We derive formulas for normalized doubly coprime factorizations over H 1 , for a class of nonexponentially stabilizable systems. Using these results, we solve the problem of robust stabilization with respect to normalized coprime factor perturbations for a class of innnite-dimensional systems (A; B; B ; D), with nite-rank inputs and outputs and dissipative A; these systems are not exponentially...

In 1980, Carl Pomerance and J. L. Selfridge proved D. J. Newman’s coprime mapping conjecture: If n is a positive integer and I is a set of n consecutive integers, then there is a bijection f :{1, 2, . . . , n}→ I such that gcd(i, f(i)) = 1 for 1 ≤ i ≤ n. The function f described in their theorem is called a coprime mapping. Around the same time, Roger Entringer conjectured that all trees are pr...

It is becoming to be well-known that an internally stabilizable transfer matrix does not necessarily admit doubly coprime factorizations. The equivalence between these two concepts is still open for important classes of plants. Hence, we may wonder whether or not it is possible to parametrize all stabilizing controllers of an internally stabilizable plant which does not necessarily admit doubly...

This paper develops a general factorization theory of transfer matrices for linear systems defined over rings. Restricting the rings to the class of unique factorization domains (UFDs) and introducing a notion of denominator sets and a generalized notion of coprimeness, a coprime factorization 1 ( ) ( ) ( ) H z W z V z − = for the transfer matrix ( ) H z of a system defined over a UFD is develo...

A closed-loop balanced truncation technique has been applied to an integral-type optimal servomechanism (IOS) expressed in graph-operator form of normalized right coprime factorization to produce a reduced-size state-feedback gain matrix, which is then converted into proportional-derivative (PD) gain matrices. On the other hand, the states of the integral of control error in the IOS are not tru...

A labeling f on a graph G on n vertices is called a prime labeling if f is a bijection from the vertex set V (G) to {1, 2, · · · , n} such that f(x) and f(y) are coprime if x and y are adjacent. It was shown by Sundaram et al. [1] that the planar grid Pm × Pn has a prime labeling if m ≤ n and n is a prime. In this paper it is proved that the following grids have a prime labeling: (i) Pn+1 × Pn+...

This paper uses combinatorial group theory to help answer some longstanding questions about the genera of orientable surfaces that carry particular kinds of regular maps. By classifying all orientably-regular maps whose automorphism group has order coprime to g − 1, where g is the genus, all orientably-regular maps of genus p+1 for p prime are determined. As a consequence, it is shown that orie...

The order of every finite group G can be expressed as a product of coprime positive integers m1, . . . ,mt such that π(mi) is a connected component of the prime graph of G. The integers m1, . . . ,mt are called the order components of G. Some nonabelian simple groups are known to be uniquely determined by their order components. As the main result of this paper, we show that the projective symp...

Abstract The problem of synchronous Fractional Sample Rate Conversion (FSRC) of a digital signal by L/M, where L and M are coprime integers [3,13], is revisited. Based on a novel approach two different efficient causal block implementations of FSRC are concurrently derived and compared with each other. While the computational load of both structures, being performed in an LTI MIMO subsystem at ...