A Converse of the Loewner–Heinz Inequality, Geometric Mean and Spectral Order

2014 ◽  
Vol 57 (2) ◽  
pp. 565-571 ◽  
Author(s):  
Mitsuru Uchiyama
Keyword(s):  
Real Number ◽  
Geometric Mean ◽  
Heinz Inequality ◽  
Spectral Order ◽  
Adjoint Operators

AbstractLet A, B be non-negative bounded self-adjoint operators, and let a be a real number such that 0 < a < 1. The Loewner–Heinz inequality means that A ≤ B implies that Aa ≦ Ba. We show that A ≤ B if and only if (A + λ)a ≦ (B + λ)a for every λ > 0. We then apply this to the geometric mean and spectral order.

scholarly journals Chebyshev-Type Integral Inequalities for Continuous Fields of Operators Concerning Khatri–Rao Products and Synchronous Properties

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 422
Author(s):  
Arnon Ploymukda ◽  
Pattrawut Chansangiam
Keyword(s):  
Radon Measure ◽  
Geometric Mean ◽  
Integral Inequalities ◽  
Counting Measure ◽  
Integrable Functions ◽  
Discrete Inequalities ◽  
Finite Space ◽  
Continuous Fields ◽  
Adjoint Operators ◽  
Locally Compact Hausdorff Space

We consider bounded continuous fields of self-adjoint operators which are parametrized by a locally compact Hausdorff space Ω equipped with a finite Radon measure μ . Under certain assumptions on synchronous Khatri–Rao property of the fields of operators, we obtain Chebyshev-type inequalities concerning Khatri–Rao products. We also establish Chebyshev-type inequalities involving Khatri–Rao products and weighted Pythagorean means under certain assumptions of synchronous monotone property of the fields of operators. The Pythagorean means considered here are three classical symmetric means: the geometric mean, the arithmetic mean, and the harmonic mean. Moreover, we derive the Chebyshev–Grüss integral inequality via oscillations when μ is a probability Radon measure. These integral inequalities can be reduced to discrete inequalities by setting Ω to be a finite space equipped with the counting measure. Our results provide analog results for matrices and integrable functions. Furthermore, our results include the results for tensor products of operators, and Khatri–Rao/Kronecker/Hadamard products of matrices, which have been not investigated in the literature.

Spectral asymmetry and Riemannian geometry. III

1976 ◽  
Vol 79 (1) ◽  
pp. 71-99 ◽  
Author(s):  
M. F. Atiyah ◽  
V. K. Patodi ◽  
I. M. Singer
Keyword(s):  
Fundamental Group ◽  
Real Number ◽  
Analytic Continuation ◽  
Riemannian Geometry ◽  
Compact Manifold ◽  
Adjoint Operator ◽  
The Real ◽  
Spectral Asymmetry ◽  
Image Position ◽  
Adjoint Operators

In Parts I and II of this paper ((4), (5)) we studied the ‘spectral asymmetry’ of certain elliptic self-adjoint operators arising in Riemannian geometry. More precisely, for any elliptic self-adjoint operator A on a compact manifold we definedwhere λ runs over the eigenvalues of A. For the particular operators of interest in Riemannian geometry we showed that ηA(s) had an analytic continuation to the whole complex s-plane, with simple poles, and that s = 0 was not a pole. The real number ηA(0), which is a measure of ‘spectral asymmetry’, was studied in detail particularly in relation to representations of the fundamental group.

Real number arithmetic for mixed behavioural and structural descriptions

1990 ◽  
Vol 137 (6) ◽  
pp. 446
Author(s):  
M.G. Hill ◽  
N.E. Peeling ◽  
I.F. Currie ◽  
J.D. Morison ◽  
E.V. Whiting ◽  
...  
Keyword(s):  
Real Number ◽  
Structural Descriptions

Radiation synovectomy of the knee joint

2006 ◽  
Vol 45 (01) ◽  
pp. 57-61
Author(s):  
M. Puille ◽  
D. Steiner ◽  
R. Bauer ◽  
R. Klett
Keyword(s):  
Knee Joint ◽  
Lymph Nodes ◽  
Geometric Mean ◽  
Radiation Synovectomy ◽  
Real Activity ◽  
Anterior View ◽  
Inguinal Lymph Nodes ◽  
The Real ◽  
Therapeutic Modalities ◽  
Real Value

Summary Aim: Multiple procedures for the quantification of activity leakage in radiation synovectomy of the knee joint have been described in the literature. We compared these procedures considering the real conditions of dispersion and absorption using a corpse phantom. Methods: We simulated different distributions of the activity in the knee joint and a different extra-articular spread into the inguinal lymph nodes. The activity was measured with a gammacamera. Activity leakage was calculated by measuring the retention in the knee joint only using an anterior view, using the geometric mean of anterior and posterior views, or using the sum of anterior and posterior views. The same procedures were used to quantify the activity leakage by measuring the activity spread into the inguinal lymph nodes. In addition, the influence of scattered rays was evaluated. Results: For several procedures we found an excellent association with the real activity leakage, shown by an r² between 0.97 and 0.98. When the real value of the leakage is needed, e. g. in dosimetric studies, simultaneously measuring of knee activity and activity in the inguinal lymph nodes in anterior and posterior views and calculation of the geometric mean with exclusion of the scatter rays was found to be the procedure of choice. Conclusion: When measuring of activity leakage is used for dosimetric calculations, the above-described procedure should be used. When the real value of the leakage is not necessary, e. g. for comparing different therapeutic modalities, several of the procedures can be considered as being equivalent.

DMSA-scintigraphy in paediatrics: Is the evaluation of the geometric mean necessary for the calculation of the differential renal function?

2001 ◽  
Vol 40 (04) ◽  
pp. 107-110 ◽  
Author(s):  
B. Roßmüller ◽  
S. Alalp ◽  
S. Fischer ◽  
S. Dresel ◽  
K. Hahn ◽  
...  
Keyword(s):  
Renal Function ◽  
Geometric Mean ◽  
Region Of Interest ◽  
Posterior View ◽  
Renal Abnormality ◽  
Anterior View ◽  
Differential Renal Function ◽  
The Mean ◽  
The Right ◽  
To Receive

SummaryFor assessment of differential renal function (PF) by means of static renal scintigraphy with Tc-99m-dimer-captosuccinic acid (DMSA) the calculation of the geometric mean of counts from the anterior and posterior view is recommended. Aim of this retrospective study was to find out, if the anterior view is necessary to receive an accurate differential renal function by calculating the geometric mean compared to calculating PF using the counts of the posterior view only. Methods: 164 DMSA-scans of 151 children (86 f, 65 m) aged 16 d to 16 a (4.7 ± 3.9 a) were reviewed. The scans were performed using a dual head gamma camera (Picker Prism 2000 XP, low energy ultra high resolution collimator, matrix 256 x 256,300 kcts/view, Zoom: 1.6-2.0). Background corrected values from both kidneys anterior and posterior were obtained. Using region of interest technique PF was calculated using the counts of the dorsal view and compared with the calculated geometric mean [SQR(Ctsdors x Ctsventr]. Results: The differential function of the right kidney was significantly less when compared to the calculation of the geometric mean (p<0.01). The mean difference between the PFgeom and the PFdors was 1.5 ± 1.4%. A difference > 5% (5.0-9.5%) was obtained in only 6/164 scans (3.7%). Three of 6 patients presented with an underestimated PFdors due to dystopic kidneys on the left side in 2 patients and on the right side in one patient. The other 3 patients with a difference >5% did not show any renal abnormality. Conclusion: The calculation of the PF from the posterior view only will give an underestimated value of the right kidney compared to the calculation of the geometric mean. This effect is not relevant for the calculation of the differntial renal function in orthotopic kidneys, so that in these cases the anterior view is not necesssary. However, geometric mean calculation to obtain reliable values for differential renal function should be applied in cases with an obvious anatomical abnormality.

On Statistics of Log-Ratio of Arithmetic Mean to Geometric Mean for Nakagami-m Fading Power

2012 ◽  
Vol E95-B (2) ◽  
pp. 647-650
Author(s):  
Ning WANG ◽  
Julian CHENG ◽  
Chintha TELLAMBURA
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Log Ratio

Milk acetone determination by the photometrical method after microdiffusion and via FT infra-red spectroscopy

2011 ◽  
Vol 28 (1) ◽  
pp. 33-48 ◽  
Author(s):  
Oto Hanuš ◽  
Václava Genčurová ◽  
Yunhai Zhang ◽  
Pavel Hering ◽  
Jaroslav Kopecký ◽  
...  
Keyword(s):  
Infrared Spectroscopy ◽  
Indirect Method ◽  
Geometric Mean ◽  
Mid Infrared ◽  
Subclinical Ketosis ◽  
Infra Red ◽  
Variation Range ◽  
Ft Ir ◽  
Infrared Method ◽  
Photometrical Method

Milk acetone determination by the photometrical method after microdiffusion and via FT infra-red spectroscopyMilk acetone (AC) and betahydroxybutyrate (BHB) are important indicators of the energy metabolism of cows (ketosis occurrence) and an effective method for their determination, with reliable results, is of great importance. The goal of this work was to investigate the infrared method MIR-FT in terms of its calibration for milk AC and to develop a usable procedure. The microdiffusion photometric (485 nm; Spekol 11) method was used with salicylaldehyde as a reference (Re) and mid infrared spectroscopy FT (MIR-FT: Lactoscope FT-IR, Delta; MilkoScan FT 6000, M-Sc) as an indirect method. The acetone addition to milk had no recovery using MIR-FT (Delta). The reference AC set must have acceptable statistics for good MIR-FT calibration (M-Sc) and they were: 10.1 ± 9.74 at a geometric mean of 7.26 mg l-1, and a variation range from 1.98 to 33.66 mg l-1. The AC correlation between Re and MIR-FT (Delta) was low at 0.32 (P>0.05 but the Log AC relationship between Re and MIR-FT (M-Sc) was markedly better at 0.80 (P<0.01). The conversion of >10 mg l-1 as an AC subclinical ketosis limit could be > -0.80 (feedback 0.158 mmol l-1 = 9.25 mg l-1) and > -1.66. This could be important for ketosis monitoring (using M-Sc).

The relationship between health effects in triathletes and microbiological quality of freshwater

1995 ◽  
Vol 31 (5-6) ◽  
pp. 19-26 ◽  
Author(s):  
G. J. Medema ◽  
I. A. van Asperen ◽  
J. M. Klokman-Houweling ◽  
A. Nooitgedagt ◽  
M. J. W. van de Laar ◽  
...  
Keyword(s):  
Water Quality ◽  
Geometric Mean ◽  
Microbiological Quality ◽  
Quality Parameters ◽  
Health Complaints ◽  
E Coli ◽  
Thermotolerant Coliforms ◽  
Microbiological Water Quality ◽  
The Relationship ◽  
Faecal Streptococci

This pilot study was carried out to determine the relationship between microbiological water quality parameters and the occurrence of health complaints among triathletes. Data were collected at an Olympic distance triathlon (n=314) and a run-bike-run (n=81; controls for exposure to fresh water). At the time of the triathlon, the concentrations of Escherichia coli , thermotolerant coliforms, faecal streptococci, entero- and reoviruses, F-specific RNA phages, Salmonella, Campylobacter, Aeromonas, Plesiomonas shigelloides, Pseudomonas aeruginosa and Staphylococcus aureus were examined over the swimming course. Information on the occurrence of health complaints during the competition and in the week thereafter was collected through a written questionnaire. The results show that triathletes and run-bike-runners are comparable with respect to factors other than water exposure (age, sex, training history, physical stress, lower intestinal health complaints during the competition) that may influence the occurrence of health complaints in the week after the competition. Triathletes and run-bike-runners reported gastro-intestinal (7.7% vs 2.5%), respiratory (5.5% vs 3.7%), skin/mucosal (2.6% vs 1.2%), general (3.5% vs 1.2%) and total symptoms (14.8% vs 7.4%) in the week after the event. The health risks for triathletes for all symptom groups are not significantly higher than for run-bike-runners. The geometric mean concentration of faecal indicator bacteria is relatively low: E. coli 170/100 ml; faecal streptococci 13/100 ml, enteroviruses were present at concentrations of 0.1/l. The group of triathletes was homogeneusly and relatively intensely exposed to water; they all swam in the same body of water at the same time and 75% reported to have swallowed freshwater. It was concluded that this study design is suitable to study the relationship between health complaints and microbiological water quality. In the summers of 1993 and 1994, a study will be carried out concerning several run-bike-runs and triathlons in freshwaters of different quality.

Assessment of the exposure of swimmers to microbiological contaminants in fresh waters

1997 ◽  
Vol 35 (11-12) ◽  
pp. 157-163 ◽  
Author(s):  
G. J. Medema ◽  
I. A. van Asperen ◽  
A. H. Havelaar
Keyword(s):  
Geometric Mean ◽  
Microbiological Quality ◽  
The Other ◽  
Pollution Level ◽  
Intestinal Infection ◽  
E Coli ◽  
Faecal Pollution ◽  
Run Off ◽  
Thermotolerant Coliforms ◽  
Mean Concentration

As part of a prospective cohort study among triathletes to determine a relationship between the microbiological quality of fresh bathing water and the risk of acquiring an intestinal infection, the exposure of the triathletes to microbiological contaminants was assessed. Waters were collected at seven triathlons (swimming course 1–1.5km) held in the summer of 1993 and 1994 to have a range of water qualities. All were influenced by sewage effluents, most also by agricultural run-off. Samples were collected several weeks before the event to establish a sampling programme (1993) and during the actual exposure of the triathletes (1993 and 1994) and examined for thermotolerant coliforms alone (samples preceding the event) and for E. coli, faecal enterococci, Staphylococcus aureus, F-specific RNAphages, enteroviruses (1993 and 1994) and for thermophilic Campylobacter, Salmonella, Aeromonas, Plesiomonas shigelloides and Pseudomonas aeruginosa (1993). The samples taken in the weeks before the exposure showed significant differences in thermotolerant coliform concentration between locations, depths and times. Also during swimmer exposure, significant differences occurred in microorganism levels at the different sampling points over the swimming course. As the triathletes swam as a group, they were exposed to approximately the same water at the same time. The geometric mean concentration was used to characterise each site. In the epidemiological study, the risk of an intestinal infection correlated with the concentration of thermotolerant coliforms and E. coli but not with the other parameters. The geometric mean concentration of thermotolerant coliforms at the triathlons ranged from 11–330/100mL and 54–1,200/100mL E. coli. Ranking of the seven sites by faecal pollution level, based on the geometric mean concentration of a faecal indicator, resulted in a different ranking for each indicator. At the fresh water sites studied, only the ratio between the geometric mean density of E. coli and thermotolerant coliforms was constant. The ratio between the other parameters related to faecal pollution (faecal enterococci, F-specific RNA phages, enteroviruses) varied considerably. Water quality standards relating to faecal pollution can only be based on parameters that show a significant correlation with risk of intestinal illness.

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