Exploring Visualization Methods for Complex Variables

Applications of complex variables and related manifolds appear throughout mathematics and science. Here we review a family of basic methods for applying visualization concepts to the study of complex variables and the properties of specific complex manifolds. We begin with an outline of the methods we can employ to directly visualize poles and branch cuts as complex functions of one complex variable. $CP^2$ polynomial methods and their higher analogs can then be exploited to produce visualizations of Calabi-Yau spaces such as those modeling the hypothesized hidden dimensions of string theory. Finally, we show how the study of N-boson scattering in dual model/string theory leads to novel cross-ratio-space methods for the treatment of analysis in two or more complex variables

A Contour Integral Representation for the Dual Five-Point Function and a Symmetry of the Genus Four Surface in R6

The invention of the "dual resonance model" N-point functions BN motivated the development of current string theory. The simplest of these models, the four-point function B4, is the classical Euler Beta function. Many standard methods of complex analysis in a single variable have been applied to elucidate the properties of the Euler Beta function, leading, for example, to analytic continuation formulas such as the contour-integral representation obtained by Pochhammer in 1890. Here we explore the geometry underlying the dual five-point function B5, the simplest generalization of the Euler Beta function. Analyzing the B5 integrand leads to a polyhedral structure for the five-crosscap surface, embedded in RP5, that has 12 pentagonal faces and a symmetry group of order 120 in PGL(6). We find a Pochhammer-like representation for B5 that is a contour integral along a surface of genus five. The symmetric embedding of the five-crosscap surface in RP5 is doubly covered by a symmetric embedding of the surface of genus four in R6 that has a polyhedral structure with 24 pentagonal faces and a symmetry group of order 240 in O(6). The methods appear generalizable to all N, and the resulting structures seem to be related to associahedra in arbitrary dimensions.Comment: 43 pages and 44 figure

Identification of HLA-A2-Restricted Mycobacterial Lipoprotein Z Peptides Recognized by T CellsFrom Patients With ActiveTuberculosis Infection

Identification of HLA-restricted peptides derived from mycobacterial antigens that are endowed with high affinity and strong antigenicity is not only of interest in tuberculosis (TB) diagnostics and treatment efficacy evaluation, but might also provide potential candidates for the development of therapeutic vaccines against drug-resistant TB. Our previous work demonstrated that lipoprotein Z (LppZ) displayed high immunogenicity and antigenicity in active TB patients. In the present study, ten HLA-A2-restricted LppZ peptides (LppZp1-10) were predicted by bioinformatics, among which LppZp7 and LppZp10 were verified to possess high affinity to HLA-A2 molecules using T2 cell-based affinity binding assay. Moreover, results from ELISpot assay showed that both LppZp7 and LppZp10 peptides were able to induce more IFN-γ producing cells upon ex vivo stimulation of PBMC from HLA-A2+ active TB (ATB) patients as compared to those from healthy controls (HCs). Also, the numbers of LppZp7 and LppZp10-specific IFN-γ producing cells exhibited positive correlations with those of ESAT-6 peptide (E6p) or CFP-10 peptide (C10p) in ATB. Interestingly, stimulation with LppZp7/p10 mixture was able to induce higher intracellular expression of IFN-γ and IL-2 cytokines in CD8+ and CD4+ T cells from ATB as compared to HC, associated with lower expression of TNF-α in both CD8+ and CD4+ T cells. Taken together, HLA-A2-restricted LppZp7 and LppZp10 peptides display high immunoreactivity in HLA-matched ATB patients demonstrated by high responsiveness in both CD8+ and CD4+ T cells. With the ability to induce strong antigen-specific cellular responses, LppZp7 and LppZp10 are of potential value for the future applications in the prevention and control of TB

Joint analysis of three genome-wide association studies of esophageal squamous cell carcinoma in Chinese populations

We conducted a joint (pooled) analysis of three genome-wide association studies (GWAS) 1-3 of esophageal squamous cell carcinoma (ESCC) in ethnic Chinese (5,337 ESCC cases and 5,787 controls) with 9,654 ESCC cases and 10,058 controls for follow-up. In a logistic regression model adjusted for age, sex, study, and two eigenvectors, two new loci achieved genome-wide significance, marked by rs7447927 at 5q31.2 (per-allele odds ratio (OR) = 0.85, 95% CI 0.82-0.88; P=7.72x10−20) and rs1642764 at 17p13.1 (per-allele OR= 0.88, 95% CI 0.85-0.91; P=3.10x10−13). rs7447927 is a synonymous single nucleotide polymorphism (SNP) in TMEM173 and rs1642764 is an intronic SNP in ATP1B2, near TP53. Furthermore, a locus in the HLA class II region at 6p21.32 (rs35597309) achieved genome-wide significance in the two populations at highest risk for ESSC (OR=1.33, 95% CI 1.22-1.46; P=1.99x10−10). Our joint analysis identified new ESCC susceptibility loci overall as well as a new locus unique to the ESCC high risk Taihang Mountain region