Search results for "Conjecture"
showing 10 items of 217 documents
On utilizing dependence-based information to enhance micro-aggregation for secure statistical databases
2011
Published version of an article in the journal: Pattern Analysis and Applications. Also available from the publisher at: http://dx.doi.org/10.1007/s10044-011-0199-9 We consider the micro-aggregation problem which involves partitioning a set of individual records in a micro-data file into a number of mutually exclusive and exhaustive groups. This problem, which seeks for the best partition of the micro-data file, is known to be NP-hard, and has been tackled using many heuristic solutions. In this paper, we would like to demonstrate that in the process of developing micro-aggregation techniques (MATs), it is expedient to incorporate information about the dependence between the random variable…
From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture
2020
Abstract In this paper we study the joint convexity/concavity of the trace functions Ψ p , q , s ( A , B ) = Tr ( B q 2 K ⁎ A p K B q 2 ) s , p , q , s ∈ R , where A and B are positive definite matrices and K is any fixed invertible matrix. We will give full range of ( p , q , s ) ∈ R 3 for Ψ p , q , s to be jointly convex/concave for all K. As a consequence, we confirm a conjecture of Carlen, Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaert and Datta and obtain the full range of ( α , z ) for α-z Renyi relative entropies to be monotone under completely positive trace preserving maps. We also give simpler proofs of many known results, including the concavity of Ψ p…
Discussions of Part I Chapters
2016
This chapter is an opportunity for one of the authors of this book to question the other two authors in the light of issues raised in Chaps. 2– 5. It constitutes both a follow-up to discussions between authors which occurred over the writing process, and emergent issues—new discussions once the book was almost complete. Some fundamental issues are addressed, about the birth of mathematics (and its deep links with the birth of writing), the relationships between mathematics and other sciences, the interactions between conjecture and proof, and the role of visualisation and of gestures. The text is kept short in order to provoke the readers to reflect on these issues rather than for the autho…
2-Groups with few rational conjugacy classes
2011
Abstract In this paper we prove the following conjecture of G. Navarro: if G is a finite 2-group with exactly 5 rational conjugacy classes, then G is dihedral, semidihedral or generalized quaternion. We also characterize the 2-groups with 4 rational classes.
Multiplicity of fixed points and growth of ε-neighborhoods of orbits
2012
We study the relationship between the multiplicity of a fixed point of a function g, and the dependence on epsilon of the length of epsilon-neighborhood of any orbit of g, tending to the fixed point. The relationship between these two notions was discovered before (Elezovic, Zubrinic, Zupanovic) in the differentiable case, and related to the box dimension of the orbit. Here, we generalize these results to non-differentiable cases introducing a new notion of critical Minkowski order. We study the space of functions having a development in a Chebyshev scale and use multiplicity with respect to this space of functions. With the new definition, we recover the relationship between multiplicity o…
On critical behaviour in generalized Kadomtsev-Petviashvili equations
2016
International audience; An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev–Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painlevé I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture. The numerical study of the long time behaviour of these examples indicates persistence of dispersive shock waves in solutions to the (subcritical) KP equations, while in the supercritical KP equations a blow-up occurs after the formation of the disp…
Hierarchies of probabilistic and team FIN-learning
2001
AbstractA FIN-learning machine M receives successive values of the function f it is learning and at some moment outputs a conjecture which should be a correct index of f. FIN learning has two extensions: (1) If M flips fair coins and learns a function with certain probability p, we have FIN〈p〉-learning. (2) When n machines simultaneously try to learn the same function f and at least k of these machines output correct indices of f, we have learning by a [k,n]FIN team. Sometimes a team or a probabilistic learner can simulate another one, if their probabilities p1,p2 (or team success ratios k1/n1,k2/n2) are close enough (Daley et al., in: Valiant, Waranth (Eds.), Proc. 5th Annual Workshop on C…
On the number of constituents of products of characters
2022
It has been conjectured that if the number of distinct irreducible constituents of the product of two faithful irreducible characters of a finite p-group, for p ≥ 5, is bigger than (p + 1)/2, then it is at least p. We give a counterexample to this conjecture.
Characterizations of Orlicz-Sobolev Spaces by Means of Generalized Orlicz-Poincaré Inequalities
2012
Let Φ be anN-function. We show that a functionu∈LΦ(ℝn)belongs to the Orlicz-Sobolev spaceW1,Φ(ℝn)if and only if it satisfies the (generalized) Φ-Poincaré inequality. Under more restrictive assumptions on Φ, an analog of the result holds in a general metric measure space setting.
Computational Aspects in Spaces of Bivariate Polynomial of w-Degree n
2005
Multivariate ideal interpolation schemes are deeply connected with H-bases. Both the definition of a H-basis and of an ideal interpolation space depend of the notion of degree used in the grading decomposition of the polynomial spaces. We studied, in the case of bivariate polynomials, a generalized degree, introduced by T. Sauer and named w-degree. This article give some theoretical results that allow us to construct algorithms for calculus of the dimension of the homogeneous spaces of bivariate polynomials of w – degree n. We implemented these algorithms in C++ language. The analysis of the results obtained, leads us to another theoretical conjecture which we proved in the end.