Degree theory for orbifolds Pasquotto, Federica; Rot, Thomas O.
Topology and its applications,
08/2020, Letnik:
282
Journal Article
Recenzirano
Odprti dostop
In 1 Borzellino and Brunsden started to develop an elementary differential topology theory for orbifolds. In this paper we carry on their project by defining a mapping degree for proper maps between ...orbifolds, which counts preimages of regular values with appropriate weights. We show that the mapping degree satisfies the expected invariance properties, under the assumption that the domain does not have a codimension one singular stratum. We study properties of the mapping degree and compute the degree in some examples.
Recently, due to its numerous applications, the spectra of the bounded operators over Banach spaces have been studied extensively. This work aims to collect some of the investigations on the spectra ...of difference operators or matrices on the Banach space
c
in the literature and provide a foundation for related problems. To the best of our investigations, the problem has been solved over the sequence space
c
so far up to maximum order 2. In the present work, the fine spectra of the difference operator
Δ
m
,
m
∈
N
on
c
have been computed. The generalized difference operator
Δ
m
on the Banach space
c
is defined by
(
Δ
m
x
)
k
=
∑
i
=
0
m
(
-
1
)
i
m
i
x
k
-
i
,
k
=
0
,
1
,
2
,
3
,
⋯
with
x
k
=
0
for
k
<
0
. Indeed, the operator
Δ
m
is represented by an
(
m
+
1
)
-th band matrix which generalizes several difference operators such as
Δ
,
Δ
2
,
B
(
r
,
s
)
and
B
(
r
,
s
,
t
) etc, under different limiting conditions. Initially, we provide some essential background results on the linearity and boundedness of the backward difference operator
Δ
m
. Finally, the sets for the spectrum and fine spectra such as the point spectrum, the continuous spectrum and the residual spectrum of the defined operator on the space
c
have been computed. The geometrical interpretation for the spectral subdivisions of the above difference operator is also incorporated.
Taking an elementary and straightforward approach, we develop the concept of a regular value for a smooth map f:O→P between smooth orbifolds O and P. We show that Sardʼs theorem holds and that the ...inverse image of a regular value is a smooth full suborbifold of O. We also study some constraints that the existence of a smooth orbifold map imposes on local isotropy groups. As an application, we prove a Borsuk no retraction theorem for compact orbifolds with boundary and some obstructions to the existence of real-valued orbifold maps from local model orbifold charts.
A note on incomplete markets Rajan, Ashvin Varada
Journal of mathematical economics,
03/1999, Letnik:
31, Številka:
2
Journal Article
Recenzirano
Odprti dostop
We adapt an elegant piece of reasoning by Y. Balasko Balasko, Y., 1979. Economies with a finite but large number of equilibria. J. Math. Econ. 6, 145–147. to the incomplete markets modelled by D. ...Duffie and W. Shafer Duffie, D., Shafer, W., 1985. Equilibrium in incomplete markets: I. A basic model of generic existence. J. Math. Econ. 14, 285–300., and prove that on compact sets of such markets, the Lebesgue measure of economies with
m equilibria is
O(
1
m
).
In cooperative game theory, the Shapley value is a central notion defining a rational way to share the total worth of a game among players. In this paper, we address a general framework leading to ...applications to games with communication graphs, where the set of feasible coalitions forms a poset where all maximal chains have the same length. We first show that previous definitions and axiomatizations of the Shapley value proprosed by Faigle and Kern, and Bilbao and Edelman still work. Our main contribution is then to propose a new axiomatization avoiding the hierarchical strength axiom of Faigle and Kern, and considering a new way to define the symmetry among players. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the classical efficiency axiom correspond actually to the two Kirchhoff’s laws in the resistor circuit associated to the Hasse diagram of feasible coalitions. We finally work out a weak form of the monotonicity axiom which is satisfied by the proposed value.
In cooperative game theory, the Shapley value is a central notion defining a rational way to share the total worth of a game among players. In this paper, we address a general framework, namely ...regular set systems, where the set of feasible coalitions forms a poset where all maximal chains have the same length. We first show that previous definitions and axiomatizations of the Shaphey value proposed by Faigle and Kern and Bilbao and Edelman still work. our main contribution is then to propose a new axiomatization avoiding the hierarchical strength axiom of Faigle and Kern, and considering a new way to define the symmetry among players. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the classical efficiency axiom correspond actually to the two Kirchhoff's laws in the resistor circuit associated to the Hasse diagram of feasible coalitions. We finally work out a weak form of the monotonicity axiom which is satisfied by the proposed value.
In cooperative game theory, the Shapley value is a central notion defining a rational way to share the total worth of a game among players. In this paper, we address a general framework, namely ...regular set systems, where the set of feasible coalitions forms a poset where all maximal chains have the same length. We first show that previous definitions and axiomatizations of the Shaphey value proposed by Faigle and Kern and Bilbao and Edelman still work. our main contribution is then to propose a new axiomatization avoiding the hierarchical strength axiom of Faigle and Kern, and considering a new way to define the symmetry among players. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the classical efficiency axiom correspond actually to the two Kirchhoff's laws in the resistor circuit associated to the Hasse diagram of feasible coalitions. We finally work out a weak form of the monotonicity axiom which is satisfied by the proposed value.
This work computes the spectrum of the backward fractional difference operator
$ \Delta ^{(\alpha )} $
Δ
(
α
)
$ (0<\alpha < 1) $
(
0
<
α
<
1
)
over various Banach spaces
$ \ell _{1},c $
ℓ
1
,
c
and
...$ c_0 $
c
0
. Note that recently, the spectrum of the fractional difference operator
$ \Delta _\nu ^{(\alpha )} $
Δ
ν
(
α
)
, (
$ \nu = (\nu _k) $
ν
=
(
ν
k
)
being either a constant or strictly decreasing sequence of positive real numbers satisfying some conditions) on the sequence space
$ \ell _{1} $
ℓ
1
has been computed by Baliarsingh P. On the spectrum of fractional difference operator. Linear Multilinear Algebra. 2021;1-13 and it has been noticed that the results obtained are unified and more general, but lack of accuracy for the case
$ 0<\alpha < 1 $
0
<
α
<
1
. In this work, we improve the results by providing the sharper estimations and some illustrative examples. Also, individual sets describing the point, the residual and the continuous spectrum of the operator
$ \Delta ^{(\alpha )} $
Δ
(
α
)
on the Banach space c and
$ c_0 $
c
0
are computed.
We address here the question of the bi-Lipschitz local triviality of a complex polynomial function over a complex value. Our main result states that a non constant complex polynomial admits a locally ...bi-Lipschitz trivial value if and only if it is a polynomial in one complex variable.
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