On density of compactly supported smooth functions in fractional Sobolev spaces

We describe some sufficient conditions, under which smooth and compactly supported functions are or are not dense in the fractional Sobolev space $$W^{s,p}(\Omega )$$ W s , p ( Ω ) for an open, bounded set $$\Omega \subset \mathbb {R}^{d}$$ Ω ⊂ R d . The density property is closely related to the lower and upper Assouad codimension of the boundary of $$\Omega$$ Ω . We also describe explicitly the closure of $$C_{c}^{\infty }(\Omega )$$ C c ∞ ( Ω ) in $$W^{s,p}(\Omega )$$ W s , p ( Ω ) under some mild assumptions about the geometry of $$\Omega$$ Ω . Finally, we prove a variant of a fractional order Hardy inequality.

UNION AND INTERSECTION OF CHAOTIC GRAPHS WITH DENSITY VARIATION

The aim of the present study is to discuss the union and intersection operations on chaotic graphs with density variation; the adjacency and incidence matrices representing the chaotic graphs induced from these operations will be introduced when physical characters of chaotic graphs have the same properties. There are several applications that have been utilized on chaotic graphs with density variation. The most practical applications of these kinds of operations on chaotic graphs with density variation are the internet signal speeds and the variation of green color for different parts of the plant. For example, in botany, in some cases, several plants suffer from a lack of chlorophyll in the damaged parts of the plant. In this case, the plant is represented by a chaotic graph, and the proportion of chlorophyll is represented by the density property, then the appropriate process is applied to increase the chlorophyll percentage in the appropriate place, so these operations help us to choose the suitable operator that satisfies our desires and requests. Keywords: adjacency matrix, incidence matrix, chaotic graph, density, union, intersection.

On the weakly-* dense subsets in $L^{\infty}(\Omega)$

In this paper we study the density property of the compactly supported smooth functions in the space $L^{\infty}(\Omega)$. We show that this set is dense with respect to the weak-* convergence in variable spaces.

A Study of The Density Property in Module Theory

In this paper, there are two main objectives. The first objective is to study the relationship between the density property and some modules in detail, for instance; semisimple and divisible modules. The Addition complement has a good relationship with the density property of the modules as this importance is highlighted by any submodule N of M has an addition complement with Rad(M)=0. The second objective is to clarify the relationship between the density property and the essential submodules with some examples. As an example of this relationship, we studied the torsion-free module and its relationship with the essential submodules in module M.

Punctured Parabolic Cylinders in Automorphisms of ℂ2

We show the existence of automorphisms $F$ of $\mathbb{C}^{2}$ with a non-recurrent Fatou component $\Omega $ biholomorphic to $\mathbb{C}\times \mathbb{C}^{*}$ that is the basin of attraction to an invariant entire curve on which $F$ acts as an irrational rotation. We further show that the biholomorphism $\Omega \to \mathbb{C}\times \mathbb{C}^{*}$ can be chosen such that it conjugates $F$ to a translation $(z,w)\mapsto (z+1,w)$, making $\Omega $ a parabolic cylinder as recently defined by L. Boc Thaler, F. Bracci, and H. Peters. $F$ and $\Omega $ are obtained by blowing up a fixed point of an automorphism of $\mathbb{C}^{2}$ with a Fatou component of the same biholomorphic type attracted to that fixed point, established by F. Bracci, J. Raissy, and B. Stensønes. A crucial step is the application of the density property of a suitable Lie algebra to show that the automorphism in their work can be chosen such that it fixes a coordinate axis. We can then remove the proper transform of that axis from the blow-up to obtain an $F$-stable subset of the blow-up that is biholomorphic to $\mathbb{C}^{2}$. Thus, we can interpret $F$ as an automorphism of $\mathbb{C}^{2}$.

NON-DENSITY OF GENERIC M-REDUCIBILITY FOR C.E. SETS

Generic-case approach to algorithmic problems was suggested by I. Kapovich, A. Myasnikov, V. Shpilrain and P. Schupp in 2003. This approach studies behavior of an al-gorithm on typical (almost all) inputs and ignores the rest of inputs. C. Jockusch and P. Schupp in 2012 began the study of generic computability in the context of classical computability theory. In particular, they defined a generic analog of Turing reducibility. A. Rybalov in 2018 introduced a generic analog of classical m-reducibility. In this paper we study the generic m-reducibility for c.e. sets and prove that unlike classical m-reducibility, generic m-reducibility does not have the density property for c.e. sets.