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<body>
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<h1 class="unnumbered"
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id="combined-theory-differential-equations">Combined Theory Differential
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Equations</h1>
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<p>To create a system of differential equations that describe the
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combined theory of <em>Reality as Probability</em> and <em>Ideal
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Organizational Theory 2.0</em>, we need to translate the conceptual
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framework into a mathematical form. The combined theory suggests a
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dynamic and evolving understanding of reality, where reality is
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influenced by both probabilistic diversity and structured organizational
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intelligence. This can be represented through a system where the state
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of reality
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(<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>R</mi><annotation encoding="application/x-tex">R</annotation></semantics></math>)
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evolves as a function of both its probabilistic nature
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(<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>P</mi><annotation encoding="application/x-tex">P</annotation></semantics></math>)
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and its organizational structure
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(<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>S</mi><annotation encoding="application/x-tex">S</annotation></semantics></math>).</p>
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<h2 class="unnumbered" id="components">Components</h2>
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<ul>
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<li><p><strong>Reality
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(<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>R</mi><annotation encoding="application/x-tex">R</annotation></semantics></math>)</strong>:
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This is our state variable that evolves over time, influenced by the
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probabilistic nature of events and the organizational
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interactions.</p></li>
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<li><p><strong>Probabilistic Nature
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(<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>P</mi><annotation encoding="application/x-tex">P</annotation></semantics></math>)</strong>:
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Represents the spectrum of possibilities or outcomes that reality can
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take, which are not fixed but are influenced by underlying
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probabilities.</p></li>
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<li><p><strong>Organizational Structure
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(<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>S</mi><annotation encoding="application/x-tex">S</annotation></semantics></math>)</strong>:
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Represents the structured interactions within reality, which could be
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influenced by intelligence, optimization, and organizational
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dynamics.</p></li>
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</ul>
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<pre><code> [Probabilistic Nature (P)] [Organizational Structure (S)]
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---> [Reality (R)] <---
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^ | ^
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| | |
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| v |
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+--------------------- [Influences] ----------------+</code></pre>
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<h2 class="unnumbered" id="differential-equations">Differential
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Equations</h2>
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<p>To model the interaction between these components, we can propose the
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following system of differential equations:</p>
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<ol>
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<li><p><strong>Equation for Reality
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(<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>R</mi><annotation encoding="application/x-tex">R</annotation></semantics></math>)</strong>:
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<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>R</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>α</mi><mi>P</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>R</mi><mo>,</mo><mi>t</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mi>β</mi><mi>S</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>R</mi><mo>,</mo><mi>t</mi><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">\frac{dR}{dt} = \alpha P(R, t) + \beta S(R, t)</annotation></semantics></math></p>
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<ul>
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<li><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfrac><mrow><mi>d</mi><mi>R</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><annotation encoding="application/x-tex">\frac{dR}{dt}</annotation></semantics></math>
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is the rate of change of reality.</p></li>
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<li><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>α</mi><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>
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and
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>β</mi><annotation encoding="application/x-tex">\beta</annotation></semantics></math>
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are coefficients representing the influence strength of probabilities
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and structure on reality.</p></li>
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<li><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>R</mi><mo>,</mo><mi>t</mi><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">P(R, t)</annotation></semantics></math>
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is a function describing the probabilistic influences on reality at time
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>t</mi><annotation encoding="application/x-tex">t</annotation></semantics></math>.</p></li>
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<li><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>R</mi><mo>,</mo><mi>t</mi><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">S(R, t)</annotation></semantics></math>
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is a function describing the structured, organizational influences on
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reality at time
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>t</mi><annotation encoding="application/x-tex">t</annotation></semantics></math>.</p></li>
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</ul></li>
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<li><p><strong>Equation for Probabilistic Nature
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(<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>P</mi><annotation encoding="application/x-tex">P</annotation></semantics></math>)</strong>:
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<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>P</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>γ</mi><mi>R</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>t</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>−</mo><mi>δ</mi><mi>P</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>R</mi><mo>,</mo><mi>t</mi><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">\frac{dP}{dt} = \gamma R(t) - \delta P(R, t)</annotation></semantics></math></p>
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<ul>
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<li><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfrac><mrow><mi>d</mi><mi>P</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><annotation encoding="application/x-tex">\frac{dP}{dt}</annotation></semantics></math>
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is the rate of change of the probabilistic nature.</p></li>
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<li><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>γ</mi><annotation encoding="application/x-tex">\gamma</annotation></semantics></math>
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and
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>δ</mi><annotation encoding="application/x-tex">\delta</annotation></semantics></math>
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are coefficients that modulate the impact of reality on probability and
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the decay of probabilistic influence.</p></li>
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</ul></li>
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<li><p><strong>Equation for Organizational Structure
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(<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>S</mi><annotation encoding="application/x-tex">S</annotation></semantics></math>)</strong>:
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<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>S</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>ϵ</mi><mi>R</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>t</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>−</mo><mi>ζ</mi><mi>S</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>R</mi><mo>,</mo><mi>t</mi><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">\frac{dS}{dt} = \epsilon R(t) - \zeta S(R, t)</annotation></semantics></math></p>
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<ul>
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<li><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfrac><mrow><mi>d</mi><mi>S</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><annotation encoding="application/x-tex">\frac{dS}{dt}</annotation></semantics></math>
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is the rate of change of organizational structure.</p></li>
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<li><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>ϵ</mi><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>
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and
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>ζ</mi><annotation encoding="application/x-tex">\zeta</annotation></semantics></math>
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are coefficients reflecting the impact of reality on organizational
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structures and the decay or adaptation rate of the structure.</p></li>
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</ul></li>
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</ol>
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<h2 class="unnumbered" id="interpretation">Interpretation</h2>
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<ul>
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<li><p>The evolution of
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>R</mi><annotation encoding="application/x-tex">R</annotation></semantics></math>
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is directly influenced by both
|
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>P</mi><annotation encoding="application/x-tex">P</annotation></semantics></math>
|
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and
|
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>S</mi><annotation encoding="application/x-tex">S</annotation></semantics></math>,
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indicating that both random and structured elements affect the state of
|
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reality.</p></li>
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<li><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>P</mi><annotation encoding="application/x-tex">P</annotation></semantics></math>
|
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evolves based on the current state of reality but has its dynamics
|
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moderated by a decay or transformation term
|
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">\delta P</annotation></semantics></math>.</p></li>
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<li><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>S</mi><annotation encoding="application/x-tex">S</annotation></semantics></math>
|
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is similarly influenced by reality but adapts or decays at a rate
|
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">\zeta S</annotation></semantics></math>.</p></li>
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</ul>
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<p>This model allows us to examine how changes in either the
|
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probabilistic or structured aspects of reality can lead to changes in
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the overall state of reality, encapsulating the concepts from the two
|
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theories into a cohesive mathematical framework.</p>
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<h2 class="unnumbered" id="jacobian-matrix">Jacobian Matrix</h2>
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<p>The Jacobian matrix
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>𝐉</mi><annotation encoding="application/x-tex">\mathbf{J}</annotation></semantics></math>
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is constructed by taking the partial derivatives of each equation with
|
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respect to each of the variables
|
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>R</mi><annotation encoding="application/x-tex">R</annotation></semantics></math>,
|
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>P</mi><annotation encoding="application/x-tex">P</annotation></semantics></math>,
|
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and
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>S</mi><annotation encoding="application/x-tex">S</annotation></semantics></math>.
|
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The matrix is defined as:</p>
|
||||
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>𝐉</mi><mo>=</mo><mrow><mo stretchy="true" form="prefix">[</mo><mtable><mtr><mtd columnalign="center" style="text-align: center"><mfrac><mrow><mi>∂</mi><mover><mi>R</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>R</mi></mrow></mfrac></mtd><mtd columnalign="center" style="text-align: center"><mfrac><mrow><mi>∂</mi><mover><mi>R</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>P</mi></mrow></mfrac></mtd><mtd columnalign="center" style="text-align: center"><mfrac><mrow><mi>∂</mi><mover><mi>R</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>S</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="center" style="text-align: center"><mfrac><mrow><mi>∂</mi><mover><mi>P</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>R</mi></mrow></mfrac></mtd><mtd columnalign="center" style="text-align: center"><mfrac><mrow><mi>∂</mi><mover><mi>P</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>P</mi></mrow></mfrac></mtd><mtd columnalign="center" style="text-align: center"><mfrac><mrow><mi>∂</mi><mover><mi>P</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>S</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="center" style="text-align: center"><mfrac><mrow><mi>∂</mi><mover><mi>S</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>R</mi></mrow></mfrac></mtd><mtd columnalign="center" style="text-align: center"><mfrac><mrow><mi>∂</mi><mover><mi>S</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>P</mi></mrow></mfrac></mtd><mtd columnalign="center" style="text-align: center"><mfrac><mrow><mi>∂</mi><mover><mi>S</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>S</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="center" style="text-align: center"></mtd></mtr></mtable><mo stretchy="true" form="postfix">]</mo></mrow></mrow><annotation encoding="application/x-tex">\mathbf{J} = \begin{bmatrix}
|
||||
\frac{\partial \dot{R}}{\partial R} & \frac{\partial \dot{R}}{\partial P} & \frac{\partial \dot{R}}{\partial S} \\
|
||||
\frac{\partial \dot{P}}{\partial R} & \frac{\partial \dot{P}}{\partial P} & \frac{\partial \dot{P}}{\partial S} \\
|
||||
\frac{\partial \dot{S}}{\partial R} & \frac{\partial \dot{S}}{\partial P} & \frac{\partial \dot{S}}{\partial S} \\
|
||||
\end{bmatrix}</annotation></semantics></math></p>
|
||||
<h2 class="unnumbered"
|
||||
id="calculating-the-partial-derivatives">Calculating the Partial
|
||||
Derivatives:</h2>
|
||||
<p>- For
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>R</mi><mo accent="true">̇</mo></mover><annotation encoding="application/x-tex">\dot{R}</annotation></semantics></math>:
|
||||
-
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>∂</mi><mover><mi>R</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>R</mi></mrow></mfrac><mo>=</mo><mi>α</mi><mfrac><mrow><mi>∂</mi><mi>P</mi></mrow><mrow><mi>∂</mi><mi>R</mi></mrow></mfrac><mo>+</mo><mi>β</mi><mfrac><mrow><mi>∂</mi><mi>S</mi></mrow><mrow><mi>∂</mi><mi>R</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial \dot{R}}{\partial R} = \alpha \frac{\partial P}{\partial R} + \beta \frac{\partial S}{\partial R}</annotation></semantics></math>
|
||||
-
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>∂</mi><mover><mi>R</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>P</mi></mrow></mfrac><mo>=</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">\frac{\partial \dot{R}}{\partial P} = \alpha</annotation></semantics></math>
|
||||
-
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>∂</mi><mover><mi>R</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>S</mi></mrow></mfrac><mo>=</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">\frac{\partial \dot{R}}{\partial S} = \beta</annotation></semantics></math></p>
|
||||
<p>- For
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>P</mi><mo accent="true">̇</mo></mover><annotation encoding="application/x-tex">\dot{P}</annotation></semantics></math>:
|
||||
-
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>∂</mi><mover><mi>P</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>R</mi></mrow></mfrac><mo>=</mo><mi>γ</mi></mrow><annotation encoding="application/x-tex">\frac{\partial \dot{P}}{\partial R} = \gamma</annotation></semantics></math>
|
||||
-
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>∂</mi><mover><mi>P</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>P</mi></mrow></mfrac><mo>=</mo><mi>−</mi><mi>δ</mi></mrow><annotation encoding="application/x-tex">\frac{\partial \dot{P}}{\partial P} = -\delta</annotation></semantics></math>
|
||||
-
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>∂</mi><mover><mi>P</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>S</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{\partial \dot{P}}{\partial S} = 0</annotation></semantics></math>
|
||||
(assuming
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>P</mi><annotation encoding="application/x-tex">P</annotation></semantics></math>
|
||||
does not depend on
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>S</mi><annotation encoding="application/x-tex">S</annotation></semantics></math>)</p>
|
||||
<p>- For
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>S</mi><mo accent="true">̇</mo></mover><annotation encoding="application/x-tex">\dot{S}</annotation></semantics></math>:
|
||||
-
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>∂</mi><mover><mi>S</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>R</mi></mrow></mfrac><mo>=</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\frac{\partial \dot{S}}{\partial R} = \epsilon</annotation></semantics></math>
|
||||
-
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>∂</mi><mover><mi>S</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>P</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{\partial \dot{S}}{\partial P} = 0</annotation></semantics></math>
|
||||
(assuming
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>S</mi><annotation encoding="application/x-tex">S</annotation></semantics></math>
|
||||
does not depend on
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>P</mi><annotation encoding="application/x-tex">P</annotation></semantics></math>)
|
||||
-
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>∂</mi><mover><mi>S</mi><mo accent="true">̇</mo></mover></mrow><mrow><mi>∂</mi><mi>S</mi></mrow></mfrac><mo>=</mo><mi>−</mi><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\frac{\partial \dot{S}}{\partial S} = -\zeta</annotation></semantics></math></p>
|
||||
<h2 class="unnumbered" id="jacobian-matrix-representation">Jacobian
|
||||
Matrix Representation:</h2>
|
||||
<p>The Jacobian matrix then is:</p>
|
||||
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>𝐉</mi><mo>=</mo><mrow><mo stretchy="true" form="prefix">[</mo><mtable><mtr><mtd columnalign="center" style="text-align: center"><mi>α</mi><mfrac><mrow><mi>∂</mi><mi>P</mi></mrow><mrow><mi>∂</mi><mi>R</mi></mrow></mfrac><mo>+</mo><mi>β</mi><mfrac><mrow><mi>∂</mi><mi>S</mi></mrow><mrow><mi>∂</mi><mi>R</mi></mrow></mfrac></mtd><mtd columnalign="center" style="text-align: center"><mi>α</mi></mtd><mtd columnalign="center" style="text-align: center"><mi>β</mi></mtd></mtr><mtr><mtd columnalign="center" style="text-align: center"><mi>γ</mi></mtd><mtd columnalign="center" style="text-align: center"><mi>−</mi><mi>δ</mi></mtd><mtd columnalign="center" style="text-align: center"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="center" style="text-align: center"><mi>ϵ</mi></mtd><mtd columnalign="center" style="text-align: center"><mn>0</mn></mtd><mtd columnalign="center" style="text-align: center"><mi>−</mi><mi>ζ</mi></mtd></mtr><mtr><mtd columnalign="center" style="text-align: center"></mtd></mtr></mtable><mo stretchy="true" form="postfix">]</mo></mrow></mrow><annotation encoding="application/x-tex">\mathbf{J} = \begin{bmatrix}
|
||||
\alpha \frac{\partial P}{\partial R} + \beta \frac{\partial S}{\partial R} & \alpha & \beta \\
|
||||
\gamma & -\delta & 0 \\
|
||||
\epsilon & 0 & -\zeta \\
|
||||
\end{bmatrix}</annotation></semantics></math></p>
|
||||
<h1
|
||||
id="correlations-based-on-cyber-space-time-thought-continuum">Correlations
|
||||
Based on Cyber-Space-Time-Thought Continuum</h1>
|
||||
<p>The cyber-space-time-thought continuum implies a complex interaction
|
||||
between cyber (machine augmentation), space (traditional and virtual),
|
||||
time (past, present, future), and thought (intellectual processes). Here
|
||||
are the suggested correlations for the coefficients:</p>
|
||||
<h2 id="correlation-between-alpha-and-gamma">Correlation Between
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>α</mi><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>
|
||||
and
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>γ</mi><annotation encoding="application/x-tex">\gamma</annotation></semantics></math></h2>
|
||||
<p><strong>Nature:</strong> Both coefficients describe the influence of
|
||||
one component on another.
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>α</mi><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>
|
||||
describes how probabilistic nature influences reality, while
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>γ</mi><annotation encoding="application/x-tex">\gamma</annotation></semantics></math>
|
||||
describes how reality influences probabilistic nature.</p>
|
||||
<p><strong>Interpretation:</strong> Since cyber interactions can
|
||||
significantly enhance the predictive power (probabilistic nature) by
|
||||
processing vast amounts of data in real-time,
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>α</mi><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>
|
||||
should be positively correlated with
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>γ</mi><annotation encoding="application/x-tex">\gamma</annotation></semantics></math>.
|
||||
A higher
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>α</mi><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>
|
||||
would mean a stronger influence of probabilistic outcomes on reality,
|
||||
which in turn enhances the influence of reality on probabilistic
|
||||
predictions
|
||||
(<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>γ</mi><annotation encoding="application/x-tex">\gamma</annotation></semantics></math>)
|
||||
through feedback loops.</p>
|
||||
<h2 id="correlation-between-beta-and-epsilon">Correlation Between
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>β</mi><annotation encoding="application/x-tex">\beta</annotation></semantics></math>
|
||||
and
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>ϵ</mi><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math></h2>
|
||||
<p><strong>Nature:</strong> Both coefficients relate to the
|
||||
organizational structure’s influence on and by reality.</p>
|
||||
<p><strong>Interpretation:</strong> In a cyber-augmented continuum,
|
||||
structured organizational data (like algorithms and AI models) directly
|
||||
impacts reality by optimizing processes and decisions. Therefore,
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>β</mi><annotation encoding="application/x-tex">\beta</annotation></semantics></math>
|
||||
(influence of structure on reality) should be positively correlated with
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>ϵ</mi><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>
|
||||
(influence of reality on structure). Enhanced organizational structures
|
||||
(better AI and machine learning models) should improve reality, which in
|
||||
turn would refine and adapt these structures.</p>
|
||||
<h2 id="correlation-between-delta-and-zeta">Correlation Between
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>δ</mi><annotation encoding="application/x-tex">\delta</annotation></semantics></math>
|
||||
and
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>ζ</mi><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></h2>
|
||||
<p><strong>Nature:</strong> Both coefficients describe decay or
|
||||
adaptation rates of probabilistic and structural influences.</p>
|
||||
<p><strong>Interpretation:</strong> In a rapidly evolving cyber
|
||||
environment, the decay or adaptation rate of probabilistic influences
|
||||
(<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>δ</mi><annotation encoding="application/x-tex">\delta</annotation></semantics></math>)
|
||||
and structural influences
|
||||
(<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>ζ</mi><annotation encoding="application/x-tex">\zeta</annotation></semantics></math>)
|
||||
should be closely linked. Faster adaptation in probabilistic models
|
||||
would necessitate quicker updates in structural models to maintain
|
||||
alignment with the current state of reality. Thus,
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>δ</mi><annotation encoding="application/x-tex">\delta</annotation></semantics></math>
|
||||
should be positively correlated with
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>ζ</mi><annotation encoding="application/x-tex">\zeta</annotation></semantics></math>.</p>
|
||||
<h1
|
||||
id="interpretation-in-cyber-space-time-thought-continuum">Interpretation
|
||||
in Cyber-Space-Time-Thought Continuum</h1>
|
||||
<p>In this continuum:</p>
|
||||
<ul>
|
||||
<li><p><strong>Cyber (Machine Augmentation):</strong> Enhances both the
|
||||
probabilistic (P) and structured (S) components by improving data
|
||||
processing and decision-making capabilities.</p></li>
|
||||
<li><p><strong>Space (Virtual and Traditional):</strong> Is influenced
|
||||
by cyber through the creation of virtual environments and augmentations
|
||||
that redefine spatial interactions.</p></li>
|
||||
<li><p><strong>Time (Past, Present, Future):</strong> Is compressed
|
||||
through real-time data processing and predictive modeling, enhancing the
|
||||
ability to respond to future states.</p></li>
|
||||
<li><p><strong>Thought (Intellectual Processes):</strong> Is augmented
|
||||
by machines, leading to higher levels of intelligence and
|
||||
decision-making capabilities.</p></li>
|
||||
</ul>
|
||||
<p>These correlations and interpretations suggest that the coefficients
|
||||
should reflect the dynamic and interconnected nature of the
|
||||
cyber-space-time-thought continuum, with positive correlations
|
||||
indicating synergistic enhancements in probabilistic and structural
|
||||
influences on reality.</p>
|
||||
<p>By ensuring these correlations, the model encapsulates the evolving
|
||||
understanding of reality influenced by both probabilistic diversity and
|
||||
structured organizational intelligence, forming a cohesive framework
|
||||
that aligns with the principles described in the "Combined Theory
|
||||
Differential Equations" document.</p>
|
||||
<h2
|
||||
id="partial-differential-system-of-coefficient-relationships">Partial
|
||||
Differential System of Coefficient Relationships</h2>
|
||||
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable><mtr><mtd columnalign="right" style="text-align: right"><mfrac><mrow><mi>∂</mi><mi>α</mi></mrow><mrow><mi>∂</mi><mi>t</mi></mrow></mfrac></mtd><mtd columnalign="left" style="text-align: left"><mo>=</mo><msub><mi>k</mi><mn>1</mn></msub><mi>β</mi><mo>−</mo><msub><mi>k</mi><mn>2</mn></msub><mi>α</mi><mi>γ</mi><mo>+</mo><msub><mi>k</mi><mn>3</mn></msub><mi>ϵ</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="right" style="text-align: right"><mfrac><mrow><mi>∂</mi><mi>β</mi></mrow><mrow><mi>∂</mi><mi>t</mi></mrow></mfrac></mtd><mtd columnalign="left" style="text-align: left"><mo>=</mo><msub><mi>k</mi><mn>4</mn></msub><mi>α</mi><mo>−</mo><msub><mi>k</mi><mn>5</mn></msub><mi>β</mi><mi>δ</mi><mo>+</mo><msub><mi>k</mi><mn>6</mn></msub><mi>ζ</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="right" style="text-align: right"><mfrac><mrow><mi>∂</mi><mi>γ</mi></mrow><mrow><mi>∂</mi><mi>t</mi></mrow></mfrac></mtd><mtd columnalign="left" style="text-align: left"><mo>=</mo><msub><mi>k</mi><mn>7</mn></msub><mi>α</mi><mi>β</mi><mo>−</mo><msub><mi>k</mi><mn>8</mn></msub><mi>γ</mi><mo>+</mo><msub><mi>k</mi><mn>9</mn></msub><mi>ϵ</mi><mi>δ</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="right" style="text-align: right"><mfrac><mrow><mi>∂</mi><mi>δ</mi></mrow><mrow><mi>∂</mi><mi>t</mi></mrow></mfrac></mtd><mtd columnalign="left" style="text-align: left"><mo>=</mo><msub><mi>k</mi><mn>10</mn></msub><mi>γ</mi><mo>−</mo><msub><mi>k</mi><mn>11</mn></msub><mi>δ</mi><mi>ζ</mi><mo>+</mo><msub><mi>k</mi><mn>12</mn></msub><mi>α</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="right" style="text-align: right"><mfrac><mrow><mi>∂</mi><mi>ϵ</mi></mrow><mrow><mi>∂</mi><mi>t</mi></mrow></mfrac></mtd><mtd columnalign="left" style="text-align: left"><mo>=</mo><msub><mi>k</mi><mn>13</mn></msub><mi>α</mi><mi>β</mi><mo>−</mo><msub><mi>k</mi><mn>14</mn></msub><mi>ϵ</mi><mi>γ</mi><mo>+</mo><msub><mi>k</mi><mn>15</mn></msub><mi>δ</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="right" style="text-align: right"><mfrac><mrow><mi>∂</mi><mi>ζ</mi></mrow><mrow><mi>∂</mi><mi>t</mi></mrow></mfrac></mtd><mtd columnalign="left" style="text-align: left"><mo>=</mo><msub><mi>k</mi><mn>16</mn></msub><mi>β</mi><mi>γ</mi><mo>−</mo><msub><mi>k</mi><mn>17</mn></msub><mi>ζ</mi><mo>+</mo><msub><mi>k</mi><mn>18</mn></msub><mi>δ</mi><mi>ϵ</mi><mo>,</mo></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
|
||||
\frac{\partial \alpha}{\partial t} &= k_1 \beta - k_2 \alpha \gamma + k_3 \epsilon, \\
|
||||
\frac{\partial \beta}{\partial t} &= k_4 \alpha - k_5 \beta \delta + k_6 \zeta, \\
|
||||
\frac{\partial \gamma}{\partial t} &= k_7 \alpha \beta - k_8 \gamma + k_9 \epsilon \delta, \\
|
||||
\frac{\partial \delta}{\partial t} &= k_{10} \gamma - k_{11} \delta \zeta + k_{12} \alpha, \\
|
||||
\frac{\partial \epsilon}{\partial t} &= k_{13} \alpha \beta - k_{14} \epsilon \gamma + k_{15} \delta, \\
|
||||
\frac{\partial \zeta}{\partial t} &= k_{16} \beta \gamma - k_{17} \zeta + k_{18} \delta \epsilon,
|
||||
\end{aligned}</annotation></semantics></math></p>
|
||||
<h1 class="unnumbered" id="incorporating-correlations">Incorporating
|
||||
Correlations</h1>
|
||||
<p>We need to modify the partial derivatives in the Jacobian to account
|
||||
for the correlations. This can be done by introducing terms that
|
||||
represent the dependencies.</p>
|
||||
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>𝐉</mi><mo>=</mo><mrow><mo stretchy="true" form="prefix">[</mo><mtable><mtr><mtd columnalign="center" style="text-align: center"><mi>α</mi><mfrac><mrow><mi>∂</mi><mi>P</mi></mrow><mrow><mi>∂</mi><mi>R</mi></mrow></mfrac><mo>+</mo><mi>β</mi><mfrac><mrow><mi>∂</mi><mi>S</mi></mrow><mrow><mi>∂</mi><mi>R</mi></mrow></mfrac></mtd><mtd columnalign="center" style="text-align: center"><mi>α</mi><mo>+</mo><msub><mi>k</mi><mn>1</mn></msub><mi>γ</mi></mtd><mtd columnalign="center" style="text-align: center"><mi>β</mi><mo>+</mo><msub><mi>k</mi><mn>2</mn></msub><mi>ϵ</mi></mtd></mtr><mtr><mtd columnalign="center" style="text-align: center"><mi>γ</mi><mo>+</mo><msub><mi>k</mi><mn>3</mn></msub><mi>α</mi></mtd><mtd columnalign="center" style="text-align: center"><mi>−</mi><mi>δ</mi></mtd><mtd columnalign="center" style="text-align: center"><msub><mi>k</mi><mn>4</mn></msub><mi>ζ</mi></mtd></mtr><mtr><mtd columnalign="center" style="text-align: center"><mi>ϵ</mi><mo>+</mo><msub><mi>k</mi><mn>5</mn></msub><mi>β</mi></mtd><mtd columnalign="center" style="text-align: center"><msub><mi>k</mi><mn>6</mn></msub><mi>δ</mi></mtd><mtd columnalign="center" style="text-align: center"><mi>−</mi><mi>ζ</mi></mtd></mtr><mtr><mtd columnalign="center" style="text-align: center"></mtd></mtr></mtable><mo stretchy="true" form="postfix">]</mo></mrow></mrow><annotation encoding="application/x-tex">\mathbf{J} = \begin{bmatrix}
|
||||
\alpha \frac{\partial P}{\partial R} + \beta \frac{\partial S}{\partial R} & \alpha + k_1 \gamma & \beta + k_2 \epsilon \\
|
||||
\gamma + k_3 \alpha & -\delta & k_4 \zeta \\
|
||||
\epsilon + k_5 \beta & k_6 \delta & -\zeta \\
|
||||
\end{bmatrix}</annotation></semantics></math></p>
|
||||
<p>Here,
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mn>1</mn></msub><mo>,</mo><msub><mi>k</mi><mn>2</mn></msub><mo>,</mo><msub><mi>k</mi><mn>3</mn></msub><mo>,</mo><msub><mi>k</mi><mn>4</mn></msub><mo>,</mo><msub><mi>k</mi><mn>5</mn></msub><mo>,</mo><msub><mi>k</mi><mn>6</mn></msub></mrow><annotation encoding="application/x-tex">k_1, k_2, k_3, k_4, k_5, k_6</annotation></semantics></math>
|
||||
are constants that represent the strength of the correlations between
|
||||
the respective coefficients.</p>
|
||||
<h1 class="unnumbered" id="energy-function-approach">Energy Function
|
||||
Approach</h1>
|
||||
<p>In systems theory, especially in dynamical systems involving
|
||||
differential equations, a Lyapunov function
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>V</mi><annotation encoding="application/x-tex">V</annotation></semantics></math>
|
||||
is used to demonstrate the stability of an equilibrium point. If we can
|
||||
define such a function where
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>V</mi><annotation encoding="application/x-tex">V</annotation></semantics></math>
|
||||
decreases over time
|
||||
(<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>V</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>≤</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{dV}{dt} \leq 0</annotation></semantics></math>),
|
||||
it suggests that the system dissipates energy, moving towards a stable
|
||||
state.</p>
|
||||
<h2 class="unnumbered"
|
||||
id="constructing-a-lyapunov-function">Constructing a Lyapunov
|
||||
Function</h2>
|
||||
<p>Given the system:</p>
|
||||
<ul>
|
||||
<li><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>R</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>α</mi><mi>P</mi><mo>+</mo><mi>β</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">\frac{dR}{dt} = \alpha P + \beta S</annotation></semantics></math></p></li>
|
||||
<li><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>P</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>γ</mi><mi>R</mi><mo>−</mo><mi>δ</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">\frac{dP}{dt} = \gamma R - \delta P</annotation></semantics></math></p></li>
|
||||
<li><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>S</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>ϵ</mi><mi>R</mi><mo>−</mo><mi>ζ</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">\frac{dS}{dt} = \epsilon R - \zeta S</annotation></semantics></math></p></li>
|
||||
</ul>
|
||||
<p>One possible Lyapunov function could be:
|
||||
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>R</mi><mo>,</mo><mi>P</mi><mo>,</mo><mi>S</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>R</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><msup><mi>P</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><msup><mi>S</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">V(R, P, S) = aR^2 + bP^2 + cS^2</annotation></semantics></math>
|
||||
where
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a, b, c</annotation></semantics></math>
|
||||
are positive constants that need to be determined based on the system’s
|
||||
parameters to ensure that
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfrac><mrow><mi>d</mi><mi>V</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><annotation encoding="application/x-tex">\frac{dV}{dt}</annotation></semantics></math>
|
||||
is negative or zero.</p>
|
||||
<h2 class="unnumbered" id="derivative-of-v">Derivative of
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>V</mi><annotation encoding="application/x-tex">V</annotation></semantics></math></h2>
|
||||
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>V</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>a</mi><mi>R</mi><mfrac><mrow><mi>d</mi><mi>R</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>+</mo><mn>2</mn><mi>b</mi><mi>P</mi><mfrac><mrow><mi>d</mi><mi>P</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>+</mo><mn>2</mn><mi>c</mi><mi>S</mi><mfrac><mrow><mi>d</mi><mi>S</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dV}{dt} = 2aR\frac{dR}{dt} + 2bP\frac{dP}{dt} + 2cS\frac{dS}{dt}</annotation></semantics></math>
|
||||
Substituting the derivatives from the system:
|
||||
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>V</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>a</mi><mi>R</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>α</mi><mi>P</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mn>2</mn><mi>b</mi><mi>P</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>γ</mi><mi>R</mi><mo>−</mo><mi>δ</mi><mi>P</mi><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mn>2</mn><mi>c</mi><mi>S</mi><mrow><mo stretchy="true" form="prefix">(</mo><mi>ϵ</mi><mi>R</mi><mo>−</mo><mi>ζ</mi><mi>S</mi><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">\frac{dV}{dt} = 2aR(\alpha P + \beta S) + 2bP(\gamma R - \delta P) + 2cS(\epsilon R - \zeta S)</annotation></semantics></math></p>
|
||||
<h2 class="unnumbered" id="simplifying-and-analyzing">Simplifying and
|
||||
Analyzing</h2>
|
||||
<p>Simplifying
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfrac><mrow><mi>d</mi><mi>V</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><annotation encoding="application/x-tex">\frac{dV}{dt}</annotation></semantics></math>
|
||||
requires choosing
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a, b, c</annotation></semantics></math>
|
||||
such that the cross terms cancel out or contribute to a negative value.
|
||||
This might look something like:
|
||||
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>V</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mrow><mo stretchy="true" form="prefix">(</mo><mi>α</mi><mi>a</mi><mi>R</mi><mi>P</mi><mo>+</mo><mi>β</mi><mi>a</mi><mi>R</mi><mi>S</mi><mo>+</mo><mi>γ</mi><mi>b</mi><mi>P</mi><mi>R</mi><mo>−</mo><mi>δ</mi><mi>b</mi><msup><mi>P</mi><mn>2</mn></msup><mo>+</mo><mi>ϵ</mi><mi>c</mi><mi>S</mi><mi>R</mi><mo>−</mo><mi>ζ</mi><mi>c</mi><msup><mi>S</mi><mn>2</mn></msup><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">\frac{dV}{dt} = 2(\alpha aR P + \beta aRS + \gamma bPR - \delta bP^2 + \epsilon cSR - \zeta cS^2)</annotation></semantics></math></p>
|
||||
<p>The coefficients and their signs must be carefully adjusted to ensure
|
||||
that
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>V</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>≤</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{dV}{dt} \leq 0</annotation></semantics></math>
|
||||
for all
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mo>,</mo><mi>P</mi><mo>,</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">R, P, S</annotation></semantics></math>
|
||||
except at the equilibrium. This might involve setting the cross term
|
||||
coefficients to balance out (e.g.,
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mi>a</mi><mo>=</mo><mi>γ</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">\alpha a = \gamma b</annotation></semantics></math>)
|
||||
and ensuring the quadratic terms are always negative or zero.</p>
|
||||
<h2 class="unnumbered" id="conclusion">Conclusion</h2>
|
||||
<p>This construction is theoretical and depends heavily on the specific
|
||||
dynamics and parameters of your model. The actual application might
|
||||
require numerical simulation or more complex analytical tools to verify
|
||||
that
|
||||
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>V</mi><annotation encoding="application/x-tex">V</annotation></semantics></math>
|
||||
decreases over time. If you can determine such a Lyapunov function, it
|
||||
can serve as a "measure of energy" in the system, showing how the system
|
||||
evolves and stabilizes over time.</p>
|
||||
</body>
|
||||
</html>
|
||||
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