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"Bernstein"
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Approximation by Stancu-type α-Bernstein-Schurer-Kantorovich operators
In the present article, we study the approximation properties of constructed operators based on the shape parameter
α
. We construct the Stancu-type operators of
α
-Bernstein–Schurer–Kantorovich operators. Here the shape parameter
α
∈
[
0
,
1
]
. We obtain the convergence of proposed operators in terms of Lipschitz-continuous functions and Peetre’s
K
-functional by using the modulus of continuity of orders one and two.
Journal Article
Experiencing Leonard Bernstein
Leonard Bernstein is a household name. Most know him for his classic musical reworking of Shakespeare's Romeo and Juliet as Broadway's West Side Story. But Bernstein accomplished so much more as a composer, and his body of work is both broad and varied. He composed ballets (Fancy Free, Facsimile, Dybbuk), operas (Trouble in Tahiti, Candide, A Quiet Place), musicals (On the Town, Wonderful Town), film scores (On the Waterfront), symphonies, choral works, chamber music pieces, art songs, and piano works. In Experiencing Leonard Bernstein: A Listener's Companion, Kenneth LaFave guides readers past Bernstein's famously tortured personal problems and into the clarity and balance of his Serenade after Plato's Symposium for Violin and Orchestra, the intense drama of his music for On the Waterfront, the existential cosmography of his three symphonies, and his vibrant works for the musical stage. Perhaps the most famous American classical musician born in the twentieth century, Bernstein divided his time between composing, conducting, writing, and teaching, a busy schedule—especially his conducting of major orchestras—that set his work as composer at a disadvantage. Often generated in short spurts, his work carries an urgency—and even an element of improvisational genius—that he flavored with his eclectic embrace of jazz, folk song, Jewish cantorial music, and innovations in contemporary classical theory. The result is a body of work that is beguilingly melodic, incomparably rhythmic, and irrepressibly individual. Experiencing Leonard Bernstein: A Listener's Companion is the ideal work for any reader seeking to learn how to listen across the spectrum of Bernstein's musical output.
eBook
Cascaded parametric amplification of lower hybrid waves using electron Bernstein waves
We propose a novel method for generating high-harmonic ion Bernstein waves in the lower hybrid (LH) range of frequencies—here referred to as LH waves—in the confined region of magnetized plasmas. The method relies on stimulating a cascaded parametric down-conversion of electron Bernstein waves (EBWs). In addition to a primary microwave source (such as a gyrotron), cascaded parametric amplification (CPA) of LH waves requires either a low-power EBW or a low-power LH wave for seeding. The CPA mechanism is demonstrated with a 1D Wentzel-Kramers-Brillouin (WKB) model, which is benchmarked against a fully kinetic particle-in-cell simulation, showing quantitative agreement. Our calculations predict attainable conversion efficiencies to LH waves around 10% in steady state, exceeding the Manley–Rowe limit more than tenfold. The CPA mechanism could provide a path for LH wave generation in magnetic confinement fusion reactors that rely on microwaves in the electron cyclotron range of frequencies for plasma heating.
Journal Article
Hardy–Littlewood and Ulyanov inequalities
We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness
The main tool is the new
Hardy–Littlewood–Nikol’skii inequalities. More precisely, we obtained the asymptotic behavior of the quantity
We also prove the
Ulyanov and Kolyada-type inequalities in the Hardy spaces. Finally, we apply the obtained estimates to derive new embedding theorems for
the Lipschitz and Besov spaces.
eBook
Leonard Bernstein : a research and information guide
Leonard Bernstein: A Research and Information Guide is an annotated bibliography and research guide on this popular American composer and conductor. It includes annotations on Bernstein's writings, performances, educational work, and major secondary sources. Also included are a biographical sketch, lists of compositions and arrangements, as well as lists of recordings and video. The second edition is updated to include research since the 1st edition was published in 2001, as well as online resources [Publisher description]
BOOK
Convergence properties related to Bézier type of λ-Bernstein Kantorovich shifted knots operators
In this article, we introduce the Lp
-spaces and create the Kantorovich-type operators of Schurer λ-Bernstein-Bézier basis functions, starting with shifted knots polynomials. We describe the convergence of our novel operators in the Lebesgue spaces and the continuous function space for any 1 ≤ p < ∞. The central moments for these operators are determined by computing the test functions. We then examine the properties of the Korovkin’s type approximation with modulus of continuity of order one and two. We also derive the convergence theorems for these new operators using Peetre’s K-functional and the fundamental conditions of Lipschitz continuous functions. Several direct approximation theorems are also derived by us. In last we given a numerical example with a graphical analysis.
Journal Article