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In this paper, we show a Marcinkiewicz type interpolation theorem for Orlicz spaces. As an application, we obtain an existence result for a parabolic equation in divergence form.

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Let E, G be Fréchet spaces and F be a complete locally convex space. It is observed that the existence of a continuous linear not almost bounded operator T on E into F factoring through G causes the existence of a common nuclear Köthe subspace of the triple (E, G, F). If, in addition, F has the property (y), then (E, G, F) has a common nuclear Köthe quotient.

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In this paper we study the sum p x   τ ( n p ) , where τ ( n ) denotes the number of divisors of n, and {np } is a sequence of integers indexed by primes. Under certain assumptions we show that the aforementioned sum is   x  as  x   . As an application, we consider the case where the sequence is given by the Fourier coefficients of a modular form.

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In this paper we derive new inequalities involving the generalized Hardy operator. The obtained results generalized known inequalities involving the Hardy operator. We also get new inequalities involving the classical Hardy–Hilbert inequality.

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Mathematica Pannonica
Authors:
Anna Bachstein
,
Wayne Goddard
, and
Michael A. Henning

The bipartite domination number of a graph is the minimum size of a dominating set that induces a bipartite subgraph. In this paper we initiate the study of this parameter, especially bounds involving the order, the ordinary domination number, and the chromatic number. For example, we show for an isolate-free graph that the bipartite domination number equals the domination number if the graph has maximum degree at most 3; and is at most half the order if the graph is regular, 4-colorable, or has maximum degree at most 5.

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This short note deals with polynomial interpolation of complex numbers verifying a Lipschitz condition, performed on consecutive points of a given sequence in the plane. We are interested in those sequences which provide a bound of the error at the first uninterpolated point, depending only on its distance to the last interpolated one.

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For a lattice L of finite length n, let RCSub(L) be the collection consisting of the empty set and those sublattices of L that are closed under taking relative complements. That is, a subset X of L belongs to RCSub(L) if and only if X is join-closed, meet-closed, and whenever {a, x, b} ⊆ S, yL, xy = a, and xy = b, then yS. We prove that (1) the poset RCSub(L) with respect to set inclusion is lattice of length n + 1, (2) if RCSub(L) is a ranked lattice and L is modular, then L is 2-distributive in András P. Huhn’s sense, and (3) if L is distributive, then RCSub(L) is a ranked lattice.

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In this paper, centralizing (semi-centralizing) and commuting (semi-commuting) derivations of semirings are characterized. The action of these derivations on Lie ideals is also discussed and as a consequence, some significant results are proved. In addition, Posner’s commutativity theorem is generalized for Lie ideals of semirings and this result is also extended to the case of centralizing (semi-centralizing) derivations of prime semirings. Further, we observe that if there exists a skew-commuting (skew-centralizing) derivation D of S, then D = 0. It is also proved that for any two derivations d 1 and d 2 of a prime semiring S with char S ≠ 2 and x d 1 x d 2 = 0, for all xS implies either d 1 = 0 or d 2 = 0.

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We offer new properties of the special Gini mean S(a, b) = aa /( a + b )bb /( a + b ), in connections with other special means of two arguments.

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