On the <i>n</i>
th Quantum Derivative

Ash, J. Marshall; Catoiu, Stefan; Ríos-Collantes-de-Terán, Ricardo

doi:10.1112/s0024610702003198

Cited by 17 publications

(15 citation statements)

References 7 publications

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“…The repeated application of (a.3) followed by the limit computation leads to the N th order derivative (Ash et al, 2002;Koornwinder, 1999):…”

Section: Discussionmentioning

confidence: 99%

ON THE LINEAR SCALE FRACTIONAL SYSTEMS - An Application of the Fractional Quantum Derivative

Ortigueira¹

2009

Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics

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The Linear Scale Invariant Systems are introduced for both integer and fractional orders. They are defined by the generalized Euler-Cauchy differential equation. It is shown how to compute the impulse responses corresponding to the two regions of convergence of the transfer function. This is obtained by using the Mellin transform. The quantum fractional derivatives are used because they are suitable for dealing with this kind of systems.

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“…The repeated application of (a.3) followed by the limit computation leads to the N th order derivative (Ash et al, 2002;Koornwinder, 1999):…”

Section: Discussionmentioning

confidence: 99%

ON THE LINEAR SCALE FRACTIONAL SYSTEMS - An Application of the Fractional Quantum Derivative

Ortigueira¹

2009

Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics

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“…LetĒ be the family of all perfect subsets P of G of positive measure such that |P ∩(C +C)| < c. 1 Let E be a maximal subfamily ofĒ of pairwise disjoint sets. Then E is at most countable, so E = E is an F σ -set and E ∩ (C + C) has cardinality less than c. For every e ∈ E ∩ (C + C) fix c e , d e ∈ C such that e = c e + d e .…”

Section: Theoremmentioning

confidence: 99%

Measure Zero Sets with Non-Measurable Sum

Ciesielski¹,

Fejzić²,

Freiling³

2002

Real Analysis Exchange

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It is not at all surprising that there should be measure zero sets, A, whose sum A+A = {x+y : x ∈ A, y ∈ A} is non-measurable. Ask a typical mathematician why this should be so and you are likely to get the following response: The Cantor middle-third set, when added to itself gives an entire interval, [0, 2]. So certainly there exists a measure zero set that when added to itself gives a non-measurable set. The intuition being that an interval has much more content than is needed for a non-measurable set. Indeed such sets do exist (in ZFC). Sierpiński (1920) seems to be the first to address this issue. Actually, he shows the existence of measure zero sets

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“…to the sequence of differences ∆ n (q, x; f ) defined in [ACR,Proposition 2]. The third sequence is d n := δ n (y, 1); here λ n becomes 2 n .…”

Section: Theorem 4 If F Has An A-quantum Symmetric Derivative Of Ordermentioning

confidence: 99%

Quantum symmetric 𝐿^{𝑝} derivatives

Ash

Catoiu

2007

Trans. Amer. Math. Soc.

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Abstract. For 1 ≤ p ≤ ∞, a one-parameter family of symmetric quantum derivatives is defined for each order of differentiation as are two families of Riemann symmetric quantum derivatives. For 1 ≤ p ≤ ∞, symmetrization holds, that is, whenever the L p kth Peano derivative exists at a point, all of these derivatives of order k also exist at that point. The main result, desymmetrization, is that conversely, for 1 ≤ p ≤ ∞, each L p symmetric quantum derivative is a.e. equivalent to the L p Peano derivative of the same order. For k = 1 and 2, each kth L p symmetric quantum derivative coincides with both corresponding kth L p Riemann symmetric quantum derivatives, so, in particular, for k = 1 and 2, both kth L p Riemann symmetric quantum derivatives are a.e. equivalent to the L p Peano derivative.

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On the n th Quantum Derivative

Cited by 17 publications

References 7 publications

ON THE LINEAR SCALE FRACTIONAL SYSTEMS - An Application of the Fractional Quantum Derivative

ON THE LINEAR SCALE FRACTIONAL SYSTEMS - An Application of the Fractional Quantum Derivative

Measure Zero Sets with Non-Measurable Sum

Quantum symmetric 𝐿^{𝑝} derivatives

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