Displaying similar documents to “Subadditive functions and partial converses of Minkowski's and Mulholland's inequalities”

Convex-like inequality, homogeneity, subadditivity, and a characterization of L p -norm

Janusz Matkowski, Marek Pycia (1995)

Annales Polonici Mathematici

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Let a and b be fixed real numbers such that 0 < mina,b < 1 < a + b. We prove that every function f:(0,∞) → ℝ satisfying f(as + bt) ≤ af(s) + bf(t), s,t > 0, and such that l i m s u p t 0 + f ( t ) 0 must be of the form f(t) = f(1)t, t > 0. This improves an earlier result in [5] where, in particular, f is assumed to be nonnegative. Some generalizations for functions defined on cones in linear spaces are given. We apply these results to give a new characterization of the L p -norm.

A problem of Galambos on Engel expansions

Jun Wu (2000)

Acta Arithmetica

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1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) x = 1 / d ( x ) + 1 / ( d ( x ) d ( x ) ) + . . . + 1 / ( d ( x ) d ( x ) . . . d n ( x ) ) + . . . , where d j ( x ) , j 1 is a sequence of positive integers satisfying d₁(x) ≥ 2 and d j + 1 ( x ) d j ( x ) for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) l i m n d n 1 / n ( x ) = e . He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. d i m H x ( 0 , 1 ] : ( 2 ) f a i l s = 1 . We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and d i m H to denote...

On infinite composition of affine mappings

László Máté (1999)

Fundamenta Mathematicae

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 Let F i = 1 , . . . , N be affine mappings of n . It is well known that if there exists j ≤ 1 such that for every σ 1 , . . . , σ j 1 , . . . , N the composition (1) F σ 1 . . . F σ j is a contraction, then for any infinite sequence σ 1 , σ 2 , . . . 1 , . . . , N and any z n , the sequence (2) F σ 1 . . . F σ n ( z ) is convergent and the limit is independent of z. We prove the following converse result: If (2) is convergent for any z n and any σ = σ 1 , σ 2 , . . . belonging to some subshift Σ of N symbols (and the limit is independent of z), then there exists j ≥ 1 such that for every σ = σ 1 , σ 2 , . . . Σ the composition (1) is a contraction....

On the preservation of separation axioms in products

Milan Z. Grulović, Miloš S. Kurilić (1992)

Commentationes Mathematicae Universitatis Carolinae

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We give sufficient and necessary conditions to be fulfilled by a filter Ψ and an ideal Λ in order that the Ψ -quotient space of the Λ -ideal product space preserves T k -properties ( k = 0 , 1 , 2 , 3 , 3 1 2 ) (“in the sense of the Łos theorem”). Tychonoff products, box products and ultraproducts appear as special cases of the general construction.

Inessentiality with respect to subspaces

Michael Levin (1995)

Fundamenta Mathematicae

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Let X be a compactum and let A = ( A i , B i ) : i = 1 , 2 , . . . be a countable family of pairs of disjoint subsets of X. Then A is said to be essential on Y ⊂ X if for every closed F i separating A i and B i the intersection ( F i ) Y is not empty. So A is inessential on Y if there exist closed F i separating A i and B i such that F i does not intersect Y. Properties of inessentiality are studied and applied to prove:  Theorem. For every countable family of pairs of disjoint open subsets of a compactum X there exists an open set G ∩ X on...

Intersection topologies with respect to separable GO-spaces and the countable ordinals

M. Jones (1995)

Fundamenta Mathematicae

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Given two topologies, T 1 and T 2 , on the same set X, the intersection topologywith respect to T 1 and T 2 is the topology with basis U 1 U 2 : U 1 T 1 , U 2 T 2 . Equivalently, T is the join of T 1 and T 2 in the lattice of topologies on the set X. Following the work of Reed concerning intersection topologies with respect to the real line and the countable ordinals, Kunen made an extensive investigation of normality, perfectness and ω 1 -compactness in this class of topologies. We demonstrate that the majority of his results...

On the cardinality and weight spectra of compact spaces, II

Istvan Juhász, Saharon Shelah (1998)

Fundamenta Mathematicae

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Let B(κ,λ) be the subalgebra of P(κ) generated by [ κ ] λ . It is shown that if B is any homomorphic image of B(κ,λ) then either | B | < 2 λ or | B | = | B | λ ; moreover, if X is the Stone space of B then either | X | 2 2 λ or | X | = | B | = | B | λ . This implies the existence of 0-dimensional compact T 2 spaces whose cardinality and weight spectra omit lots of singular cardinals of “small” cofinality.

On the positivity of the number of t-core partitions

Ken Ono (1994)

Acta Arithmetica

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A partition of a positive integer n is a nonincreasing sequence of positive integers with sum n . Here we define a special class of partitions. 1. Let t 1 be a positive integer. Any partition of n whose Ferrers graph have no hook numbers divisible by t is known as a t- core partitionof n . The hooks are important in the representation theory of finite symmetric groups and the theory of cranks associated with Ramanujan’s congruences for the ordinary partition function [3, 4, 6]. If t 1 and n 0 ,...

On B 2 k -sequences

Martin Helm (1993)

Acta Arithmetica

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Introduction. An old conjecture of P. Erdős repeated many times with a prize offer states that the counting function A(n) of a B r -sequence A satisfies l i m i n f n ( A ( n ) / ( n 1 / r ) ) = 0 . The conjecture was proved for r=2 by P. Erdős himself (see [5]) and in the cases r=4 and r=6 by J. C. M. Nash in [4] and by Xing-De Jia in [2] respectively. A very interesting proof of the conjecture in the case of all even r=2k by Xing-De Jia is to appear in the Journal of Number Theory [3]. Here we present a different, very short proof...

A preconditioner for the FETI-DP method for mortar-type Crouzeix-Raviart element discretization

Chunmei Wang (2014)

Applications of Mathematics

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In this paper, we consider mortar-type Crouzeix-Raviart element discretizations for second order elliptic problems with discontinuous coefficients. A preconditioner for the FETI-DP method is proposed. We prove that the condition number of the preconditioned operator is bounded by ( 1 + log ( H / h ) ) 2 , where H and h are mesh sizes. Finally, numerical tests are presented to verify the theoretical results.

Bing maps and finite-dimensional maps

Michael Levin (1996)

Fundamenta Mathematicae

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Let X and Y be compacta and let f:X → Y be a k-dimensional map. In [5] Pasynkov stated that if Y is finite-dimensional then there exists a map g : X 𝕀 k such that dim (f × g) = 0. The problem that we deal with in this note is whether or not the restriction on the dimension of Y in the Pasynkov theorem can be omitted. This problem is still open.  Without assuming that Y is finite-dimensional Sternfeld [6] proved that there exists a map g : X 𝕀 k such that dim (f × g) = 1. We improve this result of Sternfeld...

On ergodicity of some cylinder flows

Krzysztof Frączek (2000)

Fundamenta Mathematicae

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We study ergodicity of cylinder flows of the form    T f : T × T × , T f ( x , y ) = ( x + α , y + f ( x ) ) , where f : T is a measurable cocycle with zero integral. We show a new class of smooth ergodic cocycles. Let k be a natural number and let f be a function such that D k f is piecewise absolutely continuous (but not continuous) with zero sum of jumps. We show that if the points of discontinuity of D k f have some good properties, then T f is ergodic. Moreover, there exists ε f > 0 such that if v : T is a function with zero integral such that D k v is of bounded...