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In this paper we study the convexity and concavity properties of generalized trigonometric and hyperbolic functions in case of Logarithmic mean.
Bayesian optimisation is a popular approach for optimising expensive black-box functions. The next location to be evaluated is selected via maximising an acquisition function that balances exploitation and exploration. Gaussian processes,…
We introduce the concept of shifting distance functions, and we establish a new fixed point theorem which generalizes the Banach contraction principle. An example is provided to illustrate our result.
This paper provides an approach to establishing the calculus method from the concept of mean, i.e., average. This approach is from a statistics perspective and can help calculus learners understand calculus ideas and analyze a function…
We show that finiteness of the Lorentzian distance is equivalent to the existence of generalised time functions with gradient uniformly bounded away from light cones. To derive this result we introduce new techniques to construct and…
We study the regularity of the distance function to the boundary of a domain in $\mathbb{R}^n$, with respect to the Minkowski functional of a convex polytope. We obtain the regularity of the distance function in certain cases. We also…
Distance function is a main metrics of measuring the affinity between two data points in machine learning. Extant distance functions often provide unreachable distance values in real applications. This can lead to incorrect measure of the…
We introduce two kinds of generalized $s$-convex functions on real linear fractal sets $\mathbb{R}^{\alpha}(0<\alpha<1)$. And similar to the class situation, we also study the properties of these two kinds of generalized $s$-convex…
The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices. The transmission of a vertex $v_i$ in $G$ is the sum of the distances from $v_i$ to all other vertices and $T(G)$ is the…
The generalized circumradius of a set of points $A \subseteq \mathbb{R}^d$ with respect to a convex body $K$ equals the minimum value of $\lambda \geq 0$ such that $A$ is contained in a translate of $\lambda K$. Each choice of $K$ gives a…
Robust estimation of location is a fundamental problem in statistics, particularly in scenarios where data contamination by outliers or model misspecification is a concern. In univariate settings, methods such as the sample median and…
Metric mean dimension is a metric-depedent quantity to characterize the topological complexity of systems with infinite topological entropy. In this paper, we investigate metric mean dimension of factor maps. (1) We introduce three types of…
Sleeve functions are generalizations of the well-established ridge functions that play a major role in the theory of partial differential equation, medical imaging, statistics, and neural networks. Where ridge functions are non-linear,…
We use mixed norm estimates for the spherical averaging operator to obtain some results concerning pinned distance sets.
Gaussian processes have been widely used in cosmology to reconstruct cosmological quantities in a model-independent way. However, the validity of the adopted mean function and hyperparameters, and the dependence of the results on the choice…
Based on Colombeau's theory of algebras of generalized functions we introduce the concepts of generalized functions taking values in differentiable manifolds as well as of generalized vector bundle homomorphisms. We study their basic…
We define generalized Collatz mappings on free abelian groups of finite rank and study their iteration trajectories. Using geometric arguments we describe cones of points having a divergent trajectory and we deduce lower bounds for the…
We introduce a generalization of the b-metric we call a (b,c)-metric. We show that if $X$ is a $(b,c)$-metric space and $\psi: X \longrightarrow Y$ is a quasi-isometry then $Y$ is $(b,c)$-metrizable. We also define a particular kind of…
The normalized information distance is a universal distance measure for objects of all kinds. It is based on Kolmogorov complexity and thus uncomputable, but there are ways to utilize it. First, compression algorithms can be used to…
Monge gauge in differential geometry is generalized. The original Monge gauge is based on a surface defined as a height function $h(x,y)$ above a flat reference plane. The total curvature and the Gaussian curvature are found in terms of the…