We study detecting a boundary corrosion damage in the inaccessible part of a rectangular shaped electrostatic conductor from a one set of Cauchy data specied on an accessible boundary part of conductor. For this nonlinear ill-posed problem, we prove the uniqueness in a very general framework. Then we establish the conditional stability of Holder type based on some a priori assumptions on the unknown impedance and the electrical current input specied in the accessible part. Finally a regularizing scheme of double regularizing parameters, using the truncation of the series expansion of the solution, is proposed with the convergence analysis on the explicit regularizing solution in terms of a practical average norm for measurement data.

For the class of graph-directed self-similar measures on R, which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the spectral dimension of the Laplacians defined by these measures. For the class of finitely ramified graph-directed self-similar sets, the spectral dimension of the associated Laplace operators has been obtained by Hambly and Nyberg [6]. The main novelty of our results is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition.

LWE/RLWE-based cryptosystems require sampling error term from discrete Gaussian distribution. However, some existing samplers are somehow slow under certain circumstances therefore efficiency of such schemes is restricted. In this paper, we introduce a more efficient discretized Gaussian sampler based on ziggurat sampling algorithm. We also analyze statistical quality of our sampler to prove that it can be adopted in LWE/RLWE-based cryptosystems. Compared with ziggurat-based sampler by Buchmann et al. related samplers by Peikert, Ducas et al. and Knuth-Yao, our sampler achieves more than 2x speedup when standard deviation is large. This can benefit constructions rely on noise flooding (e.g., homomorphic encryption). We also present two applications: First, we use our sampler to optimize the RLWE-based authenticated key exchange (AKE) protocol by Zhang et al. We achieve 1.14x speedup on total runtime of this protocol over major parameter choices. Second, we give practical post-quantum Transport Layer Security (TLS) ciphersuite. Our ciphersuite inherits advantages from TLS and the optimized AKE protocol. Performance of our proof-of-concept implementation is close to TLS v1.2 ciphersuites and one post-quantum TLS construction.

In an upcoming paper by Lee and Wang, we construct examples of two-dimensional Hamiltonian stationary self-shrinkers and self-expanders for Lagrangian mean curvature flows, which are asymptotic to the union of two Schoen-Wolfson cones. These self-shrinkers and self-expanders can be glued together to yield solutions of the Brakke flow - a weak formulation of the mean curvature flow. Moreover, there is no mass loss along the Brakke flow. In this paper, we generalize these results to higher dimensions. We construct new higher-dimensional Hamiltonian stationary cones of different topology as generalizations of the Schoen-Wolfson cones. Hamiltonian stationary self-shrinkers and self-expanders that are asymptotic to these Hamiltonian stationary cones are constructed as well. They can also be glued together to produce eternal solutions of the Brakke flow without mass loss. Finally, we show that the same conclusion holds for those Lagrangian self-similar examples recently found by Joyce, Tsui and the first author.

We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system , arising from the universal family of Calabi-Yau hypersurfaces $Y_a$ in a G-variety $X$ of dimension $n$. First, we construct a natural topological correspondence between relative cycles in $H_n (X − Y_a, \cup D − Y_a)$ bounded by the union of G-invariant divisors $\cup D$ in $X$ to the solution sheaf of $\tau$, in the form of chain integrals. Applying this to a toric variety with torus action, we show that in addition to the period integrals over cycles in $Y_a$, the new chain integrals generate the full solution sheaf of a GKZ system. This extends an earlier result for hypersurfaces in a projective homogeneous variety, whereby the chains are cycles [3, 7]. In light of this result, the mixed Hodge structure of the solution sheaf is now seen as the MHS of $H_n(X − Ya, \cup D − Y_a)$. In addition, we generalize the result on chain integral solutions to the case of general type hypersurfaces. This chain integral correspondence can also be seen as the Riemann-Hilbert correspondence in one homological degree. Finally, we consider interesting cases in which the chain integral correspondence possibly fails to be bijective.