1,721,055 research outputs found
Invariance of Ideal Limit Points
Full text linkLet I be an analytic P-ideal [respectively, a summable ideal] on the positive integers and let (xn) be a sequence taking values in a metric space X. First, it is shown that the set of ideal limit points of (xn) is an Fσ-set [resp., a closet set]. Let us assume that X is also separable and the ideal I satisfies certain additional assumptions, which however includes several well-known examples, e.g., the collection of sets with zero asymptotic density, sets with zero logarithmic density, and some summable ideals. Then, it is shown that the set of ideal limit points of (xn) is equal to the set of ideal limit points of almost all its subsequences
Thinnable Ideals and Invariance of Cluster Points
No full textWe define a class of so-called thinnable ideals I on the positive integers which includes several well-known examples, e.g., the collection of sets with zero asymptotic density, sets with zero logarithmic density, and several summable ideals. Given a sequence (xn) taking values in a separable metric space and a thinnable ideal I, it is shown that the set of I-cluster points of (xn) is equal to the set of I-cluster points of almost all of its subsequences, in the sense of Lebesgue measure. Lastly, we obtain a characterization of ideal convergence, which improves the main result in [15]
Convergence Rates of Subseries
No full textLet (Formula presented.) be a positive real sequence decreasing to 0 such that the series (Formula presented.) is divergent and (Formula presented.). We show that there exists a constant (Formula presented.) such that, for each (Formula presented.), there is a subsequence (Formula presented.) for which (Formula presented.) and (Formula presented.)
A Characterization of Cesàro Convergence
Full text linkWe show that a real bounded sequence (Formula presented.) is Cesàro convergent to (Formula presented.) if and only if the sequence of averages with indices in (Formula presented.) converges to (Formula presented.) for all (Formula presented.). If, in addition, the sequence (Formula presented.) is nonnegative, then it is Cesàro convergent to 0 if and only if the same condition holds for some (Formula presented.)
Continuous projections onto ideal convergent sequences
No full textLet I⊆ P(ω) be a meager ideal. Then there are no continuous projections from l∞ onto the set of bounded sequences which are I-convergent to 0. In particular, it follows that the set of bounded sequences statistically convergent to 0 is not isomorphic to l∞
A Characterization of Convex Functions
Full text linkLet f be a real-valued radially lower semicontinuous function defined on a convex subset D of a real vector space. It is shown that f is convex if and only if, for all x, y ∈ D there exists α = α(x, y) ∈ (0, 1) such that f (αx + (1 − α)y) ≤ αf (x) + (1 − α)f (y)
A characterization of Sophie Germain primes
Full text link Let [Formula: see text] be an odd integer. It is shown that [Formula: see text] is a complete residue system modulo [Formula: see text] for some permutation [Formula: see text] of [Formula: see text] if and only if [Formula: see text] is a Sophie Germain prime. Partial results are obtained also for the case [Formula: see text] even. </jats:p
Limit points of subsequences
No full textLet x be a sequence taking values in a separable metric space and let I be an Fσδ-ideal on the positive integers (in particular, I can be any Erdős–Ulam ideal or any summable ideal). It is shown that the collection of subsequences of x which preserve the set of I-cluster points of x is of second category if and only if the set of I-cluster points of x coincides with the set of ordinary limit points of x; moreover, in this case, it is comeager. The analogue for I-limit points is provided. As a consequence, the collection of subsequences of x which preserve the set of ordinary limit points is comeager
Characterizations of the ideal core
Full text linkGiven an ideal I on ω and a sequence x in a topological vector space, we let the I-core of x be the least closed convex set containing {xn:n∉I} for all I∈I. We show two characterizations of the I-core. This implies that the I-core of a bounded sequence in Rk is simply the convex hull of its I-cluster points. As applications, we simplify and extend several results in the context of Pringsheim-convergence and e-convergence of double sequences
On small sets of integers
No full textAn upper quasi-density on H (the integers or the non-negative integers) is a real-valued subadditive function μ⋆ defined on the whole power set of H such that μ⋆(X) ≤ μ⋆(H) = 1 and μ⋆(k·X+h)=1kμ⋆(X) for all X⊆ H, k∈ N+, and h∈ N, where k· X: = { kx: x∈ X} ; and an upper density on H is an upper quasi-density on H that is non-decreasing with respect to inclusion. We say that a set X⊆ H is small if μ⋆(X) = 0 for every upper quasi-density μ⋆ on H. Main examples of upper densities are given by the upper analytic, upper Banach, upper Buck, and upper Pólya densities, along with the uncountable family of upper α-densities, where α is a real parameter ≥ - 1 (most notably, α= - 1 corresponds to the upper logarithmic density, and α= 0 to the upper asymptotic density). It turns out that a subset of H is small if and only if it belongs to the zero set of the upper Buck density on Z. This allows us to show that many interesting sets are small, including the integers with less than a fixed number of prime factors, counted with multiplicity; the numbers represented by a binary quadratic form with integer coefficients whose discriminant is not a perfect square; and the image of Z through a non-linear integral polynomial in one variable
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