Hayman conjecture and uniqueness of some delay-differential polynomials sharing a small function

In the present paper, we introduce generalized Geraghty (Proc Am Math Soc 40:604–608, 1973) mappings on partial metric spaces and give a fixed point theorem which generalizes some recent results appearing in the literature.

We consider trapped Bose gases in three dimensions in the Gross–Pitaevskii regime whose low energy states are well known to exhibit Bose–Einstein condensation. That is, the majority of the particles occupies the same condensate state. We prove exponential control of the number of particles orthogonal to the condensate state, generalizing recent results from Nam and Rademacher (Trans Am Math Soc, 2023, arXiv:2307.10622) for translation invariant systems.

In this paper, we consider the Fourier-type functionals introduced in [Chung HS, Tuan VK. Fourier-type functionals on Wiener space. Bull Korean Math Soc. 2012;49:609–619] and used in [Lee IY, Chung HS, Chang SJ. Series expansions of the analytic Feynman integral for the Fourier-type functional. J Korean Soc Math Educ Ser B. 2012;19:87–102]. We first establish the existence of the transform with respect to the Gaussian process for the Fourier-type functionals. We then obtain the various relationships for the transform with respect to the Gaussian process of the Fourier-type functionals which involved the ⋄-product and in addition, establish various integration formulas.

It is well-known that on a compact Riemannian manifold with a transverse Kahler foliation of codimension $$q=2m$$, any transverse Killing r-form $$(r\ge 2)$$ is parallel (Jung and Jung in Bull Korean Math Soc 49:445–454, 2012). In this article, we study the parallelness of $$L^2$$-transverse Killing forms on a complete foliated Riemannian manifold M with a transverse Kahler foliation $${\mathcal {F}}$$. Precisely, if all leaves of $${\mathcal {F}}$$ are compact and the mean curvature form is transversally holomorphic, coclosed and bounded, then all $$L^2$$-transverse Killing r-forms $$(r\ge 2)$$ are parallel. In addition, if the volume of M is infinite, then all $$L^2$$-transverse Killing r-forms $$(r\ge 2)$$ are trivial.

In this paper, we study the concept of R-weakly commuting of type (Ag) of Pathak et al. (Bull Korean Math Soc 34:247–257, 1997) in fuzzy metric spaces. We also establish the existence of common fixed point theorems by using the common limit in the range property and give an example to validate our the main results.

In this paper we define a modified generalized integral transform (MGIT) and a modified convolution product (MCP) for functionals on function space. We then introduce classes of functionals which are dense in L2(Ca, b[0, T]) and establish various relationships between the MGIT and the MCP involving the first variation. Note that all results in [Chang SJ, Chung HS, Skoug D. Integral transforms of functionals in L2(Ca, b[0, T]). J Fourier Anal Appl. 2009;15:441–462; Chang SJ, Chung HS, Skoug D. Some basic relationships among transforms, convolution products, first variations and inverse transforms. Central Eur J Math. 2013;11:538–551; Chang KS, Kim BS, Yoo I. Integral transform and convolution of analytic functionals on abstract Wiener space. Numer Funct Anal Optim. 2000;21:97–105; Im MK, Ji UC, Park YJ. Relations among the first variation, the convolutions and the generalized Fourier–Gauss transforms. Bull Korean Math Soc. 2011;48:291–302; Lee Y-J. Integral transforms of analytic functions on abstract Wiener spaces. J Funct Anal. 1982;47:153–164; Kim BS, Skoug D. Integral transforms of functionals in L2(C0[0, T]). Rocky Mountain J Math. 2003;33:1379–1393] and [Chung HS, Tuan VK. Generalized integral transforms and convolution products on function space. Integral Transforms Spec Funct. 2011;22:573–586] are then corollaries of the results and formulas established in this paper.

We study a Riemannian gradient method for the $L_2-$Wasserstein least squares problem of Gaussian measures under the affine-invariant geometry. The variable of $L_2-$Wasserstein least squares problem lies in the set of positive definite matrices, which, equipped with the affine-invariant metric, forms a Hadamard manifold. The same set with usual Euclidean metric is also a Hadamard manifold, with constant sectional curvature equal to 0. Hence, the gradient descent method proposed in [S. Kum, S. Yun, J. Korean Math Soc. $\mathbf{56}$ (2019)] can be considered as a Riemannian gradient method with respect to usual Euclidean inner product. This method is known to have a sublinear convergence rate and requires a singular value decomposition at each iteration, which is computationally expensive. In this paper, we adapt the Riemannian gradient method under the affine-invariant geometry for solving the $L_2-$Wasserstein least squares and prove its local linear convergence. This method does not require a singular value decomposition. We numerically show that the proposed method is more efficient than the gradient descent method mentioned.

The aim of this paper is to present a Stancu type Kantorovich modification of q-Bernstein-Schurer operators introduced by Muraru (Mathematica LVI 2:1–11, 2011) and modified by Ren and Zeng (Bull Korean Math Soc 50(4):1145–1156, 2013). Here, we obtain a convergence theorem by using the well known Bohman-Korovkin criterion and find the estimate of the rate of convergence by means of modulus of continuity and Lipschitz function for these operators. Also, we establish a Korovkin type A-statistical approximation theorem.