A Lower Bound for the Counting Function of Lucas Pseudoprimes

1988 ◽  
Vol 51 (183) ◽  
pp. 315 ◽  
Author(s):  
P. Erdos ◽  
P. Kiss ◽  
A. Sarkozy
Keyword(s):  
Lower Bound ◽  
Counting Function ◽  
Lucas Pseudoprimes

scholarly journals A lower bound for the counting function of Lucas pseudoprimes

1988 ◽  
Vol 51 (183) ◽  
pp. 315-315 ◽  
Author(s):  
P. Erdős ◽  
P. Kiss ◽  
A. S{árk{özy
Keyword(s):  
Lower Bound ◽  
Counting Function ◽  
Lucas Pseudoprimes

Supremum and infimum of subharmonic functions of order between 1 and 2

2011 ◽  
Vol 54 (3) ◽  
pp. 685-693
Author(s):  
P. C. Fenton
Keyword(s):  
Lower Bound ◽  
Convex Function ◽  
Upper Bound ◽  
Counting Function ◽  
Subharmonic Functions ◽  
Image Position ◽  
Sharp Lower Bound

AbstractFor functions u, subharmonic in the plane, letand let N(r,u) be the integrated counting function. Suppose that $\mathcal{N}\colon[0,\infty)\rightarrow\mathbb{R}$ is a non-negative non-decreasing convex function of log r for which $\mathcal{N}(r)=0$ for all small r and $\limsup_{r\to\infty}\log\mathcal{N}(r)/\4\log r=\rho$, where 1 < ρ < 2, and defineA sharp upper bound is obtained for $\liminf_{r\to\infty}\mathcal{B}(r,\mathcal{N})/\mathcal{N}(r)$ and a sharp lower bound is obtained for $\limsup_{r\to\infty}\mathcal{A}(r,\mathcal{N})/\mathcal{N}(r)$.

scholarly journals Resonances for Perturbed Periodic Schrödinger Operator

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Mouez Dimassi
Keyword(s):  
Lower Bound ◽  
Schrödinger Operator ◽  
Positive Constant ◽  
Periodic Potential ◽  
Counting Function ◽  
Schrodinger Operator ◽  
Small Positive Constant ◽  
Semiclassical Regime ◽  
Perturbed Periodic ◽  
Periodic Schrödinger Operator

In the semiclassical regime, we obtain a lower bound for the counting function of resonances corresponding to the perturbed periodic Schrödinger operatorPh=-Δ+Vx+W(hx). HereVis a periodic potential,Wa decreasing perturbation andha small positive constant.

A lower bound for the prime counting function

2011 ◽  
Vol 95 (534) ◽  
pp. 433-436
Author(s):  
Daniel Shiu ◽  
Peter Shiu
Keyword(s):  
Lower Bound ◽  
Real Number ◽  
Counting Function ◽  
Distribution Of Primes ◽  
Famous Argument ◽  
Image Position ◽  
Prime Counting Function

Let π (x) count the primes p ≤ x, where x is a large real number. Euclid proved that there are infinitely many primes, so that π (x) → ∞ as x → ∞; in fact his famous argument ([1: Section 2.2]) can be used to show thatThere was no further progress on the problem of the distribution of primes until Euler developed various tools for the purpose; in particular he proved in 1737 [1: Theorem 427] that

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.

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