On the Moore--Penrose inverse of a closed linear relation

2014 ◽  
Vol 85 (1-2) ◽  
pp. 59-72
Author(s):  
Teresa Alvarez ◽  
Qiaoling Xia
Keyword(s):  
Linear Relation ◽  
Closed Linear Relation

scholarly journals Canonical Graph Contractions of Linear Relations on Hilbert Spaces

2021 ◽  
Vol 15 (1) ◽  
Author(s):  
Zsigmond Tarcsay ◽  
Zoltán Sebestyén
Keyword(s):  
Hilbert Spaces ◽  
Linear Relation ◽  
Regular Part ◽  
Linear Relations ◽  
Closed Linear Relation ◽  
Canonical Graph

AbstractGiven a closed linear relation T between two Hilbert spaces $$\mathcal {H}$$ H and $$\mathcal {K}$$ K , the corresponding first and second coordinate projections $$P_T$$ P T and $$Q_T$$ Q T are both linear contractions from T to $$\mathcal {H}$$ H , and to $$\mathcal {K}$$ K , respectively. In this paper we investigate the features of these graph contractions. We show among other things that $$P_T^{}P_T^*=(I+T^*T)^{-1}$$ P T P T ∗ = ( I + T ∗ T ) - 1 , and that $$Q_T^{}Q_T^*=I-(I+TT^*)^{-1}$$ Q T Q T ∗ = I - ( I + T T ∗ ) - 1 . The ranges $${\text {ran}}P_T^{*}$$ ran P T ∗ and $${\text {ran}}Q_T^{*}$$ ran Q T ∗ are proved to be closely related to the so called ‘regular part’ of T. The connection of the graph projections to Stone’s decomposition of a closed linear relation is also discussed.

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.

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