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In this paper, we further Meirong Zhang, et al.'s work by computing the number of weighted eigenvalues for Sturm-Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm-Liouville equation with a general integrable potential, if its weight is a positive linear combination of $n$ Dirac Delta functions, then it has at most $n$ (may be less than $n$, or even be $0$) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly $n$. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm-Liouville problem with Dirac weights, and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichelt problems for Sturm-Liouville equations involving single Dirac distribution weights is studied., 23 pages

It has been reported that training deep neural networks is more difficult than training shallow neural networks. Hinton et al. proposed deep belief networks with a learning algorithm that trains one layer at a time. A much better generalization can be achieved when pre-training each layer with an unsupervised learning algorithm. Since then, deep neural networks have been extensively studied. On the other hand, it has been revealed that singular points affect the training dynamics of the learning models such as neural networks and cause a standstill of training. Naturally, training deep neural networks suffer singular points. As described in this paper, we present a deep neural network model that has fewer singular points than the usual one. First, we demonstrate that some singular points in the deep real-valued neural network, which is equivalent to a deep complex-valued neural network, have been resolved as its inherent property. Such deep neural networks are less likely to become trapped in local minima or plateaus caused by critical points. Results of experiments on the two spirals problem, which has an extreme nonlinearity, support our theory. Copyright © 2017 John Wiley & Sons, Ltd.

We deal with the scattering of an acoustic medium modeled by an index of refraction n varying in a bounded region Ω of and equal to unity outside Ω. This region is perforated with an extremely large number of small holes Dm's of maximum radius a, a 0 and λm,0 being constants and eventually complex numbers. Under some natural conditions on the parameters β,t, and λm,0, we characterize the equivalent medium generating approximately the same scattered waves as the original perforated acoustic medium. We give an explicit error estimate between the scattered waves generated by the perforated medium and the equivalent one, respectively, as a0. As applications of these results, we discuss the following findings: If we choose negative-valued imaginary surface impedance functions, attached to each surface of the holes, then the equivalent medium behaves as a passive acoustic medium only if it is an acoustic metamaterial with index of refraction $\tilde {n}(x)=-n(x),\; x \in \upOmega $n(x)=−n(x),x∈Ω and $\tilde {n}(x)=1,\; x \in \mathbb {R}^{3}\setminus {\overline {\upOmega }}$n(x)=1,x∈R3∖Ω¯. This means that with this process, we can switch the sign of the index of the refraction from positive to negative values. We can choose the surface impedance functions attached to each surface of the holes so that the equivalent index of refraction $\tilde {n}$n is $\tilde {n}(x)=1,\; x \in \mathbb {R}^{3}$n(x)=1,x∈R3. This means that the region Ω modeled by the original index of refraction n is approximately cloaked. Copyright © 2015 John Wiley & Sons, Ltd.

We consider a commutative algebra over the field of complex numbers with a basis {e1,e2} satisfying the conditions , . Let D be a bounded domain in the Cartesian plane xOy and Dζ={xe1+ye2:(x,y)∈D}. Components of every monogenic function Φ(xe1+ye2) = U1(x,y)e1+U2(x,y)ie1+U3(x,y)e2+U4(x,y)ie2 having the classic derivative in Dζ are biharmonic functions in D, that is, Δ2Uj(x,y) = 0 for j = 1,2,3,4. We consider a Schwarz-type boundary value problem for monogenic functions in a simply connected domain Dζ. This problem is associated with the following biharmonic problem: to find a biharmonic function V(x,y) in the domain D when boundary values of its partial derivatives ∂V/∂x, ∂V/∂y are given on the boundary ∂D. Using a hypercomplex analog of the Cauchy-type integral, we reduce the mentioned Schwarz-type boundary value problem to a system of integral equations on the real axes and establish sufficient conditions under which this system has the Fredholm property. Copyright © 2015 John Wiley & Sons, Ltd.