A three-point generalization of the ellipse

Given three points in the plane, interest is in the locus of all points for which the sum of the distances to the given points is a prescribed constant. These curves turn out to be sixth degree polynominals in x and y, and thus are complicated. However, it turns out that often there is a point, within the triangle formed by the three given points, for which the constant is a minimum. This point is determined for isosceles triangles, the general case is discussed, although all details are not revealed. Student problems, such as supplying the missing details mentioned and further generalizations, are suggested.     

Minimizing the lateral surface area of a cone

Bayes’ theorem is often applied in problems that essentially involve finding the accuracy of atesting device, given the sensitivity of the device. It is shown how linear programming can be used to determine (in a certain sense) the minimum sensitivity needed for a given prescribed minimum accuracy.     

Total synthesis of spiruchostatin A, a potent histone deacetylase inhibitor

The total synthesis of spiruchostatin A was accomplished, unambiguously confirming its structure. Key steps included the use of the Nagao thiazolidinethione auxiliary for a diastereoselective acetate aldol reaction and as an activated acylating agent for amide formation, and macrolactonization by the Yamaguchi protocol. Spiruchostatin A is shown to have biological activity similar to that of FK228, a potent histone deacetylase (HDAC) inhibitor in clinical trials. The spiruchostatin A analogue, epimeric at the beta-hydroxy acid, is inactive, highlighting the importance of stereochemistry at this position for interactions with HDACs.     

EXAFS characterization of Ti/Al2O3 supports and Co/Ti/Al2O3 catalysts

The structure of Ti/Al₂O₃ supports (0–14 wt% Ti) and Co/Ti/Al₂O₃ catalysts (3 wt% Co) was examined by EXAFS. The results indicated that the Ti was present primarily as a highly dispersed surface phase. The Ti EXAFS results indicated that the Ti species were octahedrally coordinated. Evidence of Ti—Ti interactions was found for all loadings (2–14 wt% Ti) suggesting that the Ti surface species are present as small clusters of TiO₂.The Co EXAFS results showed evidence for several structurally different Co surface phases as a function of Ti loading. Evidence of a Co species interacting with the Ti surface phase was observed for the 3% Co/2% Ti-3%Co/6%Ti catalysts. At the highest loadings studied, 3%Co/8%Ti and 3%Co/14%Ti, evidence was found for a CoTiO₃-like phase.     

Trilocal structures. I. Secular equation

The principles used with the bilocal photon are applied to the analysis of trilocal structures. From the trilocal wave equation, and the phase-space boundary condition (that the structure float on a Fermi sea filling the vacuum), a secular equation is obtained, an eighth-order polynomial in the energy of the structure. Requiring that the structure move as a particle, withw ²=k ²+m ², provides explicit expressions for two auxiliary parameters, and, to be used later in an expansion of the wave function.     

Rapid identification of stress-related fingerprint from whole bacterial cells of <Emphasis Type="Italic">Bifidobacterium lactis</Emphasis> using matrix assisted laser desorption/ionization mass spectrometry

Whole cells of Bifidobacterium lactis were analyzed by matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOFMS). Characteristic and reproducible mass spectra were obtained in the mass range from 6 to 19 kDa. After several days of bacterial cell storage at 4 degrees C (D0, D2, and D6), only minor signal differences were observed. Under identical and reproducible conditions, fourteen relevant diagnostic ions were identified. Moreover, control- and stress-related fingerprints were rapidly obtained using MALDI-TOFMS by comparison of protein patterns obtained from non-stressed (control) versus stressed cells (addition of bile salts during growth). After quantitative validation of the MALDI-MS data by a statistical approach, two and eight signals were assigned as control- and stress-specific ions, respectively. This work provides the evidence that MALDI-TOFMS can be used for the identification of stress-related fingerprint of B. lactis bacterial cells and could have a high potential for the assessment of the physiological status of the cells.     

Regular and normal closure operators and categorical compactness for groups

For a class of groupsF, closed under formation of subgroups and products, we call a subgroupA of a groupG F-regular provided there are two homomorphismsf, g: G » F, withF F, so thatA = {x G |f(x) =g(x)}.A is calledF-normal providedA is normal inG andG/A F. For an arbitrary subgroupA ofG, theF-regular (respectively,F-normal) closure ofA inG is the intersection of allF-regular (respectively,F-normal) subgroups ofG containingA. This process gives rise to two well behaved idempotent closure operators.A groupG is calledF-regular (respectively,F-normal) compact provided for every groupH, andF-regular (respectively,F-normal) subgroupA ofG × H, ₂(A) is anF-regular (respectively,F-normal) subgroup ofH. This generalizes the well known Kuratowski-Mrówka theorem for topological compactness.In this paper, theF-regular compact andF-normal compact groups are characterized for the classesF consisting of: all torsion-free groups, allR-groups, and all torsion-free abelian groups. In doing so, new classes of groups having nice properties are introduced about which little is known.