Synapse's SNAP Network Operating System

Description

• mémoire - matière potentielle : size

Synapse Wireless, Inc. 500 Discovery Drive Huntsville, AL 35806 (P) 256-852-7888 (TF) 877-982-7888 (F) 256-852-7862 Synapse's SNAP Network Operating System Today we are surrounded by tiny embedded machines – electro-mechanical systems that monitor the environment around them and issue commands to control other machines and systems. One such machine may be monitoring the ambient light and checking for motion, poised to activate a lighting system.

• applications from the physical hardware
• available communications interfaces
• complete support for remote procedure calls
• hardware abstraction layer
• wireless modules
• snappy applications
• snap
• networks

Sujets

• Server
• Abstraction
• Wireless
• Remote
• Communications
• Physical
• Support

Informations

• Publié par: upsae
• Nombre de lectures: 19
• Langue: English

Extrait

On the P-property of Z and
Lyapunov-like transformations on
Euclidean Jordan algebras
M. Seetharama Gowda
Department of Mathematics and Statistics
University of Maryland, Baltimore County
Baltimore, Maryland
[e-mail]
***************
GTORA Conference, Chennai
January 5-7,2012
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 1/20This is joint work with J. Tao and G. Ravindran.
Based on forthcoming paper in Linear Algebra and Its
Applications:
On the P-property of Z and Lyapunov-like transformations
on Euclidean Jordan algebras.
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 2/20Outline
• Motivation and a conjecture
• Euclidean Jordan algebras
• Z and Lypaunov-like transformations
• Validity of the conjecture for Lyapunov-like transformations
• A result for Z-transformations
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 3/20Motivation
Recall a result from complementarity problems:
n×n
The following are equivalent for M ∈R :
All principal minors ofM are positive.
x∗Mx≤ 0⇒x = 0.
n
LCP(M,q) has a unique solution for all q∈R .
n
LCP(M,q): Find x∈R such that
x≥ 0, Mx+q≥ 0, and hMx +q,xi = 0.
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 4/20WhenM is a Z-matrix, i.e., when all off-diagonal
entries of M are non-positive, the
above statements are further equivalent to:
LCP(M,q) has a solution for all q.
There exists ad> 0 such thatMd> 0.
M is positive stable: Real part of any
eigenvalue of M is positive.
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 5/20n
S - All n×n real symmetric matrices.
n n
S - All PSD matrices inS .
+
n
Notation: X 0 if X ∈S .
+
hX,Yi :=trace(XY).
XY+YX
X◦Y := - Jordan product.
2
Semidefinite LCP:
n n n
L :S →S linear, Q∈S .
n
SDLCP(L,Q): FindX ∈S such that
X 0, L(X)+Q 0, andhX,L(X) +Qi = 0.
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 6/20n×n
ForA∈R ,
T n
L (X) :=AX +XA - Lyapunov transformation onS .
A
T n
S (X) :=X−AXA - Stein transformation onS .
A
L denotes eitherL orS .
A A
Gowda-Song (2000), Gowda-Parthasarathy (2000):
The following are equivalent:
[XL(X) =L(X)X, X◦L(X) 0]⇒X = 0.
SDLCP(L,Q) has a solution for all Q.
There existsD≻ 0 with L(D)≻ 0.
L is positive stable.
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 7/20The above result is very similar to the matrix theory
result for Z-matrices.
Why is this happening?
DoL andS have some sort of Z-property?
A A
Can the two results be unified and extended?
n n
Note: BothR andS are Euclidean Jordan algebras!
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 8/20(V,h·,·i,◦) is a Euclidean Jordan algebra if
V is a finite dimensional real inner product space
and the bilinear Jordan productx◦y satisfies:
x◦y =y◦x
2 2
x◦(x ◦y) =x ◦(x◦y)
hx◦y,zi =hx,y◦zi
2
K ={x :x∈V} is the symmetric cone inV .
Notation: x≥ 0 if x∈K andx> 0 if x∈int(K).
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 9/20Any EJA is a product of the following:
n n×n
S = Herm(R ) - n×n real symmetric matrices.
n×n
Herm(C ) - n×n complex Hermitian matrices.
n×n
Herm(Q ) - n×n quaternion Hermitian matrices.
3×3
Herm(O ) - 3× 3 octonion Hermitian matrices.
n
L - Jordan spin algebra.
Fora∈V , L (x) :=a◦x.
a
a andb operator commute if L L =L L .
a a
b b
On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras – p. 10/20