Constant Curvature Coefficients and Exact Solutions in Fractional Gravity and Geometric Mechanics

We study fractional configurations in gravity theories and Lagrange mechanics. The approach is based on Caputo fractional derivative which gives zero for actions on constants. We elaborate fractional geometric models of physical interactions and we formulate a method of nonholonomic deformations to other types of fractional derivatives. The main result of this paper consists in a proof that for corresponding classes of nonholonomic distributions a large class of physical theories are modelled as nonholonomic manifolds with constant matrix curvature. This allows us to encode the fractional dynamics of interactions and constraints into the geometry of curve flows and solitonic hierarchies.Comment: latex2e, 11pt, 27 pages, the variant accepted to CEJP; added and up-dated reference

Fractional Hamilton formalism within Caputo's derivative

In this paper we develop a fractional Hamiltonian formulation for dynamic systems defined in terms of fractional Caputo derivatives. Expressions for fractional canonical momenta and fractional canonical Hamiltonian are given, and a set of fractional Hamiltonian equations are obtained. Using an example, it is shown that the canonical fractional Hamiltonian and the fractional Euler-Lagrange formulations lead to the same set of equations.Comment: 8 page

New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis

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Fractional order analysis of Sephadex gel structures: NMR measurements reflecting anomalous diffusion

We report the appearance of anomalous water diffusion in hydrophilic Sephadex gels observed using pulse field gradient (PFG) nuclear magnetic resonance (NMR). The NMR diffusion data was collected using a Varian 14.1 Testa imaging system with a home-built RF saddle coil. A fractional order analysis of the data was used to characterize heterogeneity in the gels for the dynamics of water diffusion in this restricted environment. Several recent studies of anomalous diffusion have used the stretched exponential function to model the decay of the NMR signal, i.e., exp[-(bD)(alpha)], where D is the apparent diffusion constant, b is determined the experimental conditions (gradient pulse separation, durations and strength), and alpha is a measure of structural complexity. In this work, we consider a different case where the spatial Laplacian in the Bloch-Torrey equation is generalized to a fractional order model of diffusivity via a complexity parameter, beta, a space constant, mu, and a diffusion coefficient, D. This treatment reverts to the classical result for the integer order case. The fractional order decay model was fit to the diffusion-weighted signal attenuation for a range of b-values (0 < b < 4000 s mm(-2)). Throughout this range of b values, the parameters beta, mu and D, were found to correlate with the porosity and tortuosity of the gel structure. (C) 2011 Elsevier B.V. All rights reserved.Radiolog

Bone marrow uptake of liposome-entrapped spin label after liver blockade with empty liposomes

Using an ESR spectrometer, we studied the time course of the uptake of the liposome-entrapped spin label 2,2,6,6-tetramethylpiperidine-N-oxyl-4-trimethylammonium in liver, spleen, and bone marrow following reticuloendothelial liver blockade. Our results show that suppression of the phagocytic activity of the liver increases the delivery of liposomes to the spleen and bone marrow without substantially altering uptake by the liver

Chaos in fractional and integer order NSG systems

The nuclear spin generator (NSG) is a high-frequency oscillator that generates and controls the oscillations of the precessional motion of a nuclear magnetization vector in a magnetic field. This nonlinear system was first described by S. Sherman in 1963, and exhibits a wide variety of chaotic behavior, but it is not as well studied as the classic Lorenz chaotic system. In this paper, chaos in the integer order nuclear spin generator system is reviewed. In addition, using fractional order stability analysis, the chaotic behavior of the fractional order NSG (FNSG) is studied. The numerical results are obtained using the Adams–Bashforth–Moulton algorithm encoded in the fde12 Matlab function. In order to confirm the numerically demonstrated chaotic behavior in the nuclear spin generator, we prepared a bifurcation diagram. The phase portrait of the FNSG is also depicted for different fractional orders to show the overall chaotic behavior of the system. These results are also verified using bifurcation analysis. Our results demonstrate a modulating effect on chaos as the fractional order decreases, which could be used to improve the design of the controller in the NSG model. This work also demonstrates how the fractional order model extends the dynamic behavior of the NSG system

Detection of bone marrow involvement in patients with cancer

Current methods for the study of bone marrow to evaluate possible primary or metastatic cancers are reviewed. Bone marrow biopsy, radionuclide scan, computed tomography and magnetic resonance imaging (MRI) are analyzed with regard to their clinical usefulness at the time of diagnosis and during the course of the disease. Bone marrow biopsy is still the examination of choice not only in hematologic malignancies but also for tumors that metastasize into the marrow. Radionuclide scans are indicated for screening for skeletal metastases, except for those from thyroid carcinoma and multiple myeloma. Computed tomography is useful for cortical bone evaluation. MRI shows a high sensitivity in finding occult sites of disease in the marrow but its use has been restricted by high cost and limited availability. However, the future of MRI in bone marrow evaluation seems assured. MRI is already the method of choice for diagnosis of multiple myeloma, when radiography is negative, and for quantitative evaluation of lymphoma when a crucial therapeutic decision (i.e. bone marrow transplantation) must be made. Finally, methods are being developed that will enhance the sensitivity and specificity of MRI studies of bone marrow