Lower Bound: Translation Method

The Direct Method was the outcome of a reaction against the Grammar Translation Method. It was based on the assumption that the learner of a foreign language should think directly in the target language. According to this method, English is taught through English. The learner learns the target language through discussion, conversation and reading in the second language. It does not take recourse to translation and foreign grammar.

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The least distance between extremums and minimal period of solutions to autonomous vector differential equations

Доклады Академии наук ◽

10.31857/s0869-56524852142-144 ◽

2019 ◽

Vol 485 (2) ◽

pp. 142-144

Author(s):

A. A. Zevin

Keyword(s):

Lower Bound ◽

Differential Equations ◽

Periodic Solutions ◽

Lipschitz Constant ◽

Minimal Period ◽

Arbitrary Vector ◽

Vector Norm ◽

Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

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A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

The Electronic Journal of Combinatorics ◽

10.37236/1188 ◽

1994 ◽

Vol 1 (1) ◽

Cited By ~ 8

Author(s):

Geoffrey Exoo

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

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On the Crossing Number of $K_{m,n}$

The Electronic Journal of Combinatorics ◽

10.37236/1748 ◽

2003 ◽

Vol 10 (1) ◽

Cited By ~ 4

Author(s):

Nagi H. Nahas

Keyword(s):

Lower Bound ◽

Bipartite Graph ◽

Complete Bipartite Graph ◽

Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

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