Abstract A lattice polytope is free (or empty) if its vertices are the only points it contains. In context of valuation theory, Klain [16] proposed to study functions $\alpha _i(P;n)$ that count number polytopes in $nP$ with $i$ vertices. For $i=1$, this famous Ehrhart polynomial; for $i> 3$, computation likely impossible; and $i=2,3$ computationally challenging. paper, we develop a theo...

This paper proposes a theory for designing stable interconnection of linear active multi-port networks at the ports. Such interconnections can lead to unstable networks even if the original networks are stable with respect to bounded port excitations. Hence such a theory is necessary for realising interconnections of active multiport networks. Stabilization theory of linear feedback systems usi...

We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers a and b giving

We show that while the number of coprime compositions a positive integer n into k parts can be expressed as Q-linear combination Jordan totient functions, this is never possible for partitions parts. also pk′(n) C-linear sufficiently large, if and only k∈{2,3} in unique way. Finally we introduce some generalizations functions always them.

In these notes, we give a short introduction to the fractional representation approach to analysis and synthesis problems [12], [14], [17], [28], [29], [50], [71], [77], [78]. In particular, using algebraic analysis (commutative algebra, module theory, homological algebra, Banach algebras), we shall give necessary and sufficient conditions for a plant to be internally stabilizable or to admit (...

In this paper, we propose a vectorized noncircular MUSIC (VNCM) algorithm based on the concept of the coarray, which can construct the difference and sum (diff-sum) coarray, for direction finding of the noncircular (NC) quasi-stationary sources. Utilizing both the NC property and the concept of the Khatri-Rao product, the proposed method can be applied to not only the ULA but also sparse arrays...

This dissertation deals with the problem of characterizing the controllers which are potentially optimal in the robustness optimization for (weighted) normalized coprime factor/gap metric uncertainty, which is the basis for the H1 Loop Shaping Design Method. This dissertation considers single-input-single-output systems. Given a plant P and a stabilizing controller C, we ask if C can be obtaine...