A power of a meromorphic function sharing two small functions with a derivative of the power

summary:In connection to a conjecture of W. Lü, Q. Li and C. Yang (2014), we prove a result on small function sharing by a power of a meromorphic function with few poles with a derivative of the power. Our results improve a number of known results

On the meromorphic solutions of a certain type of nonlinear difference-differential equation

summary:The main objective of this paper is to give the specific forms of the meromorphic solutions of the nonlinear difference-differential equation $$f^{n}(z)+P_{d}(z,f)=p_{1}(z){\rm e}^{\alpha _{1}(z)}+p_{2}(z){\rm e}^{\alpha _{2}(z)},$$ where $P_d(z,f)$ is a difference-differential polynomial in $f(z)$ of degree $d\leq n-1$ with small functions of $f(z)$ as its coefficients, $p_1$, $p_2$ are nonzero rational functions and $\alpha _1$, $\alpha _2$ are non-constant polynomials. More precisely, we find out the conditions for ensuring the existence of meromorphic solutions of the above equation

Generalizations on the results of Cao and Zhang

summary:We establish some uniqueness results for meromorphic functions when two nonlinear differential polynomials $P(f)\prod _{i=1}^{k}(f^{(i)})^{n_{i}}$ and $P(g)\prod _{i=1}^{k}(g^{(i)})^{n_{i}}$ share a nonzero polynomial with certain degree and our results improve and generalize some recent results in Y.-H. Cao, X.-B. Zhang (2012). Also we exhibit two examples to show that the conditions used in the results are sharp

Non-linear differential polynomials sharing one or two values with finite weight

The purpose of the paper is to study the uniqueness of meromorphic functions sharing a small function with finite weight. The results of the paper improve and generalize the recent results due to X. B. Zhang and J. F. Xu [21]. We also solve an open problem as posed in the last section of [21]

On the meromorphic solutions of a certain type of nonlinear difference-differential equation

The main objective of this paper is to give the specific forms of the meromorphic solutions of the nonlinear difference-differential equation f^n(z)+P_d(z,f)=p_1(z){\rm e}^{\alpha_1(z)}+p_2(z){\rm e}^{\alpha_2(z)}, where $P_d(z,f)$ is a difference-differential polynomial in $f(z)$ of degree $d\leq n-1$ with small functions of $f(z)$ as its coefficients, $p_1$, $p_2$ are nonzero rational functions and $\alpha_1$, $\alpha_2$ are non-constant polynomials. More precisely, we find out the conditions for ensuring the existence of meromorphic solutions of the above equation

Recent Advances in Research on Down Syndrome

Down syndrome (DS) or trisomy 21 is one of the most important genetic causes of mental retardation. Sincere and significant attempts have been made towards understanding the congenital diseases that affect DS patients. Better understanding of gene networks associated with such malformations will help to predict the complex genetic trait behind congenital disease in DS and will also provide the basis for tailored gene therapies that could begin to heal or prevent such malformation without the need to resort to invasive surgery. Further, susceptible mutation screening in women will also be helpful for both prenatal diagnosis of DS birth and assessing the risk of predisposition to Alzheimer’s disease and congenital heart disease. Stress condition and neurodegeneration are two important markers in Down syndrome patients and mtDNA variation can also be used as an important biomarker. It has been suggested that nutraceuticals which reduce reactive oxygen species (ROS) level may be used to treat trisomy 21 condition. As mitochondria play a crucial role in the regulation of free radicals, only a detail analysis will reveal the origin of phenotypic characteristics among trisomy 21 DS patients. On the other hand, several mechanisms are responsible for neurodegeneration as well as altered cognition. It includes impaired neurogenesis leading to hypocellularity in the cortex, hippocampus and cerebellum, altered dendritic morphology, altered synapses, increased inhibition and neurodegeneration. The new knowledge of the pathogenic mechanisms in DS individuals has been acquired from mouse model. These studies provide the basis for developing new drugs for clinical trials in DS individuals and to sustain the hope that some of these drugs will be useful in treating intellectual disability in DS individuals

A Novel Membrane Protein-Specific Serine/Threonine Kinase: Tissue Distribution and Role in Sperm Maturation

Our recent studies have described for the first time the purification of an ectoprotein kinase to apparent homogeneity using caprine sperm as the model. Purified ectokinase (CIK) is a novel membrane protein-specific kinase that phosphorylates serine and threonine residues of ectophosphoproteins. This study, using ELISA based on ecto-CIK antibody demonstrates that ecto-CIK level is remarkably higher in the sperm membrane than in the cytosol. The epididymal sperm maturational event as well as sperm vertical velocity is associated with a significant increase in the ecto-CIK level. Ecto-CIK, the membrane protein-specific kinase, is also present in all the tissues tested and is predominantly localized in the cell membrane. Ubiquitous localization of the novel kinase on the mammalian cell membrane suggests that the kinase may play pivotal role in gamete as well as somatic cell regulation by modulating membrane biology through serine/threonine phosphorylation of specific membrane proteins located in the ectodomains