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関連する概念動画

Fundamental Theorem of Algebra01:30

Fundamental Theorem of Algebra

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The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as:  with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the Complete...
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Theorems of Pappus and Guldinus: Problem Solving01:12

Theorems of Pappus and Guldinus: Problem Solving

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Pappus and Guldinus's theorems are powerful mathematical principles that are used for finding the surface area and volume of composite shapes. For example, consider a cylindrical storage tank with a conical top. Finding the surface area or volume can be challenging for such complex shapes. These theorems are particularly useful in calculating the volume and surface area of such systems. Here, the cylindrical storage tank with a conical top can be broken down into two simple shapes: a...
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Determination of Pi Terms01:15

Determination of Pi Terms

443
The Buckingham Pi theorem is a valuable method in dimensional analysis, reducing complex relationships between variables into dimensionless terms. Relevant variables in analyzing the lift force on an airplane wing include lift force, air density, wing area, aircraft velocity, and air viscosity. Expressing each variable in terms of fundamental dimensions — mass, length, and time — provides a consistent foundation for constructing these dimensionless terms.
The theorem indicates that the...
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Mathematical Induction01:29

Mathematical Induction

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Mathematical induction is a structured method of proof used to confirm the truth of statements involving natural numbers. Consider the sum of the first n natural numbers:This formula describes a pattern that appears to hold true as more terms are added. To verify that it is valid for all natural numbers, mathematical induction proceeds in two essential steps. The first is the base case, where the formula is tested for the initial value, typically n = 1. Substituting into both sides confirms the...
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Theorems of Pappus and Guldinus01:10

Theorems of Pappus and Guldinus

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The two theorems developed by Pappus and Guldinus are widely used in mathematics, engineering, and physics to find the surface area and volume of any body of revolution. This is done by revolving a plane curve around an axis that does not intersect the curve to find its surface area or revolving a plane area around a non-intersecting axis to calculate its volume.
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Euler's Formula for Pin-Ended Columns01:21

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In structural engineering, the stability of columns under compressive axial loads is a critical consideration, described as buckling. A typical example involves a column PQ, which is pin-connected at both ends and subjected to a centric axial load F applied at one end, with a reaction force of F' = -F at the other end. Here, it is crucial to understand that when an applied load exceeds the critical load, buckling occurs as the system becomes unstable.
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関連する実験動画

Updated: Nov 18, 2025

Setting Limits on Supersymmetry Using Simplified Models
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ラマヌージャン・マシンで基本定数に関する推測を生成する

Gal Raayoni1, Shahar Gottlieb1, Yahel Manor1,2

  • 1Technion-Israel Institute of Technology, Haifa, Israel.

Nature
|February 4, 2021
PubMed
まとめ

アルゴリズムは,π と e のような基本定数のための新しい数学的式を発見することができます.この体系的なアプローチは,ラマヌージャンマシンと呼ばれ,隠された構造と以前未知の方程式を明らかにします.

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Last Updated: Nov 18, 2025

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科学分野:

  • * 物理学,生物学,化学の応用がある数学と計算科学

背景:

  • 基本定数 (例えば,π,e) に関する新しい数学的公式の発見は,歴史的に稀で偶発的であった.
  • * このような発見は 体系的な方法ではなく 数学的な独創性や 深い直感に頼ったことが多い.

研究 の 目的:

  • * 基本定数のための数学式を発見するための体系的,アルゴリズム駆動的なアプローチを提案する.
  • * 基礎となる数学的構造を明らかにし,伝統的な証拠に基づく方法論を補完する.

主な方法:

  • "ラマヌージャン・マシン"の開発と応用,新しい式を特定するためのアルゴリズムを活用する.
  • * 合わせたグラデント降下最適化アルゴリズムの実装
  • * アルゴリズムは数値マッチングに基づいており,事前の構造知識なしに推測生成が可能である.

主要な成果:

  • パイ,e,カタラン定数,リマンゼータ関数の連続分数表現を含む,多くの既知の,かつ未知の式を発見した.
  • * 数学的な推測の生成,そのうちのいくつかは容易に証明可能で,いくつかは未解決の問題である.
  • * 不明な数学的基礎を持つ定数の構造を明らかにするアルゴリズムの有効性の実証.

結論:

  • * ラマヌージャン・マシンは,数学的公式の発見のための体系的な方法を提供し,人間の直感を拡張します.
  • * このアルゴリズムのアプローチは,構造を明らかにするために数値データを用いて,証明における従来の論理を逆転させます.
  • * この方法論は,特に未知の性質を持つ定数について,数学的な研究のための新しい道を開く.