Abstract In this paper, we are committed to investigating the fractal decomposition of power sets. Our main result is that every power set can be decomposed into a sum of a power set and an ...isomorphic set that does not intersect with it. For the finite power set, this property can be drawn on the ordinal line by constructing the fractal number axis of the ordinal line, and the fractal distribution and fractal graph of the finite power set can be obtained by using the parallel translation drawing method. Moreover, the distributions do not overlap or cross. The results in this paper provide a new perspective for further investigation of the fractal distribution of power sets.
We consider a differential geometric setting on power sets and Borel algebras. Our chosen framework is based on diffeologies, and we make a link between the various diffeological structures that we ...propose, having in mind set-valued maps, relations, set-valued gradients, differentiable measures, and shape analysis. This work intends to establish rigorous properties on sample diffeologies that seem of interest to us.
Let (X,
) be a topological vector space equipped with a partial order
which is induced by a nonempty pointed closed and convex cone K in X. Let
(X) = 2
X
be the power set of X and
(X) = 2
X
\{
}. In ...this paper, we consider the so called upward preordering
on
(X), which has been used by many authors (see 4
7, 9, 11
14, 18
19). We first prove some properties of this ordering relation
Let
(X) denote the collection of all
-closed subsets of X and
(X) =
(X)\{
}, which is equipped with the Fell topology
Let C be a nonempty subset of X and let F: C
(X) be a closed set-valued mapping. By applying the Fell topology
on
(X) and the Fan-KKM Theorem, we prove some existence theorems for some
-minimization and
-maximization problems with respect to F subject to the subset C for first countable topological vector spaces. These results will be applied to solve some closed ball-valued optimization problems in partially ordered Hilbert spaces. Some examples will be provided in
If
is a relation on
to
is a relation on
(
) to
, and
is a relation on
(
) to
(
), then we say that
is an ordinary relation,
is a super relation, and
is a hyper relation on
to
Motivated by an ...ingenious idea of Emilia Przemska on a unified treatment of open- and closed-like sets, we shall introduce and investigate here four reasonable notions of product relations for super relations.
In particular, for any two super relations
and
on
, we define two super relations
*
and
*
, and two hyper relations
★
and
*
on
such that :
and
for all
⊆
By using the distributivity of the operation ∩ over ∪, we can at once see that
*
⊆
*
. Moreover, if
⊆
, then we can also see that
*
=
*
. The most simple case is when
is an interior relation on
and
is the associated closure relation defined such that
(
) =
(
for all
⊆
In our super-aging society we are faced with an ever-increasing problem that needs to be resolved, namely accommodation for the elderly. To solve this problem we must first clarify its nature. We ...know that, by 2030, 10% of the elderly will have dementia and another 40% will be living alone. Therefore, it is important that, right now, we think about the emergency measures we will need to implement on a national scale to solve these problems in 2025, when the baby boomers reach the age of 75. The elderly face not only a rising incidence of diseases such as dementia and the biological factors associated with aging, but also social factors such as lack of communication. In particular, we have found that the incidence of dementia differs significantly between those with the presence or absence of a social network. Also, the number of people moving into care facilities for the elderly has been increasing in recent years, and this trend is an important social phenomenon. Here, we performed a basic study of the process used to select facilities for the elderly. First, as a basic structure, we focused on the attributes of facilities for the elderly. Next, we examined the basic attributes of the elderly as institutional tenants. We then computed a power set by using these basic attributes as elements. From the partially ordered set (poset) we derived a possible combination of elements to use in selecting facilities for the elderly. We found that the elements of the power set obtained by using the attributes of elderly people as institutional tenants constituted fuzzy measures. We showed clearly that comprehensive evaluation by using fuzzy integration was possible.
The generator vibration of the unit is sensitive to the running conditions for the low-head turbine group's low head, high flow characteristics. The generator local stress concentration often leads ...to fatigue rupture of the metal structure, which will also affect its reliability. It easily leads the generator bracket of oblique arm structure to cracking and rupture. The reason of the cracking and rupture is proved by numerical analysis, and its repair plan has been identified. On-site repair process should be strictly controlled, to ensure the greatest degree of welding quality and the elimination of welding stresses. Running for some time after repair, the main parameters, such as air-gap and other unit operation, have been analyzed. The results are satisfying, and confirm that, the analysis of the generator bracket crack repair is objective and reasonable.
It is still unknown whether three mutually orthogonal Latin squares (resp. quasigroups) of order 10 exist or whether there is a check digit system of order 10 which detects all twin errors. During ...our research on these topics we use an approach with
half quasigroups, which leads to an interesting generalization of quasigroup orthogonality. A (vertical) half quasigroup
(
H
,
∗
)
is a groupoid for which the right cancellation law
x
∗
y
=
x
′
∗
y
⇒
x
=
x
′
holds. It is close related to what is known as row or column Latin square. The set of all half quasigroups
H
n
of order
n
together with an operation
⋅
builds a group
(
H
n
,
⋅
)
and the set of quasigroups
Q
n
is a subset of
H
n
. Two half quasigroups
h
,
g
∈
H
n
are orthogonal if and only if a quasigroup
q
∈
Q
n
exists with
h
⋅
q
=
g
. We show that this is just a special case and can be generalized to arbitrary groups.
Furthermore, we prove a conjecture of Dénes, Mullen and Suchower about Latin power sets by showing that for all orders
n
≠
2
,
6
there is a quasigroup
q
of order
n
with
q
2
∈
Q
n
and
q
is orthogonal to
q
2
. Moreover, a computer search verifies a result of Wanless that there is no quasigroup
q
of order
10
having
q
2
and
q
3
∈
Q
10
.
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