Difference between revisions of "Team:SEU/Model"

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                                           <h3>Computation method</h3>
 
                                           <h3>Computation method</h3>
 
                                           <p >Addition: \(A_1 \xrightarrow{k_1} O,\quad A_2 \xrightarrow{k_2} O\)</p>
 
                                           <p >Addition: \(A_1 \xrightarrow{k_1} O,\quad A_2 \xrightarrow{k_2} O\)</p>
−
                                           <p style="font-size=36px">Proof: \(\dfrac{d [A_i](t)}{d t}=-k_i[A_i](t)\) \(\Rightarrow [A_i](t)=[A_i](0)e^{-k_it}, \) \(\dfrac{d [O](t)}{d t}=\sum_{i=1}^2 k_i[A_i](t)\) \(\Rightarrow [O](\infty)=\int_0^\infty \sum_{i=1}^2 k_i[A_i](t)dt = [A_1](0)/k_1+[A_2](0)/k_2.\) If \(k_1\approx k_2\), then addition is successfully implemented.</p>
+
                                           <p style="font-size=24px">Proof: \(\dfrac{d [A_i](t)}{d t}=-k_i[A_i](t)\) \(\Rightarrow [A_i](t)=[A_i](0)e^{-k_it}, \) \(\dfrac{d [O](t)}{d t}=\sum_{i=1}^2 k_i[A_i](t)\) \(\Rightarrow [O](\infty)=\int_0^\infty \sum_{i=1}^2 k_i[A_i](t)dt = [A_1](0)/k_1+[A_2](0)/k_2.\) If \(k_1\approx k_2\), then addition is successfully implemented.</p>
−
                                           <p style="font-size=36px">Subtraction: \(A+B \xrightarrow{k_1} \phi\)</p>
+
                                           <p style="font-size=24px">Subtraction: \(A+B \xrightarrow{k_1} \phi\)</p>
−
                                           <p style="font-size=36px">Multiplication: \(\alpha \xrightarrow{k_1} \phi, A+B+\alpha \xrightarrow{k_2} A+B+\alpha+C\)</p>
+
                                           <p style="font-size=24px">Multiplication: \(\alpha \xrightarrow{k_1} \phi, A+B+\alpha \xrightarrow{k_2} A+B+\alpha+C\)</p>
−
                                           <p style="font-size=36px">Proof: \(\dfrac{d [\alpha](t)}{d t}=-k_1[\alpha](t)\) \(\Rightarrow [\alpha](t)=[\alpha](0)e^{-k_1t},\) \(\dfrac{d [A](t)}{d t}=\dfrac{d [B](t)}{d t}=0, \dfrac{d [C](t)}{d t}=k_2[A](t)[B](t)[\alpha](t)\) \(\Rightarrow [C](\infty)=\int_0^\infty [A](0)[B](0)[\alpha](t)=k_2/k_1[\alpha](0)[A](0)[B](0)\). Hence multiplication is implemented.</p>                                         
+
                                           <p style="font-size=24px">Proof: \(\dfrac{d [\alpha](t)}{d t}=-k_1[\alpha](t)\) \(\Rightarrow [\alpha](t)=[\alpha](0)e^{-k_1t},\) \(\dfrac{d [A](t)}{d t}=\dfrac{d [B](t)}{d t}=0, \dfrac{d [C](t)}{d t}=k_2[A](t)[B](t)[\alpha](t)\) \(\Rightarrow [C](\infty)=\int_0^\infty [A](0)[B](0)[\alpha](t)=k_2/k_1[\alpha](0)[A](0)[B](0)\). Hence multiplication is implemented.</p>                                         
 
                                       </div>
 
                                       </div>
 
                                 </div>
 
                                 </div>

Revision as of 08:20, 2 October 2019





Model

Computation method

Addition: \(A_1 \xrightarrow{k_1} O,\quad A_2 \xrightarrow{k_2} O\)

Proof: \(\dfrac{d [A_i](t)}{d t}=-k_i[A_i](t)\) \(\Rightarrow [A_i](t)=[A_i](0)e^{-k_it}, \) \(\dfrac{d [O](t)}{d t}=\sum_{i=1}^2 k_i[A_i](t)\) \(\Rightarrow [O](\infty)=\int_0^\infty \sum_{i=1}^2 k_i[A_i](t)dt = [A_1](0)/k_1+[A_2](0)/k_2.\) If \(k_1\approx k_2\), then addition is successfully implemented.

Subtraction: \(A+B \xrightarrow{k_1} \phi\)

Multiplication: \(\alpha \xrightarrow{k_1} \phi, A+B+\alpha \xrightarrow{k_2} A+B+\alpha+C\)

Proof: \(\dfrac{d [\alpha](t)}{d t}=-k_1[\alpha](t)\) \(\Rightarrow [\alpha](t)=[\alpha](0)e^{-k_1t},\) \(\dfrac{d [A](t)}{d t}=\dfrac{d [B](t)}{d t}=0, \dfrac{d [C](t)}{d t}=k_2[A](t)[B](t)[\alpha](t)\) \(\Rightarrow [C](\infty)=\int_0^\infty [A](0)[B](0)[\alpha](t)=k_2/k_1[\alpha](0)[A](0)[B](0)\). Hence multiplication is implemented.