Bulletin of the
Korean Mathematical Society
BKMS
ISSN(Print) 1015-8634
ISSN(Online) 2234-3016
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2023-03-31
The third Hermitian-Toeplitz and Hankel determinants for parabolic starlike functions
Rosihan M. Ali, Sushil Kumar, Vaithiyanathan RavichandranAbstract : 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.
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2023-07-31
On the greatest common divisor of binomial coefficients
Sunben Chiu, Pingzhi Yuan, Tao ZhouAbstract : 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} $}
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2023-07-31
Semi-symmetric structure Jacobi operator for real hypersurfaces in the complex quadric
Imsoon Jeong, Gyu Jong Kim, Changhwa WooAbstract : 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.
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2023-07-31
Annihilator ideals of simple modules of restricted quantized enveloping algebra
Yu WangAbstract : Let $U$ be the restricted quantized enveloping algebra $\widetilde{U}_q(\mathfrak{sl}_2)$ over an algebraically closed field of characteristic zero, where $q$ is a primitive $l$-th root of unity (with $l$ being odd and greater than $1$). In this paper we show that any indecomposable submodule of $U$ under the adjoint action is generated by finitely many special elements. Using this result we describe all ideals of $U$. Moreover, we classify annihilator ideals of simple modules of $U$ by generators.
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2023-07-31
Products of composition, differentiation and multiplication from the cauchy spaces to the Zygmund space
Rita HibschweilerAbstract : In this paper, we study products of composition, multiplication and differentiation acting on the fractional Cauchy spaces and mapping into the Zygmund space. Characterizations are provided for boundedness and compactness of these operators.
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2023-01-31
On Chowla's hypothesis implying that $L(s,\chi)>0$ for $s>0$ for real characters $\chi$
St\'ephane R. LouboutinAbstract : Let $L(s,\chi)$ be the Dirichlet $L$-series associated with an $f$-periodic complex function $\chi$. Let $P(X)\in {\mathbb C}[X]$. We give an expression for $\sum_{n=1}^f \chi (n)P(n)$ as a linear combination of the $L(-n,\chi)$'s for $0\leq n<\deg P(X)$. We deduce some consequences pertaining to the Chowla hypothesis implying that $L(s,\chi )>0$ for $s>0$ for real Dirichlet characters $\chi$. To date no extended numerical computation on this hypothesis is available. In fact by a result of R. C. Baker and H. L. Montgomery we know that it does not hold for almost all fundamental discriminants. Our present numerical computation shows that surprisingly it holds true for at least $65\%$ of the real, even and primitive Dirichlet characters of conductors less than $10^6$. We also show that a generalized Chowla hypothesis holds true for at least $72\%$ of the real, even and primitive Dirichlet characters of conductors less than $10^6$. Since checking this generalized Chowla's hypothesis is easy to program and relies only on exact computation with rational integers, we do think that it should be part of any numerical computation verifying that $L(s,\chi )>0$ for $s>0$ for real Dirichlet characters $\chi$. To date, this verification for real, even and primitive Dirichlet characters has been done only for conductors less than $2\cdot 10^5$.
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2023-07-31
On the structure of certain subset of Farey sequence
Xing-Wang Jiang, Ya-Li LiAbstract : Let $F_n$ be the Farey sequence of order $n$. For $S\subseteq F_n$, let $\mathcal{Q}(S)$ be the set of rational numbers $x/y$ with $x,y\in S,~x\leq y$ and $y\neq 0$. Recently, Wang found all subsets $S$ of $F_n$ with $|S|=n+1$ for which $\mathcal{Q}(S)\subseteq F_n$. Motivated by this work, we try to determine the structure of $S\subseteq F_n$ such that $|S|=n$ and $\mathcal{Q}(S)\subseteq F_n$. In this paper, we determine all sets $S\subseteq F_n$ satisfying these conditions for $n\in\{p,2p\}$, where $p$ is prime.
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2023-07-31
Some relations on parametric linear Euler sums
Weiguo Lu, Ce Xu, Jianing ZhouAbstract : Recently, Alzer and Choi [2] introduced and studied a set of the four linear Euler sums with parameters. These sums are parametric extensions of Flajolet and Salvy's four kinds of linear Euler sums [9]. In this paper, by using the method of residue computations, we will establish two explicit combined formulas involving two parametric linear Euler sums $S_{p,q}^{++}(a,b)$ and $S_{p,q}^{+-}(a,b)$ defined by Alzer and Choi, which can be expressed in terms of a linear combinations of products of trigonometric functions, digamma functions and Hurwitz zeta functions.
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2023-07-31
Dynamics of random dynamical systems
Enkhbayar Azjargal, Zorigt Choinkhor, Nyamdavaa TsegmidAbstract : In this paper, we introduce the concept of $\omega$-expansive of random map on compact metric spaces $\mathcal{P}$. Also we introduce the definitions of positively, negatively shadowing property and shadowing property for two-sided RDS. Then we show that if $\varphi$ is $\omega$-expansive and has the shadowing property for $\omega$, then $\varphi$ is topologically stable for $\omega$.
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2023-07-31
Some results on 2-strongly Gorenstein projective modules and related rings
Dong Chen, Kui HuAbstract : In this paper, we give some results on 2-strongly Gorenstein projective modules and related rings. We first investigate the relationship between strongly Gorenstein projective modules and periodic modules and then give the structure of modules over strongly Gorenstein semisimple rings. Furthermore, we prove that a ring $R$ is 2-strongly Gorenstein hereditary if and only if every ideal of $R$ is Gorenstein projective and the class of 2-strongly Gorenstein projective modules is closed under extensions. Finally, we study the relationship between 2-Gorenstein projective hereditary and 2-Gorenstein projective semisimple rings, and we also give an example to show the quotient ring of a 2-Gorenstein projective hereditary ring is not necessarily 2-Gorenstein projective semisimple.
Most Read
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Semi-symmetric structure Jacobi operator for real hypersurfaces in the complex quadric
Imsoon Jeong, Gyu Jong Kim, Changhwa Woo
Bull. Korean Math. Soc. 2023; 60(4): 849-861
https://doi.org/10.4134/BKMS.b220152 -
On the greatest common divisor of binomial coefficients
Sunben Chiu, Pingzhi Yuan, Tao Zhou
Bull. Korean Math. Soc. 2023; 60(4): 863-872
https://doi.org/10.4134/BKMS.b220166 -
The third Hermitian-Toeplitz and Hankel determinants for parabolic starlike functions
Rosihan M. Ali, Sushil Kumar, Vaithiyanathan Ravichandran
Bull. Korean Math. Soc. 2023; 60(2): 281-291
https://doi.org/10.4134/BKMS.b210368 -
Products of composition, differentiation and multiplication from the cauchy spaces to the Zygmund space
Rita Hibschweiler
Bull. Korean Math. Soc. 2023; 60(4): 1061-1070
https://doi.org/10.4134/BKMS.b220471
Most Downloaded
-
The third Hermitian-Toeplitz and Hankel determinants for parabolic starlike functions
Rosihan M. Ali, Sushil Kumar, Vaithiyanathan Ravichandran
Bull. Korean Math. Soc. 2023; 60(2): 281-291
https://doi.org/10.4134/BKMS.b210368 -
On the greatest common divisor of binomial coefficients
Sunben Chiu, Pingzhi Yuan, Tao Zhou
Bull. Korean Math. Soc. 2023; 60(4): 863-872
https://doi.org/10.4134/BKMS.b220166 -
Semi-symmetric structure Jacobi operator for real hypersurfaces in the complex quadric
Imsoon Jeong, Gyu Jong Kim, Changhwa Woo
Bull. Korean Math. Soc. 2023; 60(4): 849-861
https://doi.org/10.4134/BKMS.b220152 -
Annihilator ideals of simple modules of restricted quantized enveloping algebra
Yu Wang
Bull. Korean Math. Soc. 2023; 60(4): 1025-1034
https://doi.org/10.4134/BKMS.b220460
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