scholarly journals A General Lower Bound for the Asymptotic Convergence Factor

2020 ◽  
pp. 487-507
Author(s):  
N. Tsirivas
Keyword(s):  
Lower Bound ◽  
Asymptotic Convergence ◽  
Convergence Factor

scholarly journals On the Convergence of the Benjamini–Hochberg Procedure

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2154
Author(s):  
Dean Palejev ◽  
Mladen Savov
Keyword(s):  
Lower Bound ◽  
Exponential Type ◽  
Alternative Hypothesis ◽  
Fixed Number ◽  
Multiple Comparison ◽  
Asymptotic Convergence ◽  
Test Statistic ◽  
Scientific Methods

The Benjamini–Hochberg procedure is one of the most used scientific methods up to date. It is widely used in the field of genetics and other areas where the problem of multiple comparison arises frequently. In this paper we show that under fairly general assumptions for the distribution of the test statistic under the alternative hypothesis, when increasing the number of tests, the power of the Benjamini–Hochberg procedure has an exponential type of asymptotic convergence to a previously shown limit of the power. We give a theoretical lower bound for the probability that for a fixed number of tests the power is within a given interval around its limit together with a software routine that calculates these values. This result is important when planning costly experiments and estimating the achieved power after performing them.

Emergence of bi-cluster flocking for the Cucker–Smale model

2016 ◽  
Vol 26 (06) ◽  
pp. 1191-1218 ◽  
Author(s):  
Junghee Cho ◽  
Seung-Yeal Ha ◽  
Feimin Huang ◽  
Chunyin Jin ◽  
Dongnam Ko
Keyword(s):  
Lower Bound ◽  
Asymptotic Convergence ◽  
Far Field ◽  
Differential Inequalities ◽  
Local Velocity ◽  
Velocity Fluctuations ◽  
Spatial Fluctuations ◽  
The Difference ◽  
Field Decay ◽  
Velocity Averages

We present the mathematical analysis of bi-cluster flocking phenomenon for the short-ranged Cucker–Smale model with some well-prepared initial data. For this, we derive a system of differential inequalities for the functionals measuring the local spatial and velocity fluctuations and differences of local velocity averages, and then estimate the upper bound of spatial fluctuations and the lower bound of the difference between local velocity averages. We explicitly present an admissible class of initial configurations leading to the asymptotic emergence of bi-cluster flocking phenomenon. Unlike global flocking (a mono-cluster flocking configuration in velocity), where the convergence rate is always exponential, the asymptotic convergence to bi-cluster flocking configurations is affected by the far-field decay rate of communication weights so that it can be algebraic.

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

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.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
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 $K_3$.

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

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
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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