TAD Theory and Its Applications
A Mathematical Framework for Visualizing Histories
##article.authors##
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Yasuo Matsui
Independent Researcher
https://researchmap.jp/yasuo-matsui
DOI:
https://doi.org/10.51094/jxiv.1593Keywords:
History-Dependent Dynamics, TAD Theory (Traced Allocation Dynamics), Relaxation Function and Forgetting Curve, Logistic Growth Model, Kovacs Effect (Education and Memory Model), Inventory Management and Advertising Effect, Epidemiological Model (SIR), Economic Growth ModelAbstract
本研究は、時間を通じた履歴構造を明示的に取り入れる数理フレームワーク TAD（Traced Allocation Dynamics）理論 を提案する。
TADは、入力時刻 TTT と観測時刻 ttt の二重時間軸を基盤とし、内部状態 g(t,T)g(t,T)g(t,T) の動学を割当関数 μ(t,T)\mu(t,T)μ(t,T)、自然減衰 γ(t,T)\gamma(t,T)γ(t,T)、スケーリング係数 σ(t,T)\sigma(t,T)σ(t,T)、および入力 G(t,T)G(t,T)G(t,T) により統一的に記述する。この最小原理から、従来の多様なモデル（在庫鮮度モデル、Adstockモデル、緩和関数、SIRモデル、Solow型経済成長、忘却曲線など）を再構成できることを示すとともに、それらが内包していたが顕在化していなかった履歴依存構造を可視化する。
さらに、TADは既存理論の単なる再現にとどまらず、新しい知見や予言を導く。例えば、ロジスティック成長モデルにおいて、成長率や収容力を出生率・基礎死亡率・密度依存係数といった内生パラメータに分解できることを示した。また、教育・記憶モデルに対しては「Kovacs効果」に類似するハンプ現象を予言し、数値シミュレーションによって確認した。
本稿は、TADが履歴をもつ諸現象を横断的に統一し、新しい理論的発見と実務的応用の双方に資する汎用的基盤理論となり得ることを総合的に論じる。応用領域としては、在庫管理、広告効果分析、人口動態予測、学習スケジューリング、感染症流行解析、さらには大規模言語モデルにおけるメモリ選別などを含み、TADは「過去が現在にどのように寄与するか」を数理的に可視化する新しい視座を提供する。
Conflicts of Interest Disclosure
This study was conducted independently by the author as an off-duty personal research activity. No funding, data, or facilities were provided by the author’s employer or any other organization. The paper aims to present a general theoretical framework and does not represent the work, products, or views of any specific company. The author declares no competing interests.Downloads *Displays the aggregated results up to the previous day.
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