Solving the brachistochrone and other variational problems with soap films

We show a method to solve the problem of the brachistochrone as well as other variational problems with the help of the soap films that are formed between two suitable surfaces. We also show the interesting connection between some variational problems of dynamics, statics, optics, and elasticity.Comment: 16 pages, 11 figures. This article, except for a small correction, has been submitted to the American Journal of Physic

The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups

In this paper we search for conditions on a countably compact (pseudo-compact) topological semigroup under which: (i) each maximal subgroup $H(e)$ in $S$ is a (closed) topological subgroup in $S$; (ii) the Clifford part $H(S)$(i.e. the union of all maximal subgroups) of the semigroup $S$ is a closed subset in $S$; (iii) the inversion $\operatorname{inv}\colon H(S)\to H(S)$ is continuous; and (iv) the projection $\pi\colon H(S)\to E(S)$, $\pi\colon x\longmapsto xx^{-1}$, onto the subset of idempotents $E(S)$ of $S$, is continuous

Topological monoids of monotone injective partial selfmaps of N\mathbb{N} with cofinite domain and image

In this paper we study the semigroup $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ of partial cofinal monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. We also prove that every locally compact topology $\tau$ on $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ such that $(\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N}),\tau)$ is a topological inverse semigroup, is discrete. Finally, we describe the closure of $(\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N}),\tau)$ in a topological semigroup

BRCA2 polymorphic stop codon K3326X and the risk of breast, prostate, and ovarian cancers

Background: The K3326X variant in BRCA2 (BRCA2*c.9976A>T; p.Lys3326*; rs11571833) has been found to be associated with small increased risks of breast cancer. However, it is not clear to what extent linkage disequilibrium with fully pathogenic mutations might account for this association. There is scant information about the effect of K3326X in other hormone-related cancers. Methods: Using weighted logistic regression, we analyzed data from the large iCOGS study including 76 637 cancer case patients and 83 796 control patients to estimate odds ratios (ORw) and 95% confidence intervals (CIs) for K3326X variant carriers in relation to breast, ovarian, and prostate cancer risks, with weights defined as probability of not having a pathogenic BRCA2 variant. Using Cox proportional hazards modeling, we also examined the associations of K3326X with breast and ovarian cancer risks among 7183 BRCA1 variant carriers. All statistical tests were two-sided. Results: The K3326X variant was associated with breast (ORw = 1.28, 95% CI = 1.17 to 1.40, P = 5.9x10- 6) and invasive ovarian cancer (ORw = 1.26, 95% CI = 1.10 to 1.43, P = 3.8x10-3). These associations were stronger for serous ovarian cancer and for estrogen receptor–negative breast cancer (ORw = 1.46, 95% CI = 1.2 to 1.70, P = 3.4x10-5 and ORw = 1.50, 95% CI = 1.28 to 1.76, P = 4.1x10-5, respectively). For BRCA1 mutation carriers, there was a statistically significant inverse association of the K3326X variant with risk of ovarian cancer (HR = 0.43, 95% CI = 0.22 to 0.84, P = .013) but no association with breast cancer. No association with prostate cancer was observed. Conclusions: Our study provides evidence that the K3326X variant is associated with risk of developing breast and ovarian cancers independent of other pathogenic variants in BRCA2. Further studies are needed to determine the biological mechanism of action responsible for these associations

Embedding in compact uniquely divisible semigroups

D. R. Brown and M. Friedberg have conjectured that each compact abelian semigroup can be embedded in a compact divisible semigroup. V. R. Hancock proved that each abelian algebraic semigroup can be embedded in a divisible abelian algebraic semigroup. In this paper we provide a partial solution to the conjecture of Brown and Friedberg by employing a topological version of Hancock\u27s method as part of our construction. A theorem giving sufficient conditions for the Bohr compactification of weakly reductive semigroups to be injective is proved and used in the proof of our main result. © 1972 Springer-Verlag New York Inc

The Structure of Commutative Semigroups With the Ideal Retraction Property

This paper presents a complete characterization of commutative semigroups with the ideal retraction property. These semigroups are those with the property that each ideal is a homomorphic retraction of the semigroup. The fundamental building blocks of these semigroups are the 2-cores and the semilattice of idempotents. Structure for semilattices with the ideal retraction property was discussed in an earlier paper and the structure of the 2-core is described in detail within this paper

The translational hull of a topological semigroup

This paper is concerned with three aspects of the study of topological versions of the translational hull of a topological semigroup. These include topological properties, applications to the general theory of topological semigroups, and techniques for computing the translational hull. The central result of this paper is that if S is a compact reductive topological semigroup and its translational hull &(£) is given the topology of continuous convergence (which coincides with the topology of pointwise convergence and the compact- open topology in this case), then £l(S) is again a compact topological semigroup. Results pertaining to extensions of bitranslations are given, and applications of these together with the central result to semigroup compactifications and divisibility are presented. Techniques for determining the translational hull of certain types of topological semigroups, along with numerous examples, are set forth in the final section. © 1976 American Mathematical Society

On the multilinear Hausdorff problem of moments

Given a multi-index sequence necessary and sufficient conditions are given for the existence of a regular Borel polymeasure on the unit interval such that . This problem will be called the weak multilinear Hausdorff problemof moments for . Comparison with classical results will allow us to relate the weak multilinear Hausdorff problem with the multivariate Hausdorff problem. A solution to the strong multilinear Hausdorff problem of moments will be provided by exhibiting necessary and sufficient conditions for the existence of a Radon measure on such that where is the -linear moment functional on the space of continuous functions on the unit interval defined by the sequence . Finally the previous results will be used to provide a characterization of a class of weakly harmonizable stochastic processes with bimeasures supported on compact sets