Displaying similar documents to “Choquet-like integrals with respect to level-dependent capacities and ϕ -ordinal sums of aggregation function”

Decidability and definability results related to the elementary theory of ordinal multiplication

Alexis Bès (2002)

Fundamenta Mathematicae

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The elementary theory of ⟨α;×⟩, where α is an ordinal and × denotes ordinal multiplication, is decidable if and only if α < ω ω . Moreover if | r and | l respectively denote the right- and left-hand divisibility relation, we show that Th ω ω ξ ; | r and Th ω ξ ; | l are decidable for every ordinal ξ. Further related definability results are also presented.

Ordinal indices and Ramsey dichotomies measuring c₀-content and semibounded completeness

Vassiliki Farmaki (2002)

Fundamenta Mathematicae

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We study the c₀-content of a seminormalized basic sequence (χₙ) in a Banach space, by the use of ordinal indices (taking values up to ω₁) that determine dichotomies at every ordinal stage, based on the Ramsey-type principle for every countable ordinal, obtained earlier by the author. We introduce two such indices, the c₀-index ξ ( χ ) and the semibounded completeness index ξ b ( χ ) , and we examine their relationship. The countable ordinal values that these indices can take are always of the form...

On ordinals accessible by infinitary languages

Saharon Shelah, Pauli Väisänen, Jouko Väänänen (2005)

Fundamenta Mathematicae

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Let λ be an infinite cardinal number. The ordinal number δ(λ) is the least ordinal γ such that if ϕ is any sentence of L λ ω , with a unary predicate D and a binary predicate ≺, and ϕ has a model ℳ with D , a well-ordering of type ≥ γ, then ϕ has a model ℳ ’ where D ' , ' is non-well-ordered. One of the interesting properties of this number is that the Hanf number of L λ ω is exactly δ ( λ ) . It was proved in [BK71] that if ℵ₀ < λ < κ a r e r e g u l a r c a r d i n a l n u m b e r s , t h e n t h e r e i s a f o r c i n g e x t e n s i o n , p r e s e r v i n g c o f i n a l i t i e s , s u c h t h a t i n t h e e x t e n s i o n 2λ = κ a n d δ ( λ ) < λ . W e i m p r o v e t h i s r e s u l t b y p r o v i n g t h e f o l l o w i n g : S u p p o s e < λ < θ κ a r e c a r d i n a l n u m b e r s s u c h t h a t λ < λ = λ ; ∙ cf(θ) ≥ λ⁺ and μ λ < θ whenever μ < θ; ∙ κ λ = κ . Then there...

On a special class of left-continuous uninorms

Gang Li (2018)

Kybernetika

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This paper is devoted to the study of a class of left-continuous uninorms locally internal in the region A ( e ) and the residual implications derived from them. It is shown that such uninorm can be represented as an ordinal sum of semigroups in the sense of Clifford. Moreover, the explicit expressions for the residual implication derived from this special class of uninorms are given. A set of axioms is presented that characterizes those binary functions I : [ 0 , 1 ] 2 [ 0 , 1 ] for which a uninorm U of this special...

Marcinkiewicz integrals on product spaces

H. Al-Qassem, A. Al-Salman, L. C. Cheng, Y. Pan (2005)

Studia Mathematica

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We prove the L p boundedness of the Marcinkiewicz integral operators μ Ω on n × × n k under the condition that Ω L ( l o g L ) k / 2 ( n - 1 × × n k - 1 ) . The exponent k/2 is the best possible. This answers an open question posed by Y. Ding.

Existence of discontinuous absolute minima for certain multiple integrals without growth properties

Lamberto Cesari (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In the present paper the author discusses certain multiple integrals I ( u ) of the calculus of variations satisfying convexity conditions, and no growth property, and the corresponding Serrin integrals ( u ) , to which the existence theorems in [3,4,5] do not apply. However, in the present paper, the integrals I ( u ) and ( u ) are reduced to simpler form H ( v ) and ( v ) to which the existence theorems above apply. Thus, we derive that I ( u ) ( u ) , H ( v ) ( v ) , we obtain the existence of the absolute minimum for the Serrin forms ( u ) ...

An Isomorphic Classification of C ( 2 × [ 0 , α ] ) Spaces

Elói Medina Galego (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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We present an extension of the classical isomorphic classification of the Banach spaces C([0,α]) of all real continuous functions defined on the nondenumerable intervals of ordinals [0,α]. As an application, we establish the isomorphic classification of the Banach spaces C ( 2 × [ 0 , α ] ) of all real continuous functions defined on the compact spaces 2 × [ 0 , α ] , the topological product of the Cantor cubes 2 with smaller than the first sequential cardinal, and intervals of ordinal numbers [0,α]. Consequently,...

On isomorphism classes of C ( 2 [ 0 , α ] ) spaces

Elói Medina Galego (2009)

Fundamenta Mathematicae

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We provide a complete isomorphic classification of the Banach spaces of continuous functions on the compact spaces 2 [ 0 , α ] , the topological sums of Cantor cubes 2 , with smaller than the first sequential cardinal, and intervals of ordinal numbers [0,α]. In particular, we prove that it is relatively consistent with ZFC that the only isomorphism classes of C ( 2 [ 0 , α ] ) spaces with ≥ ℵ₀ and α ≥ ω₁ are the trivial ones. This result leads to some elementary questions on large cardinals.

Locally Σ₁-definable well-orders of H(κ⁺)

Peter Holy, Philipp Lücke (2014)

Fundamenta Mathematicae

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Given an uncountable cardinal κ with κ = κ < κ and 2 κ regular, we show that there is a forcing that preserves cofinalities less than or equal to 2 κ and forces the existence of a well-order of H(κ⁺) that is definable over ⟨H(κ⁺),∈⟩ by a Σ₁-formula with parameters. This shows that, in contrast to the case "κ = ω", the existence of a locally definable well-order of H(κ⁺) of low complexity is consistent with failures of the GCH at κ. We also show that the forcing mentioned above introduces a Bernstein...

L p type mapping estimates for oscillatory integrals in higher dimensions

G. Sampson (2006)

Studia Mathematica

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We show in two dimensions that if K f = ² k ( x , y ) f ( y ) d y , k ( x , y ) = ( e i x a · y b ) / ( | x - y | η ) , p = 4/(2+η), a ≥ b ≥ 1̅ = (1,1), v p ( y ) = y ( p / p ' ) ( 1 ̅ - b / a ) , then | | K f | | p C | | f | | p , v p if η + α₁ + α₂ < 2, α j = 1 - b j / a j , j = 1,2. Our methods apply in all dimensions and also for more general kernels.

A complete characterization of R-sets in the theory of differentiation of integrals

G. A. Karagulyan (2007)

Studia Mathematica

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Let s be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis s differentiates the integral of f if s ∉ S, and D ̅ s f ( x ) = l i m s u p d i a m ( R ) 0 , x R s | R | - 1 R f = almost everywhere if s ∈ S. If the condition D ̅ s f ( x ) = holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a G δ (resp. a G δ σ ).

Cardinal sequences of length < ω₂ under GCH

István Juhász, Lajos Soukup, William Weiss (2006)

Fundamenta Mathematicae

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Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put λ ( α ) = s ( α ) : s ( 0 ) = λ = m i n [ s ( β ) : β < α ] . We show that f ∈ (α) iff for some natural number n there are infinite cardinals λ i > λ > . . . > λ n - 1 and ordinals α , . . . , α n - 1 such that α = α + + α n - 1 and f = f f . . . f n - 1 where each f i λ i ( α i ) . Under GCH we prove that if α < ω₂ then (i) ω ( α ) = s α ω , ω : s ( 0 ) = ω ; (ii) if λ > cf(λ) = ω, λ ( α ) = s α λ , λ : s ( 0 ) = λ , s - 1 λ i s ω - c l o s e d i n α ; (iii) if cf(λ) = ω₁, λ ( α ) = s α λ , λ : s ( 0 ) = λ , s - 1 λ i s ω - c l o s e d a n d s u c c e s s o r - c l o s e d i n α ; (iv) if cf(λ) > ω₁, λ ( α ) = α λ . This yields a complete characterization of the classes (α) for all...

Coloring ordinals by reals

Jörg Brendle, Sakaé Fuchino (2007)

Fundamenta Mathematicae

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We study combinatorial principles we call the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ,λ) for regular κ > ℵ₁ and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals. These principles are strengthenings of C s ( κ ) and F s ( κ ) of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that IP(ℵ₂,ℵ₁) (hence also IP(ℵ₂,ℵ₂) as well as HP(ℵ₂)) holds in a generic extension of a model of CH by Cohen forcing, and IP(ℵ₂,ℵ₂) (hence...

Semicontinuity in L for polyconvex integrals

Emilio Acerbi, Giuseppe Buttazzo, Nicola Fusco (1982)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Viene studiata la semicontinuità rispetto alla topologia di L ( Ω ; 𝐑 m ) per alcuni funzionali del Calcolo delle Variazioni dipendenti da funzioni a valori vettoriali.

Uncountable cardinals have the same monadic ∀₁¹ positive theory over large sets

Athanassios Tzouvaras (2004)

Fundamenta Mathematicae

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We show that uncountable cardinals are indistinguishable by sentences of the monadic second-order language of order of the form (∀X)ϕ(X) and (∃X)ϕ(X), for ϕ positive in X and containing no set-quantifiers, when the set variables range over large (= cofinal) subsets of the cardinals. This strengthens the result of Doner-Mostowski-Tarski [3] that (κ,∈), (λ,∈) are elementarily equivalent when κ, λ are uncountable. It follows that we can consistently postulate that the structures ( 2 κ , [ 2 κ ] > κ , < ) , ( 2 λ , [ 2 λ ] > λ , < ) are...

The method of rotation and Marcinkiewicz integrals on product domains

Jiecheng Chen, Dashan Fan, Yiming Ying (2002)

Studia Mathematica

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We give some rather weak sufficient condition for L p boundedness of the Marcinkiewicz integral operator μ Ω on the product spaces × m (1 < p < ∞), which improves and extends some known results.

Spaces of compact operators on C ( 2 × [ 0 , α ] ) spaces

Elói Medina Galego (2011)

Colloquium Mathematicae

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We classify, up to isomorphism, the spaces of compact operators (E,F), where E and F are the Banach spaces of all continuous functions defined on the compact spaces 2 × [ 0 , α ] , the topological products of Cantor cubes 2 and intervals of ordinal numbers [0,α].

Reflecting character and pseudocharacter

Lucia R. Junqueira, Alberto M. E. Levi (2015)

Commentationes Mathematicae Universitatis Carolinae

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We say that a cardinal function φ reflects an infinite cardinal κ , if given a topological space X with φ ( X ) κ , there exists Y [ X ] κ with φ ( Y ) κ . We investigate some problems, discussed by Hodel and Vaughan in Reflection theorems for cardinal functions, Topology Appl. 100 (2000), 47–66, and Juhász in Cardinal functions and reflection, Topology Atlas Preprint no. 445, 2000, related to the reflection for the cardinal functions character and pseudocharacter. Among other results, we present some new equivalences...

Characterizing the powerset by a complete (Scott) sentence

Ioannis Souldatos (2013)

Fundamenta Mathematicae

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This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing previous work of the author. A cardinal κ is characterized by a Scott sentence ϕ if ϕ has a model of size κ, but no model of size κ⁺. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if β is characterized by a Scott sentence, then 2 β + β is (homogeneously) characterized by a Scott sentence, for all 0 <...

Existence of discontinuous absolute minima for certain multiple integrals without growth properties

Lamberto Cesari (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

In the present paper the author discusses certain multiple integrals I ( u ) of the calculus of variations satisfying convexity conditions, and no growth property, and the corresponding Serrin integrals ( u ) , to which the existence theorems in [3,4,5] do not apply. However, in the present paper, the integrals I ( u ) and ( u ) are reduced to simpler form H ( v ) and ( v ) to which the existence theorems above apply. Thus, we derive that I ( u ) ( u ) , H ( v ) ( v ) , we obtain the existence of the absolute minimum for the Serrin forms ( u ) ...

Level by level equivalence and the number of normal measures over P κ ( λ )

Arthur W. Apter (2007)

Fundamenta Mathematicae

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We construct two models for the level by level equivalence between strong compactness and supercompactness in which if κ is λ supercompact and λ ≥ κ is regular, we are able to determine exactly the number of normal measures P κ ( λ ) carries. In the first of these models, P κ ( λ ) carries 2 2 [ λ ] < κ many normal measures, the maximal number. In the second of these models, P κ ( λ ) carries 2 2 [ λ ] < κ many normal measures, except if κ is a measurable cardinal which is not a limit of measurable cardinals. In this case, κ (and...

Spaces with property ( D C ( ω 1 ) )

Wei-Feng Xuan, Wei-Xue Shi (2017)

Commentationes Mathematicae Universitatis Carolinae

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We prove that if X is a first countable space with property ( D C ( ω 1 ) ) and with a G δ -diagonal then the cardinality of X is at most 𝔠 . We also show that if X is a first countable, DCCC, normal space then the extent of X is at most 𝔠 .

Infinite-Dimensionality modulo Absolute Borel Classes

Vitalij Chatyrko, Yasunao Hattori (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces X α , Y α and Z α such that (i) f X α , f Y α , f Z α = ω , where f is either trdef or ₀-trsur, (ii) A ( α ) - t r i n d X α = and M ( α ) - t r i n d X α = - 1 , (iii) A ( α ) - t r i n d Y α = - 1 and M ( α ) - t r i n d Y α = , and (iv) A ( α ) - t r i n d Z α = M ( α ) - t r i n d Z α = and A ( α + 1 ) M ( α + 1 ) - t r i n d Z α = - 1 . We also show that there exists no separable metrizable space W α with A ( α ) - t r i n d W α , M ( α ) - t r i n d W α and A ( α ) M ( α ) - t r i n d W α = , where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

A solution to Comfort's question on the countable compactness of powers of a topological group

Artur Hideyuki Tomita (2005)

Fundamenta Mathematicae

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In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number α 2 , a topological group G such that G γ is countably compact for all cardinals γ < α, but G α is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under M A c o u n t a b l e . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from M A c o u n t a b l e . However, the question has...

Internally club and approachable for larger structures

John Krueger (2008)

Fundamenta Mathematicae

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We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of P κ ( X ) , where κ ⊆ X. Suppose μ < κ, μ < μ = μ , and κ is supercompact. Then there is a generic extension in which κ = μ⁺⁺, and for all regular λ ≥ μ⁺⁺, there are stationarily many N in [ H ( λ ) ] μ which are internally club but not internally approachable.