Abstract : In this article, we study the weak and extra-weak type integral inequalities for the modified integral Hardy operators. We provide suitable conditions on the weights $\omega, \rho, \phi$ and $\psi$ to hold the following weak type modular inequality \begin{align*} \mathcal{U}^{-1} \bigg ( \int_{ \{ | \mathcal{I}f | > \gamma\}} \mathcal{U} \Big(\gamma \omega \Big ) \rho \bigg ) & \leq \mathcal{V}^{-1} \bigg ( \int_{0}^{\infty} \mathcal{V} \Big ( C |f| \phi\Big) \psi \bigg ), \end{align*} where $\mathcal{I}$ is the modified integral Hardy operators. We also obtain a necesary and sufficient condition for the following extra-weak type integral inequality \begin{align*} \omega \bigg ( \Big\{ |\mathcal{I}f| > \gamma \Big \} \bigg) &\leq \mathcal{U}\circ \mathcal{V}^{-1} \bigg ( \int_{0}^{\infty} \mathcal{V} \bigg ( \dfrac{C |f| \phi}{\gamma} \bigg) \psi \bigg ). \end{align*} Further, we discuss the above two inequalities for the conjugate of the modified integral Hardy operators. It will extend the existing results for the Hardy operator and its integral version.
Abstract : Let $T$ be a bilinear Calder\'{o}n-Zygmund operator, $$b\in \cup_{q>1}L_{loc}^{q}(G).$$ We firstly obtain a constructive proof of the weak factorisation of Hardy spaces. Then we establish the characterization of $BMO$ spaces by the boundedness of the commutator $[b, T]_{j}$ in variable Lebesgue spaces.
Abstract : We give a formula for the sizes of the dual groups. It is obtained by generalizing a size estimation of certain algebraic structure that lies in the heart of the proof of the celebrated primality test by Agrawal, Kayal and Saxena. In turn, by using our formula, we are able to give a streamlined survey of the AKS test.
Abstract : In this paper, we introduce the notion of {\it semi-symmetric structure Jacobi operator } for Hopf real hypersufaces in the complex quad\-ric $Q^m = SO_{m+2}/SO_mSO_2$. Next we prove that there does not exist any Hopf real hypersurface in the complex quadric $Q^m = SO_{m+2}/SO_mSO_2$ with semi-symmetric structure Jacobi operator. As a corollary, we also get a non-existence property of Hopf real hypersurfaces in the complex quadric $Q^m$ with either symmetric (parallel), or recurrent structure Jacobi operator.
Abstract : Let $n\geqslant 2$ be an integer, we denote the smallest integer $b$ such that $\gcd\qty{\binom nk: b<k<n-b}>1$ as $b(n)$. For any prime $p$, we denote the highest exponent $\alpha$ such that $p^\alpha\mid n$ as $v_p(n)$. In this paper, we partially answer a question asked by Hong in 2016. For a composite number $n$ and a prime number $p$ with $p\mid n$, let $n=a_mp^m+r$, $0\leqslant r<p^m$, $0<a_m<p$. Then we have\\ \resizebox{\linewidth}{4.5mm}{ $\displaystyle v_p\qty(\gcd\qty{\binom nk: b(n)<k<n-b(n),\ (n,k)>1})= \begin{cases} 1,&a_m=1\text{ and }r=b(n), \\ 0,&\text{otherwise}. \end{cases} $}
Abstract : In this paper, the concepts of $w$-linked homomorphisms, the $w_{\phi}$-operation, and DW${}_{\phi}$ rings are introduced. Also the relationships between $w_{\phi}$-ideals and $w$-ideals over a $w$-linked homomorphism $\phi: R\ra T$ are discussed. More precisely, it is shown that every $w_{\phi}$-ideal of $T$ is a $w$-ideal of $T$. Besides, it is shown that if $T$ is not a DW${}_{\phi}$ ring, then $T$ must have an infinite number of maximal $w_{\phi}$-ideals. Finally we give an application of Cohen's Theorem over $w$-factor rings, namely it is shown that an integral domain $R$ is an SM-domain with $w$-$\dim(R)\leq 1$, if and only if for any nonzero $w$-ideal $I$ of $R$, $(R/I)_w$ is an Artinian ring, if and only if for any nonzero element $a\in R$, $(R/(a))_w$ is an Artinian ring, if and only if for any nonzero element $a\in R$, $R$ satisfies the descending chain condition on $w$-ideals of $R$ containing $a$.
Abstract : It is well known that the continued fraction expansion of $\sqrt{d}$ has the form $[a_0, \overline{a_1, \ldots, a_{l-1}, 2a_0}]$ and $a_1, \ldots, a_{l-1}$ is a palindromic sequence of positive integers. For a given positive integer $l$ and a palindromic sequence of positive integers $a_1, \ldots, a_{l-1}$, we define the set $S(l;a_1,$ $\ldots, a_{l-1}) :=\{d\in \mathbb{Z} \,| \, d>0, \sqrt{d}=[a_0, \overline{a_1, \ldots, a_{l-1}, 2a_0}], \, \textup{where} \, a_0=\lfloor \sqrt{d} \rfloor\}$. In this paper, we completely determine when $S(l;a_1, \ldots, a_{l-1})$ is not empty in the case that $l$ is $4$, $5$, $6$, or $7$. We also give similar results for $(1+\sqrt{d})/2$. For the case that $l$ is $4$, $5$, or $6$, we explicitly describe the fundamental units of the real quadratic field $\mathbb{Q}(\sqrt{d})$. Finally, we apply our results to the Mordell conjecture for the fundamental units of $\mathbb{Q}(\sqrt{d})$.
Abstract : This article concerns a ring property that arises from combining one-sided duo factor rings and centers. A ring $R$ is called {\it right CIFD} if $R/I$ is right duo by some proper ideal $I$ of $R$ such that $I$ is contained in the center of $R$. We first see that this property is seated between right duo and right $\pi$-duo\textbf{,} and not left-right symmetric. We prove, for a right CIFD ring $R$, that $W(R)$ coincides with the set of all nilpotent elements of $R$; that $R/P$ is a right duo domain for every minimal prime ideal $P$ of $R$; that $R/W(R)$ is strongly right bounded; and that every prime ideal of $R$ is maximal if and only if $R/W(R)$ is strongly regular, where $W(R)$ is the Wedderburn radical of $R$. It is also proved that a ring $R$ is commutative if and only if $D_3(R)$ is right CIFD, where $D_3(R)$ is the ring of $3$ by $3$ upper triangular matrices over $R$ whose diagonals are equal. Furthermore\textbf{,} we show that the right CIFD property does not pass to polynomial rings, and that the polynomial ring over a ring $R$ is right CIFD if and only if $R/I$ is commutative by a proper ideal $I$ of $R$ contained in the center of $R$.
Abstract : In this paper, we prove that any orthogonal almost complex structure on a warped product manifold of any oriented closed surface and a round 4-sphere for a concircular warping function on the sphere is never integrable. This gives a partial answer to Calabi's problem.
Abstract : A normalized analytic function $f$ is parabolic starlike if $w(z)$ $:=zf'(z)/f(z)$ maps the unit disk into the parabolic region $\{w: \operatorname{Re} w>|w-1|\}$. Sharp estimates on the third Hermitian-Toeplitz determinant are obtained for parabolic starlike functions. In addition, upper bounds on the third Hankel determinants are also determined.
Duranta Chutia, Rajib Haloi
Bull. Korean Math. Soc. 2022; 59(3): 757-780
https://doi.org/10.4134/BKMS.b210469
Dongli Liu, Jian Tan, Jiman Zhao
Bull. Korean Math. Soc. 2022; 59(3): 547-566
https://doi.org/10.4134/BKMS.b201019
Joungmin Song
Bull. Korean Math. Soc. 2022; 59(3): 609-615
https://doi.org/10.4134/BKMS.b210096
Imsoon Jeong, Gyu Jong Kim, Changhwa Woo
Bull. Korean Math. Soc. 2023; 60(4): 849-861
https://doi.org/10.4134/BKMS.b220152
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
Rita Hibschweiler
Bull. Korean Math. Soc. 2023; 60(4): 1061-1070
https://doi.org/10.4134/BKMS.b220471
Imsoon Jeong, Gyu Jong Kim, Changhwa Woo
Bull. Korean Math. Soc. 2023; 60(4): 849-861
https://doi.org/10.4134/BKMS.b220152
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