BackgroundSleeping and crying are normal activities of infants. Infant crying and night wakings can be both distressing and exhausting for parents. At its worse it may be associated with an increased ...risk of maternal depression and psychosocial stress for both parents. Strategies for reducing crying and improving sleep include the five "S's"-swaddling, side/stomach position, sucking, swinging and shushing sounds simulating "womb-like" sensations. The "SNOO" Smart Sleeper (SNOO), a "smart" bassinet, incorporates 3 of the five "S's", swaddling, swinging (rocking) and emits soothing sounds while demonstrating safe infant sleep practices. This paper explores the effectiveness of the five "S's" and the SNOO. MethodsReferences for the five "S's" were obtained from various sources while a scoping review of publications from PubMed, Embase and Web of Science was undertaken to seek out relevant studies to document the efficacy of the SNOO. ResultsThe five "S's" appear to help soothe infants, reduce their crying and improve their sleep. In addition, infant obesity rates fell. Infants also experienced less pain following immunisations. Of the 66 papers gleaned from the database in mid-2021 for the scoping review, only those which provided clear outcomes and conclusions, were complete and related to infants were included. That resulted in only 2 studies that fitted the criteria imposed. They suggested that the SNOO incorporating 3 of the five "S's" had similar beneficial effects. ConclusionsThe five "S's" were effective non-pharmacological strategies to help reduce crying and improve sleep in infants. Confounding factors included normal crying of infants, triggers of hunger or tiredness, or recognised causes of crying. The 2 studies reviewed suggested that the SNOO was helpful in reducing crying and improving the sleep duration of normal infants. Further studies have suggested it may be used therapeutically for distressed or ill infants.
In this note the three dimensional Dirac operator
A
m
with boundary conditions, which are the analogue of the two dimensional zigzag boundary conditions, is investigated. It is shown that
A
m
is ...self-adjoint in
L
2
(
Ω
;
C
4
)
for any open set
Ω
⊂
R
3
and its spectrum is described explicitly in terms of the spectrum of the Dirichlet Laplacian in
Ω
. In particular, whenever the spectrum of the Dirichlet Laplacian is purely discrete, then also the spectrum of
A
m
consists of discrete eigenvalues that accumulate at
±
∞
and one additional eigenvalue of infinite multiplicity.
The main goal of this paper is to provide a complete description of the operator solutions (eventually multivalued) of the (multivalued) operator equation
A
∗
A
=
λ
A
n
, where
n
is a positive ...integer,
A
is a closed multivalued linear operator (a closed linear relation) on a complex Hilbert space
H
and
λ
is a complex number.
General properties of eigenvalues of
A
+
τ
u
v
∗
as functions of
τ
∈
C
or
τ
∈
R
or
τ
=
e
i
θ
on the unit circle are considered. In particular, the problem of existence of global analytic formulas for ...eigenvalues is addressed. Furthermore, the limits of eigenvalues with
τ
→
∞
are discussed in detail. The following classes of matrices are considered: complex (without additional structure), real (without additional structure), complex
H
-selfadjoint and real
J
-Hamiltonian.
We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare their number of Jordan chains of length at least
n
. ...In the operator case, it was recently proved that the difference of these numbers is independent of
n
and is at most the defect between the operators. One of the main results of this paper shows that in the case of linear relations this number has to be multiplied by
n
+
1
and that this bound is sharp. The reason for this behavior is the existence of singular chains. We apply our results to one-dimensional perturbations of singular and regular matrix pencils. This is done by representing matrix pencils via linear relations. This technique allows for both proving known results for regular pencils as well as new results for singular ones.
This short note has to make alive the overlooked work of Pedrick. It is rather a tour guide than exhausted examination of the content and intends to serve potential explorers of diverse kinds of ...reproducing kernel (Hilbert) spaces, the topic mushrooming nowadays.
Let
H
=
H
⊕
K
be the direct sum of two Hilbert spaces. In this paper we characterise the semi-projections (defined in the paper) and projections with a given kernel and a given range that can be ...described by a two by two matrix or block of relations determined by the decompositions of
H
=
H
1
⊕
H
2
and of
K
=
K
1
⊕
K
2
. This generalises the Stone - de Snoo (Oral communication to the author, 1992; J Indian Math Soc 15: 155–192, 1952) formula for the orthogonal projection on the graph of a closed linear relation, and extends the results of Mezroui (Trans AMS 352: 2789–2800, 1999) on the same subject. This requires some new results concerning blocks of linear relations as studied in (Adv Oper Theory 5: 1193–1228, 2020). Some applications are given on the product of two relations including one contained in (Complex Anal Oper Theory 6: 613–624, 2012).
A Survey of Some Norm Inequalities Gesztesy, Fritz; Nichols, Roger; Stanfill, Jonathan
Complex analysis and operator theory,
03/2021, Letnik:
15, Številka:
2
Journal Article
Recenzirano
Odprti dostop
We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type
‖
A
f
‖
X
2
≤
C
‖
f
‖
X
‖
A
2
f
‖
X
,
f
∈
dom
(
A
2
)
,
and recall that under ...exceedingly stronger hypotheses on the operator
A
and/or the Banach space
X
, the optimal constant
C
in these inequalities diminishes from 4 (e.g., when
A
is the generator of a
C
0
contraction semigroup on a Banach space
X
) all the way down to 1 (e.g., when
A
is a symmetric operator on a Hilbert space
H
). We also survey some results in connection with an extension of the Hardy–Littlewood inequality involving quadratic forms as initiated by Everitt.
The Krein–von Neumann extension is studied for Schrödinger operators on metric graphs. Among other things, its vertex conditions are expressed explicitly, and its relation to other self-adjoint ...vertex conditions (e.g. continuity-Kirchhoff) is explored. A variational characterisation for its positive eigenvalues is obtained. Based on this, the behaviour of its eigenvalues under perturbations of the metric graph is investigated, and so-called surgery principles are established. Moreover, isoperimetric eigenvalue inequalities are obtained.
Cloning of Symmetric Operators Arlinskiĭ, Yu. M.
Complex analysis and operator theory,
02/2021, Letnik:
15, Številka:
1
Journal Article
Recenzirano
Given a closed densely defined symmetric operator
S
in a separable Hilbert space, by means of a non-real and non-imaginary complex number
z
and the corresponding deficiency subspace
N
z
of
S
we ...define a new closed symmetric operator
S
(
z
) with dense domain. We prove that the operator
S
(
z
) preserves various properties of
S
. When the deficiency indices of
S
are equal a bijection of the set of all selfadjoint extensions of
S
onto the set of all selfadjoint extensions of
S
(
z
) is established. We consider in detail the case when a symmetric operator
S
is nonnegative.
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