EULER–POINCARÉ FLOWS ON THE LOOP BOTT–VIRASORO GROUP AND SPACE OF TENSOR DENSITIES AND (2 + 1)-DIMENSIONAL INTEGRABLE SYSTEMS

2010 ◽  
Vol 22 (05) ◽  
pp. 485-505
Author(s):  
PARTHA GUHA
Keyword(s):  
Kdv Equation ◽  
Virasoro Algebra ◽  
Coadjoint Orbit ◽  
Type Systems ◽  
Inner Product ◽  
Field Equations ◽  
Virasoro Group ◽  
The Right ◽  
Tensor Densities

Following the work of Ovsienko and Roger ([54]), we study loop Virasoro algebra. Using this algebra, we formulate the Euler–Poincaré flows on the coadjoint orbit of loop Virasoro algebra. We show that the Calogero–Bogoyavlenskii–Schiff equation and various other (2 + 1)-dimensional Korteweg–deVries (KdV) type systems follow from this construction. Using the right invariant H1 inner product on the Lie algebra of loop Bott–Virasoro group, we formulate the Euler–Poincaré framework of the (2 + 1)-dimensional of the Camassa–Holm equation. This equation appears to be the Camassa–Holm analogue of the Calogero–Bogoyavlenskii–Schiff type (2 + 1)-dimensional KdV equation. We also derive the (2 + 1)-dimensional generalization of the Hunter–Saxton equation. Finally, we give an Euler–Poincaré formulation of one-parameter family of (1 + 1)-dimensional partial differential equations, known as the b-field equations. Later, we extend our construction to algebra of loop tensor densities to study the Euler–Poincaré framework of the (2 + 1)-dimensional extension of b-field equations.

Hamiltonian Systems Based on a Lie Algebra

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Lie Algebra ◽  
Euler Equations ◽  
Burgers Equation ◽  
Kdv Equation ◽  
Three Dimensional ◽  
Hamiltonian Formalism ◽  
Virasoro Algebra ◽  
Inner Product ◽  
The Kdv Equation ◽  
Limiting Case

There is a remarkable analogy between Euler’s equations for a rigid body and his equations for an ideal fluid. The unifying idea is that of a Lie algebra with an inner product, which is not invariant, on it. The concepts of a vector space, Lie algebra, and inner product are reviewed. A hamiltonian dynamical system is derived from each metric Lie algebra. The Virasoro algebra (famous in string theory) is shown to lead to the KdV equation; and in a limiting case, to the Burgers equation for shocks. A hamiltonian formalism for two-dimensional Euler equations is then developed in detail. A discretization of these equations (using a spectral method) is then developed using mathematical ideas from quantum mechanics. Then a hamiltonian formalism for the full three-dimensional Euler equations is developed. The Clebsch variables which provide canonical pairs for fluid dynamics are then explained, in analogy to angular momentum.

PROJECTIVE CONNECTIONS, AGD MANIFOLD AND INTEGRABLE SYSTEMS

2000 ◽  
Vol 12 (10) ◽  
pp. 1391-1406 ◽  
Author(s):  
PARTHA GUHA
Keyword(s):  
Vector Field ◽  
Periodic Function ◽  
Kdv Equation ◽  
Quadratic Function ◽  
Virasoro Algebra ◽  
Coadjoint Orbit ◽  
Projective Connection ◽  
Order Operator ◽  
Neumann System ◽  
Projective Vector

If uis are periodic function on the line, the operator [Formula: see text], acting on periodic functions, is called a Adler–Gelfand–Dikii (or AGD) operator. In this paper we consider a projective connection as defined by this nth order operator on the circle. In particular, projective connection as defined by a second order operator can be identified with the dual of Virasoro algebra, and it is well known that the KdV equation as a Euler–Arnold equation in the coadjoint orbit of the Bott–Virasoro group. In this paper we study (formally) the evolution equation of the Adler–Gelfand–Dikii operator, Δ(n), (at least for n ≤ 4), under the action of Vect (S1). This yields a single generating equation for periodic function u. We also establish a connection between the projective vector field, a vector field leaves fixed a given (extended) projective connection, and the C. Neumann system using the idea of Knörrer and Moser. We show that certain quadratic function of a projective field satisfies C. Neumann system.

STUDY OF RADIATION CHARACTERISTICS THE RENAL FUNCTION IN PATIENTS WITH TYPE 2 DIABETES

2014 ◽  
pp. 73-77
Author(s):  
Van Chuong Nguyen ◽  
Thi Kim Anh Nguyen
Keyword(s):  
Type 2 Diabetes ◽  
Renal Function ◽  
Glomerular Filtration Rate ◽  
Glomerular Filtration ◽  
Filtration Rate ◽  
Disease Duration ◽  
Left Kidney ◽  
Cross Sectional ◽  
The Right

Background: A Research glomerular filtration rate (GFR) of 61 patients with type 2 diabetes mellitus with renal scanning 99mTc-DTPA glomerular filtration rate at the hospital 175. Objective: (1) To study characteristics of imaging of renal function. (2) Understanding the relationship between GFR with blood sugar, HbA1c, blood pressure and albuminuria in patients with type 2 diabetes. Methods: Descriptive, prospective, cross-sectional study. Clinical examination, Clinical tests and 99mTc-DTPA GFR gamma - camera renography for patients. Result: GFR of the study group was 75,4 ± 22,3 ml/phut/1,73m2, the left kidney was 35,0 ± 13,0 is lower than the right kidney and 39,8 ± 11,9; p <0,01. There is no correlation between GFR with blood glucose and HbA1c, the risk of reduced GFR in hypertensive group associated is OR = 6,5 with p<0,01; albuminuria (+) is OR = 4,2 with p <0,01; and disease duration > 10 years is OR = 3,5 with p <0.01. Conclusion: GFR of the left kidneys is lower than the right kidney; correlation decreased GFR associated with hypertension, albuminuria and disease duration. Keywords: GFR, diabetes, albuminuria

scholarly journals On local and integrated stress-tensor commutators

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Mert Besken ◽  
Jan de Boer ◽  
Grégoire Mathys
Keyword(s):  
Stress Tensor ◽  
Free Field ◽  
Virasoro Algebra ◽  
Light Sheet ◽  
Operator Product ◽  
Local Form ◽  
Two Dimensional ◽  
Conformal Embedding ◽  
General Aspects ◽  
The Right

Abstract We discuss some general aspects of commutators of local operators in Lorentzian CFTs, which can be obtained from a suitable analytic continuation of the Euclidean operator product expansion (OPE). Commutators only make sense as distributions, and care has to be taken to extract the right distribution from the OPE. We provide explicit computations in two and four-dimensional CFTs, focusing mainly on commutators of components of the stress-tensor. We rederive several familiar results, such as the canonical commutation relations of free field theory, the local form of the Poincaré algebra, and the Virasoro algebra of two-dimensional CFT. We then consider commutators of light-ray operators built from the stress-tensor. Using simplifying features of the light sheet limit in four-dimensional CFT we provide a direct computation of the BMS algebra formed by a specific set of light-ray operators in theories with no light scalar conformal primaries. In four-dimensional CFT we define a new infinite set of light-ray operators constructed from the stress-tensor, which all have well-defined matrix elements. These are a direct generalization of the two-dimensional Virasoro light-ray operators that are obtained from a conformal embedding of Minkowski space in the Lorentzian cylinder. They obey Hermiticity conditions similar to their two-dimensional analogues, and also share the property that a semi-infinite subset annihilates the vacuum.

scholarly journals Acetazolamide and SGLT2 inhibitor as potent drugs for a patient with diabetes mellitus and worsening chronic lymphedema: A case report

2020 ◽  
Vol 9 (1) ◽  
Author(s):  
Hajime Kataoka
Keyword(s):  
Body Weight ◽  
Fluid Collection ◽  
Sglt2 Inhibitor ◽  
Chloride Concentration ◽  
Tissue Fluid ◽  
Fat Deposits ◽  
Serum Chloride ◽  
The Right ◽  
Chronic Lymphedema

Treatment of lymphedema using a pharmacologic approach is reported to have limited efficacy. Here, I report a patient with type 2 diabetes (T2DM) and acute worsening of her chronic lymphedema, in whom treatment with acetazolamide and a sodiumglucose cotransporter-2 inhibitor (SGLT2i) effectively improved the lymphedema. A 94-year-old woman, who was treated for T2DM, hyperlipidemia, and hypertension for 17 years at my hospital presented to the emergency room because of acute worsening of her chronic right leg lymphedema with increased swelling, tightness, and dull aching. A pharmacologic approach was used to treat her worsening lymphedema. Acetazolamide 500 mg/d was administered to treat the acute tissue fluid collection in the right lymphedematous leg. Ten days later, the patient’s body weight was markedly reduced by 3.2 kg, pitting in the right leg was markedly improved, and the circumference of right limb was decreased. On day 11, the glucose-lowering agent vildagliptin was switched to the SGLT2i empagliflozin 10 mg/d. On day 70, her body weight had decreased further by 2.8 kg, and the circumference of her right limb was greatly reduced compared with that under treatment with acetazolamide. Her serum chloride concentration was increased after treatment, but her hemoglobin and hematocrit values did not change during the study period. In conclusion, acetazolamide and an SGLT2i have acute diuretic effects for draining the excess tissue fluid in the lymphedematous limb without vascular contraction by enhancing vascular tonicity. Additionally, an SGLT2i may have chronic effects for reducing fat deposits in the lymphedematous limb.

scholarly journals Test-Retest Reliability of Isokinetic Knee Strength Measurements in Type 2 Diabetes Mellitus Patients

2021 ◽  
Vol 13 (3) ◽  
pp. 1343
Author(s):  
José Carmelo Adsuar ◽  
Jose Parraca ◽  
Armando Raimundo ◽  
Miguel Angel Garcia-Gordillo ◽  
Patricia Polero ◽  
...  
Keyword(s):  
Diabetes Mellitus ◽  
Type 2 Diabetes ◽  
Type 2 Diabetes Mellitus ◽  
Peak Torque ◽  
Total Sample ◽  
Good For ◽  
The Right ◽  
Test Retest Reliability ◽  
Standard Error Of Measurement

Background: Reliability studies are used to verify the evaluation accuracy of a given device. Strength is an important factor for the development of daily activities and its correct management is fundamental. The objective of this study was to examine the reliability of a concentric strength test in people with type 2 diabetes mellitus (T2DM). Methods: Twenty-seven individuals with T2DM performed three repetitions of extension-flexion in concentric-concentric action at 60°/s, for both legs, using an isokinetic dynamometer. For the reliability analysis, we performed an intra-session test retest. Results: The total sample and men sub-group intra-class correlation coefficient (ICC) values were excellent for peak torque and work. In the women sub-group, ICC values were excellent for extensors in both peak torque and work; however, concerning flexor, the ICC values were good for peak torque while, for work, they were good for the right leg and moderate for the left leg. Standard error of measurement (SEM) percentage oscillated from 3.85% to 6.80%, with the smallest real difference (SRD) percentage being from 10.66% to 18.86% for peak torque. Furthermore, the SEM (%) was around 5.5% and SRD (%) was around 15% for work. Conclusions: The isokinetic dynamometry had “good” to “excellent” relative reliability for peak torque (0.862–0.983) and work (0.744–0.982) of extension-flexion in concentric-concentric action at 60°/s. In addition, our study showed that, in general, an SRD < 20% could indicate a true change in strength regarding this protocol in T2DM.

Intradural Extramedullary Lesions in Cervical Spine in Neurofibromatosis

2021 ◽  
pp. 22
Author(s):  
Rana Moshref
Keyword(s):  
Multidisciplinary Approach ◽  
Cervical Lesions ◽  
Cervical Mass ◽  
Gradual Onset ◽  
Autosomal Dominant Fashion ◽  
Intradural Extramedullary ◽  
The Right ◽  
Spastic Quadriparesis

Introduction: Neurofibromatosis (NF) is isolated into three diseases: NF type 1, type 2, and schwannoma. NF type 2 could be a disorder that's found roughly in 1/25,000–33,000 births with a mutation in gene 22q11.2, and it is passed through eras in an autosomal dominant fashion. Diagnosis is made with both clinical and radiological features. A few clinical features have been characterized in conclusion counting Manchester criteria. There is a scarce number of NF type 2 patients diagnosed with cervical lesions which are 25 in number. We report a case of an intradural extramedullary cervical lesion in a patient later diagnosed with NF type 2. Case Report: A 30-year-old male presenting with gradual onset and progressive course of spastic quadriparesis of six months’ duration was admitted through the emergency unit. MRI spine showed intradural extramedullary masses in the right side of C4 and left side of C6. The patient underwent cervical intradural excision of two masses under general anesthesia with neuromonitoring. The tumor was sent for histopathology and reported as NF type 2. Conclusion: NF is a common entity, but the diagnosis of cervical mass is judicious to avoid any complication in neurological function. It further needs a multidisciplinary approach and screening modalities.

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