Homogenization of linear transport equations. A new approach

The paper is devoted to a new approach of the homogenization of linear transport equations induced by a uniformly bounded sequence of vector fields $b_\epsilon(x)$, the solutions of which $u_\epsilon(t,x)$ agree at $t=0$ with a bounded sequence of $L^p_{\rm loc}(\mathbb{R}^N)$ for some $p\in(1,\infty)$. Assuming that the sequence $b_\epsilon\cdot\nabla w_\epsilon^1$ is compact in $L^q_{\rm loc}(\mathbb{R}^N)$ ($q$ conjugate of $p$) for some gradient field $\nabla w_\epsilon^1$ bounded in $L^N_{\rm loc}(\mathbb{R}^N)^N$, and that there exists a uniformly bounded sequence $\sigma_\epsilon>0$ such that $\sigma_\epsilon\,b_\epsilon$ is divergence free if $N\!=\!2$ or is a cross product of $(N\!-\!1)$ bounded gradients in $L^N_{\rm loc}(\mathbb{R}^N)^N$ if $N\!\geq\!3$, we prove that the sequence $\sigma_\epsilon\,u_\epsilon$ converges weakly to a solution to a linear transport equation. It turns out that the compactness of $b_\epsilon\cdot\nabla w_\epsilon^1$ is a substitute to the ergodic assumption of the classical two-dimensional periodic case, and allows us to deal with non-periodic vector fields in any dimension. The homogenization result is illustrated by various and general examples

Magneto-resistance in three-dimensional composites

In this paper we study the magneto-resistance, i.e. the second-order term of the resistivity perturbed by a low magnetic field, of a three-dimensional composite material. Extending the two-dimensional periodic framework of [4], it is proved through a H-convergence approach that the dissipation energy induced by the effective magneto-resistance is greater or equal to the average of the dissipation energy induced by the magneto-resistance in each phase of the composite. This inequality validates for a composite material the Kohler law which is known for a homogeneous conductor. The case of equality is shown to be very sensitive to the magnetic fi eld orientation. We illustrate the result with layered and columnar periodic structures.Comment: 28 page

Isotropic realizability of current fields in R^3

This paper deals with the isotropic realizability of a given regular divergence free field j in R^3 as a current field, namely to know when j can be written as sigma Du for some isotropic conductivity sigma, and some gradient field Du. The local isotropic realizability in R^3 is obtained by Frobenius' theorem provided that j and curl j are orthogonal in R^3. A counter-example shows that Frobenius' condition is not sufficient to derive the global isotropic realizability in R^3. However, assuming that (j, curl j, j x curl j) is an orthogonal basis of R^3, an admissible conductivity sigma is constructed from a combination of the three dynamical flows along the directions j/|j|, curl j/|curl j| and (j/|j|^2) x curl j. When the field j is periodic, the isotropic realizability in the torus needs in addition a boundedness assumption satisfied by the flow along the third direction (j/|j|^2) x \curl j. Several examples illustrate the sharpness of the realizability conditions.Comment: 22 page

A new div-curl result. Applications to the homogenization of elliptic systems and to the weak continuity of the Jacobian

In this paper a new div-curl result is established in an open set $\Omega$ of $\mathbb{R}^N$, $N\geq 2$, for the product of two sequences of vector-valued functions which are bounded respectively in $L^p(\Omega)^N$ and $L^q(\Omega)^N$, with ${1/p}+{1/q}=1+{1/(N-1)}$, and whose respectively divergence and curl are compact in suitable spaces. We also assume that the product converges weakly in $W^{-1,1}(\Omega)$. The key ingredient of the proof is a compactness result for bounded sequences in $W^{1,q}(\Omega)$, based on the imbedding of $W^{1,q}(S\_{N-1})$ into $L^{p'}(S\_{N-1})$ ($S\_{N-1}$ the unit sphere of $\mathbb{R}^N$) through a suitable selection of annuli on which the gradients are not too high, in the spirit of De Giorgi and Manfredi. The div-curl result is applied to the homogenization of equi-coercive systems whose coefficients are equi-bounded in $L^\rho(\Omega)$ for some \rho\textgreater{}{N-1\over 2} if N\textgreater{}2, or in $L^1(\Omega)$ if $N=2$. It also allows us to prove a weak continuity result for the Jacobian for bounded sequences in $W^{1,N-1}(\Omega)$ satisfying an alternative assumption to the $L^\infty$-strong estimate of Brezis and Nguyen. Two examples show the sharpness of the results

A statistical analysis of particle trajectories in living cells

Recent advances in molecular biology and fluorescence microscopy imaging have made possible the inference of the dynamics of single molecules in living cells. Such inference allows to determine the organization and function of the cell. The trajectories of particles in the cells, computed with tracking algorithms, can be modelled with diffusion processes. Three types of diffusion are considered : (i) free diffusion; (ii) subdiffusion or (iii) superdiffusion. The Mean Square Displacement (MSD) is generally used to determine the different types of dynamics of the particles in living cells (Qian, Sheetz and Elson 1991). We propose here a non-parametric three-decision test as an alternative to the MSD method. The rejection of the null hypothesis -- free diffusion -- is accompanied by claims of the direction of the alternative (subdiffusion or a superdiffusion). We study the asymptotic behaviour of the test statistic under the null hypothesis, and under parametric alternatives which are currently considered in the biophysics literature, (Monnier et al,2012) for example. In addition, we adapt the procedure of Benjamini and Hochberg (2000) to fit with the three-decision test setting, in order to apply the test procedure to a collection of independent trajectories. The performance of our procedure is much better than the MSD method as confirmed by Monte Carlo experiments. The method is demonstrated on real data sets corresponding to protein dynamics observed in fluorescence microscopy.Comment: Revised introduction. A clearer and shorter description of the model (section 2

On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with elasticity tensors \BC_1>0 and \BC_2=0 comprising the set U=\{\BC_1,\BC_2\} and mixed in proportions $f$ and $1-f$ is partly characterized. The material with tensor \BC_2=0 corresponds to a material which is void. (For technical reasons \BC_2 is actually taken to be nonzero and we take the limit \BC_2\to 0). Specifically, recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is easily compliant to $p\leq 5$ independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains "complementary Avellaneda material" which is a hierarchical laminate which minimizes the sum of complementary energies.Comment: 39 pages, 11 figure

Realizable response matrices of multiterminal electrical, acoustic, and elastodynamic networks at a given frequency

We give a complete characterization of the possible response matrices at a fixed frequency of n-terminal electrical networks of inductors, capacitors, resistors and grounds, and of n-terminal discrete linear elastodynamic networks of springs and point masses, both in the three-dimensional case and in the two-dimensional case. Specifically we construct networks which realize any response matrix which is compatible with the known symmetry properties and thermodynamic constraints of response matrices. Due to a mathematical equivalence we also obtain a characterization of the response matrices of discrete acoustic networks.Comment: 22 pages, 5 figure

Understanding how Notch influences the development and fate of the hemogenic endothelium using genetic mosaics

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Medicina, Departamento de Bioquímica. Fecha de lectura: 18-10-2019Esta tesis tiene embargado el acceso al texto completo hasta el 18-04-2021Notch is an important signaling pathway in the intraembryonic hematopoietic wave that occurs between E9.5-E11.5. Although most studies so far have claimed that its activity in endothelial cells (ECs) of the dorsal aorta (DA) is necessary for hematopoiesis, more recent evidence has suggested that Notch signaling must be downregulated for endothelialto-hematopoietic transition (EHT) to occur. However, the exact molecular and cellular mechanisms are still not well characterized. In this thesis we have used a wide range of novel genetic tools and imaging approaches to analyze the role of Notch in the EHT process with higher cellular, temporal, and molecular resolution. Our results show that embryos with increased Notch activity in the dorsal endothelium at E10.5 have a decrease in c-Kit+ hematopoietic stem progenitor cells (HSPCs). This is caused by a decrease in hemogenic specification, or CD31+/Runx1+ cells. Additionally, HSPCs with high Notch activation presented proliferation defects as they tended to form unicellular clusters. We also found that embryos with loss of Jag1 in DA ECs had higher Dll4/Notch activity, which may explain why the loss of this Notch ligand also induces a decrease in hematopoiesis. This differential Jagged1/Dll4 Notch activity could be due to the known inhibitory role of Fringes on Jagged1/Notch signaling. Indeed overexpression of Manic Fringe in DA ECs resulted in a decrease in Dll4/Notch signaling and an increased number of HSPCs in the DA. To determine the single cell-autonomous role of Notch during EHT, we induced Notch genetic mosaics in individual DA ECs. Single ECs with Notch loss-of-function (LOF) undergo EHT with more frequency compared to wildtype ECs. Comparative transcriptional profiling revealed that they had similar arterial endothelial identity (Sox17, Cx40) and hematopoietic progenitor (CD41) expression, however, we found that Myc and Mycn expression was significantly deregulated. Analysis of embryos with Myc and Mycn deletion in DA ECs surprisingly revealed that Mycn is a strong regulator of EHT, whereas Myc is dispensable. Mycn mutants presented a drastic loss of HSPCs in the DA whereas Myc mutants only displayed a late hematopoiesis progression phenotype. Analysis of recently published single cell transcriptomic data allowed us to validate several of the experimental findings described above and to propose a model for the regulation of EHT by Notch. During EHT, most DA ECs have high Notch signaling induced by Jagged1 and Dll4 ligands and they are quiescent (KI67-). Among these cells, some upregulate Mfng, which decreases the signaling ability of the surrounding Jagged1 ligands and lowers Notch signaling cell-autonomously. These cells with lower Notch signaling upregulate Mycn, which is necessary to induce EHT. The Mycn induced EHT is associated with an entry into cell cycle which is subsequently maintained by the expression of the homologous gene Myc in the HSPCs.Notch es una vía de señalización importante en la onda hematopoyética intraembrionaria que se produce entre E9.5-E11.5. Aunque se especula que la señalización de Notch baja en las células endoteliales (EC) para que ocurra la transición endotelial-a-hematopoyética (EHT), el mecanismo molecular debajo de Notch en la aorta dorsal (DA) aún no está bien caracterizado. En este tesis, hemos utilizado una amplia gama de herramientas genéticas novedosas.para analizar el papel de Notch en el proceso EHT con mayor resolución celular, temporal, y molecular. Nuestro resultados muestran que en en los embriones con mayor actividad de Notch en el endotelio dorsal tenían defectos en el sistema hematopoyético definitivo en la aorta gónada mesonefros (AGM) en embriones de E10.5. Vimos una pérdida prejudicial de células madres progenitoras hematopoyético c-Kit+ (HSPC) en embriones con alta señalización de Notch. Los embriones con alta señalización de Notch tenían menos especificación hemogénica y un pequeño porcentaje de células endoteliales con alta activación celular autónoma de Notch también eran Runx1+. Además, las HSPCs con alta activación de Notch presentan defectos de proliferación. También hemos visto que en los mutantes Jag1 habia una alta tinción de Dll4/Notch1 y tiene un fenotipo similar a los embriones con alta señalización de Notch. Nosotros creíamos que los resultos implican un papel para Fringe. En efecto, sobreexpresión de Manic Fringe en el aorta bajan los niveles de la señalización de Dll4/Notch y suben los números de HSPCs en el aorta dorsal. Utilizando mosaico genéticos de ratones para investigar el papel de Notch en el endotelio hemogénico (HE), hemos determinado que las ECs con pérdida de función de Notch (LOF) se someten a EHT con más frecuencia en comparación con las ECs de controles. Al asilar estas células, hemos encontrado que, aunque tenían una expresión de endotelio arterial (Sox17, CX40) y de progenitor hematopoyético (CD41+) similar a las ECs de controles, tanto Myc como Mycn se modificaron drásticamente. Utilizamos los mutantes Myc y Mycn para determinar si los dos factores de trascripción desempeñaron un papel en el desarrollo temprano del sistema hematopoyético. Los mutantes de Mycn presentaron una pérdida drástica de HSPCs en la aorta, mientras mutantes de Myc tenían defectos hematopoyéticos en adultos. El análisis de los datos transcriptómicos de las células del aorta, tanto las HSPCs como las endoteliales, nos ayuda en proponer un modelo para la regulación de EHT por Notch. Durante EHT, la mayoría de las células del aorta dorsal tienen alta señalización de Notch inducido por los ligandos Jagged1 y Dll4 y bajo niveles de proliferación (Ki67-). Dentro de estas células, algunos suben la expresión de Manic Fringe y después bajan los niveles de Notch en la célula de una forma autónoma. Estas células luego suben los niveles de Mycn, lo cual es necesaria para inducer EHT. El EHT inducido por Mycn se asocia con una entrada en el ciclo celular que posteriormente se mantiene mediante la expresión del gen homólogo Myc en las HSPCs