Nonconvexity of the Generalized Numerical Range Associated with the Principal Character

2000 ◽  
Vol 43 (4) ◽  
pp. 448-458
Author(s):  
Chi-Kwong Li ◽  
Alexandru Zaharia
Keyword(s):  
Symmetric Group ◽  
Matrix Function ◽  
Numerical Range ◽  
Complex Matrix ◽  
Normal Matrices ◽  
Principal Character ◽  
Straight Line ◽  
Generalized Matrix ◽  
Generalized Numerical Range ◽  
Special Case

AbstractSuppose m and n are integers such that 1 ≤ m ≤ n. For a subgroup H of the symmetric group Sm of degree m, consider the generalized matrix function on m × m matrices B = (bij) defined by and the generalized numerical range of an n × n complex matrix A associated with dH defined byIt is known that WH(A) is convex if m = 1 or if m = n = 2. We show that there exist normal matrices A for which WH(A) is not convex if 3 ≤ m ≤ n. Moreover, for m = 2 < n, we prove that a normal matrix A with eigenvalues lying on a straight line has convex WH(A) if and only if νA is Hermitian for some nonzero ν ∈ ℂ. These results extend those of Hu, Hurley and Tam, who studied the special case when 2 ≤ m ≤ 3 ≤ n and H = Sm.

A Note on the van Der Waerden Permanent Conjecture

1974 ◽  
Vol 26 (02) ◽  
pp. 352-354 ◽  
Author(s):  
Jacques Dubois
Keyword(s):  
Symmetric Group ◽  
Matrix Function ◽  
Combinatorial Problems ◽  
Complex Matrix ◽  
Image Position ◽  
Square Complex ◽  
Extensive Bibliography

The permanent of an n-square complex matrix P = (pij ) is defined by where the summation extends over Sn , the symmetric group of degree n. This matrix function has considerable significance in certain combinatorial problems [6; 7]. The properties and many related problems about the permanent are presented in [3] along with an extensive bibliography.

scholarly journals A remark on a conjecture of Marcus on the generalized numerical range

1983 ◽  
Vol 24 (2) ◽  
pp. 191-194 ◽  
Author(s):  
Yik-Hoi Au-Yeung ◽  
Kam-Chuen Ng
Keyword(s):  
Permutation Group ◽  
Relative Position ◽  
Numerical Range ◽  
Complete Characterization ◽  
Diagonal Matrix ◽  
Unitary Matrix ◽  
Complex Matrix ◽  
Conjugate Transpose ◽  
Generalized Numerical Range

Let A be an n × n complex matrix and c = (c1… cn) єℂn. Define the c-numerical range of A to be the set is an orthonormal set in , where * denotes the conjugate transpose. Westwick [8[ proved that if c … cn are collinear, then Wc(A) is convex. (Poon [6] gave another proof.) But in general for n ≧3, Wc(A) may fail to be convex even for normal A (for example, see Marcus [4] or Lemma 3 in this note) though it is star-shaped (Tsing [7]). In the following, we shall assume that A is normal. Let W(A) = {diag UAU*: U is unitary}. Horn [3] proved that if the eigenvalues of A are collinear, then W(A) is convex. Au-Yeung and Sing [2] showed that the converse is also true. Marcus [4] further conjectured (and proved for n = 3) that if Wc(A) is convex for all cєℂn then the eigenvalues of A are collinear. Let λ = (λ1, …, λn єℂn. We denote by the vector λ1, …, λn and by [λ] the diagonal matrix with λ1, …, λn lying on its diagonal. Since, for any unitary matrix U,. Wc(A) = Wc (UAU*), the Marcus conjecture reduces to: if Wc([λ]) is convex for all c єℂn then λ1, … λn are collinear. For the case n = 3, Au-Yeung and Poon [1] gave a complete characterization on the convexity of the set Wc([λ]) in terms of the relative position of the points , where σ є S3 the permutation group of order 3. As an example they showed that if λ1, λ2, λ3 are not collinear, then is not convex (Lemma 3 in this note gives another proof). We shall show that for the case n = 4, is not convex if λ1, λ2. λ3. λ4 are not collinear. Thus for n = 3, 4 the Marcus conjecture is answered and improved.

Mean Value and Limit Theorems for Generalized Matrix Functions

1969 ◽  
Vol 21 ◽  
pp. 982-991 ◽  
Author(s):  
Paul J. Nikolai
Keyword(s):  
Symmetric Group ◽  
Matrix Function ◽  
Limit Theorems ◽  
Mean Value ◽  
Complex Numbers ◽  
Matrix Functions ◽  
Principal Submatrix ◽  
Generalized Matrix ◽  
Image Position ◽  
Square Matrix

Let A = [aij] denote an n-square matrix with entries in the field of complex numbers. Denote by H a subgroup of Sn, the symmetric group on the integers 1, …, n, and by a character of degree 1 on H. Thenis the generalized matrix function of A associated with H and x; e.g., if H = Sn and χ = 1, then the permanent function. If the sequences ω = (ω1, …, ωm) and ϒ = (ϒ1, …, ϒm) are m-selections, m ≦ w, of integers 1, …, n, then A [ω| ϒ] denotes the m-square generalized submatrix [aωiϒj], i, j = 1, …, m, of the n-square matrix A. If ω is an increasing m-combination, then A [ω|ω] is an m-square principal submatrix of A.

scholarly journals Optical axes of catadioptric systems including visual, Purkinje and other nonvisual systems of a heterocentric astigmatic eye

2010 ◽  
Vol 69 (3) ◽  
Author(s):  
W. F. Harris
Keyword(s):  
Visual System ◽  
Optical Axis ◽  
The Other ◽  
Optical Systems ◽  
Numerical Examples ◽  
Straight Line ◽  
Optical Axes ◽  
Catadioptric Systems ◽  
Reflecting Surfaces ◽  
Special Case

For a dioptric system with elements which may be heterocentric and astigmatic an optical axis has been defined to be a straight line along which a ray both enters and emerges from the system.  Previous work shows that the dioptric system may or may not have an optical axis and that, if it does have one, then that optical axis may or may not be unique.  Formulae were derived for the locations of any optical axes.  The purpose of this paper is to extend those results to allow for reflecting surfaces in the system in addition to refracting elements.  Thus the paper locates any optical axes in catadioptric systems (including dioptric systems as a special case).  The reflecting surfaces may be astigmatic and decentred or tilted.  The theory is illustrated by means of numerical examples.  The locations of the optical axes are calculated for seven optical systems associated with a particular heterocentric astigmatic model eye.  The optical systems are the visual system, the four Purkinje systems and two other nonvisual systems of the eye.  The Purkinje systems each have an infinity of optical axes whereas the other nonvisual systems, and the visual system, each have a unique optical axis. (S Afr Optom 2010 69(3) 152-160)

scholarly journals The star-shapedness of a generalized numerical range

2016 ◽  
Vol 506 ◽  
pp. 308-315 ◽  
Author(s):  
Pan-Shun Lau ◽  
Tuen-Wai Ng ◽  
Nam-Kiu Tsing
Keyword(s):  
Numerical Range ◽  
Generalized Numerical Range

A conjecture of marcus on the generalized numerical range

1983 ◽  
Vol 14 (3) ◽  
pp. 235-239 ◽  
Author(s):  
Yik-Hoi Au-Yeung ◽  
Nam-Kiu Tsing
Keyword(s):  
Numerical Range ◽  
Generalized Numerical Range

scholarly journals Stability of a Hot Smoluchowski Fluid

2001 ◽  
Vol 08 (01) ◽  
pp. 19-27 ◽  
Author(s):  
R. F. Streater
Keyword(s):  
Boundary Conditions ◽  
Parabolic Equations ◽  
Numerical Range ◽  
Nonlinear Parabolic Equations ◽  
Stationary Solutions ◽  
Small Perturbations ◽  
Material Density ◽  
Nonlinear Parabolic ◽  
The Stability ◽  
Special Case

We study coupled nonlinear parabolic equations for a fluid described by a material density ρ and a temperature Θ, both functions of space and time. In one dimension, we find some stationary solutions corresponding to fixing the temperature on the boundary, with no-escape boundary conditions for the material. For the special case, where the temperature on the boundary is the same at both ends, the linearized equations for small perturbations about a stationary solution at uniform temperature and density are derived; they are subject to boundary conditions, Dirichlet for Θ and no-flow conditions for the material. The spectrum of the generator L of time evolution, regarded as an operator on L2[0,1], is shown to be real, discrete and non-positive, even though L is not self-adjoint. This result is necessary for the stability of the stationary state, but might not be sufficient. The problem lies in the fact that L is not a sectorial operator, since its numerical range is ℂ.

scholarly journals Congruences induced by transitive representations of inverse semigroups

1987 ◽  
Vol 29 (1) ◽  
pp. 21-40 ◽  
Author(s):  
Mario Petrich ◽  
Stuart Rankin
Keyword(s):  
General Theory ◽  
Inverse Semigroup ◽  
Symmetric Group ◽  
Transitive Group ◽  
Inverse Semigroups ◽  
Group Representations ◽  
Symmetric Inverse Semigroup ◽  
The Right ◽  
Special Case

Transitive group representations have their analogue for inverse semigroups as discovered by Schein [7]. The right cosets in the group case find their counterpart in the right ω-cosets and the symmetric inverse semigroup plays the role of the symmetric group. The general theory developed by Schein admits a special case discovered independently by Ponizovskiǐ [4] and Reilly [5]. For a discussion of this topic, see [1, §7.3] and [2, Chapter IV].

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