The present paper is devoted to the study of limit varieties of additively idempotent semirings. A limit variety is a nonfinitely based variety whose proper subvarieties are all finitely based. We ...present concrete constructions for one infinite family of limit additively idempotent semiring varieties, and one further ad hoc example. Each of these examples can be generated by a finite flat semiring, with the infinite family arising by a way of a complete characterisation of limit varieties that can be generated by the flat extension of a finite group. We also demonstrate the existence of other examples of limit varieties of additively idempotent semirings, including one further continuum-sized family, each with no finite generator, and two further ad hoc examples. While an explicit description of these latter examples is not given, one of the examples is proved to contain only trivial flat semirings.
Let
φ
:
R
→
S
be a flat local homomorphism between commutative Noetherian local rings. In this paper, the ascent and descent of Artinian module structures between R and S are investigated. For an ...Artinian R-module A, the structure of
A
⊗
R
S
is described. As an application, the Artinianess of certain local cohomology modules is clarified.
The A-Truncated K-Moment Problem Nie, Jiawang
Foundations of computational mathematics,
12/2014, Letnik:
14, Številka:
6
Journal Article
Recenzirano
Let
A
⊆
N
n
be a finite set, and
K
⊆
R
n
be a compact semialgebraic set. An
A
-truncated multisequence
(
A
-tms) is a vector
y
=
(
y
α
)
indexed by elements in
A
. The
A
-truncated
K
-moment problem ...(
A
-TKMP) concerns whether or not a given
A
-tms
y
admits a
K
-measure
μ
, i.e.,
μ
is a nonnegative Borel measure supported in
K
such that
y
α
=
∫
K
x
α
d
μ
for all
α
∈
A
. This paper proposes a numerical algorithm for solving
A
-TKMPs. It aims at finding a flat extension of
y
by solving a hierarchy of semidefinite relaxations
{
(
SDR
)
k
}
k
=
1
∞
for a moment optimization problem, whose objective
R
is generated in a certain randomized way. If
y
admits no
K
-measures and
R
x
A
is
K
-full (there exists
p
∈
R
x
A
that is positive on
K
), then
(
SDR
)
k
is infeasible for all
k
big enough, which gives a certificate for the nonexistence of representing measures. If
y
admits a
K
-measure, then for almost all generated
R
, this algorithm has the following properties: i) we can asymptotically get a flat extension of
y
by solving the hierarchy
{
(
SDR
)
k
}
k
=
1
∞
; ii) under a general condition that is almost sufficient and necessary, we can get a flat extension of
y
by solving
(
SDR
)
k
for some
k
; iii) the obtained flat extensions admit a
r
-atomic
K
-measure with
r
≤
|
A
|
. The decomposition problems for completely positive matrices and sums of even powers of real linear forms, and the standard truncated
K
-moment problems, are special cases of
A
-TKMPs. They can be solved numerically by this algorithm.
On Syros, high‐pressure metamorphism affects a lithological pile that is composed of, from base to top: (1) the Komito‐Vari granitic basement, (2) a margin sedimentary sequence that is predominantly ...made of marbles and schists (the Pyrgos and Kastri units), and (3) the Kambos metaophiolitic mélange. The tectonic history occurred in three main stages. During the first stage, in the mid‐Eocene, the Kambos oceanic unit was thrust southward on top of the sedimentary pile. Top‐to‐the‐south‐southwest ductile senses of shear are synchronous with prograde high‐pressure metamorphism and associated with this thrusting event. The second stage corresponds to a top‐to‐the‐northeast ductile shear that affects the whole metamorphic pile and is synchronous with the metamorphic retrogression from eclogite to greenschist facies. However, the Kambos oceanic unit remained partly undeformed, as shown by significant volumes containing undeformed lawsonite pseudomorphs. No major extensional detachment related to this exhumation event outcrops on the island. The localized semibrittle to brittle deformation of the third stage is associated with the postmetamorphic development of (1) a ramp‐flat extensional system at the island scale, whose southward minimum displacement is estimated at approximately 7 km, and (2) two sets of steeply dipping strike‐slip faults with a normal component, trending either east–west or around north–south, indicating that the mean stretching and shortening directions are trending NNE–SSW and ESE–WNW, respectively. This sequence of major tectonic events and their relationship to metamorphism are interpreted within the framework of the subduction of the Pindos Ocean and then of the Adria continental passive margin.
Key Points
A ramp‐flat extensional system occurs on Syros Island
There is not syn‐metamorphic detachment on Syros Island
The CBU preserved the top‐to‐the‐south‐southwest prograde deformation
In the most intuitive way, the problem of moments seeks to represent the terms of a given sequence using an integral. So, it is about determining a measure that allows this representation. In ...mathematical analysis, the problem of moments (MP) occupies an important place in the work of many researchers. Several generalizations and extensions of the original version, attributed to T. J. Stieltjes, have emerged. It has been a source of inspiration for many developments in many branches of mathematics as well as in other fields. In this article, we are interested in a class of two-dimensional MP. Precisely, we deal with the problem of the cubic real truncated moment in the case where the associated moment matrix is singular. The main target is to provide a complete solution via a minimal atomic-representative measure. This was possible by the use of an algorithmic method based on the construction of some flat extension of the associated moment matrix. The implementation of this approach is illustrated by some numerical examples using Mathematica software.
This paper studies how to solve the truncated moment problem (TMP) via homogenization and flat extensions of moment matrices. We first transform TMP to a homogeneous TMP (HTMP), and then use ...semidefinite programming (SDP) techniques to solve HTMP. Our main results are: (1) a truncated moment sequence (tms) is the limit of a sequence of tms admitting measures on Rn if and only if its homogenized tms (htms) admits a measure supported on the unit sphere in Rn+1; (2) an htms admits a measure if and only if the optimal values of a sequence of SDP problems are nonnegative; (3) under some conditions that are almost necessary and sufficient, by solving these SDP problems, a representing measure for an htms can be explicitly constructed if one exists.
Let $\gamma ^{\left( n\right) }\equiv \{\gamma _{ij}\}\,(0\leq i+j\leq 2n,\,|i-j|\leq n)$ be a sequence in the complex number set $\mathbb{C}$ and let $E\left( n\right) $ be the Embry truncated ...moment matrices corresponding from $\gamma ^{\left( n\right) }$. For an odd number $n$, it is known that $ \gamma ^{\left( n\right) }$ has a rank $E\left( n\right) $\textit{-}atomic representing measure if and only if $E(n)\geq 0$ and $E(n)$ admits a flat extension $E(n+1)$. In this paper we suggest a related problem: if $E(n)$ is positive and nonsingular, does $E(n)$ have a flat extension $E(n+1)$? and give a negative answer in the case of $E(3)$. And we obtain some necessary conditions for positive and nonsingular matrix $E\left( 3\right)$, and also its sufficient conditions. KCI Citation Count: 0
Flat extension and ideal projection Kunkle, Thomas
Journal of symbolic computation,
November-December 2018, 2018-11-00, Letnik:
89
Journal Article
Recenzirano
Odprti dostop
A generalization of the flat extension theorems of Curto and Fialkow and Laurent and Mourrain is obtained by seeing the problem as one of ideal projection.
We describe a new method to compute general cubature formulae. The problem is initially transformed into the computation of truncated Hankel operators with flat extensions. We then analyze the ...algebraic properties associated to flat extensions and show how to recover the cubature points and weights from the truncated Hankel operator. We next present an algorithm to test the flat extension property and to additionally compute the decomposition. To generate cubature formulae with a minimal number of points, we propose a new relaxation hierarchy of convex optimization problems minimizing the nuclear norm of the Hankel operators. For a suitably high order of convex relaxation, the minimizer of the optimization problem corresponds to a cubature formula. Furthermore cubature formulae with a minimal number of points are associated to faces of the convex sets. We illustrate our method on some examples, and for each we obtain a new minimal cubature formula.
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