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Based on the numerical simulation of the three-dimensional computer model of the brain, mathematical models of the parameters of potentials, such as resistance, capacitance and resistive capacity in the extracellular space are given. Development of the computer model of the brain is characterized by a high temporal accuracy. The computer model of the brain is based on the discrete finite element method. The results of the numerical simulation of the model of the brain agree qualitatively with the experimental data. The analysis of the temporal characteristics of CSD waves allows the following conclusions: the morphological features of the CSD-wave form depend on the initial geometric parameters of the model, the capacity of extracellular space (low density of interelectrode spaces between the axons) is the main factor of the delay in the spreading of the potential; the initial minimum value of potential is determined by the density of the extracellular space. The mathematical model of the potentials has a good resistance to disturbances in experimental conditions. It allows one to estimate the effects of many factors on the spreading of the potential. Using the proposed mathematical analysis of the potentials, the extracellular space and density, the density of neurons can be estimated.Stochastic Differential Equation Modeling

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As I’ve been using the conventional wisdom — we live in a hyper-partisan, hyper-polarized world — to estimate how much Donald Trump’s legitimacy as president has been undermined, I ran across an article written by a political scientist. Philip E. Converse, an American political scientist, and his nephew Steven Brams, a Canadian, make a historical argument about the need to understand turnout and polarization as different factors in the electoral process. The Converse-Brams study actually looks at the whole history of elections in the United States, except for 1864, and concludes that the differences between the election of 1864, where African American turnout was low due to the Civil War, and the election of 1912, when it was high, are due in large part to a pattern of unequal voter turnout in different states, which was exacerbated by the adoption of the national electoral college system in 1872.

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$H$ is Hermitian $implies M^H=M$

How to prove that a Hermitian $M in mathbb{R}^{ntimes n}$ implies $M^H=M$
Thank you very much for your help

A:

Since $M^H = (M^*M)^{1/2}$ and $M^*M$ is positive definite, it follows that $M^H$ is positive semidefinite. So $(M^H)^{ -1/2}$ is well-defined. Moreover, $M^H = (M^H)^{ -1/2}(M^H)^{1/2}$. It follows that $M^H$ is invertible, and that $(M^H)^{ -1} = (M^H)^{1/2}(M^*)^{1/2}$.

A:

Suppose $M$ is a hermitian matrix and $M^*=M$. Since $M$ is hermitian, $bar{x}^TMbar{x}=bar{x}^Tbar{x}$ for all $bar{x}in Bbb R^n$. Since $M^*=M$,
$bar{x}^TMbar{x}=bar{x}^Tbar{x}=||bar{x}||_2^2=||bar{x}||_1^2$, so $x^TMx=0$ for all $xin Bbb R^n$, i.e. $M^T=0$.

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