and Applied Analysis 3 2. Main Results and Proofs In this section, we give our main results and their proofs. Before stating these results, we need some auxiliary results, which are incorporated in the lemmas which follows. Lemma 2.1. Assume that g ∈ H Bn and μ : Bn → 0,∞ are normal. Then Tg : H∞ log → Bμ is compact if and only if Tg : H∞ log → Bμ is bounded and for any bounded sequence fk k∈N ...

We compute all the groups G⊗H, Tor(G, H), Hom(G, H), and Ext(G, H), where G and H can be any of the groups Z (the integers), Z/n = Z/nZ (the integers mod n), or Q (the rationals). All but one are reasonably accessible. Because all these functors are biadditive, these cases suffice to handle any finitely generated groups G and H. The emphasis here is on computation, not on the abstract definitio...

and Applied Analysis 3 2. Some Properties of the Classes ∑∗ λ p, n, α and ∑λ c p, n, α Theorem 2.1. If f z ∈ ∑ p, n satisfies ∣ ∣ ∣ ∣ ( zf z )eiλ/ p−α cosλ { e zf ′ z f z α cosλ ip sinλ } ( p − α cosλ ∣ ∣ ∣ ∣ < n √ n2 1 ( p − α cosλ z ∈ U , 2.1 then f z ∈ ∑∗λ p, n, α . Proof. Let us set a function h z by h z 1 z ( zf z )eiλ/ p−α cosλ 1 z ean ( p − α cosλ n · · · 2.2 for f z ∈ ∑ p, n . Then clea...

In this paper, we consider the problem when a differential equation y(z)=Q(z)y(z) is Fuchsian on H* and apparent H, where Q(z) meromorphic modular form of weight 4 SL(2,Z) H denotes complex upper half-plane. Such closely related to existence conformal metric with curvature 1/2 H.

Let (Y, g) denote a compact, oriented Riemann 3-manifold without boundary. Denote by π : X → Y a principal S1-bundle over Y , and by Z → Y the associated 2-disk bundle. Set c := c1(Z) ∈ H(Y,Z). Denote by tZ ∈ H2(Z, X;Z) the Thom class of Z → Y , by j the inclusion X ↪→ Z and by ζ : Y ↪→ Z the natural inclusion. Using the Thom isomorphism H•(Z) ∪tZ −→ H•+2(Z,X;Z), c = ζ∗tZ , and the long exact c...

and Applied Analysis 3 Corresponding to a function hp α1, . . . , αq; β1, . . . , βs; z defined by hp ( α1, . . . , αq; β1, . . . , βs; z ) z−pqFs ( α1, . . . , αq; β1, . . . , βs; z ) , 1.11 we now consider a linear operator Hp ( α1, . . . , αq; β1, . . . , βs ) : Σ ( p ) −→ Σp, 1.12 defined by means of the Hadamard product or convolution as follows: Hp ( α1, . . . , αq; β1, . . . , βs ) f z :...

This paper introduces an enhanced solid-state transformer topology for a medium-voltage (MV) utility grid. The main objective of the current study is to develop improved flyback converter equipped with quasi z-source (qZ_iFC) having high-voltage conversion capability integration low-input voltage DC link MV modular multilevel (MMC). system integrates and converters by operating their existing s...