We analyze spectral properties of the operator <i>H</i> = <i></i>²/<i>x</i>² <i></i>²/<i>y</i>² + <i></i>²<i>y</i>² <i>y</i>²<i>V</i>(<i>xy</i>) in <i>L</i>²(²), where <i></i> 0 and <i>V </i> 0 is a compactly supported and sufficiently regular potential. It is known that the spectrum of <i>H</i> depends on the one-dimensional Schrdinger operator <i>L</i> = <i>d</i>²/<i>dx</i>² + <i></i>² <i>V</i>(<i>x</i>) and it changes substantially as inf(<i>L</i>) switches sign. We prove that in the critical case, inf(<i>L</i>) = 0, the spectrum of <i>H</i> is purely essential and covers the interval [0, ). In the subcritical case, inf <i></i>(<i>L</i>) &gt; 0, the essential spectrum starts from <i></i> and there is a nonvoid discrete spectrum in the interval [0, <i></i>). We also derive a bound on the corresponding eigenvalue moments.

Quantitative susceptibility mapping (QSM) uses the phase data in magnetic resonance signals to visualize a three-dimensional susceptibility distribution by solving the magnetic field to susceptibility inverse problem. Due to the presence of zeros of the integration kernel in the frequency domain, QSM is an ill-posed inverse problem. Although numerous regularization-based models have been proposed to overcome this problem, incompatibility in the field data, which leads to deterioration of the recovery, has not received enough attention. In this paper, we show that the data acquisition process of QSM inherently generates a harmonic incompatibility in the measured local field. Based on this discovery, we propose a novel regularization-based susceptibility reconstruction model with an additional sparsity-based regularization term on the harmonic incompatibility. Numerical experiments show that the proposed method achieves better performance than existing approaches.

New spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel–Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Hajłasz–Sobolev spaces, Korevaar–Schoen–Sobolev spaces, Newtonian Sobolev space and Cheeger–Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy–Littlewood–Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.

We use the learning with errors (LWE) problem to build a new simple and provably secure key exchange scheme. The basic idea of the construction can be viewed as certain extension of Diffie-Hellman problem with errors. The mathematical structure behind comes from the commutativity of computing a bilinear form in two different ways due to the associativity of the matrix multiplications: (x^t \times A)\times y = x^t \times (A\times y), where x,y are column vectors and A is a square matrix. We show that our new schemes are more efficient in terms of communication and computation complexity compared with key exchange schemes or key transport schemes via encryption schemes based on the LWE problem. Furthermore, we extend our scheme to the ring learning with errors (RLWE) problem, resulting in small key size and better efficiency.

No abstract uploaded!