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    MathWorks Inc symbolic math toolbox
    Symbolic Math Toolbox, supplied by MathWorks Inc, used in various techniques. Bioz Stars score: 94/100, based on 428 PubMed citations. ZERO BIAS - scores, article reviews, protocol conditions and more
    https://www.bioz.com/product/matlab+software+function/Symbolic+Math+Toolbox/us10359513-344-10-8
    Average 94 stars, based on 428 article reviews
    symbolic math toolbox - by Bioz Stars, 2026-10
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    Article Title: Along- and cross-muscle fiber shear moduli in skeletal muscle.
    Article Snippet: The material properties of muscle play a central role in how muscle resists joint motion, transmits forces, and repairs itself.. While many studies have evaluated muscle’s tensile properties, few have investigated muscle’s shear properties.. Our objective was to quantify the shear moduli along (along-muscle fiber) and transverse (cross-muscle fiber) to muscle fibers.

    Article Title: Modeling the impact of the Revised Starling Hypothesis on vascular refilling during ultrafiltration in humans.
    Article Snippet: 1175 We used the Symbolic Toolbox in MatLab to compute the symbolic derivatives required in A.3 1176 and A.4 and used these to compute the values needed to solve A.2 for each of the parameters 1177 [PIBP, VIBP, CompSL, XJV, XJS, PC, JLmax, JLSL] to obtain the sensitivity functions for each 1178 parameter over the course of the simulation and patient data sets.

    Article Title: Multidimensional performance trade-offs in bipedal standing of common chimpanzees and implications for human bipedal evolution
    Article Snippet: Symbolic Math Toolbox version: 9.3 (R2023a) , https://www.mathworks.com , N/A.

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    Article Snippet: Analytical solutions to the equations were found using Matlab R2020a’s symbolic math toolbox.

    Article Title: Analytical Dual Flip Angle R1 Calculation Outside the Small-Angle Regime.
    Article Snippet: 6 The [1,1] Padé approximant was calculated using pade from the MATLAB Symbolic Math Toolbox and then verified and simplified by hand.

    Article Title: Analytical Dual Flip Angle R1 Calculation Outside the Small‐Angle Regime
    Article Snippet: The [1,1] Padé approximant was calculated using pade from the MATLAB Symbolic Math Toolbox and then verified and simplified by hand.

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    Article Title: Hybrid Adaptive MPC with Edge AI for 6-DoF Industrial Robotic Manipulators
    Article Snippet: .. The discrete-time system matrices Ak ( θ̂k ) ∈ R12×12 and Bk ( θ̂k ) ∈ R12×6 are computed through Jacobian linearization of the nonlinear manipulator dynamics around the current operating point: Ak ( θ̂k ) = I + Ts[∂f/∂x]x = xk, θ = θ̂k, (5a) Bk ( θ̂k ) = Ts[∂f/∂u]x = xk, θ = θ̂k, (5b) where Ts = 0.01 s is the sampling period, f represents the continuous-time nonlinear dynamics, and the partial derivatives are computed using symbolic differentiation in MATLAB R2025a Symbolic Math Toolbox [35]. ..



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    ( a ) Droplet-like DPICs were initially formed by mixing 20 μM ATTO565-labeled p53 4M ΔTAD with 0.6 μM ATTO488-labeled Random DNA and incubating for 30 minutes at room temperature. Subsequently, Cy5-labeled p21 DNA was added at varying concentrations and incubated for an additional 120 minutes: (i) 0.15 μM; (ii) 0.225 μM; (iii) 0.3 μM; (iv) 0.45 μM; (v) 0.6 μM; (vi) 0.75 μM; (vii) 0.9 μM. Representative fluorescence images at incubation time t = 4-min and t = 120-min are shown. Independent in vitro droplet experiments were repeated three times (n = 3). ( b ) <t>Boxplot</t> of characteristic time constants τ 1 and τ 2 for p21 DNA concentrations ranging from 0.3 to 0.9 μM. N indicates the number of individual biomolecule-rich condensates analyzed under each condition. In box plots, the black line denotes the median, box edges represent the 25 th and 75 th percentiles, whiskers indicate the range excluding outliers, and outliers are shown as individual dots (•). ( c ) Phase diagram showing normalized fluorescence intensities of ATTO565-labeled p53 4M ΔTAD and ATTO488-labeled Random DNA at the center of condensates under increasing concentrations of p21 DNA (0.3, 0.45, 0.6, and 0.75 μM). Values are shown both before p21 DNA addition and at the end of Stage I. Control experiments in which Random DNA was used in place of p21 DNA are also included. Error bars indicate mean ± s.d. Green dashed lines mark the estimated binodal boundary, and purple dashed lines represent the spinodal boundary, as confirmed by our phase-field model (see – ).
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    ( a ) Droplet-like DPICs were initially formed by mixing 20 μM ATTO565-labeled p53 4M ΔTAD with 0.6 μM ATTO488-labeled Random DNA and incubating for 30 minutes at room temperature. Subsequently, Cy5-labeled p21 DNA was added at varying concentrations and incubated for an additional 120 minutes: (i) 0.15 μM; (ii) 0.225 μM; (iii) 0.3 μM; (iv) 0.45 μM; (v) 0.6 μM; (vi) 0.75 μM; (vii) 0.9 μM. Representative fluorescence images at incubation time t = 4-min and t = 120-min are shown. Independent in vitro droplet experiments were repeated three times (n = 3). ( b ) <t>Boxplot</t> of characteristic time constants τ 1 and τ 2 for p21 DNA concentrations ranging from 0.3 to 0.9 μM. N indicates the number of individual biomolecule-rich condensates analyzed under each condition. In box plots, the black line denotes the median, box edges represent the 25 th and 75 th percentiles, whiskers indicate the range excluding outliers, and outliers are shown as individual dots (•). ( c ) Phase diagram showing normalized fluorescence intensities of ATTO565-labeled p53 4M ΔTAD and ATTO488-labeled Random DNA at the center of condensates under increasing concentrations of p21 DNA (0.3, 0.45, 0.6, and 0.75 μM). Values are shown both before p21 DNA addition and at the end of Stage I. Control experiments in which Random DNA was used in place of p21 DNA are also included. Error bars indicate mean ± s.d. Green dashed lines mark the estimated binodal boundary, and purple dashed lines represent the spinodal boundary, as confirmed by our phase-field model (see – ).
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    ( a ) Droplet-like DPICs were initially formed by mixing 20 μM ATTO565-labeled p53 4M ΔTAD with 0.6 μM ATTO488-labeled Random DNA and incubating for 30 minutes at room temperature. Subsequently, Cy5-labeled p21 DNA was added at varying concentrations and incubated for an additional 120 minutes: (i) 0.15 μM; (ii) 0.225 μM; (iii) 0.3 μM; (iv) 0.45 μM; (v) 0.6 μM; (vi) 0.75 μM; (vii) 0.9 μM. Representative fluorescence images at incubation time t = 4-min and t = 120-min are shown. Independent in vitro droplet experiments were repeated three times (n = 3). ( b ) <t>Boxplot</t> of characteristic time constants τ 1 and τ 2 for p21 DNA concentrations ranging from 0.3 to 0.9 μM. N indicates the number of individual biomolecule-rich condensates analyzed under each condition. In box plots, the black line denotes the median, box edges represent the 25 th and 75 th percentiles, whiskers indicate the range excluding outliers, and outliers are shown as individual dots (•). ( c ) Phase diagram showing normalized fluorescence intensities of ATTO565-labeled p53 4M ΔTAD and ATTO488-labeled Random DNA at the center of condensates under increasing concentrations of p21 DNA (0.3, 0.45, 0.6, and 0.75 μM). Values are shown both before p21 DNA addition and at the end of Stage I. Control experiments in which Random DNA was used in place of p21 DNA are also included. Error bars indicate mean ± s.d. Green dashed lines mark the estimated binodal boundary, and purple dashed lines represent the spinodal boundary, as confirmed by our phase-field model (see – ).
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    ( a ) Droplet-like DPICs were initially formed by mixing 20 μM ATTO565-labeled p53 4M ΔTAD with 0.6 μM ATTO488-labeled Random DNA and incubating for 30 minutes at room temperature. Subsequently, Cy5-labeled p21 DNA was added at varying concentrations and incubated for an additional 120 minutes: (i) 0.15 μM; (ii) 0.225 μM; (iii) 0.3 μM; (iv) 0.45 μM; (v) 0.6 μM; (vi) 0.75 μM; (vii) 0.9 μM. Representative fluorescence images at incubation time t = 4-min and t = 120-min are shown. Independent in vitro droplet experiments were repeated three times (n = 3). ( b ) <t>Boxplot</t> of characteristic time constants τ 1 and τ 2 for p21 DNA concentrations ranging from 0.3 to 0.9 μM. N indicates the number of individual biomolecule-rich condensates analyzed under each condition. In box plots, the black line denotes the median, box edges represent the 25 th and 75 th percentiles, whiskers indicate the range excluding outliers, and outliers are shown as individual dots (•). ( c ) Phase diagram showing normalized fluorescence intensities of ATTO565-labeled p53 4M ΔTAD and ATTO488-labeled Random DNA at the center of condensates under increasing concentrations of p21 DNA (0.3, 0.45, 0.6, and 0.75 μM). Values are shown both before p21 DNA addition and at the end of Stage I. Control experiments in which Random DNA was used in place of p21 DNA are also included. Error bars indicate mean ± s.d. Green dashed lines mark the estimated binodal boundary, and purple dashed lines represent the spinodal boundary, as confirmed by our phase-field model (see – ).
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    ( a ) Droplet-like DPICs were initially formed by mixing 20 μM ATTO565-labeled p53 4M ΔTAD with 0.6 μM ATTO488-labeled Random DNA and incubating for 30 minutes at room temperature. Subsequently, Cy5-labeled p21 DNA was added at varying concentrations and incubated for an additional 120 minutes: (i) 0.15 μM; (ii) 0.225 μM; (iii) 0.3 μM; (iv) 0.45 μM; (v) 0.6 μM; (vi) 0.75 μM; (vii) 0.9 μM. Representative fluorescence images at incubation time t = 4-min and t = 120-min are shown. Independent in vitro droplet experiments were repeated three times (n = 3). ( b ) <t>Boxplot</t> of characteristic time constants τ 1 and τ 2 for p21 DNA concentrations ranging from 0.3 to 0.9 μM. N indicates the number of individual biomolecule-rich condensates analyzed under each condition. In box plots, the black line denotes the median, box edges represent the 25 th and 75 th percentiles, whiskers indicate the range excluding outliers, and outliers are shown as individual dots (•). ( c ) Phase diagram showing normalized fluorescence intensities of ATTO565-labeled p53 4M ΔTAD and ATTO488-labeled Random DNA at the center of condensates under increasing concentrations of p21 DNA (0.3, 0.45, 0.6, and 0.75 μM). Values are shown both before p21 DNA addition and at the end of Stage I. Control experiments in which Random DNA was used in place of p21 DNA are also included. Error bars indicate mean ± s.d. Green dashed lines mark the estimated binodal boundary, and purple dashed lines represent the spinodal boundary, as confirmed by our phase-field model (see – ).
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    ( a ) Droplet-like DPICs were initially formed by mixing 20 μM ATTO565-labeled p53 4M ΔTAD with 0.6 μM ATTO488-labeled Random DNA and incubating for 30 minutes at room temperature. Subsequently, Cy5-labeled p21 DNA was added at varying concentrations and incubated for an additional 120 minutes: (i) 0.15 μM; (ii) 0.225 μM; (iii) 0.3 μM; (iv) 0.45 μM; (v) 0.6 μM; (vi) 0.75 μM; (vii) 0.9 μM. Representative fluorescence images at incubation time t = 4-min and t = 120-min are shown. Independent in vitro droplet experiments were repeated three times (n = 3). ( b ) <t>Boxplot</t> of characteristic time constants τ 1 and τ 2 for p21 DNA concentrations ranging from 0.3 to 0.9 μM. N indicates the number of individual biomolecule-rich condensates analyzed under each condition. In box plots, the black line denotes the median, box edges represent the 25 th and 75 th percentiles, whiskers indicate the range excluding outliers, and outliers are shown as individual dots (•). ( c ) Phase diagram showing normalized fluorescence intensities of ATTO565-labeled p53 4M ΔTAD and ATTO488-labeled Random DNA at the center of condensates under increasing concentrations of p21 DNA (0.3, 0.45, 0.6, and 0.75 μM). Values are shown both before p21 DNA addition and at the end of Stage I. Control experiments in which Random DNA was used in place of p21 DNA are also included. Error bars indicate mean ± s.d. Green dashed lines mark the estimated binodal boundary, and purple dashed lines represent the spinodal boundary, as confirmed by our phase-field model (see – ).
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    ( a ) Droplet-like DPICs were initially formed by mixing 20 μM ATTO565-labeled p53 4M ΔTAD with 0.6 μM ATTO488-labeled Random DNA and incubating for 30 minutes at room temperature. Subsequently, Cy5-labeled p21 DNA was added at varying concentrations and incubated for an additional 120 minutes: (i) 0.15 μM; (ii) 0.225 μM; (iii) 0.3 μM; (iv) 0.45 μM; (v) 0.6 μM; (vi) 0.75 μM; (vii) 0.9 μM. Representative fluorescence images at incubation time t = 4-min and t = 120-min are shown. Independent in vitro droplet experiments were repeated three times (n = 3). ( b ) <t>Boxplot</t> of characteristic time constants τ 1 and τ 2 for p21 DNA concentrations ranging from 0.3 to 0.9 μM. N indicates the number of individual biomolecule-rich condensates analyzed under each condition. In box plots, the black line denotes the median, box edges represent the 25 th and 75 th percentiles, whiskers indicate the range excluding outliers, and outliers are shown as individual dots (•). ( c ) Phase diagram showing normalized fluorescence intensities of ATTO565-labeled p53 4M ΔTAD and ATTO488-labeled Random DNA at the center of condensates under increasing concentrations of p21 DNA (0.3, 0.45, 0.6, and 0.75 μM). Values are shown both before p21 DNA addition and at the end of Stage I. Control experiments in which Random DNA was used in place of p21 DNA are also included. Error bars indicate mean ± s.d. Green dashed lines mark the estimated binodal boundary, and purple dashed lines represent the spinodal boundary, as confirmed by our phase-field model (see – ).
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    ( a ) Droplet-like DPICs were initially formed by mixing 20 μM ATTO565-labeled p53 4M ΔTAD with 0.6 μM ATTO488-labeled Random DNA and incubating for 30 minutes at room temperature. Subsequently, Cy5-labeled p21 DNA was added at varying concentrations and incubated for an additional 120 minutes: (i) 0.15 μM; (ii) 0.225 μM; (iii) 0.3 μM; (iv) 0.45 μM; (v) 0.6 μM; (vi) 0.75 μM; (vii) 0.9 μM. Representative fluorescence images at incubation time t = 4-min and t = 120-min are shown. Independent in vitro droplet experiments were repeated three times (n = 3). ( b ) <t>Boxplot</t> of characteristic time constants τ 1 and τ 2 for p21 DNA concentrations ranging from 0.3 to 0.9 μM. N indicates the number of individual biomolecule-rich condensates analyzed under each condition. In box plots, the black line denotes the median, box edges represent the 25 th and 75 th percentiles, whiskers indicate the range excluding outliers, and outliers are shown as individual dots (•). ( c ) Phase diagram showing normalized fluorescence intensities of ATTO565-labeled p53 4M ΔTAD and ATTO488-labeled Random DNA at the center of condensates under increasing concentrations of p21 DNA (0.3, 0.45, 0.6, and 0.75 μM). Values are shown both before p21 DNA addition and at the end of Stage I. Control experiments in which Random DNA was used in place of p21 DNA are also included. Error bars indicate mean ± s.d. Green dashed lines mark the estimated binodal boundary, and purple dashed lines represent the spinodal boundary, as confirmed by our phase-field model (see – ).
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    ( a ) Droplet-like DPICs were initially formed by mixing 20 μM ATTO565-labeled p53 4M ΔTAD with 0.6 μM ATTO488-labeled Random DNA and incubating for 30 minutes at room temperature. Subsequently, Cy5-labeled p21 DNA was added at varying concentrations and incubated for an additional 120 minutes: (i) 0.15 μM; (ii) 0.225 μM; (iii) 0.3 μM; (iv) 0.45 μM; (v) 0.6 μM; (vi) 0.75 μM; (vii) 0.9 μM. Representative fluorescence images at incubation time t = 4-min and t = 120-min are shown. Independent in vitro droplet experiments were repeated three times (n = 3). ( b ) <t>Boxplot</t> of characteristic time constants τ 1 and τ 2 for p21 DNA concentrations ranging from 0.3 to 0.9 μM. N indicates the number of individual biomolecule-rich condensates analyzed under each condition. In box plots, the black line denotes the median, box edges represent the 25 th and 75 th percentiles, whiskers indicate the range excluding outliers, and outliers are shown as individual dots (•). ( c ) Phase diagram showing normalized fluorescence intensities of ATTO565-labeled p53 4M ΔTAD and ATTO488-labeled Random DNA at the center of condensates under increasing concentrations of p21 DNA (0.3, 0.45, 0.6, and 0.75 μM). Values are shown both before p21 DNA addition and at the end of Stage I. Control experiments in which Random DNA was used in place of p21 DNA are also included. Error bars indicate mean ± s.d. Green dashed lines mark the estimated binodal boundary, and purple dashed lines represent the spinodal boundary, as confirmed by our phase-field model (see – ).
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    ( a ) Droplet-like DPICs were initially formed by mixing 20 μM ATTO565-labeled p53 4M ΔTAD with 0.6 μM ATTO488-labeled Random DNA and incubating for 30 minutes at room temperature. Subsequently, Cy5-labeled p21 DNA was added at varying concentrations and incubated for an additional 120 minutes: (i) 0.15 μM; (ii) 0.225 μM; (iii) 0.3 μM; (iv) 0.45 μM; (v) 0.6 μM; (vi) 0.75 μM; (vii) 0.9 μM. Representative fluorescence images at incubation time t = 4-min and t = 120-min are shown. Independent in vitro droplet experiments were repeated three times (n = 3). ( b ) Boxplot of characteristic time constants τ 1 and τ 2 for p21 DNA concentrations ranging from 0.3 to 0.9 μM. N indicates the number of individual biomolecule-rich condensates analyzed under each condition. In box plots, the black line denotes the median, box edges represent the 25 th and 75 th percentiles, whiskers indicate the range excluding outliers, and outliers are shown as individual dots (•). ( c ) Phase diagram showing normalized fluorescence intensities of ATTO565-labeled p53 4M ΔTAD and ATTO488-labeled Random DNA at the center of condensates under increasing concentrations of p21 DNA (0.3, 0.45, 0.6, and 0.75 μM). Values are shown both before p21 DNA addition and at the end of Stage I. Control experiments in which Random DNA was used in place of p21 DNA are also included. Error bars indicate mean ± s.d. Green dashed lines mark the estimated binodal boundary, and purple dashed lines represent the spinodal boundary, as confirmed by our phase-field model (see – ).

    Journal: bioRxiv

    Article Title: Hollow condensates emerge from gelation-induced spinodal decomposition

    doi: 10.1101/2025.06.25.661497

    Figure Lengend Snippet: ( a ) Droplet-like DPICs were initially formed by mixing 20 μM ATTO565-labeled p53 4M ΔTAD with 0.6 μM ATTO488-labeled Random DNA and incubating for 30 minutes at room temperature. Subsequently, Cy5-labeled p21 DNA was added at varying concentrations and incubated for an additional 120 minutes: (i) 0.15 μM; (ii) 0.225 μM; (iii) 0.3 μM; (iv) 0.45 μM; (v) 0.6 μM; (vi) 0.75 μM; (vii) 0.9 μM. Representative fluorescence images at incubation time t = 4-min and t = 120-min are shown. Independent in vitro droplet experiments were repeated three times (n = 3). ( b ) Boxplot of characteristic time constants τ 1 and τ 2 for p21 DNA concentrations ranging from 0.3 to 0.9 μM. N indicates the number of individual biomolecule-rich condensates analyzed under each condition. In box plots, the black line denotes the median, box edges represent the 25 th and 75 th percentiles, whiskers indicate the range excluding outliers, and outliers are shown as individual dots (•). ( c ) Phase diagram showing normalized fluorescence intensities of ATTO565-labeled p53 4M ΔTAD and ATTO488-labeled Random DNA at the center of condensates under increasing concentrations of p21 DNA (0.3, 0.45, 0.6, and 0.75 μM). Values are shown both before p21 DNA addition and at the end of Stage I. Control experiments in which Random DNA was used in place of p21 DNA are also included. Error bars indicate mean ± s.d. Green dashed lines mark the estimated binodal boundary, and purple dashed lines represent the spinodal boundary, as confirmed by our phase-field model (see – ).

    Article Snippet: The function of “boxplot” in MATLAB software (R2015a, 64-bit, February 12, 2015) was used to plot the boxplots in , , and Supplementary Fig. 3.

    Techniques: Labeling, Incubation, Fluorescence, In Vitro, Control