Basics in free probability

2016 ◽  
pp. 7-16 ◽  
Author(s):  
Moritz Weber
Keyword(s):  
Free Probability

scholarly journals Plate-like structure damage location identification based on Lamb wave baseline-free probability imaging method

2017 ◽  
Vol 9 (1) ◽  
pp. 168781401668570 ◽  
Author(s):  
Dongsheng Li ◽  
Zihao Jing ◽  
Mengdao Jin
Keyword(s):  
Lamb Wave ◽  
Time Window ◽  
Free Probability ◽  
Structural Response ◽  
Guided Wave ◽  
Time Of Arrival ◽  
Imaging Method ◽  
Detection Techniques ◽  
Damage Location ◽  
Ultrasonic Guided Wave

Damage-scattering signal extraction using conventional ultrasonic guided wave–based damage detection techniques requires the measurement of baseline data under pristine condition. This study proposes a baseline-free ultrasonic guided wave damage localization and imaging method based on Lamb wave baseline-free probability imaging method. Although traditional Lamb wave probability imaging can monitor damage location in plate-like structures, the absolute time of arrival and magnitude of the signal are affected by several factors and are therefore difficult to obtain. This study also proposes a probability-based hyperbola diagnostic imaging method that is based on different times of arrival and has no magnitude information. A distributed active sensor network conforming to a pulse-echo configuration and time window functions is developed to separate damage-scattering signals from structural response signals. Continuous wavelet transform is used to calculate the time of flight of damage signal waves. The numerical simulation and experiments validate the effectiveness of the proposed method in identifying damage.

scholarly journals A rigidity result for crossed products of actions of Baumslag–Solitar groups

2015 ◽  
Vol 26 (14) ◽  
pp. 1550117
Author(s):  
Niels Meesschaert
Keyword(s):  
Probability Measure ◽  
Free Probability ◽  
Crossed Product ◽  
Crossed Products ◽  
Orbit Equivalence ◽  
Rigidity Result ◽  
Profinite Completions ◽  
Abelian Subgroups ◽  
Measure Equivalence ◽  
Measure Preserving Actions

Let [Formula: see text] and [Formula: see text] be two ergodic essentially free probability measure preserving actions of nonamenable Baumslag–Solitar groups whose canonical almost normal abelian subgroups act aperiodically. We prove that an isomorphism between the corresponding crossed product II1 factors forces [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. This improves an orbit equivalence rigidity result obtained by Houdayer and Raum in [Baumslag–Solitar groups, relative profinite completions and measure equivalence rigidity, J. Topol. 8 (2015) 295–313].

scholarly journals Dual Lukacs regressions of negative orders for noncommutative variables

2014 ◽  
Vol 17 (03) ◽  
pp. 1450021 ◽  
Author(s):  
Kamil Szpojankowski
Keyword(s):  
Free Probability ◽  
Random Variable ◽  
Probability Measures ◽  
Classical Probability ◽  
Binomial Distributions ◽  
Beta Distributions

In the paper we study characterizations of probability measures in free probability. By constancy of regressions for random variable 𝕍1/2(𝕀 - 𝕌)𝕍1/2 given by 𝕍1/2𝕌𝕍1/2, where 𝕌 and 𝕍 are free, we characterize free Poisson and free binomial distributions. Our paper is a free probability analogue of results known in classical probability,3 where gamma and beta distributions are characterized by constancy of 𝔼((V(1 - U))i|UV), for i ∈ {-2, -1, 1, 2}. This paper together with previous results18 exhaust all cases of characterizations from Ref. 3.

scholarly journals $L_∞$-Khintchine-Bonami inequality in free probability

1998 ◽  
Vol 43 (1) ◽  
pp. 105-109
Author(s):  
Artur Buchholz
Keyword(s):  
Free Probability

scholarly journals R-Diagonal Elements and Freeness With Amalgamation

2001 ◽  
Vol 53 (2) ◽  
pp. 355-381 ◽  
Author(s):  
Alexandru Nica ◽  
Dimitri Shlyakhtenko ◽  
Roland Speicher
Keyword(s):  
Free Probability ◽  
Diagonal Element ◽  
New Approach ◽  
Probability Spaces

AbstractThe concept of R-diagonal element was introduced in [5], and was subsequently found to have applications to several problems in free probability. In this paper we describe a new approach to R-diagonality, which relies on freeness with amalgamation. The class of R-diagonal elements is enlarged to contain examples living in non-tracial *-probability spaces, such as the generalized circular elements of [7].

scholarly journals LOW ENTROPY OUTPUT STATES FOR PRODUCTS OF RANDOM UNITARY CHANNELS

2013 ◽  
Vol 02 (01) ◽  
pp. 1250018 ◽  
Author(s):  
BENOÎT COLLINS ◽  
MOTOHISA FUKUDA ◽  
ION NECHITA
Keyword(s):  
Free Probability ◽  
Bell State ◽  
Quantum Channels ◽  
Unitary Operators ◽  
Graphical Calculus ◽  
Additivity Problems ◽  
Communication Capacity ◽  
The Difference ◽  
Low Entropy ◽  
Math Phys

In this paper, we study the behavior of the output of pure entangled states after being transformed by a product of conjugate random unitary channels. This study is motivated by the counterexamples by Hastings [Superadditivity of communication capacity using entangled inputs, Nat. Phys.5 (2009) 255–257] and Hayden–Winter [Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1, Comm. Math. Phys.284(1) (2008) 263–280] to the additivity problems. In particular, we study in depth the difference of behavior between random unitary channels and generic random channels. In the case where the number of unitary operators is fixed, we compute the limiting eigenvalues of the output states. In the case where the number of unitary operators grows linearly with the dimension of the input space, we show that the eigenvalue distribution converges to a limiting shape that we characterize with free probability tools. In order to perform the required computations, we need a systematic way of dealing with moment problems for random matrices whose blocks are i.i.d. Haar distributed unitary operators. This is achieved by extending the graphical Weingarten calculus introduced in [B. Collins and I. Nechita, Random quantum channels I: Graphical calculus and the Bell state phenomenon, Comm. Math. Phys.297(2) (2010) 345–370].

An Application of Free Probability to Arithmetic Functions

2014 ◽  
Vol 9 (7) ◽  
pp. 1457-1489 ◽  
Author(s):  
Ilwoo Cho ◽  
Palle E. T. Jorgensen
Keyword(s):  
Free Probability ◽  
Arithmetic Functions
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