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...

The fast transformation of technologies into new products or processes is one of the core challenges for any technology-based enterprise. Within the innovation process, we believe, the early phases (“fuzzy front end”) to have the highest impact on the whole process and the result (Input-Output Process), since it will influence the design and total costs of the innovation extremely. However the ...

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

Multipoles are the pieces we obtain by cutting some edges of a cubic graph in one or more points. As a result of the cut, a multipole M has vertices attached to a dangling edge with one free end, and isolated edges with two free ends. We refer to such free ends as semiedges, and to isolated edges as free edges. Every 3-edge-coloring of a multipole induces a coloring or state of its semiedges, w...

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 . . . . . . . . ....

The Gale–Nikaido–Debreu lemma plays an important role in establishing the existence of competitive equilibrium. In this paper, we use Sperner’s and basic elements topology to prove lemma.

This article tries to answer the question: How far can the algebro-geometric theory of rational descendant Gromov-Witten invariants be carried over to the tropical world? Given the fact that our moduli spaces are non-compact, the answer is surprisingly positive: We discuss universal families and the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection wi...