81 research outputs found
On the Egoroff property of pointwise convergent sequences of functions
No full textThe space
L
(
x
)
\mathcal {L}(x)
of real-valued functions on
X
X
has the Egoroff property if for any
{
f
n
k
}
\{ {f_{nk}}\}
such that
0
⩽
f
n
k
↑
k
f
0 \leqslant {f_{nk}}{ \uparrow _k}f
(for every
n
n
), there exists
g
m
↑
f
{g_m} \uparrow f
such that, for each
m
m
and
n
n
,
g
m
⩽
n
k
{g_m}{ \leqslant _{nk}}
for some
k
k
. We show that
L
(
X
)
\mathcal {L}(X)
has the Egoroff property if and only if the cardinality of
X
X
is smaller than the minimum cardinality of an unbounded family of functions from the set of natural numbers to itself. Therefore, the statement that there is an uncountable set
X
X
such that
L
(
X
)
\mathcal {L}(X)
has the Egoroff property is independent of the axioms of set theory.</p
The Egoroff Property and Related Properties in the Theory of Riesz Spaces
No full textNOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
A Riesz space L is said to be Egoroff, if, whenever [...] and [...], there is a sequence [...] in L such that [...] and, for each n,m, there exists an index k(n,m) such that [...]. This notion was introduced, in rather a different form, by Nakano. Banach function spaces are Egoroff, and Lorentz showed that, for any function seminorm [...], the maximal seminorm [...] among those which are dominated by [...] and which are [...] (a monotone seminorm [...] is [...] if [...]) is precisely the "Lorentz seminorm" [...], where [...]. In this thesis the extent to which [...] holds in general Riesz spaces is determined. In fact, [...] for every monotone seminorm [...] on a Riesz space L if, and only if, L is "almost-Egoroff". The almost-Egoroff property is closely related to the Egoroff property and, indeed, coincides with it in the case of Archimedean spaces. Analogous theorems for Boolean algebras are discussed. The almost-Egoroff property is shown to yield a number of results which ensure that, under certain conditions, a monotone seminorm is [...] when restricted to an appropriate super order dense ideal. Riesz spaces L possessing an integral, Riesz norm [...](i.e., a Riesz norm such that [...] are considered also, since in many cases these are known to be Egoroff. In particular if [...] is normal on L (i.e., [...] a directed system, [...] ), then L is Egoroff. In this connection, a pathological space, possessing an integral Riesz norm which is nowhere normal, is constructed
On construction of symmetries and recursion operators from zero-curvature representations and the Darboux-Egoroff system
No full textThe Darboux–Egoroff system of PDEs with any number n ≥ 3 of independent variables plays an essential role in the problems of describing n-dimensional flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a recursion operator and its inverse for symmetries of the Darboux–Egoroff system and describe some symmetries generated by these operators. The constructed recursion operators are not pseudodifferential, but are Bäcklund autotransformations for the linearized system whose solutions correspond to symmetries of the Darboux–Egoroff system. For some other PDEs, recursion operators of similar types were considered previously by Papachristou, Guthrie, Marvan, Pobořil, and Sergyeyev. In the structure of the obtained third and fifth order symmetries of the Darboux–Egoroff system, one finds the third and fifth order flows of an (n − 1)-component vector modified KdV hierarchy. The constructed recursion operators generate also an infinite number of nonlocal symmetries. In particular, we obtain a simple construction of nonlocal symmetries that were studied by Buryak and Shadrin in the context of the infinitesimal version of the Givental– van de Leur twisted loop group action on the space of semisimple Frobenius manifolds. We obtain these results by means of rather general methods, using only the zerocurvature representation of the considered PDEs
The Egoroff Property and its Relation to the Order Topology in the Theory of Riesz Spaces
No full textNOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
A sequence(f[subscript]n : n = 1, 2, ...) in a Riesz space L is order convergent to an element [...] whenever there exists a sequence [...] in L such that [...] holds for all n. Sequential order convergence defines the order topology on L. The closure of a subset S in this topology is denoted by cl(S). The pseudo order closure S' of a subset S is the set of all [...] such that there exists a sequence in S which is order convergent to f. If S' = cl(S) for every convex subset S, then S' = cl(S) for every subset S. L has the Egoroff property if and only if S' = cl(S) for every order bounded subset S of L. A necessary and sufficient condition for L to have the property that S' = cl(S) for every subset S of L is that L has the strong Egoroff property.
A sequence(f[subscript]n : n = 1, 2, ...) in a Riesz space L is ru-convergent to an element [...] whenever there exists a real sequence [...] and an element [...] such that [...] holds for all n. Sequential ru-convergence defines the ru-topology on L. The closure of a subset S in this topology is denoted by [...]. The pseudo ru-closure S'[subscript ru] of a subset S is the set of all [...] such that there exists a sequence in S which is ru-convergent to f. If L is Archimedean, then [...] for every convex subset S implies that [...] for every subset S. A characterization of those Archimedean Riesz spaces L with the property that [...] for every subset S of L is obtained.
If [...] is a monotone seminorm on a Riesz space L, then a necessary and sufficient condition for [...] in L implies [...] is that the set [...] is order closed. For every monotone seminorm [...] on L, the largest [...]-Fatou monotone serninorm bounded by [...] is the Minkowski functional of the order closure of [...].
A monotone seminorm p on a Riesz space L is called strong Fatou whenever [...]. A characterization of those Riesz spaces L which have the following property is given: "For every monotone seminorm [...], the largest strong Fatou monotone seminorm bounded by [...] : [...]." A similar characterization for Boolean algebras is also obtained
The construction of Frobenius manifolds from KP tau-functions
No full textFrobenius manifolds (solutions of WDVV equations) in canonical coordinates are determined by the system of Darboux–Egoroff equations. This system of partial differential equations appears as a specific subset of the n-component KP hierarchy. KP representation theory and the related Sato infinite Grassmannian are used to construct solutions of this Darboux–Egoroff system and the related Frobenius manifolds. Finally we show that for these solutions Dubrovin's isomonodromy tau-function can be expressed in the KP tau-function
The construction of Frobenius manifolds from KP tau-functions
No full textFrobenius manifolds (solutions of WDVV equations) in canonical coordinates are determined by the system of Darboux--Egoroff equations. This system of partial differential equations appears as a specific subset of the -component KP hierarchy. KP representation theory and the related Sato infinite Grassmannian are used to construct solutions of this Darboux--Egoroff system and the related Frobenius manifolds. Finally we show that for these solutions Dubrovin's isomonodromy tau-function can be expressed in the KP tau-function
The -property in
No full textsummary:The -property of a Riesz space (real vector lattice) is: For each sequence of positive elements of , there is a sequence of positive reals, and , with for each . This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when ``'' obtains for a Riesz space of continuous real-valued functions . A basic result is: For discrete , has iff the cardinal , Rothberger's bounding number. Consequences and generalizations use the Lindelöf number : For a -space , if , then has . For paracompact , if has , then , and conversely if is also locally compact. For metrizable , if has , then \textit{is} locally compact
N.N.: Foto von N. Egoroff, Petersburg
No full textOriginalfotografie
Umfang: 8,5 x 5 cm
Diese Fotografie ist eines von rund 340 Fotos von bedeutenden Wissenschaftlern, die dem Wiener Physiker Viktor von Lang zu dessen 70. Geburtstag am 2. März 1908 geschenkt wurden
Le patrimoine géologique en France
No full textInitiée dans La Lettre de l’OCIM en 2015 à l’occasion des 30 ans de l'OCIM, la rubrique "Quoi de neuf ?" propose un retour sur un article marquant de l’histoire de la revue. À partir d’un corpus d’articles sélectionnés par la rédaction de La Lettre de l’OCIM, les membres du comité des Publications de l'OCIM ont sélectionné plusieurs contributions. Dans cette perspective, il a été demandé à l’auteur ou à un expert du domaine de revisiter la problématique exposée dans l’article à la lueur des changements intervenus, notamment dans les pratiques professionnelles, depuis son écriture et de proposer des éléments prospectifs sur la question. Les auteurs font un retour sur l’article de Patrick De Wever Un inventaire du patrimoine géologique pour la France publié en 2009 dans le n° 121 de La Lettre de l’OCIM et qui présentait les principaux éléments de la démarche engagée pour la réalisation de l’inventaire du patrimoine géologique français
Measure and Integration: A Concise Introduction to Real Ananysis
No full textA uniquely accessible book for general measure and integration, emphasizing the real line, Euclidean space, and the underlying role of translation in real analysis. Measure and Integration: A Concise Introduction to Real Analysis presents the basic concepts and methods that are important for successfully reading and understanding proofs. Blending coverage of both fundamental and specialized topics, this book serves as a practical and thorough introduction to measure and integration, while also facilitating a basic understanding of real analysis. The author develops the theory of measure and integration on abstract measure spaces with an emphasis of the real line and Euclidean space. Additional topical coverage includes: Measure spaces, outer measures, and extension theorems. Lebesgue measure on the line and in Euclidean space. Measurable functions, Egoroff\u27s theorem, and Lusin\u27s theorem. Convergence theorems for integrals. Product measures and Fubini\u27s theorem. Differentiation theorems for functions of real variables. Decomposition theorems for signed measures. Absolute continuity and the Radon-Nikodym theorem. Lp spaces, continuous-function spaces, and duality theorems. Translation-invariant subspaces of L2 and applications. The book\u27s presentation lays the foundation for further study of functional analysis, harmonic analysis, and probability, and its treatment of real analysis highlights the fundamental role of translations. Each theorem is accompanied by opportunities to employ the concept, as numerous exercises explore applications including convolutions, Fourier transforms, and differentiation across the integral sign. Providing an efficient and readable treatment of this classical subject, Measure and Integration: A Concise Introduction to Real Analysis is a useful book for courses in real analysis at the graduate level. It is also a valuable reference for practitioners in the mathematical sciences. © 2009 John Wiley & Sons, Inc. All rights reserved
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