Narrow Results

In $1964$, Auslander conjectured that every crystallographic subgroup $\mathrm{\Gamma }$ of the affine group $G{L}_n(ℝ)⋉{ℝ}^n$ acting properly discontinuously on ${ℝ}^n$ is virtually solvable, i.e. contains a solvable subgroup of finite index. One of the major difficulties to prove such a conjecture is to generate a criterion of the proper action of a given discrete group acting properly discontinuously on ${ℝ}^n$. In this paper, we focus on this question in the setting where $\mathrm{\Gamma }$ is abelian. We show that $\mathrm{\Gamma }\simeq {\mathbb{Z}}^n$ acts properly on ${ℝ}^n$ if and only if any $\mathrm{\Gamma }$-orbit is discrete (or equivalelntly closed). We further generate an equivalent criterion in the setting where $\mathrm{\Gamma }$ stands for a crystallographic discontinuous affine group for ${ℝ}^n$ based upon a generating family.

We investigate the time-dependent behaviour of brane-antibrane orbital systems, mainly concentrating on the stability of the "Branonium" configuration.

We devise an algorithm which, given a bounded automaton $A$, decides whether the group generated by $A$ is finite. The solution comes from a description of the infinite sequences having an infinite $A$-orbit using a deterministic finite-state acceptor. This acceptor can also be used to decide whether the bounded automaton acts level-transitively.

In 2006, Nelson and Wong introduced that every finite quandle can be decomposed into a disjoint union of some quandles and for given $n$ quandles satisfying certain conditions, there is a quandle which is a disjoint union of the $n$ quandles. In 2008, Ehrman, Gurpinar, Thibault and Yetter also introduced similar results about the decomposition of quandles. In this paper, we observe an operation table $Q$ which consists of four sub-operation tables, where diagonal tables are quandle operations. We study which conditions make $Q$ a quandle operation table.

Let $D=(V,A)$ be a simple strongly connected digraph and let Aut$(D)$ be an automorphism of $V(D)$. For $x\in V(D)$, the set $\{x^g:g\in \mathrm{\text{Aut}}(D)\}$ is called an orbit of Aut$(D)$. In this paper, first, we show that if $D$ is a 2-regular strongly connected digraph with two orbits then $\lambda (D)=2$, if $D$ is a $k$-regular strongly connected digraph with two orbits and $\lambda (G)<k$$(k\ge 3)$, then $\lambda (G)\ge 2$. Second, we prove that if $g(D)\ge 3$, then $\lambda (D)=3$. Last, we characterize the arc atoms of 3-regular strongly connected digraphs with two orbits and $\lambda (D)=2$.

Surprisingly, skew derivations rather than ordinary derivations are more basic (important) object in study of the Grassmann algebras. Let Λ_n = K ⌊x₁, …, x_n⌋ be the Grassmann algebra over a commutative ring K with ½ ∈ K, and δ be a skew K-derivation of Λ_n. It is proved that δ is a unique sum δ = δ^ev + δ^od of an even and odd skew derivation. Explicit formulae are given for δ^ev and δ^od via the elements δ (x₁), …, δ (x_n). It is proved that the set of all even skew derivations of Λ_n coincides with the set of all the inner skew derivations. Similar results are proved for derivations of Λ_n. In particular, Der_K(Λ_n) is a faithful but not simple Aut_K(Λ_n)-module (where K is reduced and n ≥ 2). All differential and skew differential ideals of Λ_n are found. It is proved that the set of generic normal elements of Λ_n that are not units forms a single Aut_K(Λ_n)-orbit (namely, Aut_K(Λ_n)x₁) if n is even and two orbits (namely, Aut_K(Λ_n)x₁ and Aut_K(Λ_n)(x₁ + x₂ ⋯ x_n)) if n is odd.

Let $G$ be a permutation group of degree $n$ and let $s(G)$ denote the number of set-orbits of $G$. We determine $inf(\frac{{log}_{2}s(G)}{n})$ over all groups $G$ that satisfy certain restrictions on composition factors.

Here, we study the action of the groups $H(\lambda )$ generated by the linear fractional transformations $x:z\mapsto -\frac{1}{z}\text{ and }w:z\mapsto z+\lambda$ and $\lambda$ is a positive integer, on the subsets ${\mathbb{Q}}^{\ast }(\sqrt{n})=\{\frac{a+\sqrt{n}}{c}|a,b=\frac{a^2-n}{c},c\in \mathbb{Z}\}\cup \{0,1,\infty \}$ and $\left|n\right|$ is a square-free positive integer. We prove that this action has a finite number of orbits if and only if $\lambda =1$ or $\lambda =2$. Moreover, we give an upper bound for the number of orbits for $\lambda =2$.

Let G be a nonabelian finite group. Then Irr(G/G') is an abelian group under the multiplication of characters and acts on the set of non-linear irreducible characters of G via the multiplication of characters. The purpose of this paper is to establish some facts about the action of linear character group on non-linear irreducible characters and determine the structures of groups G for which either all the orbit kernels are trivial orthe number of orbits is at most two. Using the established results on this action, it is very easy to classify groups G having at most three non-linear irreducible characters.

In this paper, we introduce some Fuchsian groups and calculate their parabolic class numbers. In addition, we give two conjectures concerning with the parabolic class number.

The paper shows a mathematical model for vibration analysis of soft mounted two-pole induction motors regarding electromagnetic excitation due to static rotor eccentricity. A static rotor eccentricity causes an electromagnetic force, acting on the rotor and on the stator and oscillating with the double supply frequency. This magnetic force is implemented into a simplified analytical machine dynamic model and the correlations between the rotor dynamics, electromagnetic, oil film characteristics of the sleeve bearings, and the stiffness and damping of a soft foundation are mathematically described. The derived results are clarified using an example that shows the influence of the rotor speed and the direction of the magnetic force on the vibration behavior. On one hand the aim of the paper is to show the mathematical correlations, based on a simplified model. On the other hand, the aim is to derive a method for calculating the forced vibrations — as a worst case — caused by a static rotor eccentricity. Therefore, the paper shall prepare the basis for implementing this method in more detailed numerical programs, e.g., finite element programs.

Let $b\ge 2$ be an integer, and write the base $b$ expansion of any non-negative integer $n$ as $n=x_0+x_{1}b+\cdots +x_d{b}^d$, with $x_d>0$ and $0\le x_i<b$ for $i=0,\dots ,d$. Let $\varphi (x)$ denote an integer polynomial such that $\varphi (n)>0$ for all $n>0$. Consider the map $S_{\varphi ,b}:{\mathbb{Z}}_{\ge 0}\to {\mathbb{Z}}_{\ge 0}$, with $S_{\varphi ,b}(n):=\varphi (x_0)+\cdots +\varphi (x_d)$. It is known that the orbit set $\{n,S_{\varphi ,b}(n),S_{\varphi ,b}(S_{\varphi ,b}(n)),\dots ,\}$ is finite for all $n>0$. Each orbit contains a finite cycle, and for a given $b$, the union of such cycles over all orbit sets is finite.

Fix now an integer $\ell \ge 1$ and let $\varphi (x)=x^2$. We show that the set of bases $b\ge 2$ which have at least one cycle of length $\ell$ always contains an arithmetic progression and thus has positive lower density. We also show that a 1978 conjecture of Hasse and Prichett on the set of bases with exactly two cycles needs to be modified, raising the possibility that this set might not be finite.

Based on Bohr’s model of hydrogen atom, the main purpose of this paper is to propose a new hypothesis for the model of hydrogen atom and perform mathematical calculations and verifications using experimental values and physical constants. The new model can not only provide the explanations that the existing theories failed to provide, but also can be used to construct neutrons, atomic nuclei, and individual atoms and molecules by involving only electrons and protons and their interactions. This means we only need to know how single electron and proton interact with each other, then we can establish a full model of hydrogen atom, and then we can establish other atomic nuclei. After all, all matter in the universe is made from hydrogen atoms through nuclear fusion.

Because of their negligible mass and size, asteroids act as particle test in the gravitational field of the Sun (or the solar system at large); hence the knowledge of their orbit can provide local tests of general relativity. In addition to the "3D census of the Galaxy" the space-mission Gaia will enable, this ESA astrometric mission will provide highly accurate positions of a large number of solar systems objects. Given the expected nominal precision — ranging from a few milli-arcseconds for the faintest bodies down to sub-mas precision for the brightest ones — one can expect to perform the classical perihelion precession test of the GR from the motion of the asteroids. We present preliminary results of a variance analysis involving realistic simulations of a subset of asteroids including Near-Earth objects and main-belt asteroids that will be observed by Gaia. These show the formal precision achievable for the joint determination of the Solar J₂ together with the PPN parameter β, as well as the precision for and the link of the dynamical reference frame to the kinematically non-rotating conventional ICRS.

Let n be any positive integer and ℤ_n = {0, 1, …, n - 1} be the ring of integers modulo n. Let X_n be the set of all nonzero, nonunits of ℤ_n, and G_n the group of all units of ℤ_n. In this paper, by considering the regular representation π : G_n → Sym(X_n), the following are investigated as follows: (1) G_n is not fixed-point free; (2) If Fix(g) = {x ∈ X_n : gx = x }≠ ∅ for some g ∈ G_n, then Fix(g) is a union of orbits under the regular action of G_n on X_n; (3) B(⊆ o(x₁) ∪ ⋯ ∪ o(x_ℓ)) is a set of imprimitivity under π for some orbits o(x₁), …, o(x_ℓ) if and only if B = g₁H₁x₁∪⋯∪g_ℓH_ℓx_ℓ for some subgroups H₁, …, H_ℓ of G_n and some elements g₁, …, g_ℓ ∈ G_n satisfying that if (gB)∩B ≠ ∅ for some g ∈ G, then g ∈ stab(x_i)H_i for each i = 1, …, ℓ.

We prove that each residuated commutative ℓ-monoid is a monoid of all residuated endomorphisms of some universal algebra. This result demonstrates that similarly to groups and semigroups, a monoidal operation of a residuated commutative ℓ-monoid is represented by a composition of endomorphisms that are moreover, residuated.