Linear Relation (or Correlation)

2015 ◽  
Keyword(s):  
Linear Relation

On compressions of self-adjoint extensions of a symmetric linear relation with unequal deficiency indices

2019 ◽  
Vol 16 (4) ◽  
pp. 567-587
Author(s):  
Vadim Mogilevskii
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Linear Relation ◽  
Deficiency Indices ◽  
Space Extension ◽  
Limit Value ◽  
Abstract Boundary ◽  
Symmetric Linear Relation ◽  
Symmetric Operators

Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with unequal deficiency indices $n_-A <n_+(A)$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an (exit space) extension of $A$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of extensions $\wt A=\wt A^*$. Our main result is a description of compressions $C (\wt A)$ by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter $\tau(\l)$ from the Krein formula for generalized resolvents. We describe also all extensions $\wt A=\wt A^*$ of $A$ with the maximal symmetric compression $C (\wt A)$ and all extensions $\wt A=\wt A^*$ of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators $A$ with equal deficiency indices $n_+(A)=n_-(A)$.

scholarly journals Prospective study of dynamic whole-body 68Ga-DOTATOC-PET/CT acquisition in patients with well-differentiated neuroendocrine tumors

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Philippe Thuillier ◽  
David Bourhis ◽  
Jean Philippe Metges ◽  
Romain Le Pennec ◽  
Karim Amrane ◽  
...  
Keyword(s):  
Neuroendocrine Tumors ◽  
Linear Relation ◽  
Whole Body ◽  
Pet Ct ◽  
Non Linear ◽  
Well Differentiated ◽  
Analog Systems ◽  
Pet System ◽  
Dynamic Acquisition ◽  
Linear Regressions

AbstractTo present the feasibility of a dynamic whole-body (DWB) 68Ga-DOTATOC-PET/CT acquisition in patients with well-differentiated neuroendocrine tumors (WD-NETs). Sixty-one patients who underwent a DWB 68Ga-DOTATOC-PET/CT for a histologically proven/highly suspected WD-NET were prospectively included. The acquisition consisted in single-bed dynamic acquisition centered on the heart, followed by the DWB and static acquisitions. For liver, spleen and tumor (1–5/patient), Ki values (in ml/min/100 ml) were calculated according to Patlak's analysis and tumor-to-liver (TLR-Ki) and tumor-to-spleen ratios (TSR-Ki) were recorded. Ki-based parameters were compared to static parameters (SUVmax/SUVmean, TLR/TSRmean, according to liver/spleen SUVmean), in the whole-cohort and according to the PET system (analog/digital). A correlation analysis between SUVmean/Ki was performed using linear and non-linear regressions. Ki-liver was not influenced by the PET system used, unlike SUVmax/SUVmean. The regression analysis showed a non-linear relation between Ki/SUVmean (R2 = 0.55,0.68 and 0.71 for liver, spleen and tumor uptake, respectively) and a linear relation between TLRmean/TLR-Ki (R2 = 0.75). These results were not affected by the PET system, on the contrary of the relation between TSRmean/TSR-Ki (R2 = 0.94 and 0.73 using linear and non-linear regressions in digital and analog systems, respectively). Our study is the first showing the feasibility of a DWB 68Ga-DOTATOC-PET/CT acquisition in WD-NETs.

Effects of Deformation Mode on Resistivity Curve Used for Estimating Martensite Induced in Stainless Steel

2010 ◽  
Vol 638-642 ◽  
pp. 2992-2997 ◽  
Author(s):  
Hidefumi Date
Keyword(s):  
Stainless Steel ◽  
Austenitic Stainless Steel ◽  
Linear Relation ◽  
Volume Fraction ◽  
Tensile Deformation ◽  
Deformation Mode ◽  
Transformation Process ◽  
Martensite Formation ◽  
Austenitic Phase ◽  
Ε Phase

The martensite induced in three types of austenitic stainless steel, which indicate the different stability of the austenitic phase (γ), were estimated by the resistivity measured during the tensile deformation or compressive deformation at the temperatures 77, 187 and 293 K. The resistivity curves were strongly dependent on the deformation mode. The volume fraction of the martensite (α’) was also affected by the deformation mode. The ε phase, which is the precursor of the martensite and is induced from the commencement of the deformation, decreased the resistivity. However, lots of defects generated by the deformation-induced martensite increased the resistivity. The experimental facts and the results shown by the modified parallelepiped model suggested a complicated transformation process depending on each deformation mode. The results shown by the model also suggested a linear relation between the resistivity and the martensite volume at the region of the martensite formation. The fact denoted that the resistivity is mostly not controlled by the austenite, ε phase and martensite, but by the defects induced due to the deformation-induced martensite.

35Cl NQR Spectroscopy on Salts and Molecular Compounds of Trichloroacetic Acid

1986 ◽  
Vol 41 (1-2) ◽  
pp. 215-224 ◽  
Author(s):  
Winfried Fichtner ◽  
Axel Markworth ◽  
Norbert Weiden ◽  
Alarich Weiss
Keyword(s):  
Frequency Shift ◽  
Temperature Dependence ◽  
Linear Relation ◽  
Trichloroacetic Acid ◽  
Mean Frequency ◽  
35Cl Nqr ◽  
The Mean ◽  
Molecular Compounds

The temperature dependence of salts M(1)H(Cl3CCOO)2 and molecular compounds of trichloroacetic acid with amines and benzaldehydes, TCA · X, was studied,The data fit rather well to the known dependence of the mean frequency shift Δ <v(35Cl)> on the pkadifference of X with respect to TCA. A linear relation is observed between the bleaching out temperature Tb of the 35Cl NQR lines and Δ <v(35Cl)> for M(1)H(Cl3CCOO)2 and for TCA · X, X = benzaldehydes.

THE ISOPIESTIC METHOD APPLIED TO SORPTION ISOTHERMS

1949 ◽  
Vol 27b (2) ◽  
pp. 87-100 ◽  
Author(s):  
S. Barnartt ◽  
J. B. Ferguson
Keyword(s):  
Carbon Tetrachloride ◽  
Linear Relation ◽  
Sorption Isotherms ◽  
Sorption Process ◽  
The Other ◽  
Isopiestic Method ◽  
Simple Criterion ◽  
Water Vapors ◽  
Linear Sections ◽  
Wide Pressure

The isopiestic method has been applied to the sorption of carbon tetrachloride and water vapors by activated coconut shell charcoals. The isopiestic charges were found to be linearly related over wide pressure ranges. Isotherms formed by plotting the isopiestic charges of two charcoals one against the other consisted of three linear sections for both carbon tetrachloride and water. If the pressure isotherm of one charcoal be known, those of other charcoals may be computed from it by weighing relatively few isopiestic charges. Errors inherent in the measurement of equilibrium pressures, as well as those caused by the drift of the pressure isotherms towards higher sorption capacities at a given pressure, are eliminated in the isopiestic method of comparing charcoals. The linear relation between the isopiestic charges affords a simple criterion of rejection for equations proposed to fit the pressure isotherms. It also throws into relief the structural regularities in activated charcoals. The existence of discontinuities m the sorption process, reported by previous experimenters, is supported by the isopiestic data.

Interactions in block copolymers in solution: The effect of hetero-contacts on the segmental interactions in block copolymers of poly(isoprene : styrene) in solution

1969 ◽  
Vol 22 (8) ◽  
pp. 1649 ◽  
Author(s):  
JR Urwin
Keyword(s):  
Block Copolymer ◽  
Block Copolymers ◽  
Linear Relation ◽  
Sequence Block ◽  
Cluster Integrals ◽  
Excluded Volumes

Binary cluster integrals or excluded volumes for chemically different segment pairs in block copolymers of poly(isoprene : styrene) have been calculated from the equation derived by Froelich and Benoit for a two- sequence block copolymer. Expansion factors have been recalculated assuming a linear relation for [η]θ with respect to composition employing published values for polystyrene and polyisoprene. The results are discussed in relation to possible conformations of block copolymers.

Relation of algal biovolume to chlorophyll a in selected lakes and wetlands in the north-central United States

1995 ◽  
Vol 52 (2) ◽  
pp. 416-424 ◽  
Author(s):  
James W. LaBaugh
Keyword(s):  
United States ◽  
North Dakota ◽  
Chlorophyll A ◽  
Linear Relation ◽  
Algal Biomass ◽  
Open Water ◽  
Light Penetration ◽  
North Central ◽  
The North ◽  
Central United States

Algal chlorophyll a is commonly used as a surrogate for algal biomass. Data from three lakes in western Nebraska, five wetlands in north-central North Dakota, and two lakes in north-central Minnesota represented a range in algal biovolume of over four orders of magnitude and a range in chlorophyll a from less than 1 to 380 mg∙m−3. Analysis of these data revealed that there was a linear relation, log10 algal biovolume = 5.99 + 0.09 chlorophyll a (r2 = 0.72), for cases in which median values of chlorophyll a for open-water periods were less than 20 mg∙m−3. There was no linear relation in cases in which median chlorophyll a concentrations were larger than 20 mg∙m−3 for open-water periods, an occurrence found only in shallow prairies lakes and wetlands for years in which light penetration was the least.

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