2 Glivenko-Cantelli Classes 5 2.1 The classical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 The symmetrization procedure . . . . . . . . . . . . . . . . . . . . . 7 2.1.2 Covering numbers and complexity estimates . . . . . . . . . . . . . . 9 2.2 Combinatorial parameters and covering numbers . . . . . . . . . . . . . . . 12 2.2.1 Uniform entropy and the VC dimen...

We prove Hausdorff-Young inequality for the Fourier transform connected with Riemann-Liouville operator. We use this inequality to establish the uncertainty principle in terms of entropy. Next, we show that we can derive the Heisenberg-Pauli-Weyl inequality for the precedent Fourier transform.

In this paper, we prove that every F ∗ space (i.e., Hausdorff topological vector space satisfying the first countable axiom) can be characterized by means of its “standard generating family of pseudo-norms”. By using the standard generating family of pseudo-norms P, the concepts of P-bounded set and γ-maxpseudo-norm-subadditive operator in F ∗ space are introduced. The uniform boundedness princ...

In 1969, Arhangel’skĭı proved that for every Hausdorff space X, |X| ≤ 2χ(X)L(X). This beautiful inequality solved a nearly fifty-year old question raised by Alexandroff and Urysohn. In this paper we survey a wide range of generalizations and variations of Arhangel’skĭı’s inequality. We also discuss open problems and an important legacy of the theorem: the emergence of the closure method as a fu...

We formulate three new error disturbance relations, one of which is free from explicit dependence upon intrinsic fluctuations of observables. The first error-disturbance relation is tighter than the one provided by the Branciard inequality and the Ozawa inequality for some initial states. Other two error disturbance relations provide a tighter bound to Ozawa’s error disturbance relation and one...

We call a nonempty subset A of a topological space X finitely non-Urysohn if for every nonempty finite subset F of A and every family {Ux : x ∈ F} of open neighborhoods Ux of x ∈ F , ∩{cl(Ux) : x ∈ F} 6 = ∅ and we define the non-Urysohn number of X as follows: nu(X) := 1 + sup{|A| : A is a finitely non-Urysohn subset of X}. Then for any topological space X and any subset A of X we prove the fol...