Zeros and the functional equation of the quadrilateral zeta function

In this paper, we show that all real zeros of the bilateral Hurwitz zeta function $Z(s,a):=\zeta (s,a) + \zeta (s,1-a)$ with $1/4 \le a \le 1/2$ are on only the non-positive even integers exactly same as in the case of $(2^s-1) \zeta (s)$. We also prove that all real zeros of the bilateral periodic zeta function $P(s,a):={\rm{Li}}_s (e^{2\pi ia}) + {\rm{Li}}_s (e^{2\pi i(1-a)})$ with $1/4 \le a \le 1/2$ are on only the negative even integers just like $\zeta (s)$. Moreover, we show that all real zeros of the quadrilateral zeta function $Q(s,a):=Z(s,a) + P(s,a)$ with $1/4 \le a \le 1/2$ are on only the negative even integers. On the other hand, we prove that $Z(s,a)$, $P(s,a)$ and $Q(s,a)$ have at least one real zero in $(0,1)$ when $0<a<1/2$ is sufficiently small. The complex zeros of these zeta functions are also discussed when $1/4 \le a \le 1/2$ is rational or transcendental. As a corollary, we show that $Q(s,a)$ with rational $1/4 < a < 1/3$ or $1/3 < a < 1/2$ does not satisfy the analogue of the Riemann hypothesis even though $Q(s,a)$ satisfies the functional equation that appeared in Hamburger's or Hecke's theorem and all real zeros of $Q(s,a)$ are located at only the negative even integers again as in the case of $\zeta (s)$.Comment: 12 pages. We changed the title. Some typos are correcte

Identification of Endogenous Control miRNAs for RT-qPCR in T-Cell Acute Lymphoblastic Leukemia

Optimal endogenous controls enable reliable normalization of microRNA (miRNA) expression in reverse-transcription quantitative PCR (RT-qPCR). This is particularly important when miRNAs are considered as candidate diagnostic or prognostic biomarkers. Universal endogenous controls are lacking, thus candidate normalizers must be evaluated individually for each experiment. Here we present a strategy that we applied to the identification of optimal control miRNAs for RT-qPCR profiling of miRNA expression in T-cell acute lymphoblastic leukemia (T-ALL) and in normal cells of T-lineage. First, using NormFinder for an iterative analysis of miRNA stability in our miRNA-seq data, we established the number of control miRNAs to be used in RT-qPCR. Then, we identified optimal control miRNAs by a comprehensive analysis of miRNA stability in miRNA-seq data and in RT-qPCR by analysis of RT-qPCR amplification efficiency and expression across a variety of T-lineage samples and T-ALL cell line culture conditions. We then showed the utility of the combination of three miRNAs as endogenous normalizers (hsa-miR-16-5p, hsa-miR-25-3p, and hsa-let-7a-5p). These miRNAs might serve as first-line candidate endogenous controls for RT-qPCR analysis of miRNAs in different types of T-lineage samples: T-ALL patient samples, T-ALL cell lines, normal immature thymocytes, and mature T-lymphocytes. The strategy we present is universal and can be transferred to other RT-qPCR experiments

Additional file 4 of Transcriptomic population markers for human population discrimination

: Infinium Human OmniExpressExome microarray. (DOCX 11 kb

Additional file 10 of Transcriptomic population markers for human population discrimination

: Table S4. List of mRNA transcripts and TLDA probes. (DOCX 14 kb

Additional file 7 of Transcriptomic population markers for human population discrimination

: RNA isolation procedure. (DOCX 10 kb

Additional file 3 of Transcriptomic population markers for human population discrimination

: Figure S2. A binary Decision-Tree classifier built based on UTS2 and UGT2B17 data (left Panel) and for UTS2 (Right Panel) obtained from Caucasian (n = 37), and Chinese (n = 29) blood samples. (DOCX 87 kb

Additional file 11 of Transcriptomic population markers for human population discrimination

: Table S5. Primer design for qRT-PCR validation experiment. (DOCX 13 kb