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Register a new userEvolutio producti infiniti (1-x)(1-xx)(1-x^3)(1-x^4)(1-x^5) etc. in seriem simplicem
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This paper does exactly what the title says it does. It expands the given series to arrive at the familiar pentagonal number expansion, also known as the pentagonal number theorem, and recalls its application to partition numbers. The paper is translated from Eulers Latin original into German.
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The scope of this article is to report very detailed results of the measurements of magnetic relaxation phenomena in the new Cu$_{0.5}$Fe$_{2.5}$O$_{4}$ nanoparticles and known CuFe$_{2}$O$_{4}$ nanoparticles. The size of synthesized particles is (6.5$pm $1.5)nm. Both samples show the superparamagnetic behaviour, with the well-defined phenomena of blocking of magnetic moment. This includes the splitting of zero-field-cooled and field-cooled magnetic moment curves, dynamical hysteresis, slow quasi-logarithmic relaxation of magnetic moment below blocking temperature. The scaling of the magnetic moment relaxation data at different temperatures confirms the applicability of the simple thermal relaxation model. The two copper-ferrites with similar structures show significantly different magnetic anisotropy density and other magnetic properties. Investigated systems exhibit the consistency of all obtained results.
Stripe order in La{2-x}Sr{x}NiO4 beyond x = 1/3 was studied with neutron scattering technique. At low temperatures, all the samples exhibit hole stripe order. Incommensurability epsilon of the stripe order is approximately linear in the hole concentration n_h = x + 2delta up to x = 1/2, where delta denotes the off-stoichiometry of oxygen atoms. The charge and spin ordering temperatures exhibit maxima at n_h = 1/3, and both decrease beyond n_h > 1/3. For 1/3 < n_h < 1/2, the stripe ordering consists of the mixture of the epsilon = 1/3 stripe order and the n_h = 1/2 charge/spin order.
The evolution of the Fermi surface of CeRh$_{1-x}$Co$_x$In$_5$ was studied as a function of Co concentration $x$ via measurements of the de Haas-van Alphen effect. By measuring the angular dependence of quantum oscillation frequencies, we identify a Fermi surface sheet with $f$-electron character which undergoes an abrupt change in topology as $x$ is varied. Surprisingly, this reconstruction does not occur at the quantum critical concentration $x_c$, where antiferromagnetism is suppressed to T=0. Instead we establish that this sudden change occurs well below $x_c$, at the concentration x ~ 0.4 where long range magnetic order alters its character and superconductivity appears. Across all concentrations, the cyclotron effective mass of this sheet does not diverge, suggesting that critical behavior is not exhibited equally on all parts of the Fermi surface.
The polynomial $f_{2n}(x)=1+x+cdots+x^{2n}$ and its minimizer on the real line $x_{2n}=operatorname{arg,inf} f_{2n}(x)$ for $ninBbb N$ are studied. Results show that $x_{2n}$ exists, is unique, corresponds to $partial_x f_{2n}(x)=0$, and resides on the interval $[-1,-1/2]$ for all $n$. It is further shown that $inf f_{2n}(x)=(1+2n)/(1+2n(1-x_{2n}))$ and $inf f_{2n}(x)in[1/2,3/4]$ for all $n$ with an exact solution for $x_{2n}$ given in the form of a finite sum of hypergeometric functions of unity argument. Perturbation theory is applied to generate rapidly converging and asymptotically exact approximations to $x_{2n}$. Numerical studies are carried out to show how many terms of the perturbation expansion for $x_{2n}$ are needed to obtain suitably accurate approximations to the exact value.
Magnetic susceptibility of the isostructural Ce(Ni{1-x}Cu{x})5 alloys (0< x <0.9) was studied as a function of the hydrostatic pressure up to 2 kbar at fixed temperatures 77.3 and 300 K, using a pendulum-type magnetometer. A pronounced magnitude of the pressure effect is found to be negative in sign and to depend strongly and non-monotonously on the Cu content, showing a sharp maximum in vicinity of x = 0.4. The experimental results are discussed in terms of the Ce valence change under pressure. It has been concluded that the fractional occupation of the f-states, which corresponds to the half-integer valence of Ce ion (3.5), is favorable for the valence instability in alloys studied. For the reference CeNi5 compound the main contributions to magnetic susceptibility and their volume dependence are calculated ab initio within the local spin density approximation (LSDA), and appeared to be in close agreement with experimental data.
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