认知神经科学研究报告【20260096】

文章目录

• [Technical Documentation: `limsup_inf` -- Limit Superior and Limit Inferior of Set Sequences](#Technical Documentation: limsup_inf – Limit Superior and Limit Inferior of Set Sequences)
• • Overview
  • [Mathematical Background](#Mathematical Background)
  • Dependencies
  • [Module Functions](#Module Functions)
  • • `format_interval(obj)`
    • [`get_approach(expr, var, L)`](#get_approach(expr, var, L))
    • [`subsequence_limset(expr, left, var, name="subsequence", verbose=True)`](#subsequence_limset(expr, left, var, name="subsequence", verbose=True))
    • [`analyze_general(expr_list, modulus, left=5, var=n, verbose=True)`](#analyze_general(expr_list, modulus, left=5, var=n, verbose=True))
  • [Usage Examples](#Usage Examples)
  • • [Example 1: Odd/Even Segmentation (m=2)](#Example 1: Odd/Even Segmentation (m=2))
    • [Example 2: Three Subsequences (m=3)](#Example 2: Three Subsequences (m=3))
    • [Example 3: Four Subsequences (m=4)](#Example 3: Four Subsequences (m=4))
  • [Interpreting the Output](#Interpreting the Output)
  • [Limitations and Assumptions](#Limitations and Assumptions)
  • [How to Extend](#How to Extend)
  • Source

Technical Documentation: limsup_inf -- Limit Superior and Limit Inferior of Set Sequences

Overview

limsup_inf is a Python module for computing the limit superior (limsup) and limit inferior (liminf) of a sequence of sets { A n } n = 1 ∞ \{A_n\}_{n=1}^\infty {An}n=1∞ that is defined by modular segmentation . It is particularly useful in real analysis and measure theory , where set sequences are often given by different formulas for different residue classes modulo m m m. The program handles:

• Arbitrary modulus m m m (number of subsequences).
• Left‑hand endpoint constant L L L (common to all sets).
• Right‑hand endpoints that are rational, algebraic, or simple transcendental functions of n n n.
• Automatic detection of convergence from above/below and correct interval openness.
• Empty sets and singleton limits.

---

Mathematical Background

For a sequence of sets { A n } n = 1 ∞ \{A_n\}_{n=1}^\infty {An}n=1∞, the upper limit and lower limit are defined as:

lim sup ⁡ n → ∞ A n = ⋂ n = 1 ∞ ⋃ k = n ∞ A k , lim inf ⁡ n → ∞ A n = ⋃ n = 1 ∞ ⋂ k = n ∞ A k . \limsup_{n\to\infty} A_n = \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty A_k, \qquad \liminf_{n\to\infty} A_n = \bigcup_{n=1}^\infty \bigcap_{k=n}^\infty A_k. n→∞limsupAn=n=1⋂∞k=n⋃∞Ak,n→∞liminfAn=n=1⋃∞k=n⋂∞Ak.

If the sequence is segmented into m m m subsequences by the residue of the index modulo m m m:

A m n + r = L , f r ( n ) , r = 0 , 1 , ... , m − 1 , n = 1 , 2 , ... A_{m n + r} = L,\\ f_r(n), \quad r = 0,1,\dots,m-1, \quad n=1,2,\dots Amn+r=L, fr(n),r=0,1,...,m−1,n=1,2,...

then the limit set of each subsequence is:

lim ⁡ n → ∞ A m n + r = { L , lim ⁡ f r ( n ) if f r ( n ) → c from above , L , c ) if f r ( n ) → c from below , { L } if c = L and approached from above , ∅ if c \< L or ( c = L approached from below ) . \\lim_{n\\to\\infty} A_{m n + r} = \\begin{cases} \[L,\\ \\lim f_r(n) & \text{if } f_r(n) \to c \text{ from above},\\ L,\\ c) \& \\text{if } f_r(n) \\to c \\text{ from below},\\\\ \\{L\\} \& \\text{if } c = L \\text{ and approached from above},\\\\ \\emptyset \& \\text{if } c \< L \\text{ or } (c = L \\text{ approached from below}). \\end{cases} n→∞limAmn+r=⎩ ⎨ ⎧\[L, limfr(n)[L, c){L}∅if fr(n)→c from above,if fr(n)→c from below,if c=L and approached from above,if c<L or (c=L approached from below).

Finally:

lim inf ⁡ A n = ⋂ r = 0 m − 1 ( lim ⁡ n → ∞ A m n + r ) , lim sup ⁡ A n = ⋃ r = 0 m − 1 ( lim ⁡ n → ∞ A m n + r ) . \liminf A_n = \bigcap_{r=0}^{m-1} \left(\lim_{n\to\infty} A_{m n + r}\right), \qquad \limsup A_n = \bigcup_{r=0}^{m-1} \left(\lim_{n\to\infty} A_{m n + r}\right). liminfAn=r=0⋂m−1(n→∞limAmn+r),limsupAn=r=0⋃m−1(n→∞limAmn+r).

---

Dependencies

• Python 3.8 or higher
• SymPy (>= 1.9) -- symbolic computation

Install SymPy via pip if not already installed:

pip install sympy

---

Module Functions

format_interval(obj)

Converts SymPy set objects (Interval, FiniteSet, EmptySet) to human‑readable strings like [a, b], [a, b), {c}, or ∅.

get_approach(expr, var, L)

Determines whether the expression expr(var) approaches the limit L from above ('above'), from below ('below'), equals L exactly ('equal'), or unknown ('unknown').

subsequence_limset(expr, left, var, name="subsequence", verbose=True)

Computes the limit set of a single subsequence whose right endpoint is expr(var) and left endpoint is left. Returns a SymPy set and prints detailed steps if verbose=True.

analyze_general(expr_list, modulus, left=5, var=n, verbose=True)

Main entry point.

Parameters:

• expr_list : list of SymPy expressions of length modulus; expr_list[r] corresponds to f r ( n ) f_r(n) fr(n) for A m n + r A_{m n + r} Amn+r.
• modulus : integer m ≥ 2 m \ge 2 m≥2.
• left : constant left endpoint L L L (default 5).
• var : symbolic variable (default n from sympy.abc).
• verbose : if True, prints the full calculation process for each subsequence and the final intersection/union.

Returns : a tuple (liminf_set, limsup_set) as SymPy objects.

---

Usage Examples

Example 1: Odd/Even Segmentation (m=2)

from sympy import symbols
from sympy.abc import n
from limsup_inf import analyze_general

expr_list = [
    7 + n/(n+1) + n/(n**2),   # A_{2n}
    10 - 3*n/(n**2) - 1/n     # A_{2n+1}
]
liminf, limsup = analyze_general(expr_list, modulus=2, left=5)

Output (abbreviated):

--- 汇总 ---
A_{2n + 0} 的极限集合: [5, 8]
A_{2n + 1} 的极限集合: [5, 10)

下限集 (liminf) = [5, 8] ∩ [5, 10) = [5, 8]
上限集 (limsup) = [5, 8] ∪ [5, 10) = [5, 10)

Example 2: Three Subsequences (m=3)

expr_list = [
    5 + 1/n,        # A_{3n}   → {5}
    6 + n**(-2),    # A_{3n+1} → [5,6]
    7 + 2/n         # A_{3n+2} → [5,7)
]
liminf, limsup = analyze_general(expr_list, modulus=3, left=5)

Output:

下限集 (liminf) = {5} ∩ [5, 6] ∩ [5, 7) = {5}
上限集 (limsup) = {5} ∪ [5, 6] ∪ [5, 7) = [5, 7)

Example 3: Four Subsequences (m=4)

expr_list = [
    5 + 1/n,        # A_{4n}   → {5}
    6 + 1/(n**2),   # A_{4n+1} → [5,6]
    7 + 2/n,        # A_{4n+2} → [5,7)
    8 + 3/n         # A_{4n+3} → [5,8)
]
liminf, limsup = analyze_general(expr_list, modulus=4, left=5)

Output:

下限集 (liminf) = {5} ∩ [5,6] ∩ [5,7) ∩ [5,8) = {5}
上限集 (limsup) = {5} ∪ [5,6] ∪ [5,7) ∪ [5,8) = [5,8)

---

Interpreting the Output

The program prints every step:

1. For each subsequence, it shows:

   • Right‑hand expression.
   • Limit value.
   • Difference limit f r ( n ) − L f_r(n) - L fr(n)−L.
   • Approach direction (above/below).
   • Resulting limit set with proper openness.

2. The summary lists all subsequence limit sets.

3. Finally, the liminf and limsup are computed as the intersection and union of these sets, respectively, displayed with full symbolic notation.

---

Limitations and Assumptions

• The left endpoint L L L is assumed constant (not depending on n n n).
• The approach direction detection relies on multiplying the difference f r ( n ) − L f_r(n) - L fr(n)−L by n k n^k nk and taking limits. This works well for polynomials, rational functions, and combinations of rational functions with 1 / n 1/n 1/n or 1 / n 2 1/n^2 1/n2. For more exotic functions (e.g., exp ⁡ ( − n ) \exp(-n) exp(−n), log ⁡ ( n ) / n \log(n)/n log(n)/n), the detection may fall back to a conservative closed‑interval assumption; manual verification is advised.
• The sequence indices are assumed to start at n = 1 n=1 n=1 for all subsequences.
• Infinite limits ( ± ∞ \pm\infty ±∞) are treated as empty sets.

---

How to Extend

• Different starting index : If your sequence starts at n = 0 n=0 n=0, redefine var accordingly or shift the expressions.
• Variable left endpoint : Modify subsequence_limset to accept a left expression left_expr and compute its limit.
• More than one variable: The current design is for single‑variable sequences. For multivariate sequences, additional logic is needed.

---

Source

https://gitee.com/waterruby/ANNA.git

============================================================
示例 1：模 2（奇偶分段）
============================================================
============================================================
集合序列定义（模 2 分段）:
A_{2n + 0} = [5, n/(n + 1) + 7 + 1/n]
A_{2n + 1} = [5, 10 - 4/n]
============================================================

--- 分析 A_{2n + 0} ---
右端点表达式: n/(n + 1) + 7 + 1/n
左端点: 5
右端点极限: 8
右端点 - 左端点 的极限: 3
趋近方向: above
从上方趋近，极限区间为 闭区间: [5, 8]

--- 分析 A_{2n + 1} ---
右端点表达式: 10 - 4/n
左端点: 5
右端点极限: 10
右端点 - 左端点 的极限: 5
趋近方向: below
从下方趋近，极限区间为 右开区间: [5, 10)

--- 汇总 ---
A_{2n + 0} 的极限集合: [5, 8]
A_{2n + 1} 的极限集合: [5, 10)

下限集 (liminf) = [5, 8] ∩ [5, 10) = [5, 8]
上限集 (limsup) = [5, 8] ∪ [5, 10) = [5, 10)

============================================================
示例 2：模 3（三项分段）
============================================================
============================================================
集合序列定义（模 3 分段）:
A_{3n + 0} = [5, 5 + 1/n]
A_{3n + 1} = [5, 6 + n**(-2)]
A_{3n + 2} = [5, 7 + 2/n]
============================================================

--- 分析 A_{3n + 0} ---
右端点表达式: 5 + 1/n
左端点: 5
右端点极限: 5
右端点 - 左端点 的极限: 0
趋近方向: above
从上方趋近，极限区间为 单点集: {5}

--- 分析 A_{3n + 1} ---
右端点表达式: 6 + n**(-2)
左端点: 5
右端点极限: 6
右端点 - 左端点 的极限: 1
趋近方向: above
从上方趋近，极限区间为 闭区间: [5, 6]

--- 分析 A_{3n + 2} ---
右端点表达式: 7 + 2/n
左端点: 5
右端点极限: 7
右端点 - 左端点 的极限: 2
趋近方向: above
从上方趋近，极限区间为 闭区间: [5, 7]

--- 汇总 ---
A_{3n + 0} 的极限集合: {5}
A_{3n + 1} 的极限集合: [5, 6]
A_{3n + 2} 的极限集合: [5, 7]

下限集 (liminf) = {5} ∩ [5, 6] ∩ [5, 7] = {5}
上限集 (limsup) = {5} ∪ [5, 6] ∪ [5, 7] = [5, 7]

============================================================
示例 3：模 4（四项分段）
============================================================
============================================================
集合序列定义（模 4 分段）:
A_{4n + 0} = [5, 5 + 1/n]
A_{4n + 1} = [5, 6 + n**(-2)]
A_{4n + 2} = [5, 7 + 2/n]
A_{4n + 3} = [5, 8 + 3/n]
============================================================

--- 分析 A_{4n + 0} ---
右端点表达式: 5 + 1/n
左端点: 5
右端点极限: 5
右端点 - 左端点 的极限: 0
趋近方向: above
从上方趋近，极限区间为 单点集: {5}

--- 分析 A_{4n + 1} ---
右端点表达式: 6 + n**(-2)
左端点: 5
右端点极限: 6
右端点 - 左端点 的极限: 1
趋近方向: above
从上方趋近，极限区间为 闭区间: [5, 6]

--- 分析 A_{4n + 2} ---
右端点表达式: 7 + 2/n
左端点: 5
右端点极限: 7
右端点 - 左端点 的极限: 2
趋近方向: above
从上方趋近，极限区间为 闭区间: [5, 7]

--- 分析 A_{4n + 3} ---
右端点表达式: 8 + 3/n
左端点: 5
右端点极限: 8
右端点 - 左端点 的极限: 3
趋近方向: above
从上方趋近，极限区间为 闭区间: [5, 8]

--- 汇总 ---
A_{4n + 0} 的极限集合: {5}
A_{4n + 1} 的极限集合: [5, 6]
A_{4n + 2} 的极限集合: [5, 7]
A_{4n + 3} 的极限集合: [5, 8]

下限集 (liminf) = {5} ∩ [5, 6] ∩ [5, 7] ∩ [5, 8] = {5}
上限集 (limsup) = {5} ∪ [5, 6] ∪ [5, 7] ∪ [5, 8] = [5, 8]