A Criterion for Finite Topological Determinacy of Map-Germs

Let f, g: (Rⁿ, 0) (R^p, 0) be two C map-germs. Then f and gare C⁰-equivalent if there exist homeomorphism-germs h and lof (Rⁿ, 0) and (R^p, 0) respectively such that g = l f h^–1.Let k be a positive integer. A germ f is k-C⁰-determined ifevery germ g with j^k g(0) = j^k f(0) is C⁰-equivalent to f. Moreover,we say that f is finitely topologically determined if f is k-C⁰-determinedfor some finite k. We prove a theorem giving a sufficient conditionfor a germ to be finitely topologically determined. We explainthis condition below. Let N and P be two C manifolds. Consider the jet bundle J^k(N,P) with fiber J^k(n, p). Let z in J^k(n, p) and let f be suchthat z = j^kf(0). Define Whether (f) < k depends only on z, not on f. We can thereforedefine the set Let W^k(N, P) be the subbundle of J^k(N, P) with fiber W^k(n, p).Mather has constructed a finite Whitney (b)-regular stratificationS^k(n, p) of J^k(n, p) – W^k(n, p) such that all strata aresemialgebraic and K-invariant, having the property that if S^k(N,P) denotes the corresponding stratification of J^k(N, P) –W^k(N, P) and f C(N, P) is a C map such that j^kf is multitransverseto S^k(N, P), j^kf(N) W^k(N, P) = and N is compact (or f is proper),then f is topologically stable. For a map-germ f: (Rⁿ, 0) (R^p, 0), we define a certain ojasiewiczinequality. The inequality implies that there exists a representativef: U R^p such that j^kf(U – 0) W^k (Rⁿ, R^p = and suchthat j^kf is multitransverse to S^k (Rⁿ, R^p) at any finite setof points S U – 0. Moreover, the inequality controlsthe rate j^kf becomes non-transverse as we approach 0. We showthat if f satisfies this inequality, then f is finitely topologicallydetermined. 1991 Mathematics Subject Classification: 58C27.     

Haplotyping of putative microRNA-binding sites in the SNAP-25 gene

Synaptosomal-associated protein 25 (SNAP-25) plays a crucial role in exocitosis. Single nucleotide polymorphisms (rs3746544 and rs1051312) in the 3' un-translated region of the SNAP-25 gene have been described to be in association with attention-deficit hyperactivity disorder. As the disease affects millions of children world-wide, understanding the genetic background of attention-deficit hyperactivity disorder is of crucial importance. Efficient and reliable PCR-RFLP protocols were elaborated for the genotyping of the rs3746544 and rs1051312 SNPs employing a high-throughput capillary electrophoresis method for fragment analysis. A novel real-time PCR-based technique was used applying sequence specific TaqMan probes to haplotype the two SNPs, and the G-C haplotype could not be detected in a large Caucasian population (N=1376). These findings have been confirmed by molecular biology tools as well as by the PHASE Bayesian computational approach. In silico analyses have suggested that the two SNPs might alter microRNA binding and thus have an effect on SNAP-25 production. We have demonstrated that this biological information can be revealed only by direct haplotype analysis emphasizing the importance of our novel molecular haplotye analysis protocol. Results of the study of the two SNPs might shed light on the association of SNAP-25 variants and pathological phenotypes at the molecular level.     

Preparation and characterization of MgB2 superconductor

The MgB₂ superconductor, synthesized using solid-state and liquid-phase sintering methods, have been characterized for various properties. The upper critical field, irreversibility line and critical current density have been determined using magnetization data. The current-voltage characteristics recorded under an applied magnetic field revealed the existence of vortex glass transition. The surface analysis using X-ray photoelectron spectroscopy shows that MgB₂ is sensitive to atmospheric degradation.