Numerical range
  1. M. T. Chien, Boundedness of the numerical range, Linear Algebra and Its Applications, 134(1990), 25-30.
  2. M. T. Chien and B. S. Tam, Circularity of the numerical range, Linear Algebra and Its Applications 201(1994), 113-133.
  3. M. T. Chien, Numerical computation and circularity of higher numerical range, Soochow Journal Mathematics 20(1994), 555-567.
  4. M. T. Chien, On the numerical range of tridiagonal operators, Linear Algebra and Its Applications 246(1996), 203-214.
  5. M. T. Chien, The envelope of the generalized numerical range, Linear and Multilinear Algebra 43(1998), 363-376.
  6. M. T. Chien, Lina Yeh, Y. T. Yeh and F. Z. Lin, On geometric properties of the numerical range, Linear Algebra and Its Applications 274(1998), 389-410.
  7. M. T. Chien and M. Neumann, Positive definiteness of tridiagonal matrices via the numerical range, Electronic Journal of Linear Algebra 3(1998), 93-102.
  8. M. T. Chien and H. Nakazato, Boundary generating curves of the c-numerical range, Linear Algebra and Its Applications, 294(1999), 67-84.
  9. M. T. Chien and Y. H. Lin, On the area of numerical range, Soocchow Journal of Mathematics, 26(2000), 255-269.
  10. M. T. Chien and J. M. Huang, Numerical range of a continuant matrix, Applied Mathematics Letters, 14(2001), 213-216.
  11. M. T. Chien, The c-numerical range of a rank one matrix, Applied Mathematics Letters, 14(2001), 167-170.
  12. M. T. Chien and H. Nakazato, The c-numerical range of tridiagonal matrices, Linear Algebra and Its Applications, 335(2001), 55-61.
  13. M. T. Chien and H. Nakazato, Davis-Wielandt shell and q-numerical range, Linear Algebra and Its Applications, 340(2002), 15-31.
  14. M. T. Chien and H. Nakazato, The numerical range of linear pencils of 2-by-2 matrices, Linear Algebra and Its Applications, 341(2002), 69-100.
  15. M. T. Chien, H. Nakazato and P. Psarrakos, Point equation of the boundary of the numerical range of a matrix polynomial, Linear Algebra and Its Applications, 347(2002), 205-217.
  16. Y. A. Alpin, M. T. Chien and L. Yeh, The numerical radius and bounds for zeros of a polynomial, Proceedings American Mathematical Society, 131(2003), 725-730.
  17. M. T. Chien, S. H. Tso and P. Y. Wu, Higher-dimensional numerical ranges of quadratic operators, Journal of Operator Theory, 49(2003), 155-173.
  18. M. T.  Chien and H. Nakazato, Complex formulation of Poncelet property in numerical range, Linear and Multilinear Algebra, 52(2004), 159-175.
  19. M. T.  Chien, H. Nakazato and P. Psarrakos, On the q-numerical range of matrices and matrix polynomials,  Linear and Multilinear Algebra, 53(2005), 357-374.
  20. M. T. Chien and H. Nakazato, The q-numerical ranges of normal operators, Linear and Multilinear Algebra, 53(2005), 393-416.
  21. M. T. Chien, H. Nakazato and P. Psarrakos, The q-numerical range and the Davis-Wielandt shell of reducible 3-by-3 matrices, Linear and Multilinear Algebra, 54(2006), 79-112.
  22. M. T. Chien and H. Nakazato, Elliptic Curves Arising from Numerical Ranges, International Journal of Pure and Applied Mathematics, 30(2006), 441-465.
  23. M. T. Chien and H. Nakazato, The q-numerical range of a reducible matrix via a normal operator, Linear Algebra and Its Applications, 419(2006), 440-465.
  24. M. T. Chien and H. Nakazato, The q-numerical radius of weighted shift operators with periodic weights, Linear Algebra and Its Applications, 422(2007), 198-218.
  25. M. T. Chien and H. Nakazato, The boundary of the q-numerical range of a reducible matrix, Linear and Multilinear Algebra, 55(2007), 275-292.
  26. M. T. Chien and H. Nakazato, Flat portions on the boundary of the numerical ranges of certain Toeplitz matrices, Linear and Multilinear Algebra, 56(2008), 143-162.
  27. M. T. Chien and H. Nakazato, The q-numerical range of unitarily irreducible 3-by-3 matrices, International Journal of Contemporary Mathematical Sciences, 3(2008), 339-355.
  28. M. T. Chien and H. Nakazato, Perturbation of the q-numerical radius of a weighted shift operator, Journal of Mathematical Analysis and Applications, 345(2008), 954-963.
  29. M. T. Chien and H. Nakazato, The q-numerical range of 3-by-3 weighted shift matrices, Applied Mathematics Letter, 21( 2008), 1199-1203.
  30. M. T. Chien and H. Nakazato, Flat portions on the boundary of the Davis-Wielandt shell of 3-by-3 matrices, Linear Algebra and Its Applications, 430(2009), 204-214.
  31. M. T. Chien and H. Nakazato, The numerical radius of a weighted shift operator with geometric weights, Electronic Journal of Linear Algebra, 18(2009), 58-63.
  32. M. T. Chien and H. Nakazato, Joint numerical range and its generating hypersurface, Linear Algebra and Its Applications, 432(Jan 2010), 173-179.
  33. M. T. Chien and L. Yeh, On the boundary of the numerical range of a matrix, Applied Mathematics Letters, 23(June 2010), 725-727.
  34. M. T. Chien, C. L. Ko and H. Nakazato, On the numerical ranges of matrix products, Applied Mathematics Letters, 23(June 2010), 732-737.
  35. M. T. Chien and H. Nakazato, The q-numerical range of 3×3 tridiagonal matrices, Electronic Journal of Linear Algebra, 20(July, 2010), 376-390.
  36. M. T. Chien and H. Nakazato, Numerical range for orbits under a central force, Mathematical Physics Analysis and Geometry, 13(Dec 2010), 315-330.
  37. M. T. Chien and H. Nakazato, The numerical range of a tridiagonal operator, Journal of Mathematical Analysis and Applications, 373 (January, 2011), 297-304.
  38. M. T. Chien and H. Nakazato, Reduction of the c-numerical range to the classical numerical range, Linear Algebra and Its Applications, 434(Feb 2011), 615-624.
  39. M. T. Chien, Lucas' theorem and numerical range, Linear and Multilinear Algebra, 59( June 2011), 687-691.
  40. M. T. Chien and H. Nakazato, Construction of determinantal representation of trigonometric polynomials, Linear Algebra and Its Applications, 435(Sept 2011), 1277-1284.
  41. M. T. Chien and H. Nakazato, The boundary of higher rank numerical ranges, Linear Algebra and Its Applications, 435( Dec 2011), 2971-2985.
  42. M. T. Chien and H. Nakazato, Central force and the higher rank numerical range, Journal of Mathematical Analysis and Applications, 389(May 2012), 531-540.
  43. M. T. Chien and K. H. Hung, Elliptic numerical ranges of bordered matrices, Taiwanese  Journal Mathematics, 16(June 2012), 1007-1016.
  44. M. T. Chien and H. Nakazato, Determinantal representations of closed orbits, Linear Algebra and Its Applications, 437(Aug 2012), 992-1002.
  45. M. T. Chien and H. Nakazato, Singular points of the ternary polynomials associated with 4 by 4 matrices, Electronic Journal of Linear Algebra, 23(Aug 2012), 755-769.
  46. M. T. Chien and H. Nakazato, Critical values for higher rank numerical ranges associated with roulette curves, Linear Algebra and Its Applications,  437(Nov 2012), 2117-2127.
  47. M. T. Chien and H. Nakazato, Strict convexity of the joint c-numerical range, Linear Algebra and Its Applications,  438(Feb 2013), 1305-1321.
  48. M. T. Chien and H. A. Sheu, The numerical radii of weighted shift matrices and operators, Operators and Matrices, 7(March 2013), 197-204.
  49. M. T. Chien and H. Nakazato, Cubic surfaces and q-numerical ranges, Mathematical Communications, 18(May 2013), 133-141.
  50. M. T. Chien and H. Nakazato, Hyperbolic forms associated with cyclic weighted shift matrices, Linear Algebra and Its Applications, 439(Dec 2013), 3541-3554.
  51. M. T. Chien and H. Nakazato, Singular points of cyclic weighted shift matrices, Linear Algebra and Its Applications, 439(Dec 2013), 4090-4100.
  52. M. T. Chien and H. Nakazato, Determinantal representation of trigonometric polynomial curves via Sylvester method, Banach Journal of Mathematical Analysis, 8(Jan 2014), 269-278.
  53. M. T. Chien and H. Nakazato, Reduction of joint c-numerical ranges, Applied Mathematics and Computation, 232(Apr 2014), 178-182.
  54. M. T. Chien and H. Nakazato, Numerical range of a nilpotent matrix and related central force,  Linear and Multilinear Algebra, 62(May 2014), 595-613.
  55. M. T. Chien and H. Nakazato, Singular points of the algebraic curves of symmetric hyperbolic forms, Linear Algebra and Its Applications, 470(April 2015), 40-50.
  56. M. T. Chien and H. Nakazato, Determinantal representations of hyperbolic forms via weighted shift matrices, Applied Mathematics and Computation, 258(May 2015), 172-181.
  57. M. T. Chien and H. Nakazato, Elliptic modular invariants and numerical ranges,  Linear and Multilinear Algebra, 63(Aug 2015), 1501-1519.
  58. B. Undrakh, H. Nakazato, A. Vandanjav, M. T. Chien*, The numerical radius of a weighted shift operator, Electronic Journal of Linear Algebra, 30(Dec 2015), 944-963
  59. M. T. Chien and H. Nakazato, A new Poncelet curve for the boundary generating curve of a numerical range, Linear Algebra and Its Applications, 487(Dec 2015), 1-21.
  60. M. T. Chien, H. Nakazato, B. Undrakh and A. Vandanjav, Determinantal polynomials of a weighted shift operator, Linear and Multilinear Algebra, 64(Jan 2016), 2-13.
  61. M. T. Chien, H.-L. Gau*, C.-K. Li, M.-C. Tsai, K.-Z. Wang, Product of operators and numerical range, Linear and Multilinear Algebra, 64(Jan 2016), 58-67.
  62. M. T. Chien, S.-T. Fang, Y.-X. Su, A generalization of Gershgorin circles, Applied and Computational Mathematics, 15(Feb 2016), 106-111.
  63. M. T. Chien and H. Nakazato, Computing the determinantal representations of hyperbolic forms, Czechoslovak Mathematical Journal, 66(Sept 2016), 633-651.
  64. M. T. Chien and H. Nakazato, Singular points of the algebraic curves associated to unitary bordering matrices, Linear Algebra and Its Applications, 513(Jan 2017), 224-239.
  65. M. T. Chien and H. Nakazato, Computation of Riemann matrices for the hyperbolic curves of determinantal polynomials, Annals of Functional Analysis, 8(May 2017), 152-167.
  66. M. T. Chien and H. Nakazato, Reducibility of the ternary forms of unitary bordering matrices, Linear Algebra and Its Applications, 527(Aug 2017), 73-86.

 

    c*-algebra

  1. M. T. Chien, C*-algebra fibre bundles, Transactions American Mathematical Society, 273(1982), 795-801.
  2. M. T. Chien, Duality of C*-algebra fibre bundles, Mathematical Reports, Academy Science Canada, 4(1982), 331-336.
  3. M. T. Chien, Tensor products of C*-algebra fibre bundles, Mathematical Reports, Academy Science Canada, 6(1984), 211-216.
  4. M. T. Chien, Algebra actions of semigroup algebras, Soochow Journal of Mathematics, 10(1984), 33-48.
  5. M. T. Chien, Closed derivations on the unit square, Bulletin Australian Mathematics Society, 11(1986), 55-67.

    Miscellanea

  1. M. T. Chien, Generalized convolutions, Chinese Journal of Mathematics, 10(1982), 31-38.
  2. M. T. Chien, On the adjoint of spectral operators, Bulletin Institute of Mathematics, Academia Sinica, 11(1983), 257-260
  3. M. T. Chien, The Fourier transforms of almost periodic functions on groups, Chinese Journal of Mathematics, 19(1991), 1-10.
  4. W. H Chen and M. T. Chien, Tree sign pattern matrices that require zero eigenvalues, Bulletin Australian Mathematical Society 55(1997), 81-88.
  5. M. T. Chien, Independent edges of certain graphs, Combinatorial Mathematics and Combinatorial Computing, 30(1999), 231-236.
  6. M. T. Chien and H. Nakazato, Circumscribed sphere of a convex polyhedron, Applied Mathematics Letters, 18(2005), 1199-1203.



 

    c-numerical range plotting

    The following codes plot the boundary of the c-numerical range of a matrix


       

    The image is a sample run for A=[1 0 1;0 -1 2;0 0 -1]; c=[1 0 0]; m=50

 clg;clc;
 disp('');disp('');disp('');disp('');
 disp('Programmer: Mao-Ting Chien');
disp(' Department of Mathematics');
disp(' Soochow University');
disp(' Taipei, TAIWAN');
disp(' e-mail: mtchien@scu.edu.tw');
disp(' ');
disp(' Save this file  as cnumg.m');
disp('1. run matlab');
disp('2. input this file name  cnumg');
disp(' ')
disp(' This program produces the boundary of the c-numerical range of a ');
disp('complex matrix by using MATLAB package, where c is a real n-tuple.');
disp('Enter a complex matrix A, an n-tuple c and the number of ');
disp('boundary points m');
disp('eg. A=[1 2;3 4] enter, c=[5 6] enter, m=50 enter ');
disp(' ');
disp(' ****** Press any key to run the program ******');
pause;
axis;
axis('square')
i=sqrt(-1);
Tol=10^(-5);
A=input('enter a matrix A= ');
C=input('enter an n-tuple c= ');
m=input('enter steps m= ');
% The begin of progran
C=-(sort(-C));
p=[];
for j=1:m+1
theta=(j-1)*2*pi/m;
H=(exp(i*theta)*A+exp(-i*theta)*A')/2;
[X,L]=eig(H);
[lambda,s]=sort(-diag(real(L)));
lambda=-lambda;
X=X(:,s);
k=1;
Y=[];
while k<size(A)
E=[];
if abs(lambda(k)-lambda(k+1)) > Tol
E=X(:,k)/norm(X(:,k));
Y=[Y E];
if k==size(A)-1
E=X(:,k+1)/norm(X(:,k+1));
Y=[Y E];
end
k=k+1;
else
E=[E X(:,k)];
while abs(lambda(k)-lambda(k+1)) <Tol
E=[E X(:,k+1)];
k=k+1;
if (k>=size(A)),break,end
end
E=orth(E);
Y=[Y E];
if k==size(A)-1
E=X(:,k+1)/norm(X(:,k+1));
Y=[Y E];
end
k=k+1;
end
end
for k=1:size(A)
S=Y(:,k)'*A*Y(:,k);
Q(k)=C(k)*S;
end
p(j)=sum(Q);
end
plot(real(p),imag(p));
meta cnumg
% end of the program