On the local fractional derivative

We present the necessary conditions for the existence of the Kolwankar-Gangal local fractional derivatives (KG-LFD) and introduce more general but weaker notions of LFDs by using limits of certain integral averages of the difference-quotient. By applying classical results due to Stein and Zygmund (1965) [16] we show that the KG-LFD is almost everywhere zero in any given intervals. We generalize some of our results to higher dimensional cases and use integral approximation formulas obtained to design numerical schemes for detecting fractional dimensional edges in signal processing.     

On the computational complexity of cut-reduction

Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations. Explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theories can be reobtained in a uniform way.     

A free-volume hole-filling model for the solubility of liquid molecules in glassy polymers 1: Model derivation

The use of hole-filling models is quite common for the sorption of gases into glassy polymers, but these have yet to be explicitly applied to the case of liquid immersion, where extensive use of Flory-Huggins theory dominates. This paper explores how the models based on the idea of sequential filling of a Gaussian distribution of pre-existing free-volume holes within the structure of the glassy polymer can be modified to allow the prediction of the equilibrium solubility of a liquid penetrant. For liquids, the driving force for sorption is more subtle than for gases, with more emphasis on molecular interactions rather than external pressure. For this reason, terms relating to the molecular interactions of a liquid molecule filling a hole were developed, including the effects of elastic constraint for small holes. Consideration of thermal fluctuations show that configurational entropy provides much of the driving force for sorption. Some comparisons with experimental data show a reasonable agreement, and one which is far better than the Flory-Huggins theory.     

Log-Harnack inequality for stochastic Burgers equations and applications

By proving an L²-gradient estimate for the corresponding Galerkin approximations, the log-Harnack inequality is established for the semigroup associated to a class of stochastic Burgers equations. As applications, we derive the strong Feller property of the semigroup, the irreducibility of the solution, the entropy-cost inequality for the adjoint semigroup, and entropy upper bounds of the transition density.     

A polynomial chaos approach to the robust analysis of the dynamic behaviour of friction systems

This paper presents a dynamic behaviour study of non-linear friction systems subject to uncertain friction laws. The main aspects are the analysis of the stability and the associated non-linear amplitude around the steady-state equilibrium. As friction systems are highly sensitive to the dispersion of friction laws, it is necessary to take into account the uncertainty of the friction coefficient to obtain stability intervals and to estimate the extreme magnitudes of oscillations. Intrusive and non-intrusive methods based on the polynomial chaos theory are proposed to tackle these problems. The efficiency of these methods is investigated in a two degree of freedom system representing a drum brake system. The proposed methods prove to be interesting alternatives to the classic methods such as parametric studies and Monte Carlo based techniques.     

Blow-up for Stochastic Reaction-Diffusion Equations with Jumps

In this paper, we focus on stochastic reaction-diffusion equations with jumps. By a new auxiliary function, we investigate non-negative property of the local strong (variational) solutions, which applies to stochastic reaction-diffusion equations with highly nonlinear noise terms. As a byproduct, we study the problem of non-existence of global strong solutions by imposing appropriate conditions on the drift terms, which can cover many more models than the existing literature. Moreover, we also investigate the subject of Lévy-type noise-induced explosion by bringing some plausible assumptions to bear on the noise terms, which, however, need not guarantee local strong (variational) solutions to enjoy the non-negative property. Meanwhile, several examples are presented to illustrate the theory established.     

Equilibrium Diffusion on the Cone of Discrete Radon Measures

Let \({\mathbb {K}(\mathbb {R}^{d})}\) denote the cone of discrete Radon measures on \(\mathbb {R}^{d}\). There is a natural differentiation on \(\mathbb {K}(\mathbb {R}^{d})\): for a differentiable function \(F:\mathbb {K}(\mathbb {R}^{d})\to \mathbb {R}\), one defines its gradient \(\nabla ^{\mathbb {K}}F\) as a vector field which assigns to each \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) an element of a tangent space \(T_{\eta }(\mathbb {K}(\mathbb {R}^{d}))\) to \(\mathbb {K}(\mathbb {R}^{d})\) at point η. Let \(\phi :\mathbb {R}^{d}\times \mathbb {R}^{d}\to \mathbb {R}\) be a potential of pair interaction, and let μ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on \(\mathbb {R}^{d}\). In particular, μ is a probability measure on \(\mathbb {K}(\mathbb {R}^{d})\) such that the set of atoms of a discrete measure \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) is μ-a.s. dense in \(\mathbb {R}^{d}\). We consider the corresponding Dirichlet form

$$\mathcal{E}^{\mathbb{K}}(F,G)={\int}_{\mathbb K(\mathbb{R}^{d})}\langle\nabla^{\mathbb{K}} F(\eta), \nabla^{\mathbb{K}} G(\eta)\rangle_{T_{\eta}(\mathbb{K})}\,d\mu(\eta). $$

Integrating by parts with respect to the measure μ, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If d ≥ 2, there exists a conservative diffusion process on \(\mathbb {K}(\mathbb {R}^{d})\) which is properly associated with the Dirichlet form \(\mathcal {E}^{\mathbb {K}}\).

     

Sodiation Of Hard Carbon: How Separating Enthalpy And Entropy Contributions Can Find Transitions Hidden In The Voltage Profile

Sodium-ion batteries (NIBs) utilize cheaper materials than lithium-ion batteries (LIBs) and can thus be used in larger scale applications. The preferred anode material is hard carbon, because sodium cannot be inserted into graphite. We apply experimental entropy profiling (EP), where the cell temperature is changed under open circuit conditions. EP has been used to characterize LIBs; here, we demonstrate the first application of EP to any NIB material. The voltage versus sodiation fraction curves (voltage profiles) of hard carbon lack clear features, consisting only of a slope and a plateau, making it difficult to clarify the structural features of hard carbon that could optimize cell performance. We find additional features through EP that are masked in the voltage profiles. We fit lattice gas models of hard carbon sodiation to experimental EP and system enthalpy, obtaining: 1. a theoretical maximum capacity, 2. interlayer versus pore filled sodium with state of charge.     

Singlet and Triplet State Properties and Singlet Oxygen Yield of an Amino-Acyl-Quinoxalinone Yellow Dye

Amino-acyl-quinoxalinone yellow dyes are cyclised analogues of the yellow azomethine dyes developed for, and still used in, silver halide colour photography. Unlike image azomethine dyes, which are rapidly deactivated in their excited states by torsion about the azomethine bond, amino-acyl-quinoxalinone dyes have an interesting photophysics because torsion is not possible due to their cyclised structure. We report results from studies on singlet and triplet state properties, and singlet oxygen yields, of the yellow dye, 7-diethylamino-3-(2,2-dimethyl-propionyl)-5-methyl-1-phenyl-1H-quinoxalin-2-one, in polar and nonpolar solvents. The dye photophysics is characterised by a weak fluorescence, with a solvent dependent emission yield (Φ_F?≈?0.002–0.004), and short singlet state lifetime (τ_expt?≈?20–50 ps), both increasing by a factor of ≈2 in going from polar acetonitrile to non-polar dioxane as solvent. DFT ZINDO calculations show a transition involving significant electron transfer from the diethyl-amino group into the carbonyl region of the molecule. In solution, in the presence of oxygen, the triplet state decays almost exclusively by oxygen quenching, and singlet oxygen is produced in high yield (Φ_??≈?0.5–0.55). The triplet state absorbs across the 450–750 nm region with maxima around 480 and 650 nm, and moderate molar absorption coefficients (ca. 6000–8000 M^?1 cm^?1). In a glass at 77 K, triplet decay gives a red phosphorescence, with λ_max?≈?640–650 nm, and a ?≈?0.25 s lifetime. If singlet oxygen yields are a good indication of triplet yields, then internal conversion and intersystem crossing occur with roughly equal efficiency.

     

Eigenvalue density of linear stochastic dynamical systems: A random matrix approach

Eigenvalue problems play an important role in the dynamic analysis of engineering systems modeled using the theory of linear structural mechanics. When uncertainties are considered, the eigenvalue problem becomes a random eigenvalue problem. In this paper the density of the eigenvalues of a discretized continuous system with uncertainty is discussed by considering the model where the system matrices are the Wishart random matrices. An analytical expression involving the Stieltjes transform is derived for the density of the eigenvalues when the dimension of the corresponding random matrix becomes asymptotically large. The mean matrices and the dispersion parameters associated with the mass and stiffness matrices are necessary to obtain the density of the eigenvalues in the frameworks of the proposed approach. The applicability of a simple eigenvalue density function, known as the Marenko–Pastur (MP) density, is investigated. The analytical results are demonstrated by numerical examples involving a plate and the tail boom of a helicopter with uncertain properties. The new results are validated using an experiment on a vibrating plate with randomly attached spring–mass oscillators where 100 nominally identical samples are physically created and individually tested within a laboratory framework.