The aim of this note is to generalize the Nakayama lemma to a class of multiplication modules over commutative rings with identity. In this note, by considering the notion of multiplication modules and the product of submodules of them, we state and prove two versions of Nakayama lemma for such modules. In the first version we give some equivalent conditions for faithful finitely generated mult...

We present a constructive proof of Ky Fan’s combinatorial lemma concerning labellings of triangulated spheres. Our construction works for triangulations of Sn that contain a flag of hemispheres. As a consequence, we produce a constructive proof of Tucker’s lemma that holds for a larger class of triangulations than previous constructive proofs.

The approximation lemma is a simplification of the well-known take lemma, and is used to prove properties of programs that produce lists of values. We show how the approximation lemma, unlike the take lemma, can naturally be generalised from lists to a large class of datatypes, and present a generic approximation lemma that is parametric in the datatype to which it applies. As a useful by-produ...

– in this paper we consider some (α ,β ) -metrics such as generalized kropina, matsumoto and f (α β )2α = + metrics, and obtain necessary and sufficient conditions for them to be einstein metrics when βis a constant killing form. then we prove with this assumption that the mentioned einstein metrics must beriemannian or ricci flat.

Abstract In this work the Brouwer fixed point theorem for a triangle was proved by two methods based on Sperner Lemma. One of proofs Sperner’s Lemma given in paper carried out using so-called index.

B Omitted Proofs 2 B.1 How to make individual and aggregate distributions coincide . . . . . . . . . . . . . 2 B.2 Proof of Lemma 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 B.3 Proof of Lemma A1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 B.4 Proof of Lemma A2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

I N THE continuous-time world, the description of passivity properties of linear time-invariant systems in terms of minimal state-space descriptions is well known. The application of such results in system theory, network synthesis, and in stability analysis is also well established [1],[2]. For example, given a Lossless Positive Real (LPR) matrix Z(s), any minimal state-space description satis...

A Declarative Subtyping 6 A.1 Properties of Well-Formedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1 Proposition (Weakening) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Proposition (Substitution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 A.2 Reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

Does there exist a Lipschitz injection of Z into the open set of a site percolation process on Z, if the percolation parameter p is sufficiently close to 1? We prove a negative answer when d = D, and also when d ≥ 2 if the Lipschitz constant M is required to be 1. Earlier work of Dirr, Dondl, Grimmett, Holroyd, and Scheutzow yields a positive answer for d < D and M = 2. As a result, the above q...

B Proofs 3 B.1 Well-Formed Types and Environments . . . . . . . 3 B.2 Subtyping Inversion Lemma . . . . . . . . . . . . 3 B.3 Store Typing . . . . . . . . . . . . . . . . . . . . . 4 B.4 Values Inversion Lemma . . . . . . . . . . . . . . 6 B.5 Substitution . . . . . . . . . . . . . . . . . . . . . 9 B.6 Free Variables Lemma . . . . . . . . . . . . . . . 12 B.7 Well-Form Lemmas . . . . . . . . ....