Let L be a one-to-one operator of type ω in L2(ℝn), with ω∈[0, π/2), which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. Let p(·): ℝn→(0, 1] be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable Hardy space $H_L^{p(\cdot )} ({\open R}^n)$ associated with L. By means of variable tent spaces, the authors establish the molecular characterization of $H_L^{p(\cdot )} ({\open R}^n)$. Then the authors show that the dual space of $H_L^{p(\cdot )} ({\open R}^n)$ is the bounded mean oscillation (BMO)-type space ${\rm BM}{\rm O}_{p(\cdot ),{\kern 1pt} L^ * }({\open R}^n)$, where L* denotes the adjoint operator of L. In particular, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of $H_L^{p(\cdot )} ({\open R}^n)$ and show that the fractional integral L−α for α∈(0, (1/2)] is bounded from $H_L^{p(\cdot )} ({\open R}^n)$ to $H_L^{q(\cdot )} ({\open R}^n)$ with (1/p(·))−(1/q(·))=2α/n, and the Riesz transform ∇ L−1/2 is bounded from $H_L^{p(\cdot )} ({\open R}^n)$ to the variable Hardy space Hp(·)(ℝn).

A systematic study, using ion trap time-of-flight mass spectrometry, is presented for the photo-dissociation processes of Bisanthenquinone (Bq) cations, C28H12O2+, a ketone substituted Polycyclic Aromatic Hydrocarbon (PAH). The Bq cation fragments through sequential loss of the two neutral carbonyl (CO) units upon laser (626nm) irradiation, resulting in a PAH-like derivative C26H12+. Upon further irradiation, C26H12+ exhibits both stepwise dehydrogenation and C2/C2H2 loss fragmentation channels. Quantum chemistry calculations reveal a detailed picture for the first CO-loss, which involves a transition state with a barrier of ∼ 3.4 eV, which is lower than the energy required for the lowest H-loss pathway (∼ 5.0 eV). The barrier for the second CO-loss is higher (∼ 4.9 eV). The subsequent loss of this unit changes the Bq geometry from a planar to a bent one. It is concluded that the photodissociation mechanism of the substituted PAH cations studied here is site selective in the substituted subunit. This work also shows that an acetone substituted PAH cation is not photo-stable upon irradiation.

In this work, we utilize a method based on Wang-Landau Monte Carlo sampling for studying the temperature effects of astrophysically relavant molecules. Anharmonic effects, e.g., resonances, overtones, and combination bands, are fully incoportated in this method. The calculated infrared (IR) spectra are consistent with the experimental data measured by National Institute of Standards and Technology (NIST).

Let $w$ be either in the Muckenhoupt class of ${{A}_{2}}\left( {{\mathbb{R}}^{n}} \right)$ weights or in the class of $QC\left( {{\mathbb{R}}^{n}} \right)$ weights, and let ${{L}_{w}}\,:=\,-{{w}^{-1}}\,\text{div}\left( A\nabla \right)$ be the degenerate elliptic operator on the Euclidean space ${{\mathbb{R}}^{n}}$, $n\,\ge \,2$. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space $H_{{{L}_{w}}}^{p}\,\left( {{\mathbb{R}}^{n}} \right)$ associated with ${{L}_{w}}$ for $p\,\in \,(0,\,1]$, and when $p\,\in \,(\frac{n}{n+1},\,1]$ and $w\,\in \,{{A}_{{{q}_{0}}}}\left( {{\mathbb{R}}^{n}} \right)$ with ${{q}_{0}}\,\in \,[1,\,\frac{p(n+1)}{n})$, the authors prove that the associated Riesz transform $\nabla L_{w}^{-1/2}$ is bounded from $H_{{{L}_{w}}}^{p}\,\left( {{\mathbb{R}}^{n}} \right)$ to the weighted classical Hardy space $H_{w}^{p}\left( {{\mathbb{R}}^{n}} \right)$.

The stochastic model for breaking of molecular segments proposed by Bithell is analysed and some results relating to the distribution of the number of fragments are obtained by using a slightly more general model which allows multiple ruptures. The product density technique is employed to derive the mean and mean square number of segments at any time t and the number of segments with length greater than y at time of production.