Newman’s identity and infinite families of congruences modulo 7 for broken 3-diamond partitions

In 2007, Andrews and Paule introduced the notion of broken k-diamond partitions. Let \(\Delta _k(n)\) denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, Paule and Radu presented some conjectures on congruences modulo 7 for \(\Delta _3(n)\) which were proved by Jameson and Xiong based on the theory of modular forms. Very recently, Xia proved several infinite families of congruences modulo 7 for \(\Delta _3(n)\) using theta function identities. In this paper, many new infinite families of congruences modulo 7 for \(\Delta _3(n)\) are derived based on an identity of Newman and the (p; k)-parametrization of theta functions due to Alaca, Alaca and Williams. In particular, some non-standard congruences modulo 7 for \(\Delta _3(n)\) are deduced. For example, we prove that for \(\alpha \ge 0\), \(\Delta _3\left( \frac{14\times 757^{\alpha }+1}{3}\right) \equiv 6 -\alpha \ (\mathrm{mod}\ 7)\).     

C0-estimates and smoothness of solutions to the parabolic equation defined by Kimura operators

Kimura diffusions serve as a stochastic model for the evolution of gene frequencies in population genetics. Their infinitesimal generator is an elliptic differential operator whose second-order coefficients matrix degenerates on the boundary of the domain. In this article, we consider the inhomogeneous initial-value problem defined by generators of Kimura diffusions, and we establish $C^0$-estimates, which allows us to prove that solutions to the inhomogeneous initial-value problem are smooth up to the boundary of the domain where the operator degenerates, even when the initial data is only assumed to be continuous.     

An inequality concerning the growth bound of an evolution family and the norm of a convolution operator

Let \(\mathcal {U}=\{U(t,s)\}_{t\ge s\ge 0}\) be a strongly continuous and exponentially bounded evolution family acting on a complex Banach space X and let \(\mathcal {X}\) be a certain Banach function space of X-valued functions. We prove that the growth bound of the family \(\mathcal {U}\) is less than or equal to \(-\frac{1}{c(\mathcal {U}, \mathcal {X})}\) provided that the convolution operator \(f\mapsto \mathcal {U}*f\) acts on \(\mathcal {X}.\) It is well known that under the latter assumption, the convolution operator is bounded and then \(c(\mathcal {U}, \mathcal {X})\) denotes (ad-hoc) its norm in \(\mathcal {L}(\mathcal {X}).\) As a consequence, we prove that if \(\sup \nolimits _{s\ge 0}\int \nolimits _{s}^\infty \Vert U(t,s)\Vert dt=u_1(\mathcal {U})<\infty ,\) then \(\omega _0(\mathcal {U})u_1(\mathcal {U})\le -1.\) Finally, we give an example showing that the accuracy of the estimates may be quite accurate.     

Facile synthesis of silver core - silica shell composite nanoparticles

Combining metal nanoparticles and dielectrics (e.g. silica) to produce composite materials with high dielectric constant is motivated by application in energy storage. Control over dielectric properties and their uniformity throughout the composite material is best accomplished if the composite is comprised of metal core - dielectric shell structured nanoparticles with tunable dimensions. We have synthesized silver nanoparticles in the range of 40-100nm average size using low concentration of saccharide simultaneously as the reducing agent and electrostatic stabilizer. Coating these silver particles with silica from tetraalkoxysilanes has different outcomes depending on the alcoholic solvent and the silver particle concentration. A common issue in solution-based synthesis of core-shell particles is heterogeneous nucleation whereupon two populations are formed: the desired core-shell particles and undesired coreless particles of the shell material. We report the formation of Ag@SiO(2) core-shell particles without coreless silica particles as the byproduct in 2-propanol. In ethanol, it depends on the silver surface area available whether homogeneous nucleation of silica on silver is achieved. In methanol and 1-butanol, core-shell particles did not form. This demonstrates the significance of controlling the tetraalkoxysilane hydrolysis rate when growing silica shells on silver nanoparticles.     

Effect of pH on H2O2 production in the radiolysis of water

The yields of hydrogen peroxide have been measured in the radiolysis of aqueous solutions of acrylamide, bromide, nitrate, and air in the pH range of 1-13. Hydrogen peroxide is the main stable oxidizing species formed in the radiolysis of water, and its long-term yield is found to be very sensitive to the system used in the measurements. Experiments with γ-irradiation combined with model calculations show that the primary yields of hydrogen peroxide are nearly independent of pH in the range of 2-12. Slightly higher primary yields are suggested at very low pH in particular when O(2) is present, while the yields seem to decrease at very high pH. Irradiations were performed with 5 MeV H ions, 5 MeV He ions, and 10 MeV C ions to evaluate the intratrack and homogeneous kinetic contributions to H(2)O(2) formation with different ions. Many of the trends in hydrogen peroxide yields with pH observed with γ-irradiations are observed with irradiation by the heavy ions. The lower yields of radicals in the homogeneous phase with the heavier ions tend to minimize the effects of radicals on the hydrogen peroxide yields at long times.     

Unsteady MHD free convection flow past a vertical permeable flat plate in a rotating frame of reference with constant heat source in a nanofluid

The unsteady magnetohydrodynamic flow of a nanofluid past an oscillatory moving vertical permeable semi-infinite flat plate with constant heat source in a rotating frame of reference is theoretically investigated. The velocity along the plate (slip velocity) is assumed to oscillate on time with a constant frequency. The analytical solutions of the boundary layer equations are assumed of oscillatory type and they are obtained by using the small perturbation approximations. The influence of various relevant physical characteristics are presented and discussed.     

Free convection flow over a truncated cone embedded in a porous medium saturated with pure or saline water at low temperatures

Steady free convection boundary layer about a truncated cone embedded in a porous medium saturated with pure or saline water at low temperatures has been studied in this paper. The governing coupled partial differential equations are solved numerically using a very efficient finite-difference method. Several new parameters arise and the results are given for some specific values of these parameters. The obtained results for a Boussinesq fluid are compared with known results from the open literature and it is shown that the agreement between these results is very good.     

Itinerant antiferromagnetism of α-Titanium

The present study reports magnetic susceptibility, electrical resistivity and specific heat measurements on α-titanium. The experimental results indicate the existence of an anomaly in the temperature dependence of these parameters. The temperature dependence of the specific heat indicates that the anomaly observed in these parameters is a second order phase transition. The similarity between the temperature dependence of the magnetic susceptibility of α-titanium and of chromium suggests that α-titanium is an itinerant-electron antiferromagnet, such as chromium, with the Néel temperature at 276 ± 4 K.     

The magnetic behaviour of diluted CrAl alloys

Magnetic susceptibility as a function of temperature of binary CrAl alloys containing 0.7,1.07,1.35,1.59, 1.79,1.98, 2.18, 2.84, 3.74 and 4.36 at %A1 is investigated. Aluminium, by alloying, acts on the Néel temperature of pure chromium giving rise to a complicated phase diagram. The bahaviour of the magnetic susceptibility as a function of temperature, in the ordered phase, for the samples containing 1.28, 2.18 and 2.84 at %A1 is found to be temperature independent. In the paramagnetic region, for the samples containing between 1.07 and 2.84 at %A1, it is found a decrease of the magnetic susceptibility as a function of temperature.