J. Math. , 3 : 133 – 143 .
   Google Scholar
• Rizza , GB . 1993 . A topology for the set of nonnegative integers . Riv. Mat. Univ. Parma , 2 ( 5 ) : 179 – 185 .
   Google Scholar
• Sándor , J and Crstici , B . 2004 . Handbook of Number Theory II , The Netherlands : Kluwer .
   Google Scholar
• Sivaramakrishnan , R . Classical Theory of Arithmetic Functions, Monographs and Text Books in Pure and Applied Mathematics, Vol. 126, Marcel Dekker, New York, 1989
   Google Scholar
• Smith , HJS . 1861 . On systems of linear indeterminate equations and congruences . Philos. Trans. Royal Soc. London. , 151 : 293 – 326 .
   Google Scholar
• Smith , HJS . 1876 . On the value of a certain arithmetical determinant . Proc. London Math. Soc. , 7 : 208 – 212 .
   Google Scholar
• Stanley , RP . 1997 . “ Enumerative Combinatorics ” . In Cambridge Studies in Advanced Mathematics, 49 , Vol. 1 , Cambridge : Cambridge University Press . (Corrected reprint of the 1986 original)
   Google Scholar
• Tan , Q . 2009 . Notes on the non-divisibility of determinants of power GCD and power LCM matrices . Southeast Asian Bull. Math. , 33 : 563 – 567 .
   Google Scholar
• Tan , Q . 2010 . Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains . Linear Multilinear Algebra , 58 : 659 – 671 .
   Web of Science ®Google Scholar
• Tan , Q and Lin , Z . 2010 . Divisibility of determinants of power GCD matrices and power LCM matrices on finitely many quasi-coprime divisor chains . Appl. Math. Comput. , 217 : 3910 – 3915 .
   Web of Science ®Google Scholar
• Tan , Q , Lin , Z and Liu , L . 2009 . Divisibility properties of the determinants of power GCD matrices and power LCM matrices on two coprime divisor chains (Chinese) . Sichuan Daxue Xuebao , 46 : 1581 – 1584 .
   Google Scholar
• Thompson , RC . 1986 . Left multiples and right divisors of integral matrices . Linear Multilinear Algebra , 19 : 287 – 295 .
   Google Scholar
• Zhao , J , Hong , S , Liao , Q and Shum , KP . 2007 . On the divisibility of power LCM matrices by power GCD matrices . Czechoslovak Math. J. , 57 ( 132 ) : 115 – 125 .
   Google Scholar
• Zhao , W and Zhao , J . 2009 . A characterization for the gcd-closed set S with |S| = 5 such that (S e ) divides [S e ] . Southeast Asian Bull. Math. , 33 : 1023 – 1028 .
   Google Scholar