Abstract:
We prove a new selection theorem for multivalued mappings of C-space. Using this theorem we prove extension dimensional version of Hurewicz theorem for a closed mapping $f\colon X\to Y$ of $k$-space $X$ onto paracompact $C$-space $Y$: if for finite $CW$-complex $M$ we have $\ed Y\le [M]$ and for every point $y\in Y$ and every compactum $Z$ with $\ed Z\le [M]$ we have $\ed(f^{-1}(y)\times Z)\le [L]$ for some $CW$-complex $L$, then $\ed X\le [L]$.

Abstract:
Let $L$ be a countable CW-complex and $F\colon X\to Y$ be upper semicontinuous $UV^{[L]}$-valued mapping of a paracompact space $X$ to a complete metric space $Y$. We prove that if $X$ is a C-space of extension dimension $\ed X \le [L]$, then $F$ admits single-valued graph approximations. For $L=S^n$ our result implies well-known approximation theorem for $UV^{n-1}$-valued mappings of $n$-dimensional spaces. And for $L=\{\rm point\}$ our theorem implies a theorem of Ancel on approximations of $UV^\infty$-valued mappings of C-spaces.

Abstract:
It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. Results of this paper extend Whitney theorem to the case when all fibers are homeomorphic to a given compact two-dimensional manifold.

Abstract:
It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. An extension of the Whitney's theorem to the case when all fibers are homeomorphic to some fixed compact two-dimensional manifold was proved by the authors \cite{BCS}. The main result of this paper proves the existence of local sections in a Serre fibration with all fibers homeomorphic to some fixed compact three-dimensional manifold.

Abstract:
We prove extension-dimensional versions of finite dimensional selection and approximation theorems. As applications, we obtain several results on extension dimension.

Abstract:
It was proved by H. Whitney in 1933 that it is possible to mark a point in all curves in a continuous way. The main result of this paper extends the Whitney theorem to dimensions 2 and 3. Namely, we prove that it is possible to choose a point continuously in all two-dimensional surfaces sufficiently close to a given surface, and in all 3-manifolds sufficiently close to a given 3-manifold.

Abstract:
Plasmodium falciparum causes the most virulent form of malaria and encodes a large number of molecular chaperones. Because the parasite encounters radically different environments during its lifecycle, many members of this chaperone ensemble may be essential for P. falciparum survival. Therefore, Plasmodium chaperones represent novel therapeutic targets, but to establish the mechanism of action of any developed therapeutics, it is critical to ascertain the functions of these chaperones. To this end, we report the development of a yeast expression system for PfHsp70-1, a P. falciparum cytoplasmic chaperone. We found that PfHsp70-1 repairs mutant growth phenotypes in yeast strains lacking the two primary cytosolic Hsp70s, SSA1 and SSA2, and in strains harboring a temperature sensitive SSA1 allele. PfHsp70-1 also supported chaperone-dependent processes such as protein translocation and ER associated degradation, and ameliorated the toxic effects of oxidative stress. By introducing engineered forms of PfHsp70-1 into the mutant strains, we discovered that rescue requires PfHsp70-1 ATPase activity. Together, we conclude that yeast can be co-opted to rapidly uncover specific cellular activities mediated by malarial chaperones.

Abstract:
We construct new tests of perturbative QCD by considering a hypothetical tau lepton of arbitrary mass, which decays hadronically through the electromagnetic current. We can explicitly compute its hadronic width ratio directly as an integral over the e^+ e^- annihilation cross section ratio, R_{e^+e^-}. Furthermore, we can design a set of commensurate scale relations and perturbative QCD tests by varying the weight function away from the form associated with the V-A decay of the physical tau. This method allows the wide range of the R_{e^+e^-} data to be used as a probe of perturbative QCD.

Abstract:
We review our recent works on tests of perturbative QCD, inspired by the relation between the hadronic decay of the tau lepton and the e+ e- annihilation into hadrons. First, we present a set of commensurate scale relations that probe the self-consistency of leading-twist QCD predictions for any observable which defines an effective charge. These tests are independent of the renormalization scheme and scale, and are applicable over wide data ranges. As an example we apply this approach to R_{e+ e-}. Second, using a differential form of these conmensurate scale relations, we present a method to measure the QCD Gell-Mann--Low Psi function.

Abstract:
Inspired by the relation between the hadronic decay of the tau lepton and the electron-positron annihilation into hadrons, we derive new tests of perturbative QCD. We design a set of commensurate scale relations to test the self-consistency of leading-twist QCD predictions for any observable which defines an effective charge. This method provides renormalization scheme and scale invariant probes of QCD which can be applied over wide data ranges.