In the paper, a global optimization problem is considered where the objective function f(x) is univariate, black-box, and its first derivative f (x) satisfies the Lipschitz condition with an unknown Lipschitz constant K. In the literature, there exist methods solving this problem by using an a priori given estimate of K, its adaptive estimates, and adaptive estimates of local Lipschitz constant...

We show that if Y is a separable subspace of a Banach space X such that both X and the quotient X/Y have Cp-smooth Lipschitz bump functions, and U is a bounded open subset of X, then, for every uniformly continuous function f : Y ∩U → R and every ε > 0, there exists a Cp-smooth Lipschitz function F : X → R such that |F (y)− f(y)| ≤ ε for every y ∈ Y ∩U . If we are given a separable subspace Y o...

We discuss recent work of Chambers, Dotterrer, Ferry, Manin, and Weinberger, which resolved a fundamental question in quantitative topology: if f : Sm → Sn is a contractible map with Lipschitz constant L, what can we say about the Lipschitz constant of a null-homotopy of f? In the mid-90s, Gromov wrote an article on quantitative topology [G], raising a number of interesting questions. We will f...

We consider the problem of Lp-testing of class of bounded derivative properties over hypergrid domain with points distributed according to some product distribution. This class includes monotonicity, the Lipschitz property, (α, β)-generalized Lipschitz and many more properties. Previous results for Lp testing on [n] d for this class were known for monotonicity and c-Lipschitz properties over un...

Let T be an orientation-preserving Lipschitz expanding map of the circle T. A pre-image selector is a map τ : T → T with finitely many discontinuities, each of which is a jump discontinuity, and such that τ (x) ∈ T(x) for all x ∈ T. The closure of the image of a pre-image selector is called a flower, and a flower with p connected components is called a p-flower. We say that a real-valued Lipsch...

In this paper‎, ‎using a generalized Dunkl translation operator‎, ‎we obtain a generalization of Titchmarsh's Theorem for the Dunkl transform for functions satisfying the$(psi,p)$-Lipschitz Dunkl condition in the space $mathrm{L}_{p,alpha}=mathrm{L}^{p}(mathbb{R},|x|^{2alpha+1}dx)$‎, ‎where $alpha>-frac{1}{2}$.