Closure of dilates of shift-invariant subspaces

Let $V$ be any shift-invariant subspace of square summable functions. We prove that if for some $A$ expansive dilation $V$ is $A$-refinable, then the completeness property is equivalent to several conditions on the local behaviour at the origin of the spectral function of $V$, among them the origin is a point of $A^*$-approximate continuity of the spectral function if we assume this value to be one. We present our results also in the more general setting of $A$-reducing spaces. We also prove that the origin is a point of $A^*$-approximate continuity of the Fourier transform of any semiorthogonal tight frame wavelet if we assume this value to be zero

Estudio de las propiedades de aproximación de espacios invariantes por traslaciones a través de la función espectral

Tesis Doctoral inédita. Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura 18-06-201

Vector-borne disease risk indexes in spatially structured populations

There are economic and physical limitations when applying prevention and control strategies for urban vector borne diseases. Consequently, there are increasing concerns and interest in designing efficient strategies and regulations that health agencies can follow in order to reduce the imminent impact of viruses like Dengue, Zika and Chikungunya. That includes fumigation, abatization, reducing the hatcheries, picking up trash, information campaigns. A basic question that arise when designing control strategies is about which and where these ones should focus. In other words, one would like to know whether preventing the contagion or decrease vector population, and in which area of the city, is more efficient. In this work, we propose risk indexes based on the idea of secondary cases from patch to patch. Thus, they take into account human mobility and indicate which patch has more chance to be a corridor for the spread of the disease and which is more vulnerable. They can also indicate the neighborhood where hatchery control will reduce more the number of potential cases. In order to illustrate the usefulness of these indexes, we run a set of numerical simulations in a mathematical model that takes into account the urban mobility and the differences in population density among the areas of a city. If i is a particular neighborhood, the transmission risk index TR_i measures the potential secondary cases caused by a host in that neighborhood. The vector transmission risk index VTR_i measures the potential secondary cases caused by a vector. Finally, the vulnerability risk index VR_i measures the potential secondary cases in the neighborhood. Transmission indexes can be used to give geographical priority to some neighborhoods when applying prevention and control measures. On the other hand, the vulnerability index can be useful to implement monitoring campaigns or public health investment.Comment: 16 pages, 5 figure

Anisotropic dilations of shift-invariant subspaces and approximation properties in L2(Rd)

Let A be an expansive linear map in Rd. Approximation properties of shift-invariant subspaces of L2(Rd) when they are dilated by integer powers of A are studied. Shift-invariant subspaces providing approximation order α or density order α associated to A are characterized. These characterizations impose certain restrictions on the behavior of the spectral function at the origin expressed in terms of the concept of point of approximate continuity. The notions of approximation order and density order associated to an isotropic dilation turn out to coincide with the classical ones introduced by de Boor, DeVore and Ron. This is no longer true when A is anisotropic. In this case the A-dilated shift-invariant subspaces approximate the anisotropic Sobolev space associated to A and α. Our main results are also new when S is generated by translates of a single function. The obtained results are illustrated by some examplesThe first author is supported in part by MEC/MICINN grants MTM2010-16518 and MTM2013-40945-P (Spain). The second author was partially supported by MEC/MICINN grant MTM2011-27998 (Spain). The third author is supported in part by MEC/MlClNN grant MTM2013-40945-P (Spain)

Description and estimated value of the model parameters ([30, 31]).

<p>Description and estimated value of the model parameters ([<a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0006234#pntd.0006234.ref030" target="_blank">30</a>, <a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0006234#pntd.0006234.ref031" target="_blank">31</a>]).</p

Time series of the model (8)–(12) for different values of the carrying capacity <i>C</i>, and the parameters values given in Table 1 with <i>β</i> = 0.67.

<p>Time series of the model <a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0006234#pntd.0006234.e020" target="_blank">(8)</a>–<a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0006234#pntd.0006234.e024" target="_blank">(12)</a> for different values of the carrying capacity <i>C</i>, and the parameters values given in <a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0006234#pntd.0006234.t001" target="_blank">Table 1</a> with <i>β</i> = 0.67.</p

Schematic draw of a metapopulation network with human mobility.

<p>At the right we represent the measure of the Transmission Risk index <a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0006234#pntd.0006234.e008" target="_blank">(2)</a> and, at the left, we represent the Vulnerability Risk index <a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0006234#pntd.0006234.e010" target="_blank">(4)</a>.</p

Dynamics of the infected population system-wide with control applied in different patches.

<p>(a) In a fully connected network the vector transmission index <i>VTR</i>_<i>i</i> is greater in patch 2. (b) In a Barabasi-Albert network the patch with largest vector transmissibility is number 4 as indicated by its <i>VTR</i>_<i>i</i>. The most effective strategy at the beginning of the outbreak is to reduce the carrying capacity in the patch with greatest <i>VTR</i>_<i>i</i>.</p

Dynamics of the number of infected humans per patch in a network of 5 nodes for a fully connected network (a) and a Barabasi-Albert network (b).

<p>In (a) the vulnerability index <i>VR</i>_<i>i</i> indicates patch 3 (red line) as the most vulnerable and in (b) the most vulnerable as indicated by <i>VR</i>_<i>i</i> is patch 1 (red line). The most vulnerable patch corresponds to the one with more cases at the beginning of the epidemic (let say <i>t</i> ≤ 50).</p

Carrying capacity <i>C</i>_<i>i</i> and human population <i>N</i>_<i>hi</i> used in the simulations within each patch.

<p><i>R</i>₀ if patches where isolated and risk index values for two types of networks.</p