(1.1) 0 −→ N f −−→M g −−→ P −→ 0 which is exact at N , M , and P . That means f is injective, g is surjective, and im f = ker g. Example 1.1. For an R-module M and submodule N , there is a short exact sequence 0 // N // M // M/N // 0, where the map N →M is the inclusion and the map M →M/N is reduction modulo N . Example 1.2. For R-modules N and P , the direct sum N ⊕ P fits into the short exact...

We study the pure braid groups Pn(RP) of the real projective plane RP2, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence 1 −→ Pm(RP \ {x1, . . . , xn}) −֒→ Pn+m(RP 2) p∗ −→ Pn(RP) −→ 1, where n ≥ 2 and m ≥ 1, and p∗ is the homomorphismwhich corresponds geometrically to forgetting the last m strings. This problem is equivalent to that of the existence of a sect...

A linear deformation of a Meyer set M in R is the image of M under a group homomorphism of the group [M ] generated by M into R. We provide a necessary and sufficient condition for such a deformation to be a Meyer set. In the case that the deformation is a Meyer set and the deformation is injective, the deformation is pure point diffractive if the orginal set M is pure point diffractive. AMS Cl...

Seawater-soluble nutrient fractions of aerosols better represent the contribution of aerosol dry deposition to the nutrient load from this source to the ocean than any estimates based on aerosol nutrients leached in pure water or acidic solutions. To understand the solubility difference between seawater and pure water, 31 pairs of aerosol samples collected from the Gulf of Aqaba were extracted ...

In the present paper, an exact analytic solution for the optimal unambiguous state discrimination(OPUSD) problem involving an arbitrary number of pure linearly independent quantum states with real and complex inner product is presented. Using semidefinite programming and Karush-Kuhn-Tucker convex optimization method, we derive an analytical formula which shows the relation between optimal solut...