Conditional Values in Quantum Mechanics.

We consider the local value of an operator for a given position or momentum and, more generally on the value of another arbitrary observable. We develop a general approach that is based on breaking up Aψ(x) as Aψ(x)ψ(x)=Aψ(x)ψ(x)R+iAψ(x)ψ(x)I where A is the operator whose local value we seek and ψ(x) is the position wave function. We show that the real part is related to the conditional value for a given position and the imaginary part is related to the standard deviation of the conditional value. We show that the uncertainty of an operator can be expressed in two parts that depend on the real and imaginary parts. In the case of the position representation, the expression for the uncertainty of an operator shows that there are two fundamental contributions, one due to the amplitude of the wave function and the other due to the phase. We obtain the equation of motion for the conditional values, and in particular, we generalize the Ehrenfest theorem by deriving a local version of the theorem. We give a number of examples, including the local value of momentum, kinetic energy, and Hamiltonian. We also discuss other approaches for obtaining a conditional value in quantum mechanics including using quasi-probability distributions and the characteristic function approach, among others.

Quantum Mechanical Approach to the Khintchine and Bochner Criteria for Characteristic Functions.

While it is generally accepted that quantum mechanics is a probability theory, its methods differ radically from standard probability theory. We use the methods of quantum mechanics to understand some fundamental aspects of standard probability theory. We show that wave functions and operators do appear in standard probability theory. We do so by generalizing the Khintchine and Bochner criteria for a complex function to be a characteristic function. We show that quantum mechanics clarifies these criteria and suggests generalizations of them.

The eigenvalue problem in phase space.

We formulate the standard quantum mechanical eigenvalue problem in quantum phase space. The equation obtained involves the c-function that corresponds to the quantum operator. We use the Wigner distribution for the phase space function. We argue that the phase space eigenvalue equation obtained has, in addition to the proper solutions, improper solutions. That is, solutions for which no wave function exists which could generate the distribution. We discuss the conditions for ascertaining whether a position momentum function is a proper phase space distribution. We call these conditions psi-representability conditions, and show that if these conditions are imposed, one extracts the correct phase space eigenfunctions. We also derive the phase space eigenvalue equation for arbitrary phase space distributions functions. © 2017 Wiley Periodicals, Inc.

Contracted Schrödinger equation in quantum phase-space.

The phase space formulation of quantum mechanics is equivalent to standard quantum mechanics where averages are calculated by way of phase space integration as in the case of classical statistical mechanics. We derive the quantum hierarchy equations, often called the contracted Schrödinger equation, in the phase space representation of quantum mechanics which involves quasi-distributions of position and momentum. We use the Wigner distribution for the phase space function and the Moyal phase space eigenvalue formulation to derive the hierarchy. We show that the hierarchy equations in the position, momentum, and position-momentum representations are very similar in structure. © 2017 Wiley Periodicals, Inc.

Statistics of a space-time random pulse train.

A general approach for obtaining the statistics of a pulse train is developed, where the pulses are space and time dependent, and where the initial distribution of the pulses may be non-uniform. Explicit expressions for the moments up to fourth order, in terms of the moments of the elementary signals constituting the pulse train, are obtained. This is done for elementary signals that are waves. The expressions obtained are totally general as long as the elementary signals are not correlated. In contrast to the standard scintillation index, it may be space and time dependent, in which case it is called the dynamical scintillation index. This allows one to understand how a space-time pulse train approaches stationarity at different positions and times. The scintillation index using both the raw and central moments is defined. For the raw moments, both exact expressions and expansions in 1/N, where N is the number of elementary signals are obtained. For central moments, particularly simple exact expressions are given.

Equations of motion for rays in a Snell's law medium.

The equations of motion for a ray in a Snell's law medium with a varying index of refraction are derived. A stratified medium is considered. Explicit expressions are given for the velocity and acceleration components of the ray. These are derived directly from Snell's law. It is further shown that the propagation of a ray can be modeled in terms of Newtonian-like equations of motion and that momentum is conserved along the interface. It is shown that Snell's law follows from this conservation law. Properties of the motion are studied and an example is given.

A phase space approach to wave propagation with dispersion.

A phase space approximation method for linear dispersive wave propagation with arbitrary initial conditions is developed. The results expand on a previous approximation in terms of the Wigner distribution of a single mode. In contrast to this previously considered single-mode case, the approximation presented here is for the full wave and is obtained by a different approach. This solution requires one to obtain (i) the initial modal functions from the given initial wave, and (ii) the initial cross-Wigner distribution between different modal functions. The full wave is the sum of modal functions. The approximation is obtained for general linear wave equations by transforming the equations to phase space, and then solving in the new domain. It is shown that each modal function of the wave satisfies a Schrödinger-type equation where the equivalent "Hamiltonian" operator is the dispersion relation corresponding to the mode and where the wavenumber is replaced by the wavenumber operator. Application to the beam equation is considered to illustrate the approach.

Computational biomechanics for a standing human body: Modal analysis and simulation.

We develop computational mechanical modeling and methods for the analysis and simulation of the motions of a human body. This type of work is crucial in many aspects of human life, ranging from comfort in riding, the motion of aged persons, sports performance and injuries, and many ergonomic issues. A prevailing approach for human motion studies is through lumped parameter models containing discrete masses for the parts of the human body with empirically determined spring, mass, damping coefficients. Such models have been effective to some extent; however, a much more faithful modeling method is to model the human body as it is, namely, as a continuum. We present this approach, and for comparison, we choose two digital CAD models of mannequins for a standing human body, one from the versatile software package LS-DYNA and another from open resources with some of our own adaptations. Our basic view in this paper is to regard human motion as a perturbation and vibration from an equilibrium position which is upright standing. A linear elastodynamic model is chosen for modal analysis, but a full nonlinear viscoelastoplastic extension is possible for full-body simulation. The motion and vibration of these two mannequin models is analyzed by modal analysis, where the normal vibration modes are determined. LS-DYNA is used as the supercomputing and simulation platform. Four sets of low-frequency modes are tabulated, discussed, visualized, and compared. Higher frequency modes are also selectively displayed. We have found that these modes of motion and vibration form intrinsic basic modes of biomechanical motion of the human body. This view is supported by our finding of the upright walking motion as a low-frequency mode in modal analysis. Such a "walking mode" shows the in-phase and out-of-phase movements between the legs and arms on the left and right sides of a human body, implying that this walking motion is spontaneous, likely not requiring any directives from the brain. Dynamic motions of CAD mannequins are also simulated by drop tests for comparisons and the validity of the models is discussed through Fourier frequency analysis. All computed modes of motion are collected in several sets of video animations for ease of visualization. Samples of LS-DYNA computer codes are also included for possible use by other researchers.

Acute obstetric coagulopathy during postpartum hemorrhage is caused by hyperfibrinolysis and dysfibrinogenemia: an observational cohort study.

BACKGROUND: Postpartum hemorrhage (PPH) may be exacerbated by hemostatic impairment. Information about PPH-associated coagulopathy is limited, often resulting in treatment strategies based on data derived from trauma studies. OBJECTIVES: To investigate hemostatic changes associated with PPH. PATIENTS/METHODS: From a population of 11 279 maternities, 518 (4.6%) women were recruited with PPH ≥ 1000 mL or placental abruption, amniotic fluid embolism, or concealed bleeding. Routine coagulation and viscoelastometric results were collated. Stored plasma samples were used to investigate women with bleeds > 2000 mL or those at increased risk of coagulopathy defined as placenta abruption, amniotic fluid embolism, or need for blood components. Procoagulant factors were assayed and global hemostasis was assessed using thrombin generation. Fibrinolysis was investigated with D-dimer and plasmin/antiplasmin complexes. Dysfibrinogenemia was assessed using the Clauss/antigen ratio. RESULTS: At 1000 mL blood loss, Clauss fibrinogen was ≤2 g/L in 2.4% of women and 6/27 (22.2%) cases of abruption. Women with very large bleeds (>3000 mL) had evidence of a dilutional coagulopathy, although hemostatic impairment was uncommon. A subgroup of 12 women (1.06/1000 maternities) had a distinct coagulopathy characterized by massive fibrinolysis (plasmin/antiplasmin > 40 000 ng/mL), increased D-dimer, hypofibrinogenemia, dysfibrinogenemia, reduced factor V and factor VIII, and increased activated protein C, termed acute obstetric coagulopathy. It was associated with fetal or neonatal death in 50% of cases and increased maternal morbidity. CONCLUSIONS: Clinically significant hemostatic impairment is uncommon during PPH, but a subgroup of women have a distinct and severe coagulopathy characterized by hyperfibrinolysis, low fibrinogen, and dysfibrinogenemia associated with poor fetal outcomes.

Local virial and tensor theorems.

We show that for any wave function and potential the local virial theorem can always be satisfied 2K(r) = r·ΔV by choosing a particular expression for the local kinetic energy. In addition, we show that for each choice of local kinetic energy there are an infinite number of quasi-probability distributions which will generate the same expression. We also consider the local tensor virial theorem.

Dispersion-invariant features for classification.

In dispersive propagation, waves from the same source will generally differ depending on how far they have traveled. Accordingly, it is desirable for classification in such environments to either account for propagation effects, or to obtain features that are invariant to such effects. The latter approach is taken in this paper, and features are derived that are unaffected by channel dispersion, per mode. A "local" view of pulse propagation in time-frequency phase space is considered. It is shown that the local duration of a wave, obtained from its time-frequency Wigner distribution, is invariant to dispersion. While higher moments of the Wigner distribution are not invariant to dispersion, the phase space considerations suggest an approach for defining "dispersion-invariant moments" (DIMs) of any order. This approach is also used to define a dispersion-invariant correlation coefficient that can be used for classification. The classification utility of the DIMs, and of the dispersion-invariant correlation coefficient, is evaluated via simulations of acoustic scattering from steel shells in a dispersive channel model (Pekeris waveguide). Receiver operating characteristic curves quantify the improved discriminability of the DIMs versus ordinary temporal moments, and of the dispersion-invariant correlation coefficient versus the ordinary correlation coefficient.

Revision of failed vertical banded gastroplasty to non-resectional Scopinaro biliopancreatic diversion: early experience.

BACKGROUND: The most commonly performed revision operation following failed vertical banded gastroplasty (VBG) is Roux-en-Y gastric bypass, although revision to biliopancreatic diversion (BPD) with duodenal switch is now another common option. We describe the surgical technique for revision of a failed VBG to a non-resectional Scopinaro BPD in a series of patients, as well as the outcome in terms of complications and mean % excess weight loss (%EWL). METHODS: A retrospective review was conducted on all patients who underwent revision to BPD at Mercy Bariatrics, Western Australia, between June 2001 and April 2005. This yielded 20 patients who had revision to BPD, 9 of whom had VBG as their initial operation. The mean %EWL was measured at regular intervals postoperatively (3, 6, 12, and 24 months). RESULTS: Mean %EWL at 12 and 24 months was 69.5 and 76.7, respectively. These results are comparable to %EWL after a primary BPD. Nutritional manifestations were found to be the most common of the minor complications. CONCLUSION: Our technique for revision of a failed restrictive operation to a non-resectional Scopinaro BPD is described. The preliminary results in terms of %EWL and complications are comparable to other revisional malabsorptive operations. Prospective randomized controlled trials are needed to further evaluate effects of revision to a non-resectional Scopinaro BPD and to ensure that the results (in terms of %EWL) are reproducible.

Improved obesity reduction and co-morbidity resolution in patients treated with 40-French bougie versus 50-French bougie four years after laparoscopic sleeve gastrectomy. Analysis of 294 patients.

BACKGROUND: We compared percent excess body mass index loss (%EBMIL) and resolution of dyslipidaemia, hypertension, and type 2 diabetes mellitus in the 4 years following laparoscopic sleeve gastrectomy (LSG) between patients calibrated with a 40-French (40F) or a 50-French (50F) bougie. METHODS: We conducted a longitudinal retrospective descriptive study of routinely collected pre- and post-surgical data from 294 patients who underwent LSG at a single surgical centre (50F--n = 106, 40F--n = 185). Obesity measurements were taken prior to surgery and at regular intervals until 48 months post-surgery. Co-morbidity resolution was also assessed across the 48-month observation period. Multivariate regression modelling was used to control analyses for baseline obesity and sociodemographic variables. RESULTS: At 48 months post-surgery mean (±SD) %EBMIL was 60.2 ± 27.6% and 45.4 ± 38.4% for those treated with the 40F and 50F bougie, respectively. After controlling for sociodemographic variables and baseline excess weight, mean %EBMIL was 15.5% greater with a 40F bougie compared with a 50F bougie at the end of follow-up. The likelihood of dyslipidaemia resolution within 48 months post-LSG was 19.0 times greater (p = 0.006), hypertension resolution 3.6 times greater (p = 0.005) and type 2 diabetes mellitus resolution 5.2 times greater (p = 0.034) by 4 years post-surgery in patients treated with the 40F bougie compared with a 50F bougie. CONCLUSION: Improved obesity reduction and resolution of dyslipidaemia, hypertension and type 2 diabetes mellitus is experienced during the 4 years following surgery by patients treated with a 40F bougie compared with the 50F. These findings remain when controlling for potential confounding clinical and sociodemographic factors.

Effect of a single 'megadose' intramuscular vitamin D (600,000 IU) injection on vitamin D concentrations and bone mineral density following biliopancreatic diversion surgery.

BACKGROUND: Vitamin D (VitD) deficiency is common following biliopancreatic diversion (BPD). We conducted a prospective open-label study to evaluate the efficacy of a single intramuscular injection with 600,000 IU of cholecalciferol in an arachis oil depot formulation (VitD3, Arachitol Solvay Pharmacia) as an adjunct to regular oral VitD supplementation (Citrical+D) for a period of 12 months following BPD surgery. METHODS: Some 29 patients who had undergone BPD during 2000-2005 were recruited and received a single injection of 600,000 IU of cholecalciferol. Venous blood VitD, parathyroid hormone (PTH), alkaline phosphatase (ALP), ionised calcium and urinary N-telopeptide (NTX) were assessed at baseline and at 1.5, 3, 6, 9 and 12 months post-injection. Bone mineral density (BMD) was determined at baseline and 12 months post-injection. RESULTS: VitD concentrations (mean +/- SD) were significantly increased from baseline values (61.5 +/- 18.8 nmol/L) at 1.5 months (92.4 +/- 21.5, p < 0.001), 3 months (100.5 +/- 24.4, p < 0.001) and 6 months (79.1 +/- 20.9, p = 0.014) post-injection, with non-significant elevations at 9 months (73.3 +/- 15.1, p = 0.248) and 12 months (73.4 +/- 17.3, p = 0.278). The proportion of patients with 'normalised' VitD levels was significantly higher at all post-injection time points (range, 93-100%) compared with baseline (71.4%; p < 0.01). Ionised calcium and ALP remained within normal levels at baseline and all follow-up time points, although ionised calcium decreased by 3.4% (p = 0.015) and ALP increased by 14.6% (p = 0.021) at 12 months compared with baseline. No significant change in PTH, NTX or BMD was observed. CONCLUSIONS: Intramuscular cholecalciferol injection, as an adjunct to oral supplementation, appears a safe and effective method to increase and maintain VitD levels after BPD.