Remarks on Ramanujan’s inequality concerning the prime counting function

In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that π ( x 2 ) < e x log x π ( x e ) \pi \left( {{x^2}} \right) < {{ex} \over {\log x}}\pi \left( {{x \over e}} \right) for x sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor x log x {x \over {\log x}} on its right hand side by the factor x log x - h {x \over {\log x - h}} for a given h, and by replacing the numerical factor e by a given positive α. Finally, we introduce and study inequalities analogous to Ramanujan’s inequality.

Modified Newton identities

The sums of powers with identical exponents of natural, real, or complex numbers, considered as roots of algebraic equation, are expressed directly through the products of coefficients of that equation, starting from the well-known Newton identities. The final Eq. (6) includes the same power of sum of all numbers ± a sum over all partitions of the exponent. Each term of the last sum is the equation coefficients product with the net power keeping the “dimensionality” of the exponent and having a numerical factor, equal to a proper polynomial coefficient, built of exponents of equation coefficients entering the product. The revers Eq. (43) for equation coefficients is also a sum over all partitions of the same exponent with known numerical coefficients. The entering products are built of “commutators-anticommutators of power of sum and sum of powers” (C-A) of the initial sum addends. The numerous identities Eq. (44) for a C-A with an exponent, exceeding the number of C-A sum terms by 2, and similar C-A-s with lower exponents are established.

Dynamics of ferrofluid drop deformations under spatially uniform magnetic fields

We systematically study the shape and dynamics of a Newtonian ferrofluid drop immersed in an immiscible, Newtonian and non-magnetic viscous fluid under the action of a uniform external magnetic field. We obtain the exact equilibrium drop shapes for arbitrary ferrofluids, characterize the extent of deviations of the exact shape from the commonly assumed ellipsoidal shape, and analyse the smoothness of highly curved tips in elongated drops. We also present a comprehensive study of drop deformation for a Langevin ferrofluid. Using a computational scheme that allows fast and accurate simulations of ferrofluid drop dynamics, we show that the dynamics of drop deformation by an applied magnetic field is described up to a numerical factor by the same time scale as drop relaxation in the absence of any magnetic field. The numerical factor depends on the ratio of viscosities and the ratio of magnetic to capillary stresses, but is independent of the nature of the ferrofluid in most practical cases.

The Visibilization of Religious Minorities in Spain

The author presents the increasing visibilization of the multitude of religious possibilities that are currently on offer in Spain and that form such a clear counterpoint to what for centuries was the one and only religion in this country: Catholicism. He proposes a presentation divided into three parts, which analyse the following visibilization factors: 1) The heritage factor, as seen in the multiplication of visible alternatives of all kinds that minority religions put forward; 2) The legal factor, which in Spain divides religious groups into five categories, characterized by differences, inequalities and privileges; 3) The numerical factor, a problematic aspect that relates to the difficulties of religious quantification in Spain, reflective of the complexity of religious identity in this country and the struggle to obtain reliable statistical data.

The stimulated emission of Bremsstrahlung in a fully ionized plasma

The process of induced bremsstrahlung is studied and it is found that anisotropy in the electron distribution can lead to the amplification of radiation. The gain rate is ,Λ where ν is the electron-ion collision frequency, ωp and ω the plasma and radiation frequency, and Λ a numerical factor which is positive only for anisotropic distributions.

LOCAL AND GLOBAL INTEGRABILITY OF SUPERSYMMETRIC ANOMALIES

The consistency conditions are proved to be necessary and sufficient for local integrability of supersymmetric anomalies. For the global integrability to hold the overall numerical factor in the action is found to be discrete. The explicit form of supersymmetric anomalous action depending on superfields of various types is presented. The parity invariance is discussed and the Abelian limit of the action obtained is shown.

On the definition of a close-to-convex function

The standard definition of a close-to-convex function involves a complex numerical factoreiβwhich is on occasion erroneously replaced by1. While it is known to experts in the field that this replacement cannot be made without essentially changing the class, explicit reasons for this fact seem to be lacking in the literature. Our purpose is to fill this gap, and in so doing we are lead to a new coefficient problem which is solved forn=2, but is open forn>2.

The International System of Units (SI)

Le Système International d'Unités (officially designated SI in all languages) provides a logical, interconnected framework for measurements in commerce, industry, and science, including the textile and allied fields. SI is based on only nine elemental units. Seventeen important derived units have special names. Any number of derived units is possible to meet particular needs. SI has only one unit for each type of physical quantity. Prefixes cover a range of 1036 to form multiples and submultiples. SI has explicitly distinct units for mass (the kilogram) and force (the newton). Numerous older units of pressure, energy, and power are superseded by the pascal, the joule, and the watt, respectively. Each equation defining a derived unit contains only the number 1 as the numerical factor. SI has salient advantages because it is a system of units coherent with respect to the system of physical quantities and the equations relating them.

A NUMERICAL FACTOR FOR THE TAXONOMIC SEPARATION OF PTEROSTICHUS PENSYLVANICUS AND P. ADSTRICTUS (COLEOPTERA: CARABIDAE)

A discriminant function is derived which effectively distinguishes Pterostichus pensylvanicus Lec. from P. adstrictus Eschz. The function is Z = −0.0861X + 0.1657Y, where X = total length of the pronotum at the midline, and Y = distance from widest part of the pronotum to its base. If Z is equal to or less than a critical value, Z0 = 0.0607, the specimen is identified as pensylvanicus; if Z is greater than Z0, it is adstrictus. The rate of error of the function for 182 specimens tested was 2%.