Abstract : In this paper, we consider the differential equation \begin{equation*} f''+Af'+Bf=0, (*) \end{equation*} where $A(z)$ and $B(z)\not\equiv 0$ are entire functions. Assume that $A(z)$ is a non-trivial solution of $\omega''+P(z)\omega=0$, where $P(z)$ is a polynomial. If $B(z)$ satisfies extremal for Yang's inequality and other conditions, then every non-trivial solution $f$ of equation (*) has $\mu(f)=\infty$. We also investigate the relation between a small function and a differential polynomial of $f$.
Abstract : Von Neumann regular rings are studied by ring theorists and functional analysts in connection with operator algebra theory. In particular, the concept of idempotent in algebra is a generalization of projection in analysis. We study the structure of idempotents in $\pi$-regular rings, right AI rings (i.e., for every element $a$, $ab$ is an idempotent for some nonzero element $b$), NI rings, and generalized regular rings (i.e., every nonzero principal right ideal contains a nonzero idempotent). We obtain a well-known fact, proved by Menal, Nicholson and Zhou, that idempotents can be lifted modulo every ideal in $\pi$-regular rings, as a corollary of one of main results of this article. It is shown that the $\pi$-regularity is seated between right AI and regularity. We also show that from given any $\pi$-regular ring, we can construct a right AI ring but not $\pi$-regular. In addition, we study the structure of idempotents of $\pi$-regular rings and right AI rings in relation to the ring properties of Abelian and NI, giving simpler proofs to well-known results for Abelian $\pi$-regular rings.
Abstract : We confirm the conjecture proposed by ourselves and J. Lovejoy that for all $n>9$ \[ p'_e(n) > p'_o(n) \] holds, where $p'_e(n)$ (respectively, $p'_o(n)$) is the number of partitions of $n$ having an even (respectively, odd) number of odd parts larger than twice of the number of even parts. Moreover, we examine the connections between the number of partitions weighted by the number of two types of parts and partition functions from the literature on the theory of partitions.
Abstract : In the present paper we establish the boundedness and continuity of the higher order maximal commutators with Lipschitz symbols on the Sobolev spaces, Triebel–Lizorkin spaces and Besov spaces. More precisely, let 0 ≤ α < d and Mkb,α(k ≥ 1) be the k-th order fractional maximal commutator. When α = 0, we denote Mkb,α = Mkb. We prove that Mkb,α maps the first order Sobolev spaces W1,p(Rd) boundedly and continuously to W1,q(Rd) for 1 < p < q < ∞ and 1/q = 1/p−α/d if b belongs to the inhomogeneous Lipschitz space Lip(Rd). We also show that if 0 < γ ≤ 1,0 < s < γ, 1 < p,q < ∞ and b ∈ Lipγ(Rd), then Mkbis bounded and continuous from the fractionalSobolev spaces Ws,p(Rd) to itself, from the inhomogeneous Triebel–Lizorkin spaces Fp,qs (Rd) to itself and from the inhomogeneous Besov spaces Bp,qs (Rd) to itself.
Abstract : In [16], we obtain two Lichnerowicz type formulas for the generalized Zhang’s operator. And we give the proof of the Kastler-Kalau-Walze type theorem for the generalized Zhang’s operator on 4-dimensional oriented compact manifolds with (resp. without) boundary. In this paper, we give the proof of the Kastler-Kalau-Walze type theorem for the generalized Zhang’s operator on 6-dimensional oriented compact manifolds with boundary.
Imsoon Jeong, Gyu Jong Kim, Changhwa Woo
Bull. Korean Math. Soc. 2023; 60(4): 849-861
https://doi.org/10.4134/BKMS.b220152
Sunben Chiu, Pingzhi Yuan, Tao Zhou
Bull. Korean Math. Soc. 2023; 60(4): 863-872
https://doi.org/10.4134/BKMS.b220166
Rosihan M. Ali, Sushil Kumar, Vaithiyanathan Ravichandran
Bull. Korean Math. Soc. 2023; 60(2): 281-291
https://doi.org/10.4134/BKMS.b210368
Rita Hibschweiler
Bull. Korean Math. Soc. 2023; 60(4): 1061-1070
https://doi.org/10.4134/BKMS.b220471
Rosihan M. Ali, Sushil Kumar, Vaithiyanathan Ravichandran
Bull. Korean Math. Soc. 2023; 60(2): 281-291
https://doi.org/10.4134/BKMS.b210368
Sunben Chiu, Pingzhi Yuan, Tao Zhou
Bull. Korean Math. Soc. 2023; 60(4): 863-872
https://doi.org/10.4134/BKMS.b220166
Imsoon Jeong, Gyu Jong Kim, Changhwa Woo
Bull. Korean Math. Soc. 2023; 60(4): 849-861
https://doi.org/10.4134/BKMS.b220152
Rita Hibschweiler
Bull. Korean Math. Soc. 2023; 60(4): 1061-1070
https://doi.org/10.4134/BKMS.b220471
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