Narrow Results

In this paper, we prove that every matrix over a division ring is representable as a product of at most 10 traceless matrices as well as a product of at most four semi-traceless matrices. By applying this result and the obtained so far other results, we show that elements of some algebras possess some rather interesting and nontrivial decompositions into products of images of non-commutative polynomials.

We study structures on the fields of characteristic zero obtained by introducing (multi-valued) operations of raising to power. Using Hrushovski–Fraisse construction we single out among the structures exponentially-algebraically closed once and prove, under certain Diophantine conjecture, that the first order theory of such structures is model complete and every its completion is superstable.

Let F be a field and G an Abelian group. For every prime number q and every ordinal number α we compute only in terms of F and G the Warfield q-invariants W_{α, q}(VF[G]) of the group VF[G] of all normed units in the group algebra F[G] under some minimal restrictions on F and G.

This expands own recent results from (Extracta Mathematicae, 2005) and (Collectanea Mathematicae, 2008).

We determine entirely which Artinian rings have maximal subring. In particular, we show that an Artinian ring without maximal subring is integral over some finite subring and in particular that every Artinian ring which is uncountable or of characteristic zero has a maximal subring. We also determine when a finite direct product of rings has a maximal subring. Finally, we show that if a ring R has an Artinian maximal subring then R itself is Artinian.

Let $R$ be a ring and let $n$ be an arbitrary but fixed positive integer. We characterize those rings $R$ whose elements $a$ satisfy at least one of the relations that $a^n+a$ or $a^n-a$ is a nilpotent whenever $n\in \mathbb{N}\setminus \{1\}$. This extends results from the same branch obtained by Danchev [A characterization of weakly J(n)-rings, J. Math. Appl. 41 (2018) 53–61], Koşan et al. [Rings with $x^n-x$ nilpotent, J. Algebra Appl. 19 (2020)] and Abyzov and Tapkin [On rings with $x^n-x$ nilpotent, J. Algebra Appl. 21 (2022)], respectively.

In this paper, we will consider the birth and evolution of fields during formation of $N$-dimensional manifolds from joining point-like ones. We will show that at the beginning, only there are point-like manifolds which some strings are attached to them. By joining these manifolds, 1-dimensional manifolds are appeared and gravity, fermion, and gauge fields are emerged. By coupling these manifolds, higher dimensional manifolds are produced and higher orders of fermion, gauge fields and gravity are emerged. By decaying $N$-dimensional manifold, two child manifolds and a Chern–Simons one are born and anomaly is emerged. The Chern–Simons manifold connects two child manifolds and leads to the energy transmission from the bulk to manifolds and their expansion. We show that $F$-gravity can be emerged during the formation of $N$-dimensional manifold from point-like manifolds. This type of $F$-gravity includes both type of fermionic and bosonic gravity. $G$-fields and also $C$-fields which are produced by fermionic strings produce extra energy and change the gravity.

A Pythagorean (k, l)-tuple over a commutative ring A is a vector x = (x_i) ∈ A^k+l, where k, l ∈ ℕ, k ≥ l which satisfies . In this paper, a polynomial parametrization of Pythagorean (k, l)-tuples over the ring F[t] is given, for l ≥ 2. In the case where l = 1, solutions of the above equation are provided for k = 2, 3, 4, 5, and 9.

We make some attempts to define a general notion of groups and fields of dimension one, and to determine their algebraic properties.

We address the question of point particle motion coupled to classical fields, in the context of scalar fields derived from higher-order Lagrangians and BLTP electrodynamics.

In complex scalar fields, singularities of the phase (optical vortices, wavefront dislocations) are lines in space, or points in the plane, where the wave amplitude vanishes. Phase singularities are illustrated by zeros in edge diffraction and amphidromies in the heights of the tides. In complex vector waves, there are two sorts of polarization singularity. The polarization is purely circular on lines in space or points in the plane (C singularities); these singularities have index ±1/2. The polarization is purely linear on lines in space for general vector fields, and surfaces in space or lines in the plane for transverse fields (L singularities); these singularities have index ±1. Polarization singularities (C points and L lines) are illustrated in the pattern of tidal currents.