to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

The mTOR Complex 1 (mTORC1) protein kinase promotes growth and is the target of rapamycin, a clinically useful drug that also prolongs lifespan in model organisms A persistent mystery is why the phosphorylation of many bona fide mTORC1 substrates is resistant to rapamycin We find that the in vitro kinase activity of mTORC1 toward peptides encompassing established phosphorylation sites varies widely and correlates strongly with the resistance of the sites to rapamycin as well as to nutrient and growth factor starvation within cells Slight modifications of the sites were sufficient to alter mTORC1 activity toward them in vitro and to cause concomitant changes within cells in their sensitivity to rapamycin and starvation Thus, the intrinsic capacity of a phosphorylation site to serve as an mTORC1 substrate, a property we call substrate quality, is a major determinant of its sensitivity to modulators of the pathway Our results reveal a mechanism through which mTORC1 effectors can respond differentially to the same signals

mTORC1 Phosphorylation Sites Encode Their Sensitivity to Starvation and Rapamycin

Cross-sectional imaging of an electrical conductivity distribution inside the human body has been an active research goal in impedance imaging. By injecting current into an electrically conducting object through surface electrodes, we induce current density and voltage distributions. Based on the fact that these are determined by the conductivity distribution as well as the geometry of the object and the adopted electrode configuration, electrical impedance tomography (EIT) reconstructs cross-sectional conductivity images using measured current-voltage data on the surface. Unfortunately, there exist inherent technical difficulties in EIT. First, the relationship between the boundary current-voltage data and the internal conductivity distribution bears a nonlinearity and low sensitivity, and hence the inverse problem of recovering the conductivity distribution is ill posed. Second, it is difficult to obtain accurate information on the boundary geometry and electrode positions in practice, and the inverse problem is sensitive to these modeling errors as well as measurement artifacts and noise. These result in EIT images with a poor spatial resolution. In order to produce high-resolution conductivity images, magnetic resonance electrical impedance tomography (MREIT) has been lately developed. Noting that injection current produces a magnetic as well as electric field inside the imaging object, we can measure the induced internal magnetic flux density data using an MRI scanner. Utilization of the internal magnetic flux density is the key idea of MREIT to overcome the technical difficulties in EIT. Following original ideas on MREIT in early 1990s, there has been a rapid progress in its theory, algorithm and experimental techniques. The technique has now advanced to the stage of human experiments. Though it is still a few steps away from routine clinical use, its potential is high as a new impedance imaging modality providing conductivity images with a spatial resolution of a few millimeters or less. This paper reviews MREIT from the basics to the most recent research outcomes. Focusing on measurement techniques and experimental methods rather than mathematical issues, we summarize what has been done and what needs to be done. Suggestions for future research directions, possible applications in biomedicine, biology, chemistry and material science are discussed.

https://iopscience.iop.org/article/10.1088/0967-3334/29/10/R01/pdf

Magnetic resonance electrical impedance tomography (MREIT) for high-resolution conductivity imaging

This paper presents a new algorithm for conductivity imaging. Our idea is to extract more information about the conductivity distribution from data that have been enriched by coupling impedance electrical measurements to localized elastic perturbations. Using asymptotics of the fields in the presence of small volume inclusions, we relate the pointwise values of the energy density to the measured data through a nonlinear PDE. Our algorithm is based on this PDE and takes full advantage of the enriched data. We give numerical examples that illustrate the performance and the accuracy of our approach.

/pdf/electrical-impedance-tomography-by-elastic-deformation-27132osihw.pdf

Electrical Impedance Tomography by Elastic Deformation

We propose a simple numerical method for capturing the steady state solution of hyperbolic systems with geometrical source terms. We use the interface value, rather than the cell-averages, for the source terms that balance the nonlinear convection at the cell interface, allowing the numerical capturing of the steady state with a formal high order accuracy. This method applies to Godunov or Roe type upwind methods but requires no modification of the Riemann solver. Numerical experiments on scalar conservation laws and the one dimensional shallow water equations show much better resolution of the steady state than the conventional method, with almost no new numerical complexity.

https://www.esaim-m2an.org/articles/m2an/pdf/2001/04/m2an0095.pdf

A steady-state capturing method for hyperbolic systems with geometrical source terms

We are concerned with the following density-suppressed motility model: $u_t=\Delta (\gamma(v) u)+\mu u(1-u); v_t=\Delta v+ u-v,$ in a bounded smooth domain $\Omega\subset \mathbb{R}^2$ with homogen...

Boundedness, stabilization, and pattern formation driven by density-suppressed motility

The global existence and the instability of constant steady states are obtained together for a Keller-Segel type chemotactic aggregation model. Organisms are assumed to change their motility depending only on the chemical density but not on its gradient. However, the resulting model is closely related to the logarithmic model, ut=Δ(ź(v)u)=źź(ź(v)(źuźkvuźv)),vt=źΔvźv+u, $$\begin{aligned} u_{t}=\Delta \bigl(\gamma (v)u\bigr)=\nabla \cdot \biggl(\gamma (v) \biggl(\nabla u- \frac{k}{v}u\nabla v \biggr) \biggr),\quad v_{t}={\varepsilon }\Delta v-v+u, \end{aligned}$$ where ź(v):=c0vźk$\gamma (v):=c_{0}v^{-k}$ is the motility function. The global existence is shown for all chemosensitivity constant k>0$k>0$ with a smallness assumption on c0>0$c_{0}>0$ . On the other hand constant steady states are shown to be unstable only if k>1$k>1$ and ź>0${\varepsilon }>0$ is small. Furthermore, the threshold diffusivity ź1>0${\varepsilon }_{1}>0$ is found that, if ź<ź1${\varepsilon }<{\varepsilon }_{1}$, any constant steady state is unstable and an aggregation pattern appears. Numerical simulations are given for radial cases.

Global Existence and Aggregation in a Keller---Segel Model with Fokker---Planck Diffusion

Magnetic resonance electrical impedance tomography (MREIT) is a new medical imaging modality providing high resolution conductivity images based on the current injection MRI technique. In contrast to electrical impedance tomography (EIT), the MREIT system utilizes the internal information of current density distribution which plays an important role in eliminating the ill-posedness of the inverse problem in EIT. It has been shown that the J-substitution algorithm in MREIT reconstructs conductivity images with higher spatial resolution. However, fundamental mathematical questions, including the uniqueness of the MREIT problem itself and the convergence of the algorithm, have not yet been answered. This paper provides a rigorous proof of the uniqueness of the MREIT problem and analyses the convergence behaviour of the J-substitution algorithm.

/pdf/uniqueness-and-convergence-of-conductivity-image-3j6un5svdf.pdf

Uniqueness and convergence of conductivity image reconstruction in magnetic resonance electrical impedance tomography

http://amath.kaist.ac.kr/papers/Kim/23.pdf

Starvation driven diffusion as a survival strategy of biological organisms.

The global existence of a chemotaxis model for cell aggregation phenomenon is obtained. The model system belongs to the class of logarithmic models and takes a Fokker–Planck type diffusion for the equation of cell density. We show that weak solutions exist globally in time in dimensions n ∈ { 1 , 2 , 3 } and for large initial data. The proof covers the parameter regimes that constant steady states are linearly stable. It also partially covers the other parameter regimes that constant steady states are unstable. We also find the sharp instability condition of constant steady states and provide numerical simulations which illustrate the formation of aggregation patterns.

Yong-Jung Kim

Papers

Boundedness, stabilization, and pattern formation driven by density-suppressed motility

Global Existence and Aggregation in a Keller---Segel Model with Fokker---Planck Diffusion

Uniqueness and convergence of conductivity image reconstruction in magnetic resonance electrical impedance tomography

Starvation driven diffusion as a survival strategy of biological organisms.

A logarithmic chemotaxis model featuring global existence and aggregation