We consider the coorbit theory associated to a square-integrable, irreducible quasi-regular representation of a semidirect product group G=Rd⋊H. The existence of coorbit spaces for this very general ...setting has been recently established, together with concrete vanishing moment criteria for analyzing vectors and atoms that can be used in the coorbit scheme. These criteria depend on fairly technical assumptions on the dual action of the dilation group, and it is one of the chief purposes of this paper to considerably simplify these assumptions.
We then proceed to verify the assumptions for large classes of dilation groups, in particular for all abelian dilation groups in arbitrary dimensions, as well as a class called generalized shearlet dilation groups, containing and extending all known examples of shearlet dilation groups employed in dimensions two and higher. We explain how these groups can be systematically constructed from certain commutative associative algebras of the same dimension, and give a full list, up to conjugacy, of shearing groups in dimensions three and four. In the latter case, three previously unknown groups are found.
As a result, the existence of Banach frames consisting of compactly supported wavelets, with simultaneous convergence in a whole range of coorbit spaces, is established for all groups involved.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
On sheaves in finite group representations Xiong, Tengfei; Xu, Fei
Journal of pure and applied algebra,
October 2022, 2022-10-00, Volume:
226, Issue:
10
Journal Article
Peer reviewed
Open access
Given a general finite group G, we consider several categories built on it, their Grothendieck topologies and resulting sheaf categories. For a certain class of transporter categories and their ...quotients, equipped with atomic topology, we explicitly compute their sheaf categories via sheafification. This enables us to identify G-representations with various fixed-point sheaves. As a consequence, it provides an intrinsic new proof to the equivalence of M. Artin between the category of sheaves on the orbit category and that of group representations.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK
This paper presents a simple and efficient method to determine the self-equilibrated configurations of prismatic tensegrity structures, nodes and members of which have dihedral symmetry. It is ...demonstrated that stability of this class of structures is not only directly related to the connectivity of members, but is also sensitive to their geometry (height/radius ratio), and is also dependent on the level of self-stress and stiffness of members. A catalogue of the structures with relatively small number of members is presented based on the stability investigations.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Let G be an arbitrary group. We define a gain-line graph for a gain graph (Γ,ψ) through the choice of an incidence G-phase matrix inducing ψ. We prove that the switching equivalence class of the gain ...function on the line graph L(Γ) does not change if one chooses a different G-phase inducing ψ or a different representative of the switching equivalence class of ψ. In this way, we generalize to any group some results proven by N. Reff in the abelian case. The investigation of the orbits of some natural actions of G on the set HΓ of G-phases of Γ allows us to characterize gain functions on Γ, gain functions on L(Γ), their switching equivalence classes and their balance property. The use of group algebra valued matrices plays a fundamental role and, together with the matrix Fourier transform, allows us to represent a gain graph with Hermitian matrices and to perform spectral computations. Our spectral results also provide some necessary conditions for a gain graph to be a gain-line graph.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Let
O
(
3
)
denote the group of orthogonal
3
×
3
real matrices, and
M
the 5-dimensional real vector space of all
3
×
3
real symmetric matrices with trace zero. Let
λ
1
(
A
)
≤
λ
2
(
A
)
≤
λ
3
(
A
)
...be the eigenvalues of
A
∈
M
and
Ξ
-
=
{
A
∈
M
∣
λ
1
(
A
)
=
λ
2
(
A
)
}
.
M
is an inner product space with the inner product
⟨
A
,
B
⟩
=
trace
(
A
B
)
. Let
G
3
(
M
)
be the set of all 3-dimensional subspaces of
M
, a 6-dimensional Grassman manifold.
O
(
3
)
acts on
M
on the left by conjugation via inner product preserving linear isomorphisms, which map any 3-dimensional subspace into another 3-dimensional subspace; thus
G
3
(
M
)
also has a left action of
O
(
3
)
.
G
3
(
M
)
becomes a category, an action groupoid, with morphisms
(
V
,
M
,
W
)
∈
G
3
(
M
)
×
O
(
3
)
×
G
3
(
M
)
, where
W
=
M
V
M
T
. Composition of morphisms is
(
V
1
,
N
,
V
2
)
∘
(
V
0
,
M
,
V
1
)
=
(
V
0
,
N
M
,
V
2
)
. Let
C
be a category whose objects (
V
,
S
) consist of a real inner product space
V
and
S
⊂
V
, and whose arrows
(
V
,
S
)
→
(
W
,
T
)
consist of
f
:
V
→
W
, an inner product preserving real linear mapping such that
f
(
S
)
⊂
T
. We have the functor
V
↓
(
V
,
M
,
W
)
W
G
3
(
M
)
⟶
F
-
(
V
,
Ξ
-
∩
V
)
↓
F
-
(
M
)
(
W
,
Ξ
-
∩
W
)
C
where
F
-
(
M
)
:
V
→
W
:
A
↦
M
A
M
T
. Suppose further that
S
3
denotes the group of permutations of
{
1
,
2
,
3
}
, and
ρ
:
S
3
→
O
(
3
)
denotes a group homomorphism which is isomorphic as a group representation to the natural representation of
S
3
on
R
3
(which permutes the coordinates). Let
Obj
(
L
S
)
denote the set of all
V
∈
G
3
(
M
)
whose isotropy subgroup contains
S
=
ρ
(
S
3
)
as a subgroup. This paper completely describes the full subcategory
L
S
of
G
3
(
M
)
with object set
Obj
(
L
S
)
, as well as the details of the above functor restricted to
L
S
. Thus all the members
V
∈
Obj
(
L
S
)
are determined, as well as the smooth manifold structure on
Obj
(
L
S
)
; it is embedded as a one-dimensional submanifold of
G
3
(
M
)
. The isotropy subgroups of all
V
∈
Obj
(
L
S
)
are computed and all pairs
V
,
W
∈
Obj
(
L
S
)
which are isomorphic via some
M
∈
O
(
3
)
are determined. The sets
Ξ
-
∩
V
are all determined, and the functorial mappings on morphism sets are computed. However,
L
S
is not a Lie groupoid. The image of
Obj
(
L
S
)
under the functor
π
1
F
-
is the collection of fibres of the smooth manifold
∐
V
∈
Obj
(
L
S
)
V
, which is the total space of the canonical vector bundle over the base manifold
Obj
(
L
S
)
. The bifurcation points of the family of subsets
Ξ
-
∩
V
as
V
ranges over
Obj
(
L
S
)
(within this total space) are seen to be the points of
Obj
(
L
S
)
with infinite isotropy subgroups. We also show how this mathematical problem arises naturally from a problem in mathematical chemistry. Hence certain features of numerical calculations of energy eigenvalue intersection patterns of the simple chemical system H3 are rationalized through linearization about the triple intersection point.
Full text
Available for:
EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
The challenge of grid security assessment with N - k contingencies lies in a huge amount of transmission security constraints and even for the modest values of N and k, the computational complexity ...would be very high. A new method for fast grid security assessment is presented in this paper for DC grid. The key idea is to construct a small number of representative constraints to equivalently "represent" the huge number of original security constraints. It is proved that applying the representative constraints is sufficient to assess grid security as the original security constraints do. An important feature of the new method is that the representative constraints need to be constructed only once offline. They are only related to the parameters of transmission network and do not change with power injections. Numerical testing is performed for IEEE-RTS 24-bus system, IEEE 118-bus system, and the Polish 2383-bus system. The number of representative constraints to be assessed is only 1.1%, 0.32%, and 0.27% of that of the original constraints, respectively, and the computational time of security assessment is greatly reduced.
The paper is devoted to perfect and almost perfect homogeneous polytopes in Euclidean spaces. We have classified perfect and almost perfect polytopes among all regular polytopes and all semiregular ...polytopes except Archimedean solids and two four-dimensional Gosset polytopes. Also we have constructed some non-regular homogeneous polytopes that are (or are not) perfect and posed some unsolved questions.
Full text
Available for:
DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, SIK, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
In this paper we study the characteristic polynomial of multiparameter pencil z1A1+z2A2+⋯+zsAs. The main theorem states that a unitary representation of a finitely generated group contains a ...one-dimensional representation if and only if the characteristic polynomial of its generators contains a linear factor. It follows that a two or three dimensional unitary representation of a finitely generated group is irreducible if and only if the characteristic polynomial of the pencil of its generators is irreducible. The result is of kin to the Dedekind and Frobenius theorem on finite group determinant.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
PDF
