The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions

Motivated from studies on anomalous relaxation and diffusion, we show that the memory function M(t) of complex materials, that their creep compliance follows a power law, J(t)∼tq with q∈R+, is proportional to the fractional derivative of the Dirac delta function, dqδ(t−0)dtq with q∈R+. This leads to the finding that the inverse Laplace transform of sq for any q∈R+ is the fractional derivative of the Dirac delta function, dqδ(t−0)dtq. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of sqsα∓λ where α<q∈R+, which is the fractional derivative of order q of the Rabotnov function εα−1(±λ,t)=tα−1Eα,α(±λtα). The fractional derivative of order q∈R+ of the Rabotnov function, εα−1(±λ,t) produces singularities that are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of q in association with the recurrence formula of the two-parameter Mittag–Leffler function.

Fractional Schrödinger equation and anomalous relaxation: Nonlocal terms and delta potentials

We review and extend some results for the fractional Schrödinger equation by considering nonlocal terms or potential given in terms of delta functions. For each case, we have obtained the solution in terms of the Green function approach.

An observation of the evolution of equilibrium stress on poly(lactic acid) and poly(lactic acid)/hydroxyapatite nanocomposites

Relaxation behavior provides specific information that indicates stress development with elapsed time for the determination of material characterization and constituting of material modeling. In addition, anomalous relaxation behavior of polymeric materials enables experimental analysis of some theoretical variables in constitutive equation. Equilibrium stress is one of the most prevailing variables in material modeling and it is generally considered as unmeasurable. In this study, evolution of equilibrium stress was examined with relaxation tests of two semi crystalline polymer materials, one of which was found to reach precise equilibrium stress levels. In accordance with this purpose, extensive relaxation tests were conducted on poly(lactic acid) and poly(lactic acid)/hydroxyapatite nanocomposites specimens at a wide range of temperatures from 23 ℃ and 55 ℃, and stress levels from 1 MPa to 50 MPa for poly(lactic acid) and 51 MPa for poly(lactic acid)/hydroxyapatite nanocomposites. All the specimens were subjected tensile loading–unloading and partial retraction process. The starting points of the relaxation test were chosen on unloading segment of stress–strain curves. Evolution of stress may decay, increase or decay then increase depending on the test point on unloading curves. Relaxation behavior of poly(lactic acid) and poly(lactic acid)/hydroxyapatite nanocomposites was simulated using viscoplasticity theory based on overstress for polymeric materials. Experimental results of poly(lactic acid) and poly(lactic acid)/hydroxyapatite nanocomposites were matched with numerical results of viscoplasticity theory based on overstress for polymeric materials, and viscoplasticity theory based on overstress for polymeric material model was found to have an aptitude for predicting anomalous relaxation behavior of poly(lactic acid) and poly(lactic acid)/hydroxyapatite nanocomposites. Additionally, the effect of temperature on relaxation time was investigated with using Kohlraush–Williams–Watts time-decay function. Modeling capability of Kohlrausch–Williams–Watts model was reasonable to predict short-term simple relaxation of poly(lactic acid) and poly(lactic acid)/hydroxyapatite nanocomposites. Responses of the model showed that interaction between relaxation time and environmental temperature was related to transition from glass state to rubbery state.

Stretched-Exponential Modeling of Anomalous T1ρ and T2 Relaxation in the Intervertebral Disc In Vivo

ABSTRACTPurposeIntervertebral disc degeneration (IVDD), resulting in the depletion of hydrophilic glycosaminoglycans (GAGs) located in the nucleus pulposus (NP), can lead to debilitating neck and back pain. Magnetic Resonance Imaging (MRI) is a promising means of IVD assessment due to the sensitivity of MRI tissue relaxation properties to matrix composition. Furthermore, anomalous (i.e. non-monoexponential) relaxation models have shown higher sensitivity to specific matrix components compared to conventional monoexponential models. Here, we extend the use of the stretched exponential model, an anomalous relaxation model, to IVD relaxometry data.Theory and MethodsT1ρ and T2 relaxation data were measured in the cervical IVDs of healthy volunteers and IVDs adjacent to cervical fusion, and analyzed using both conventional and stretched-exponential (SE) models. Model differences were evaluated via goodness of fit in the healthy data. Normalized histograms of the resultant quantitative MRI (qMRI) maps were described using stable distributions, and data were compared across adjacent disc segments.ResultsIn the healthy IVDs, we found lower mean squared error in the SE relaxation model fitting behavior compared to monoexponential models, supporting anomalous relaxation behavior in healthy IVDs. SE model parameter αT1ρ increased level-wise in the caudal direction, especially in the nucleus pulposus, while conventional T1ρ and T2 monoexponential measures did not vary. For IVDs adjacent to cervical fusion, SE parameters deviated near the fusion site compared with those in the healthy population.ConclusionSE modeling of T1ρ relaxation provides greater sensitivity to level-wise variation in IVD matrix properties compared with conventional relaxation modeling, and could provide improved sensitivity to early stages of IVD degeneration. The improved model fit and correlation between the SE αT1ρ parameter with IVD level suggests SE modeling may be a more sensitive method for detection of GAG content variation.

Site-selective 1H/2H labeling enables artifact-free 1H CPMG relaxation dispersion experiments in aromatic side chains

Aromatic side chains are often key residues in enzyme active sites and protein binding sites, making them attractive probes of protein dynamics on the millisecond timescale. Such dynamic processes can be studied by aromatic 13C or 1H CPMG relaxation dispersion experiments. Aromatic 1H CPMG relaxation dispersion experiments in phenylalanine, tyrosine and the six-ring moiety of tryptophan, however, are affected by 3J 1H–1H couplings which are causing anomalous relaxation dispersion profiles. Here we show that this problem can be addressed by site-selective 1H/2H labeling of the aromatic side chains and that artifact-free relaxation dispersion profiles can be acquired. The method has been further validated by measuring folding–unfolding kinetics of the small protein GB1. The determined rate constants and populations agree well with previous results from 13C CPMG relaxation dispersion experiments. Furthermore, the CPMG-derived chemical shift differences between the folded and unfolded states are in excellent agreement with those obtained directly from the spectra. In summary, site-selective 1H/2H labeling enables artifact-free aromatic 1H CPMG relaxation dispersion experiments in phenylalanine and the six-ring moiety of tryptophan, thereby extending the available methods for studying millisecond dynamics in aromatic protein side chains.