Ricci almost solitons with associated projective vector field

A Ricci almost soliton whose associated vector field is projective is shown to have vanishing Cotton tensor, divergence-free Bach tensor and Ricci tensor as conformal Killing. For the compact case, a sharp inequality is obtained in terms of scalar curvature.We show that every complete gradient Ricci soliton is isometric to the Riemannian product of a Euclidean space and an Einstein space. A complete K-contact Ricci almost soliton whose associated vector field is projective is compact Einstein and Sasakian.

A Schwarz lemma for hyperbolic harmonic mappings in the unit ball

Assume that $p\in [1,\infty ]$ and $u=P_{h}[\phi ]$, where $\phi \in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^n)$ and $u(0) = 0$. Then we obtain the sharp inequality $\lvert u(x) \rvert \le G_p(\lvert x \rvert )\lVert \phi \rVert_{L^{p}}$ for some smooth function $G_p$ vanishing at $0$. Moreover, we obtain an explicit form of the sharp constant $C_p$ in the inequality $\lVert Du(0)\rVert \le C_p\lVert \phi \rVert \le C_p\lVert \phi \rVert_{L^{p}}$. These two results generalize and extend some known results from the harmonic mapping theory (D. Kalaj, Complex Anal. Oper. Theory 12 (2018), 545–554, Theorem 2.1) and the hyperbolic harmonic theory (B. Burgeth, Manuscripta Math. 77 (1992), 283–291, Theorem 1).

Doubly Warped Product Submanifolds of a Riemannian Manifold of Nearly Quasi-constant Curvature

In the present paper, we form a sharp inequality for a doubly warped product submanifold of a Riemannian manifold of nearly quasi-constant curvature.

On sharp inequality of Kolmogorov type for functions of low smoothness from the class $L_1^r(T)$

We obtain new sharp inequality of Kolmogorov type for differentiable periodic functions $x \in L_1^3$.

Inequalities of Bernstein type for splines in $L_2(\mathbb{R})$ space

We obtain sharp inequality of Bernstein type in $L_2(\mathbb{R})$ space for non-periodic spline functions of degree $m$, of minimal defect, with equidistant knots.

Inequalities of Kolmogorov type for fractional derivatives of multivariable functions

We prove new sharp inequality of Kolmogorov type that estimates the norm of mixed fractional Marchaud derivative of n-variable function by C-norm of this function and its norms in Lipschitz spaces.

Sharp Nagy type inequalities for the classes of functions with given quotient of the uniform norms of positive and negative parts of a function

For any $p\in (0, \infty],$ $\omega > 0,$ $d \ge 2 \omega,$ we obtain the sharp inequality of Nagy type$$\|x_{\pm}\|_\infty \le\frac{\|(\varphi+c)_{\pm}\|_\infty}{\|\varphi+c\|_{L_p(I_{2\omega})}} \left\|x \right\|_{L_{p} \left(I_d \right)}$$on the set $S_{\varphi}(\omega)$ of $d$-periodic functions $x$ having zeros with given the sine-shaped $2\omega$-periodiccomparison function $\varphi$, where $c\in [-\|\varphi\|_\infty, \|\varphi\|_\infty]$ is such that$$ \|x_{+}\|_\infty \cdot\|x_{-}\|^{-1}_\infty = \|(\varphi+c)_{+}\|_\infty \cdot\|(\varphi+c)_{-}\|^{-1}_\infty .$$In particular, we obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and polynomial splines with given quotient of the norms $\|x_{+}\|_\infty / \|x_-\|_\infty$.

A hierarchical cosimulation algorithm integrated with an acceptance–rejection method for the geostatistical modeling of variables with inequality constraints

This work addresses the problem of the cosimulation of cross-correlated variables with inequality constraints. A hierarchical sequential Gaussian cosimulation algorithm is proposed to address this problem, based on establishing a multicollocated cokriging paradigm; the integration of this algorithm with the acceptance–rejection sampling technique entails that the simulated values first reproduce the bivariate inequality constraint between the variables and then reproduce the original statistical parameters, such as the global distribution and variogram. In addition, a robust regression analysis is developed to derive the coefficients of the linear function that introduces the desired inequality constraint. The proposed algorithm is applied to cosimulate Silica and Iron in an Iron deposit, where the two variables exhibit different marginal distributions and a sharp inequality constraint in the bivariate relation. To investigate the benefits of the proposed approach, the Silica and Iron are cosimulated by other cosimulation algorithms, and the results are compared. It is shown that conventional cosimulation approaches are not able to take into account and reproduce the linearity constraint characteristics, which are part of the nature of the dataset. In contrast, the proposed hierarchical cosimulation algorithm perfectly reproduces these complex characteristics and is more suited to the actual dataset.