many results were proved on the structure of finite groups with some‎ ‎restrictions on their real elements and on their conjugacy classes‎. ‎we‎ ‎generalize a few of these to some classes of infinite groups‎. ‎we study groups in which real elements are central‎, ‎groups in which real elements are $2$-elements‎, ‎groups in which all non-trivial classes have the same finite size and $fc$-groups w...

We prove that the statement ‘For all Borel ideals I and J on ω, every isomorphism between Boolean algebras P(ω)/I and P(ω)/J has a continuous representation’ is relatively consistent with ZFC. In this model every isomorphism between P(ω)/I and any other quotient P(ω)/J over a Borel ideal is trivial for a number of Borel ideals I on ω. We can also assure that the dominating number, d, is equal t...

Solovay showed that there are noncomputable reals α such that H(α ↾ n) 6 H(1) + O(1), where H is prefix-free Kolmogorov complexity. Such H-trivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an H-trivial real. We also analyze various computability-theoretic properties of the H-trivial reals, showing f...

It is pointed out that one-component φ theory in four dimensions has a non-perturbative sector that can be studied by means of an exact duality transformation of its Ising limit. This duality maps it to a membrane model. As a consequence, the φ-theory turns out to have, in the broken symmetry phase, a remarkably rich spectrum of physical states corresponding to membrane excitations. A numerical...

Using a construction similar to an iterated function system, but with functions changing at each step of iteration, we provide a natural example of a continuous one-parameter family of holomorphic functions of infinitely many variables. This family is parametrized by the compact space of positive integer sequences of prescribed growth and hence it can also be viewed as a parametric description ...

‎Triangular matrix rings are examples of trivial extensions‎. ‎In this article we determine the structure of derivations and biderivations of the trivial extensions‎, ‎and thereby we describe the derivations and biderivations of the upper triangular matrix rings‎. ‎Some related results are also obtained‎.

It is usually said that two-dimensional (1+1) gauge theories are trivial. In the continuum there are no transverse degrees of freedom to sustain any dynamics. Likewise, in the lattice formulation, the system factorizes (after gauge fixing) and a partition function reduces to a simple one-plaquette integral [1–3]. This picture neglects boundary conditions. It is well known [4] that, e.g., Quantu...