Some Normal Criteria about Shared Values with Their Multiplicity Zeros

Let F be a family of meromorphic functions in the domain D, all of whose zeros are multiple. Let n  (n≥2) be an integer and let a, b be two nonzero finite complex numbers. If f+a(f')n and g+a(g')n share b in D for every pair of functions f,g∈F, then F is normal in D

Entire solutions for a nonlinear differential equation

In this article, we study the existence of solutions to the differential equation $$f^n(z)+P(f)= P_1e^{h_1}+ P_2e^{h_2},$$ where $ngeq 2$ is an positive integer, f is a transcendental entire function, $P(f)$ is a differential polynomial in f of degree less than or equal n-1, $P_1, P_2$ are small functions of $e^z$, $h_1$, $h_2$ are polynomials, and $z$ is in the open complex plane $mathbb{C}$. Our results extend those obtained by Li [6,7,8]

The Natural Boundary Element Method of the Uniform Transmission Line Equation in 2D Unbounded Region

Herein, we are mainly concerned with the natural boundary element (NBE) method of the uniform transmission line (UTL) equation defined in the two-dimensional (2D) boundless region, which has a real physical background. We first create the time semi-discretized scheme of the UTL equation, as well as analyze the convergence and stability for the series of time semi-discretized solutions. Then, we create a fully discretized NBE format by means of a natural boundary reduction and analyze the stability and errors between the fully discretized NBE solutions and the analytical solution. Lastly, we employ two numerical examples to verify the effectiveness of the NBE method

The Natural Boundary Element Method of the Uniform Transmission Line Equation in 2D Unbounded Region

Herein, we are mainly concerned with the natural boundary element (NBE) method of the uniform transmission line (UTL) equation defined in the two-dimensional (2D) boundless region, which has a real physical background. We first create the time semi-discretized scheme of the UTL equation, as well as analyze the convergence and stability for the series of time semi-discretized solutions. Then, we create a fully discretized NBE format by means of a natural boundary reduction and analyze the stability and errors between the fully discretized NBE solutions and the analytical solution. Lastly, we employ two numerical examples to verify the effectiveness of the NBE method

Normality Criteria of Meromorphic Functions That Share a Holomorphic Function

Let F be a family of meromorphic functions defined in D, let ψ(≢0), a0,a1,...,ak-1 be holomorphic functions in D, and let k be a positive integer. Suppose that, for every function f∈F, f≠0, P(f)=f(k)+ak-1f(k-1)+⋯+a1f'+a0f≠0 and, for every pair functions (f,g)∈F, P(f), P(g) share ψ, then F is normal in D

Some results about a special nonlinear difference equation and uniqueness of difference polynomial

<p>Abstract</p> <p>In this paper, we continue to study a special nonlinear difference equation solutions of finite order entire function. We also continue to investigate the value distribution and uniqueness of difference polynomials of meromorphic functions. Our results which improve the results of Yang and Laine [Proc. Jpn. Acad. Ser. A Math. Sci. 83:50-55 (2007)]; Qi et al. [Comput. Math. Appl. 60:1739-1746 (2010) ].</p> <p><b>Mathematics Subject Classification (2000): </b>30D35, 39B32, 34M05.</p

Cadherin-17 and SATB2 are sensitive and specific immunomarkers for medullary carcinoma of the large intestine.

ContextDistinction of medullary carcinoma of the large intestine from other cytokeratin (CK) 7⁻/CK20⁻ carcinomas can be challenging when working on a tumor of unknown primary because the majority of medullary carcinomas are negative for CK7, CK20, and CDX2.ObjectiveTo investigate the expression of cadherin-17 and SATB-2 and other markers in medullary carcinomas of the large intestine and cadherin-17 and SATB2 in a large number of carcinomas and normal tissues from various organs to further test their diagnostic specificity.DesignThis study evaluated cadherin-17 and SATB2 expression in 18 medullary carcinoma cases and 1941 tumors and 358 normal tissues from various organs. Other immunomarkers, including MLH1, PMS2, MSH2, MSH6, CDX2, CK7, CK20, TFF3, MUC4, calretinin, p504S, villin, and synaptophysin, were also tested on the 18 medullary carcinoma cases.ResultsThe results demonstrated (1) loss of MLH1 and PMS2 in more than 80% of medullary carcinomas; (2) expression of cadherin-17 and SATB2 in 89% of medullary carcinomas; (3) focal expression of TFF3, MUC4, calretinin, CDX2, CK20, and synaptophysin in 72%, 72%, 67%, 67%, 28%, and 17% of 18 medullary carcinoma cases, respectively; and (4) expression of SATB2 and cadherin-17 in 97% and 98% of the colorectal adenocarcinomas, respectively, whereas their expression was seen in 3.6% and 3.3% of nongastrointestinal tumors, respectively.ConclusionWe concluded that SATB2 and cadherin-17 were highly sensitive and specific markers for colorectal carcinomas and propose including MLH1, cadherin-17, and SATB2 in a routine immunostaining panel when working on a tumor of unknown primary, especially in an elderly patient with a CK7⁻/CK20⁻ carcinoma