Scott conjectured in [6] that the class of graphs with no induced subdivision of a given graph is χ-bounded. We verify his conjecture for maximal triangle-free graphs. Let F be a graph. We denote by Forb∗(F ) the class of graphs with no induced subdivision of F . A class G of graphs is χ-bounded if there exists a function f such that every graph G of G satisfies χ(G) ≤ f(ω(G)), where χ and ω re...

We discuss optimal constants in a recent result of Rudelson and Vershynin on marginal densities. We show that if f is a probability density on R of the form f(x) = ∏n i=1 fi(xi), where each fi is a density on R, say bounded by one, then the density of any marginal πE(f) is bounded by 2 , where k is the dimension of E. The proof relies on an adaptation of Ball’s approach to cube slicing, carried...

A mapping f : Z → Z is called residue-class-wise affine if there is a positive integer m such that f is affine on residue classes (mod m). The smallest such m is called the modulus of f . In this article it is shown that if the mapping f is surjective but not injective, then the set of moduli of its powers is not bounded. Further it is shown by giving examples that the three other combinations ...

We show that the equality m1(f(x)) = m2(g(x)) for x in a neighborhood of a point a remains valid for all x provided that f and g are open holomorphic maps, f(a) = g(a) = 0 and m1, m2 are Minkowski functionals of bounded balanced domains. Moreover, a polynomial relation between f and g is obtained. Next we generalize these results to bounded quasi-balanced domains. Moreover, the main results of ...

Given Ω,Λ ⊂ R two bounded open sets, and f and g two probability densities concentrated on Ω and Λ respectively, we investigate the regularity of the optimal map ∇φ (the optimality referring to the Euclidean quadratic cost) sending f onto g. We show that if f and g are both bounded away from zero and infinity, we can find two open sets Ω′ ⊂ Ω and Λ′ ⊂ Λ such that f and g are concentrated on Ω′ ...

MR2864818 39A05 37B99 Sacker, Robert J. (1-SCA; Los Angeles, CA) An invariance theorem for mappings. (English summary) J. Difference Equ. Appl. 18 (2012), no. 1, 163–166. The following theorem is proved. Theorem 2.1. LetD ⊂ R be a bounded subset and f :D→ R be continuous. Suppose f : ◦ D→ R is injective (one-to-one) and f(∂D)⊂D. If D := R rD has no bounded components, then f(D)⊂D. Here D is the...

For a Banach space X we demonstrate the equivalence of following two properties: (1) is B-convex (that is, possesses nontrivial infratype), and (2) if F:[0,1]→2X∖{∅} multifunction with bounded values, convF denotes t↦conv(F(t)), then Riemann integrability equivalent to F. multifunctions relatively norm compact values F without any restrictions on X.

Recent research in theoretical physics on ‘Malament-Hogarth space-times’ indicates that so-called relativistic computers can be conceived that can carry out certain classically undecidable queries in finite time. We observe that the relativistic Turing machines which model these computations recognize precisely theΔ2−sets of the Arithmetical Hierarchy. In a complexity-theoretic analysis, we sho...

This paper shows that the proof complexity (minimum computational complexity of proving formally or asymptotically) of \P 6=NP" is super-polynomial-time with respect to a theory T , which is a consistent extension of Peano Arithmetic (PA), and PTM-!-consistent, where the PTM-!-consistency is a polynomial-time Turing machine (PTM) version of !consistency. In other words, to prove \P 6=NP" (by an...

This paper defines a notion of binding trees that provide a suitable model for second-order type systems with F-bounded quantifiers and equirecursive types. It defines a notion of regular binding trees that correspond in the right way to notions of regularity in the first-order case. It defines a notion of subtyping on these trees and proves various properties of the subtyping relation. It defi...