Abstract : In this paper, we prove $L^{p}$ estimates of a class of singular integral operators on product domains along surfaces defined by mappings that are more general than polynomials and convex functions. We assume that the kernels are in $L(\log L)^{2}(\mathbb{S}^{n-1}\times \mathbb{S}^{m-1})$. Furthermore, we\ prove $L^{p}$ estimates of the related class of Marcinkiewicz integral operators. Our results extend as well as improve previously known results.
Abstract : The purpose of this paper is to introduce the concept of joint essential numerical spectrum $\sigma_{en}(\cdot)$ of $q$-tuple of operators on a Banach space and to study its properties. This notion generalize the notion of the joint essential numerical range.
Abstract : This paper is concerned with the following Schr\"{o}dinger-\linebreak Poisson system$$\left\{\begin{array}{ll} -{\Delta}u+V(x)u+K(x){\phi}u=a(x)|u|^{p-2}u &\mbox{in}\ \mathbb{R}^3, \\[0.1cm] -{\Delta}{\phi}=K(x)u^{2}&\mbox{in}\ \mathbb{R}^3, \\[0.1cm]\end{array}\right.$$where $4<p<6$. For the case that $K$ is nonnegative, $V$ and $a$ are indefinite, we prove the above problem possesses one ground state sign-changing solutionwith exactly two nodal domains by constraint variational method and quantitative deformation lemma. Moreover, we show that the energy of sign-changing solutions islarger than that of the ground state solutions. The novelty of this paper is that the potential $a$ is indefinite and allowed to vanish at infinity. In this sense, we complementthe existing results obtained by Batista and Furtado \cite{BF18}.
Abstract : In this paper, we attempt to study several topological properties for the function space ${H(X)}$, space of self-homeomorphisms on a metric space endowed with the regular topology. We investigate its metrizability and countability and prove their coincidence at $X$ compact. Furthermore, we prove that the space ${H(X)}$ endowed with the regular topology is a topological group when $X$ is a metric, almost $P$-space. Moreover, we prove that the homeomorphism spaces of increasing and decreasing functions on $\mathbb R$ under regular topology are open subspaces of $H(\mathbb R)$ and are homeomorphic.
Abstract : The main intention of the current paper is to characterize certain properties of $\star$-conformal Ricci solitons on non-coK\"ahler $(\kappa,\mu)$-almost coK\"{a}hler manifolds. At first, we find that there does not exist $\star$-conformal Ricci soliton if the potential vector field is the Reeb vector field $\theta$. We also prove that the non-coK\"ahler $(\kappa,\mu)$-almost coK\"ahler manifolds admit $\star$-conformal Ricci solitons if the potential vector field is the infinitesimal contact transformation. It is also studied that there does not exist $\star$-conformal gradient Ricci solitons on the said manifolds. An example has been constructed to verify the obtained results.
Abstract : In this paper we prove the existence of nontrivial weak solutions to the boundary value problem \begin{align*} - G_1 u & =u^3 + f(x,y,u) \quad \text{ in } \Omega ,\\ u &\geq 0 \quad \text{ in } \Omega ,\\ u & =0 \quad \text{ on } \partial\Omega , \end{align*} where $\Omega $ is a bounded domain with smooth boundary in $\mathbb{R}^3$, $G_1 $ is a Grushin type operator, and $f(x,y,u)$ is a lower order perturbation of $u^3$ with $f(x,y,0)=0$. The nonlinearity involved is of critical exponent, which differs from the existing results in \cite{Tri:2018,TriLuyen:2020}.
Abstract : The main purpose of this paper is to give some new identities and properties related to Bernoulli type numbers and polynomials associated with the Bessel function of the first kind. We give symmetric properties of the Bernoulli type numbers and polynomials. Moreover, using generating functions and the Fa \`{a} di Bruno's formula, we derive some new formulas and relations related to not only these polynomials, but also the Bernoulli numbers and polynomials and the Euler numbers and polynomials.
Abstract : Magnetic curves are the trajectories of charged particals \linebreak which are influenced by magnetic fields and they satisfy the Lorentz equation. It is important to find relationships between magnetic curves and other special curves. This paper is a study of magnetic curves and this kind of relationships. We give the relationship between $\beta $-magnetic curves and Mannheim, Bertrand, involute-evolute curves and we give some geometric properties about them. Then, we study this subject for $\gamma $-magnetic curves. Finally, we give an evaluation of what we did.
Abstract : The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus to make an approximation of a continuous probability distribution by a discrete distribution. It has broad application in signal processing and data compression. In this paper, first we define the uniform distributions on different curves such as a line segment, a circle, and the boundary of an equilateral triangle. Then, we give the exact formulas to determine the optimal sets of $n$-means and the $n$th quantization errors for different values of $n$ with respect to the uniform distributions defined on the curves. In each case, we further calculate the quantization dimension and show that it is equal to the dimension of the object; and the quantization coefficient exists as a finite positive number. This supports the well-known result of Bucklew and Wise \cite{BW}, which says that for a Borel probability measure $P$ with non-vanishing absolutely continuous part the quantization coefficient exists as a finite positive number.
Abstract : In the present paper, we consider a kind of generalized hyperbolic geometric flow which has a gradient form. Firstly, we establish the existence and uniqueness for the solution of this flow on an $n$-dimensional closed Riemannian manifold. Then, we give the evolution of some geometric structures of the manifold along this flow.
Gour Gopal Biswas, Uday Chand De
Commun. Korean Math. Soc. 2022; 37(3): 825-837
https://doi.org/10.4134/CKMS.c210046
Praveena Manjappa Mundalamane, Bagewadi Channabasappa Shanthappa, Mallannara Siddalingappa Siddesha
Commun. Korean Math. Soc. 2022; 37(3): 813-824
https://doi.org/10.4134/CKMS.c200471
Wafa Selmi, Mohsen Timoumi
Commun. Korean Math. Soc. 2022; 37(3): 693-703
https://doi.org/10.4134/CKMS.c210008
Mohd Aquib, Mohd Aslam, Michel Nguiffo Boyom, Mohammad Hasan Shahid
Commun. Korean Math. Soc. 2023; 38(1): 179-193
https://doi.org/10.4134/CKMS.c210026
Najib Mahdou, El Houssaine Oubouhou
Commun. Korean Math. Soc. 2024; 39(1): 45-58
https://doi.org/10.4134/CKMS.c230065
MOHAMED CHHITI, SALAH EDDINE MAHDOU
Commun. Korean Math. Soc. 2023; 38(3): 705-714
https://doi.org/10.4134/CKMS.c220260
Daisuke Shiomi
Commun. Korean Math. Soc. 2023; 38(3): 715-723
https://doi.org/10.4134/CKMS.c220271
Rachida EL KHALFAOUI, Najib Mahdou
Commun. Korean Math. Soc. 2023; 38(4): 983-992
https://doi.org/10.4134/CKMS.c220332
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