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we-have-a-system-made-up-of-two-cells-x-and-y-Both-of-the-cell-types-are-dividing-and-dying-X-type-cells-also-differentiate-into-Y-type-cells-The-dynamics-of-this-system-interms-of-size-of-X-and-Y

Question Number 207154 by Wuji last updated on 07/May/24 $$\mathrm{we}\:\mathrm{have}\:\mathrm{a}\:\mathrm{system}\:\mathrm{made}\:\mathrm{up}\:\mathrm{of}\:\mathrm{two}\:\mathrm{cells} \\ $$$$\mathrm{x}\:\mathrm{and}\:\mathrm{y}.\:\mathrm{Both}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cell}\:\mathrm{types}\:\mathrm{are} \\ $$$$\mathrm{dividing}\:\mathrm{and}\:\mathrm{dying}.\:\mathrm{X}\:\mathrm{type}\:\mathrm{cells}\:\mathrm{also} \\ $$$$\mathrm{differentiate}\:\mathrm{into}\:\mathrm{Y}\:\mathrm{type}\:\mathrm{cells}.\:\mathrm{The} \\ $$$$\mathrm{dynamics}\:\mathrm{of}\:\mathrm{this}\:\mathrm{system}\:\mathrm{interms}\:\mathrm{of}\:\:\mathrm{size} \\ $$$$\mathrm{of}\:\mathrm{X}\:\mathrm{and}\:\mathrm{Y}\:\mathrm{population}\:\mathrm{is}\:\mathrm{given}\:\mathrm{below}. \\ $$$$\mathrm{calculate}\:\mathrm{the}\:\mathrm{steady}\:\mathrm{state}\:\mathrm{of}\:\mathrm{this}\:\mathrm{system} \\ $$$$\frac{\mathrm{dx}}{\mathrm{dt}}=−\mathrm{3x}\:\:\:\frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{2x}−\mathrm{2y} \\…

Expression-of-protein-is-controlled-by-an-external-S-the-protein-also-controls-its-own-exprexxion-by-a-negative-feedback-The-fol-lt-lwing-system-of-ODEs-represents-the-dynamics-of-the-system-with-m

Question Number 207150 by Wuji last updated on 07/May/24 $$\mathrm{Expression}\:\mathrm{of}\:\mathrm{protein}\:\mathrm{is}\:\mathrm{controlled} \\ $$$$\mathrm{by}\:\mathrm{an}\:\mathrm{external}\:\mathrm{S}.\:\mathrm{the}\:\mathrm{protein}\:\mathrm{also}\:\mathrm{controls} \\ $$$$\mathrm{its}\:\mathrm{own}\:\mathrm{exprexxion}\:\mathrm{by}\:\mathrm{a}\:\mathrm{negative}\:\mathrm{feedback}. \\ $$$$\mathrm{The}\:\mathrm{fol}<\mathrm{lwing}\:\mathrm{system}\:\mathrm{of}\:\mathrm{ODEs}\:\mathrm{represents} \\ $$$$\mathrm{the}\:\mathrm{dynamics}\:\mathrm{of}\:\mathrm{the}\:\mathrm{system},\:\mathrm{with} \\ $$$$\mathrm{m}\:\mathrm{and}\:\mathrm{p}\:\mathrm{representing}\:\mathrm{mRNA}\:\mathrm{and} \\ $$$$\mathrm{protein}\:\mathrm{respectively},\:\mathrm{prove}\:\mathrm{that}\:\mathrm{for} \\ $$$$\mathrm{any}\:\mathrm{value}\:\mathrm{of}\:\mathrm{S}\geqslant\mathrm{0},\:\mathrm{the}\:\mathrm{steady}\:\mathrm{state}\:\mathrm{of}\:\mathrm{the} \\…

Two-proteins-x-and-y-control-each-other-through-mutual-repression-The-dynamic-model-for-the-system-consists-of-the-following-system-of-ODEs-calculate-the-steady-state-values-of-these-two-protein

Question Number 207151 by Wuji last updated on 07/May/24 $$\mathrm{Two}\:\mathrm{proteins}\:\left(\mathrm{x}\:\mathrm{and}\:\mathrm{y}\right)\:\mathrm{control}\:\mathrm{each} \\ $$$$\mathrm{other}\:\mathrm{through}\:\mathrm{mutual}\:\mathrm{repression}.\:\mathrm{The} \\ $$$$\mathrm{dynamic}\:\mathrm{model}\:\mathrm{for}\:\mathrm{the}\:\mathrm{system}\:\mathrm{consists} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{system}\:\mathrm{of}\:\mathrm{ODEs}.\: \\ $$$$\mathrm{calculate}\:\mathrm{the}\:\mathrm{steady}\:\mathrm{state}\:\mathrm{values}\:\mathrm{of}\: \\ $$$$\mathrm{these}\:\mathrm{two}\:\mathrm{proteins}.\:\mathrm{comment}\:\mathrm{on}\:\mathrm{the}\: \\ $$$$\mathrm{stability}\:\mathrm{of}\:\mathrm{the}\:\mathrm{steady}\:\mathrm{state}\:\mathrm{and}\:\mathrm{the}\:\mathrm{type}\:\mathrm{of}\: \\ $$$$\mathrm{phase}\:\mathrm{portrait}\:\mathrm{expected}\:\mathrm{for}\:\mathrm{this}\:\mathrm{system}. \\…

Two-proteins-x-amp-y-control-each-other-through-mutual-repression-the-dynamic-model-for-the-system-consists-of-the-following-system-of-ODEs-calculate-the-steady-state-values-of-these-two-peoteins

Question Number 207139 by Wuji last updated on 07/May/24 $$ \\ $$$$\mathrm{Two}\:\mathrm{proteins}\:\left(\mathrm{x\&y}\right)\:\mathrm{control}\:\mathrm{each} \\ $$$$\mathrm{other}\:\mathrm{through}\:\mathrm{mutual}\:\mathrm{repression}.\:\mathrm{the} \\ $$$$\mathrm{dynamic}\:\mathrm{model}\:\mathrm{for}\:\mathrm{the}\:\mathrm{system}\:\mathrm{consists}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{following}\:\mathrm{system}\:\mathrm{of}\:\mathrm{ODEs}. \\ $$$$\mathrm{calculate}\:\mathrm{the}\:\mathrm{steady}\:\mathrm{state}\:\mathrm{values}\:\mathrm{of}\:\mathrm{these}\:\mathrm{two}\:\mathrm{peoteins}.\: \\ $$$$\mathrm{comment}\:\mathrm{of}\:\mathrm{the}\:\mathrm{stabilityof}\:\mathrm{the}\:\mathrm{steady}\:\mathrm{state} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{type}\:\mathrm{of}\:\mathrm{phase}\:\mathrm{of}\:\mathrm{portrait}\:\mathrm{expected}\:\mathrm{foe}\:\mathrm{this} \\…

for-the-given-system-of-ODEs-calculate-the-eigenvalues-and-corresponding-eigenvectors-of-the-coefficient-matrix-dx-dt-2x-y-dy-dt-x-2y-

Question Number 207096 by Wuji last updated on 06/May/24 $${for}\:{the}\:{given}\:{system}\:{of}\:{ODEs},\:{calculate}\:{the} \\ $$$${eigenvalues}\:{and}\:{corresponding}\:{eigenvectors}\:{of}\:{the}\: \\ $$$${coefficient}\:{matrix} \\ $$$$\frac{{dx}}{{dt}}=\mathrm{2}{x}+{y}\:\:\:\frac{{dy}}{{dt}}={x}+\mathrm{2}{y} \\ $$ Commented by Wuji last updated on 07/May/24…

Let-g-x-be-the-inverse-function-of-gt-f-x-x-3-3x-2-4x-5-lt-Evaluate-Lim-n-4n-g-1-1-n-g-1-2-n-

Question Number 207082 by SEKRET last updated on 06/May/24 $$\:\:\:\boldsymbol{\mathrm{Let}}\:\:\boldsymbol{\mathrm{g}}\left(\boldsymbol{\mathrm{x}}\right)\:\:\boldsymbol{\mathrm{be}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{inverse}}\:\boldsymbol{\mathrm{function}}\:\:\:\boldsymbol{\mathrm{of}} \\ $$$$ \\ $$$$\:\:\:\:−−>\:\:\:\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)=\:\boldsymbol{\mathrm{x}}^{\mathrm{3}} +\mathrm{3}\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{4}\boldsymbol{\mathrm{x}}+\mathrm{5}\:\:\:\:\:\:<−− \\ $$$$\:\: \\ $$$$ \\ $$$$\boldsymbol{\mathrm{Evaluate}}\:\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\boldsymbol{\mathrm{Lim}}}\:\mathrm{4}\boldsymbol{\mathrm{n}}\centerdot\left(\:\boldsymbol{\mathrm{g}}\left(\mathrm{1}+\frac{\mathrm{1}}{\boldsymbol{\mathrm{n}}}\right)\:−\boldsymbol{\mathrm{g}}\left(\mathrm{1}−\frac{\mathrm{2}}{\boldsymbol{\mathrm{n}}}\right)\:\right)=? \\ $$$$…

Calculate-the-generalized-solution-for-the-following-system-of-ODEs-dx-dt-1-2-x-dy-dt-1-2-x-1-4-y-dz-dt-1-4-y-1-6-z-

Question Number 207106 by Wuji last updated on 07/May/24 $${Calculate}\:{the}\:{generalized}\:{solution}\:{for}\:{the}\:{following} \\ $$$${system}\:{of}\:{ODEs}: \\ $$$$\frac{{dx}}{{dt}}=−\frac{\mathrm{1}}{\mathrm{2}}{x},\:\frac{{dy}}{{dt}}=\frac{\mathrm{1}}{\mathrm{2}}{x}−\frac{\mathrm{1}}{\mathrm{4}}{y},\:\:\frac{{dz}}{{dt}}=\frac{\mathrm{1}}{\mathrm{4}}{y}−\frac{\mathrm{1}}{\mathrm{6}}{z} \\ $$ Answered by mr W last updated on 07/May/24 $$\begin{vmatrix}{−\frac{\mathrm{1}}{\mathrm{2}}−\lambda}&{\mathrm{0}}&{\mathrm{0}}\\{\frac{\mathrm{1}}{\mathrm{2}}}&{−\frac{\mathrm{1}}{\mathrm{4}}−\lambda}&{\mathrm{0}}\\{\mathrm{0}}&{\frac{\mathrm{1}}{\mathrm{4}}}&{−\frac{\mathrm{1}}{\mathrm{6}}−\lambda}\end{vmatrix}=\mathrm{0}…

k-1-1-k-2-n-0-1-2-1-n-n-1-k-1-H-n-k-1-n-k-2-

Question Number 206861 by MrGHK last updated on 28/Apr/24 $$\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\infty} {\boldsymbol{\sum}}}\frac{\mathrm{1}}{\boldsymbol{\mathrm{k}}^{\mathrm{2}} }\underset{\boldsymbol{\mathrm{n}}=\mathrm{0}} {\overset{\infty} {\boldsymbol{\sum}}}\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{1}+\boldsymbol{\mathrm{n}}} }\left(\frac{\boldsymbol{\Gamma}\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)\boldsymbol{\Gamma}\left(\boldsymbol{\mathrm{k}}+\mathrm{1}\right)\boldsymbol{\mathrm{H}}_{\boldsymbol{\mathrm{n}}+\boldsymbol{\mathrm{k}}+\mathrm{1}} }{\boldsymbol{\Gamma}\left(\boldsymbol{\mathrm{n}}+\boldsymbol{\mathrm{k}}+\mathrm{2}\right)}\right)=??? \\ $$ Terms of Service Privacy Policy Contact:…