Browsing by Author "Gheondea, Aurelian"

Now showing 1 - 13 of 13

Open Access
Bifurcation in the evolution of certainty in a small decision-making group by consensus
(2021-07-06) Gheondea-Eladi, A.; Gheondea, Aurelian
In a previous paper, the evolution of certainty measured during a consensus-based small-group decision process was shown to oscillate to an equilibrium value for about two-thirds of the participants in the experiment. Starting from the observation that experimental participants are split into two groups, those for whom the evolution of certainty oscillates and those for whom it does not, in this paper we perform an analysis of this bifurcation with a more accurate model and answer two main questions: what is the meaning of this bifurcation, and is this bifurcation amenable to the approximation method or numerical procedure? Firstly, we have to refine the mathematical model of the evolution of certainty to a function explicitly represented in terms of the model parameters and to verify its robustness to the variation of parameters, both analytically and by computer simulation. Then, using the previous group decision experimental data, and the model proposed in this paper, we run the curve-fitting software on the experimental data. We also review a series of interpretations of the bifurcated behaviour. We obtain a refined mathematical model and show that the empirical results are not skewed by the initial conditions, when the proposed model is used. Thus, we reveal the analytical and empirical existence of the observed bifurcation. We then propose that sensitivity to the absolute value of certainty and to its rate of change are considered as potential interpretations of this split in behaviour, along with certainty/uncertainty orientation, uncertainty interpretation, and uncertainty/certainty-related intuition and affect.
Open Access
The classical SIR model in epidemiology
(Romanian Mathematical Society, 2020) Gheondea, Aurelian
This is a survey note in which we describe the classical SIR model in mathematical epidemiology, a bit of qualitative analysis, its Euler discretisation, and some simulations.
Open Access
Corrigendum to “representations of ⁎-semigroups associated to invariant kernels with values adjointable operators” [Linear Algebra Appl. 486 (2015) 361–388]
(2020) Ay, Serdar; Gheondea, Aurelian
We correct a lemma by adding the assumption that the ordered ⁎-space is Archimedean and show by counter-examples and examples that this is needed.
Open Access
Editorial Preface
(The Theta Foundation, 2003) Gheondea, Aurelian; Şabac, M.; Gheondea, Aurelian; Şabac, M.
Open Access
Interpolation for completely positive maps: Numerical solutions
(Societatea de Stiinte Matematice din Romania, 2018) Ambrozie, C.; Gheondea, Aurelian
We present a few techniques to find completely positive maps between full matrix algebras taking prescribed values on given data, based on semidefinite programming, convex minimization supported by a numerical example, as well as representations by linear functionals. The particular case of commutative data is also discussed.
Open Access
Invariant weakly positive semidefinite kernels with values in topologically ordered ∗-spaces
(Instytut Matematyczny PAN, 2019) Ay, Serdar; Gheondea, Aurelian
We consider weakly positive semidefinite kernels valued in ordered ∗-spaces with or without certain topological properties, and investigate their linearisations (Kolmogorov decompositions) as well as their reproducing kernel spaces. The spaces of realisations are of VE (Vector Euclidean) or VH (Vector Hilbert) type, more precisely, vector spaces that possess gramians (vector valued inner products). The main results refer to the case when the kernels are invariant under certain actions of ∗-semigroups and show under which conditions ∗-representations on VE-spaces, or VH-spaces in the topological case, can be obtained. Finally, we show that these results unify most of dilation type results for invariant positive semidefinite kernels with operator values as well as recent results on positive semidefinite maps on ∗-semigroups with values operators from a locally bounded topological vector space to its conjugate Z-dual space, for Z an ordered ∗-space.
Open Access
Kreǐn spaces induced by symmetric operators
(Academia Romana * Institutul de Matematica, 2009) Cojuhari P.; Gheondea, Aurelian
We introduce the notion of Kreǐn space induced by a densely defined symmetric operator in a Hilbert space, as an abstract notion of indefinite energy spaces. Characterizations of existence and uniqueness, as well as certain canonical representations, are obtained. We exemplify these by the free and certain perturbed Dirac operators.
Open Access
On generalised triplets of Hilbert spaces
(Editura Academiei Romane, 2020) Cojuhari, P.; Gheondea, Aurelian
We compare the concept of triplet of closely embedded Hilbert spaces with that of generalised triplet of Hilbert spaces in the sense of Berezanskii by showing when they coincide, when they are different, and when starting from one of them one can naturally produce the other one that essentially or fully coincides.
Open Access
Probability error bounds for approximation of functions in reproducing kernel Hilbert spaces
(Hindawi Limited, 2021-05-03) Aydın, Ata Deniz; Gheondea, Aurelian; Hassi, Seppo
We find probability error bounds for approximations of functions f in a separable reproducing kernel Hilbert space H with reproducing kernel K on a base space X, firstly in terms of finite linear combinations of functions of type Kxi and then in terms of the projection πxn on spanKxii=1n, for random sequences of points x=xii in X. Given a probability measure P, letting PK be the measure defined by dPKx=Kx,xdPx, x∈X, our approach is based on the nonexpansive operator L2X;PK∋λ→LP,Kλ≔∫XλxKxdPx∈H, where the integral exists in the Bochner sense. Using this operator, we then define a new reproducing kernel Hilbert space, denoted by HP, that is the operator range of LP,K. Our main result establishes bounds, in terms of the operator LP,K, on the probability that the Hilbert space distance between an arbitrary function f in H and linear combinations of functions of type Kxi, for xii sampled independently from P, falls below a given threshold. For sequences of points xii=1∞ constituting a so-called uniqueness set, the orthogonal projections πxn to spanKxii=1n converge in the strong operator topology to the identity operator. We prove that, under the assumption that HP is dense in H, any sequence of points sampled independently from P yields a uniqueness set with probability 1. This result improves on previous error bounds in weaker norms, such as uniform or Lp norms, which yield only convergence in probability and not almost certain convergence. Two examples that show the applicability of this result to a uniform distribution on a compact interval and to the Hardy space H2D are presented as well.
Open Access
Reproducing kernel kreĭn spaces
(Springer Basel, 2015) Gheondea, Aurelian
This chapter is an introduction to reproducing kernel Kre?in spaces and their interplay with operator valued Hermitian kernels. Existence and uniqueness properties are carefully reviewed. The approach used in this survey involves the more abstract, but very useful, concept of linearization or Kolmogorov decomposition, as well as the underlying concepts of Kre?in space induced by a selfadjoint operator and that of Kre?in space continuously embedded. The operator range feature of reproducing kernel spaces is emphasized. A careful presentation of Hermitian kernels on complex regions that point out a universality property of the Szegö kernels with respect to reproducing kernel Kre?in spaces of holomorphic functions is included. © Springer Basel 2015. All rights are reserved.
Open Access
The spectral theorem for locally normal operators
(AGH University of Science and Technology, 2018) Gheondea, Aurelian
We prove the spectral theorem for locally normal operators in terms of a locally spectral measure. In order to do this, we first obtain some characterisations of local projections and we single out and investigate the concept of a locally spectral measure.
Open Access
Symmetries versus conservation laws in dynamical quantum systems: a unifying approach through propagation of fixed points
(Birkhauser Verlag AG, 2018) Gheondea, Aurelian
We unify recent Noether-type theorems on the equivalence of symmetries with conservation laws for dynamical systems of Markov processes, of quantum operations, and of quantum stochastic maps, by means of some abstract results on propagation of fixed points for completely positive maps on C∗-algebras. We extend most of the existing results with characterisations in terms of dual infinitesimal generators of the corresponding strongly continuous one-parameter semigroups. By means of an ergodic theorem for dynamical systems of completely positive maps on von Neumann algebras, we show the consistency of the condition on the standard deviation for dynamical systems of quantum operations, and hence of quantum stochastic maps as well, in case the underlying Hilbert space is infinite dimensional.
Open Access
The forcing effect on the evolution of certainty in a small decision making group by consensus
(Bilkent University, 2024-02) Gheondea-Eladi, Alexandra ; Gheondea, Aurelian