Quotient sets have attracted the attention of mathe- maticians in the past three decades. The set of quotients of primes is dense in the positive real numbers and the set of all quotients of Gauss- ian primes is also dense in the complex plane. Sittinger has proved that the set of quotients of primes in an imaginary quadratic ring is dense in the complex plane and the set of quotients of primes in a real quadratic number ring is dense in R: An interesting open question is introduced by Sittinger: Is the set of quotients of Hurwitz primes dense in the quaternions? In this paper, we answer the question and prove that the set of all quotients of Hurwitz primes is dense in the quaternions.

In this paper, using the concept of attractive points of a nonlinear mapping, we obtain a strong convergence theorem of Halperns type [<i>Bull. Amer. Math. Soc</i>. <b>73</b> (1967), 957961] for a wide class of nonlinear mappings which contains nonexpansive mappings, nonspreading mappings and hybrid mappings in a Hilbert space. Using this result, we obtain well-known and new strong convergence theorems of Halperns type in a Hilbert space. In particular, we solve a problem posed by Kurokawa and Takahashi [<i>Nonlinear Anal</i>. <b>73</b> (2010, 15621568].

In this paper we construct a new time-periodic solution of the vacuum Einsteins field equations, this solution possesses physical singularities, i.e., the norm of the solutions Riemann curvature tensor takes the infinity at some points. We show that this solution is intrinsically time-periodic and describes a time-periodic universe with the time-periodic physical singularity. By calculating the Weyl scalars of this solution, we investigate new physical phenomena and analyze new singularities for this universal model.

General relativity explains gravitational radiation from binary black hole or neutron star mergers, from core-collapse supernovae and even from the inflation period in cosmology. These waves exhibit a unique effect called memory or Christodoulou effect, which in a detector like LIGO or LISA shows as a permanent displacement of test masses and in radio telescopes like NANOGrav as a change in the frequency of pulsars' pulses. It was shown that electromagnetic fields and neutrino radiation enlarge the memory. Recently it has been understood that the two types of memory addressed in the literature aslinear'andnonlinear'are in fact two different phenomena. The former is due to fields that do not and the latter is due to fields that do reach null infinity.

We define new energy momentum surface density and quasilocal mass for the boundary surface of a spacelike region in spacetime. Isometric embeddings of such a surface into the Minkowski space are used as reference surfaces in the definition. When the reference surface lies in a flat three-dimensional space slice of the Minkowski space, our expression recovers the Brown-York and Liu-Yau quasilocal mass. Moreover, we prove the positivity of the new quasilocal mass under the dominant energy condition and show that it is zero for surfaces in the Minkowski space.