A new stability theorem of the direct Lyapunov’s method is proposed for neutral-type systems. The main contribution of the proposed theorem is to remove the condition that the D operator is stable. In order to demonstrate the effectiveness, the proposed theorem is used to determine the stability of a neutral-type system in a critical case, i.e. the dominant eigenvalues of the principal neutral ...

In the spirit of Chern’s proof of the Gauss-Bonnet theorem, we show that Sha’s secondary Chern-Euler form Ψ is exact away from the outward and inward unit normal vectors by constructing a form Γ such that dΓ = Ψ. Using Stokes’ theorem, this evaluates the boundary term α∗(Ψ)[M ] in Sha’s relative Poincaré-Hopf theorem in terms of more classical local indices, Ind ∂+V and Ind ∂−V , for the tangen...

We consider the problem of sampling signals which are not bandlimited, but still have a finite number of degrees of freedom per unit of time, such as, for example, piecewise polynomials. We demonstrate that by using an adequate sampling kernel and a sampling rate greater or equal to the number of degrees of freedom per unit of time, one can uniquely reconstruct such signals. This proves a sampl...

We denote by ∆ the closed unit disk, by Γ the unit circle, and by A0 the disk algebra, which consists of all functions holomorphic in int(∆) and continuous on ∆. By a module over A0 we mean a vector space M of continuous complex-valued functions on ∆ such that the constant 1 lies in M, and for every a0 ∈ A0 and φ ∈ M, one has a0 · φ ∈ M. In his book “Real and Complex Analysis” (1966) Walter Rud...

where P̂ is a linear projection onto the space of circulations, i.e., of all the flows f with Bf = 0, and ĥ is some fixed unit s-t flow. As we showed last time, once we construct a projection P (·) then we obtain an (1 − ε)-approximate maximum flow algorithm whose performance is described in the following theorem. Theorem 1 If P (·) is an α-approximate affine `∞-projection P (·) onto the space F...

In this paper we prove two di erent generalizations of Kummer's Lemma that describes when a unit of a cyclotomic eld is a p-th power of another unit. One of these results is then used to prove a theorem about the Picard group of the integer group ring ZC, where C is a cyclic group of prime power order. The theorem was rst proved by Kervaire and Murthy; we use a more elementary method to reprove...

In this paper we prove two di erent generalizations of Kummer's Lemma that describes when a unit of a cyclotomic eld is a p-th power of another unit. One of these results is then used to prove a theorem about the Picard group of the integer group ring ZC, where C is a cyclic group of prime power order. The theorem was rst proved by Kervaire and Murthy; we use a more elementary method to reprove...

For n ≥ 3 distinct points in the d-dimensional unit sphere S ⊂ R , there exists a Möbius transformation such that the barycenter of the transformed points is the origin. This Möbius transformation is unique up to post-composition by a rotation. We prove this lemma and apply it to prove the uniqueness part of a representation theorem for 3-dimensional polytopes as claimed by Ziegler (1995): For ...

this paper gives a short survey of some of the known results generalizing the theorem, credited to i. schur, that if the central factor group is finite then the derived subgroup is also finite.

A theorem of Lusin is proved in the non-ordered context of JB∗-triples. This is applied to obtain versions of a general transitivity theorem and to deduce refinements of facial structure in closed unit ballls of JB∗-triples and duals.