Generalized Normal Derivatives and Their Applications in DDMs with Nonmatching Grids and DG Methods

A class of normal-like derivatives for functions with low regularity defined on Lipschitz domains are introduced and studied.It is shown that the new normal-like derivatives,which are called the generalized normal derivatives,preserve the major prop- erties of the existing standard normal derivatives.The generalized normal derivatives are then applied to analyze the convergence of domain decomposition methods (DDMs) with nonmatching grids and discontinuous Galerkin (DG) methods for second-order el- liptic problems.The approximate solutions generated by these methods still possess the optimal energy-norm error estimates,even if the exact solutions to the underlying elliptic problems admit very low regularities.

Normalized edge detection, and the horizontal extent and depth of geophysical anomalies

Edge detection is an image processing technique for finding the boundaries of objects within images. It is typically used to interpret gravity and magnetic data, and find the horizontal boundaries of geological bodies. Large deviations between model and true edges are common because of the interference of depth and errors in computing the derivatives; thus, edge detection methods cannot provide information about the depth of the source. To simultaneously obtain the horizontal extent and depth of geophysical anomalies, we use normalized edge detection filters, which normalize the edge detection function at different depths, and the maxima that correspond to the location of the source. The errors between model and actual edges are minimized as the depth of the source decreases and the normalized edge detection method recognizes the extent of the source based on the maxima, allowing for reliable model results. We demonstrate the applicability of the normalized edge detection filters in defining the horizontal extent and depth using synthetic and actual aeromagnetic data.

A new edge recognition technology based on the normalized vertical derivative of the total horizontal derivative for potential field data

Edge detection and enhancement techniques are commonly used in recognizing the edge of geologic bodies using potential field data. We present a new edge recognition technology based on the normalized vertical derivative of the total horizontal derivative which has the functions of both edge detection and enhancement techniques. First, we calculate the total horizontal derivative （THDR） of the potential-field data and then compute the n-order vertical derivative （VDRn） of the THDR. For the n-order vertical derivative, the peak value of total horizontal derivative （PTHDR） is obtained using a threshold value greater than 0. This PTHDR can be used for edge detection. Second, the PTHDR value is divided by the total horizontal derivative and normalized by the maximum value. Finally, we used different kinds of numerical models to verify the effectiveness and reliability of the new edge recognition technology.

On Results the Growth of Meromorphic Solutions of Algebraic Diferential Equations

In this paper, we give an estimate result of Gol＇dberg＇s theorem concern- ing the growth of meromorphic solutions of Mgebraic differential equations by using Zalcman Lemma. It is an extending result of the corresponding theorem by Yuan et al. （Yuan W J, Xiao B, Zhang J J. The general theorem of Gol＇dberg concerning the growth of meromorphic solutions of algebraic differential equations. Comput. Math. Appl., 2009, 58：1788 1791）. Meanwhile, we also take some examples to show that our estimate is sharp.

Hepatocytes in a normal adult liver are derived solely from the embryonic hepatocytes

In vertebrates,body weight increases many folds as a consequence of body growth from the childhood to adulthood（e.g.,～20 folds for a mouse）.Considering the fact that the liver-to-body weight ratio（LBR）stays relatively constant within species（Weglarz and Sandgren,2000;Kan et al.,2009）,the cell number in a liver must therefore keep increasing along with the growth of an individual organism. For example, during the growth of the zebrafish from 5 days post-fertilization （dpf） to 1.5 years old, the number of liver cells increased -900 folds （Figs. 1A and S1 ）.

Identification of the Caroline Plate boundary:constraints from magnetic anomaly

The Caroline Plate is located among the Pacific Plate,the Philippine Sea Plate,and the India Australia Plate,and plays a key role in controlling the spreading direction of the Philippine Sea Plate.The Caroline Submarine Plateau(or Caroline Ridge)and the Eauripik Rise on the south formed a remarkable T-shaped large igneous rock province,which covered the northern boundary between the Caroline Plate and the Pacific Plate.However,relationship between these tectonic units and magma evolution remains unclear.Based on magnetic data from the Earth Magnetic Anomaly Grid(2-arc-minute resolution)(V2),the normalized vertical derivative of the total horizontal derivative(NVDR-THDR)technique was used to study the boundary of the Caroline Plate.Results show that the northern boundary is a transform fault that runs 1400 km long in approximately 28 km wide along the N8°in E-W direction.The eastern boundary is an NNW-SSE trending fault zone and subduction zone with a width of tens to hundreds of kilometers;and the north of N4°is a fracture zone of dense faults.The southeastern boundary may be the Lyra Trough.The area between the southwestern part of the Caroline Plate and the Ayu Trough is occupied by a wide shear zone up to 100 km wide in nearly S-N trending in general.The Eauripik transform fault(ETF)in the center of the Caroline Plate and the fault zones in the east and west basins are mostly semi-parallel sinistral NNW-SSE–trending faults,which together with the eastern boundary Mussau Trench(MT)sinistral fault,the northern Caroline transform fault(CTF),and the southern shear zone of the western boundary,indicates the sinistral characteristics of the Caroline Plate.The Caroline hotspot erupted in the Pacific Plate near the CTF and formed the west Caroline Ridge,and then joined with the Caroline transform fault at the N8°.A large amount of magma erupted along the CTF,by which the east Caroline Ridge was formed.At the same time,a large amount of magma developed southward via the eastern branch of the ETF,forming the northern segment of the Eauripik Rise.Therefore,the magmatic activity of the T-shaped large igneous province is obviously related to the fault structure of the boundary faults between the Caroline Plate and Pacific Plate,and the active faults within the Caroline Plate.

Generation of Airy beams using a phase-only Fresnel holographic lens

A method to generate Airy beam by combining the Fresnel holographic lens and the cubic phase of Airy beam is proposed. The detailed theoretical derivation to express the optical transform principle of the proposed method is presented. And excellent experimental results are demonstrated. It is shown that this approach works well and simplifies the experimental facility effectively, especially reducing the optical system length to half of that of the conventional method. In addition, the proposed method can realize the beam propagation trajectory control of Airy beam and generate Airy beam array.