|
@@ -122,6 +122,18 @@
|
|
|
<h3>Submitted</h3>
|
|
|
<ol reversed>
|
|
|
<li>
|
|
|
+ <a class="anchor" id="Blasten-Isozaki-Lassas-Lu_2021"></a>
|
|
|
+ <span class="ref-authors">E. Blåsten, H. Isozaki,
|
|
|
+ M. Lassas, J. Lu,</span>
|
|
|
+ <span class="ref-title">The Gel'fand's inverse problem
|
|
|
+ for the graph Laplacian.</span>
|
|
|
+ <div class="ref-links">
|
|
|
+ <a href="#desc_Blasten-Isozaki-Lassas-Lu_2021">Description</a>
|
|
|
+ <a href="https://arxiv.org/abs/2101.10026">Preprint</a>
|
|
|
+ </div>
|
|
|
+ </li>
|
|
|
+
|
|
|
+ <li>
|
|
|
<a class="anchor" id="Blasten-Liu-Xiao-2019"></a>
|
|
|
<span class="ref-authors">E. Blåsten, H. Liu, J. Xiao,</span>
|
|
|
<span class="ref-title">On an electromagnetic problem
|
|
@@ -581,6 +593,51 @@
|
|
|
<dl>
|
|
|
<dt>
|
|
|
<a class="anchor"
|
|
|
+ id="desc_Blasten-Isozaki-Lassas-Lu_2021"></a>
|
|
|
+ <a href="#Blasten-Isozaki-Lassas-Lu_2021">
|
|
|
+ <span class="ref-authors">E. Blåsten, H. Isozaki,
|
|
|
+ M. Lassas, J. Lu,</span>
|
|
|
+ <span class="ref-title">The Gel'fand's inverse problem
|
|
|
+ for the graph Laplacian.</span>
|
|
|
+ </a>
|
|
|
+ </dt>
|
|
|
+ <dd>
|
|
|
+ <p>
|
|
|
+ We consider the graph Laplacian, a discrete operator
|
|
|
+ defined on simple graphs, and its spectrum and
|
|
|
+ eigenfunctions on a weighted simple graph. We prove
|
|
|
+ that knowing the spectrum and Neumann boundary data of
|
|
|
+ the associated Neumann eigenfunctions determines the
|
|
|
+ structure of the graph and also the unknown weights
|
|
|
+ and potential function for a class of graph that
|
|
|
+ satisfy a so-called "two-points condition."
|
|
|
+ </p>
|
|
|
+ <p>
|
|
|
+ The above assumes that the graph has
|
|
|
+ a <em>boundary</em>. This is simply a subset of the
|
|
|
+ graph's vertices. For our result, it cannot be an
|
|
|
+ arbitrary subset, but must be "large enough". It must
|
|
|
+ at least be a <em>resolving set</em> for the
|
|
|
+ graph. This means that no two different points in the
|
|
|
+ graph have the exact same distances to every single
|
|
|
+ boundary point.
|
|
|
+ </p>
|
|
|
+ <p>
|
|
|
+ The two-point condition means that any subset of
|
|
|
+ interior vertices that has at least two elements must
|
|
|
+ have two <em>extreme points</em>. An extreme point of
|
|
|
+ a set <i>S</i> is a point which is the unique closest
|
|
|
+ point of <i>S</i> to some boundary point.
|
|
|
+ </p>
|
|
|
+ <p>
|
|
|
+ Cases which satisfy our assumptions include standard
|
|
|
+ lattices and their perturbations. In particular loops
|
|
|
+ are not a problem for our method.
|
|
|
+ </p>
|
|
|
+ </dd>
|
|
|
+
|
|
|
+ <dt>
|
|
|
+ <a class="anchor"
|
|
|
id="desc_Blasten-Paivarinta-Sadique-2020"></a>
|
|
|
<a href="#Blasten-Paivarinta-Sadique-2020">
|
|
|
<span class="ref-authors">E. Blåsten, L. Päivärinta,
|