论文标题
基本优化问题中具有对数周期术语的精确渐近造型
Precise asymptotics with log-periodic term in an elementary optimization problem
论文作者
论文摘要
函数$ \ inf_n nx^{1/n} $具有渐近性$ eu+e+e d^2（u）/（2U）/（2U）+O（1/u^2）$ as $ x \ to \ infty $，其中$ u = \ u = \ log x $ and $ d（$ d（u）是从$ u $ to $ u $ to $ u $ to y $ $ to the最近的integer。我们概括了这一观察结果。 首先，曲线$ y = nx^{1/n} $可以参数写为$ \ log x = nt $，$ y = nt $。通常，令$（u_n（t），v_n（t））$为参数曲线的家族，具有渐近性$ u_n = n p_1（t）+q_1（t）+q_1（t）+r_1（t）/n+o（1/n^2）$和$ v_n = n p_2（t）+q_2（t）+q_2（t）+q_2（t）+q_2（t）$ r_2（t）$ r_2（t）+r_2（t）+r_2（t）+r_2（t）+r_2（t）+r_2（t）+o。假设功能$ p_1（t）/p_0（t）$在参数域中具有唯一的非等级最小值。结果表明，其下部信封的渐近学$ v（u）= \ inf_ {n，t} v_n（t）$，其中$ u = u_n（t）$，具有form $ v（u）= a_0 u+a_1+a_1+φ（u）的渐近学$ d^2（\ cdot）$。 其次，请注意，$ nx^{1/n} $是总和$ t_1+t_2/t_1+\ dots+t_ {n}/t_ {n-1} $的最小值。我们考虑了总和$ t_1+t_2/（t_1+1）+\ dots+t_n/（t_ {n-1} +1）$的类似渐近问题。让$ f_n（x）$是约束$ t_n = x $下的$ n $ term总和的最小值。定义$ f（x）= \ inf_n f_n（x）$。我们表明$ f（x）= eu-a+e d^2（u+b）/（2U）+o（1/u^2）$，带有某些数值常数$ a $ a $和$ b $。我们提出了此优化问题的替代形式，尤其是``最小动作''公式。另外，我们还会发现$ f_n^{（p）}（x）= e \ log n-a（p）+o（1/\ log n）$（1/\ log n）$是由与表单$ t_j+p $的分母一起产生的功能，该函数$ t_j+p $带有任意$ p> 0 $的$ t_j+p $，并建立了有关功能$ a（p）$的事实。
The function $\inf_n nx^{1/n}$ has the asymptotics $eu+e d^2(u)/(2u)+O(1/u^2)$ as $x\to\infty$, where $u=\log x$ and $d(u)$ is the distance from $u$ to the nearest integer. We generalize this observation. First, the curves $y=nx^{1/n}$ can be written parametrically as $\log x=nt$, $y=nt$. In general, let $(u_n(t),v_n(t))$ be a family of parametric curves with asymptotics $u_n=n p_1(t)+q_1(t)+r_1(t)/n+O(1/n^2)$ and $v_n=n p_2(t)+q_2(t)+r_2(t)/n+O(1/n^2)$. Suppose the function $p_1(t)/p_0(t)$ has a unique nondegenerate minimum in the parameter domain. It is shown that the asymptotics of their lower envelope $v(u)=\inf_{n,t} v_n(t)$, where $u=u_n(t)$, has the asymptotics of the form $v(u)=a_0 u+a_1+Φ(u)/u+O(1/u^2)$, where $Φ$ is an affinely transformed function $d^2(\cdot)$. Second, note that $nx^{1/n}$ is the minimum of the sum $t_1+t_2/t_1+\dots+t_{n}/t_{n-1}$ subject to the constraint $t_n=x$. We consider a similar asymptotic problem for the sums $t_1+t_2/(t_1+1)+\dots+t_n/(t_{n-1}+1)$. Let $F_n(x)$ is the minimum value of the $n$-term sum under the constraint $t_n=x$. Define $F(x)=\inf_n F_n(x)$. We show that $F(x)=eu-A+e d^2(u+b)/(2u)+O(1/u^2)$ with certain numerical constants $A$ and $b$. We present alternative forms of this optimization problem, in particular, a ``least action'' formulation. Also we find the asymptotics $F_n^{(p)}(x)=e\log n-A(p)+O(1/\log n)$ for the function arising from the sums with denominators of the form $t_j+p$ with arbitrary $p>0$ and establish some facts about the function $A(p)$.
