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2024-11-30
Cyclic and constacyclic codes over $\mathbb{Z}_{4}+w\mathbb{Z}_{4}$
Nuh Aydin; Yasemin Cengellenmis; Abdullah DertliAbstract : In this paper, we study the structure of cyclic, quasi-cyclic and constacyclic codes over the finite ring $R=\mathbb{Z}_{4}+w\mathbb{Z}_{4}$ of order 16, where $w^{2}=2$. Additionally, self-dual codes and their properties are discussed. Two Gray maps are defined to obtain linear codes over $\mathbb{Z}_4$. Gray images of cyclic, quasi-cyclic and constacyclic codes over $R$ are examined under these maps. We present new linear codes over $\mathbb{Z}_4$ obtained from codes over $R$ via each Gray map.
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2024-11-30
Rational homology balls in Milnor fibers of cyclic quotient surface singularities
Heesang Park; Dongsoo ShinAbstract : We show the existence of infinitely many rational homology balls embedded in Milnor fibers of certain cyclic quotient surface singularities, where the rational blowup surgeries performed on these balls cannot be achieved through ordinary rational blow-ups from the Milnor fibers in which they are embedded.
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2024-11-30
The proportion of monogenic orders of prime power indices of a pure cubic field
Minchan Kang
; Dohyeong Kim Abstract : In this paper, we investigate the proportion of monogenic orders among the orders whose indices are powers of a fixed prime in a pure cubic field. We prove that the proportion is zero for a prime number that is not equal to 2 or 3. To do this, we first count the number of orders whose indices are powers of a fixed prime. This is done by considering every full rank submodules of the ring of integers, and establishing the condition to be closed under multiplication. Next, we derive the index form of arbitrary orders based on the index form of the ring of integers. Then, we obtain an upper bound of the number of monogenic orders with prime power indices by applying the finiteness of the number of primitive solutions of a Thue-Mahler equation.
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2024-11-30
Screen isotropic leaves and Simons' formula for null hypersurfaces
Théophile Kemajou Mbiakop; Oscar PalmasAbstract : We study the null hypersurfaces of a Lorentzian manifold through the rigging technique, which uses a convenient choice of a vector field in the ambient space to define a Riemannian metric. With this machinery at hand, we study the geometry of null hypersurfaces admitting screen isotropic leaves. In the second part of the paper, we establish a formula for the Laplacian of the null second fundamental form of a null hypersurface of a Lorentzian manifold with constant curvature. Based on this formula, we characterize null hypersurfaces for which the screen sectional curvature is non-negative. We also prove two de Rham-like decomposition theorems, showing that under suitable conditions, the null hypersurface has a triple product structure.
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2024-11-30
Complex symmetry of Toeplitz operators and composition operators
Xuke Hu; Yanyue Shi; Changguo Wei Abstract : Denote by $H_{\gamma}(\mathbb{D})$ the Hilbert space of holomorphic functions over the open unit disk $\mathbb{D}$ with the reproducing kernel $K_{\omega}^{(\gamma)}(z)=\frac{1}{(1-\overline{\omega}z)^{\gamma}}$, $\gamma>0$. Let $J_{\alpha}: J_{\alpha}f(z)=\overline{f(\overline{\alpha z})}$ for $z\in \mathbb{D}$ and $\alpha \in \partial \mathbb{D}$, and let $W_{u,v}$ be the weighted composition operator. In this paper, we prove that if $b\in \mathbb{R}$ and $b\neq \frac{k\pi}{2}$, for any integer $k$, then $(\cos b+i\sin bW_{u,v})J_{\alpha}$ is a conjugation if and only if $v(z)=\frac{c-z}{1-\overline{c}z}$, $u(z)=\pm k_{c}^{(\gamma)}(z)$, $c\in\mathbb{D}$ and $\overline{c}=c\alpha$, where $k_c^{(\gamma)}(z)$ is the normalized reproducing kernel, or $(\cos b+i\sin bW_{u,v})J_{\alpha}=\lambda J_{\alpha}$ for some $\lambda\in \mathbb{C}$ with $|\lambda|=1$. We then derive a necessary and sufficient condition for a Toeplitz operator to be complex symmetric with respect to the conjugation $(\cos b+i\sin bW_{u,v})J_{\alpha}$ on $H_{\gamma}(\mathbb{D})$ with $\gamma \geq 1$. Similarly, we also conduct a similar study for composition operators on $H_{\gamma}(\mathbb{D})$ with $\gamma >0$.
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2024-11-30
Infinitely many solutions for a fractional $p$-Kirchhoff type equation on the Heisenberg group
Yanhong WangAbstract : In this work, we deal with the fractional $p$-Kirchhoff type equation with critical growth on the Heisenberg group, we make a truncation on the function $M$ and define an auxiliary problem. By using Krasnoselskii's genus theory, we obtain that the auxiliary problem has infinitely many nontrivial solutions. With aid of the size of $\lambda$, we prove that each solution of the auxiliary problem corresponds to a solution of the original problem.
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2024-11-30
On the characteristic Jacobi operator of the unit tangent sphere bundles over surfaces
Jun-ichi Inoguchi; Ji-Eun Lee Abstract : We prove that the characteristic Jacobi operator $\ell$ of the unit tangent sphere bundle over a surface is properly pseudo-parallel when and only when the base surface is of constant curvature $1$ or $4/3$. The latter case only occurs for $2$-dimensional base manifolds.
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2024-11-30
Proper classes generated by isosimple modules
Salahattin Özdemir; Zübeyir Türkoğlu Abstract : The proper class projectively generated by isosimple modules and that injectively generated by isosimple modules do not coincide in general, even for abelian groups. We characterize the commutative principal ideal domains in which these two proper classes coincide. We determine some closure properties of the coprojectives of the latter proper class, and characterize the rings all of whose modules are such coprojectives: right strongly $V$-rings (i.e., rings all of whose isosimple modules are injective).
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2024-11-30
On powers of some classes of monomial ideals
Hafsa Bibi; Asia Rauf; Ayesha UmarAbstract : The depth of edge ideal of centipede and related graphs is computed. It also discussed the persistence property of the path ideal of length three in the centipede-related graphs. In particular, it is proved that these ideals satisfy the strong persistence property.
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2024-11-30
A generalization of conducive domains
Hwankoo Kim; Rahul KumarAbstract : A domain $R$ is called conducive if every conductor ideal $(R:T)$ is nonzero for all overrings $T$ of $R$ other than the quotient field of $R$. Let $\mathcal{H}$ denote the set of all commutative rings $R$ for which the set of all nilpotent elements forms a divided prime ideal. We extend the concept of conducive domains to the rings in the class $\mathcal{H}$. Initially, we explore the basic properties of $\phi$-conducive rings and rings closely related to them. Subsequently, we study these properties in the context of a specific pullback construction and a trivial ring extension.
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2024-11-30
Some double series for $\pi$ and their $q$-analogues
Guoping Gu; Xiaoxia WangAbstract : Inspired by the recent work on double series for $\pi$ of Wei, in this paper, by applying the partial derivative operator on several summation formulas of hypergeometric series and basic hypergeometric series, we establish some double series for $\pi$ and their $q$-analogues.
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2024-11-30
On convergence rate in the weak law of large numbers for FGM random sequences
Mohammad Amini; Omolbanin Hadianfar; Habib NaderiAbstract : In this paper, we establish the weak law of large numbers for FGM random sequences by extending the classic Kolmogorov--Feller weak law of large numbers. In addition, we make a simulation study for the asymptotic behavior in the sense of convergence in probability for FGM random sequences.
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2024-11-30
Integers not of the form $2^n+p^{\alpha} q^{\beta}$
Jun-Jia ZhaoAbstract : Let $\mathbb{N}$ and $\mathcal{P}$ denote the set of nonnegative integers and the set of odd primes, respectively. In this paper, we prove that if $r$ is a positive integer with $(r,30)=1$, then there is a positive proportion of primes $p$ such that $p^r\neq2^n+q_1^{\alpha} q_2^{\beta}$ for any $n,\alpha,\beta\in\mathbb{N},q_1,q_2\in\mathcal{P}$. This generalizes a result of Chen.
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2024-11-30
Finite sum of weighted differentiation composition operators from the fractional Cauchy spaces to the Zygmund-type space
Rita A. Hibschweiler; Ajay K. SharmaAbstract : In this paper, we characterize the boundedness and compactness of a finite sum of weighted differentiation composition operators acting from the fractional Cauchy spaces to Zygmund-type spaces.
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2024-11-30
Idempotents in $\pi$-regular rings, right AI rings, NI rings and generalized regular rings
Sera Kim; Chang Ik Lee; Zhelin PiaoAbstract : 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.
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2024-11-30
Partitions weighted by the number of two types of parts
Byungchan Kim; Eunmi KimAbstract : 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.
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2024-11-30
Some classes of composition operators on Orlicz spaces
Yousef Estaremi; Zhidong HuangAbstract : The notions of expansivity and positive expansivity for composition operators on Orlicz spaces are investigated. In particular, necessary and sufficient conditions are given for a composition operator to be expansive, positively expansive, and uniformly expansive. Additionally, equivalent conditions for these concepts are provided in the case that the system is dissipative.
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2024-11-30
Regularity properties of higher order maximal commutators with Lipschitz symbols
Yuan MaAbstract : 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\leq\alpha
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2024-11-30
$L^p$-boundedness of the Cauchy-Leray-Fantappi\`e integral on certain class of weakly pseudoconvex domains in $\mathbb{C}^2$
Ly Kim Ha; Pham Huu TrungAbstract : Let $\mathbb{H}_k=\{(z_1,z_2) \in \mathbb{C}^2 : \operatorname{Im}z_2 > \vert z_1 \vert^{2k}\}$, for $k\in\mathbb{N}, k\ge 2$. In this paper we are interested in the $L^p$-boundedness of the Cauchy-Leray-Fantappi\`e integral. Although this operator is simpler than the Cauchy-Szeg\"o projection, it maps all $L^2(b\mathbb{H}_k)$-functions to holomorphic functions in $\mathbb{H}_k$.
November, 2024 | Volume 61, No. 6
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Semi-symmetric structure Jacobi operator for real hypersurfaces in the complex quadric
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Products of composition, differentiation and multiplication from the cauchy spaces to the Zygmund space
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The third Hermitian-Toeplitz and Hankel determinants for parabolic starlike functions
Rosihan M. Ali, Sushil Kumar, Vaithiyanathan Ravichandran
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