Physics use of mathematics

Orthodox physics makes extensive use of number relation mathematics such as mapping, probability, and infinite series. This mathematics is devoid of causative relations. Other scientific disciplines such as medicine and chemistry use causative models. Using causative models would advance physics. Causation should be overtly stated in the mathematics. Causation is linked with emergence philosophy and not reductionism.

Is the F10.7cm - Sunspot Number relation linear and stable?

The F 10.7cm radio flux and the Sunspot Number are the most widely used long-term indices of solar activity. They are strongly correlated, which led to the publication of many proxy relations allowing to convert one index onto the other. However, those existing proxies show significant disagreements, in particular at low solar activity. Moreover, a temporal drift was recently found in the relative scale of those two solar indices. Our aim is to bring a global clarification of those many issues. We compute new polynomial regressions up to degree 4, in order to obtain a more accurate proxy over the whole range of solar activity. We also study the role of temporal averaging on the regression, and we investigate the issue of the all-quiet F 10.7 background flux. Finally, we check for any change in the quiet-Sun F 10.7 - sunspot number relation over the entire period 1947--2015. We find that, with a 4 th -degree polynomial, we obtain a more accurate proxy relation than all previous published ones, and we derive a formula giving standard errors. The relation is different for daily, monthly and yearly mean values, and it proves to be fully linear for raw non-averaged daily data. By a simple two-component model for daily values, we show how temporal averaging leads to non-linear proxy relations. We also show that the F 10.7 background is not absolute and actually depends on the duration of the spotless periods. Finally, we find that the F 10.7cm time series is inhomogeneous, with an abrupt 10.5% upward jump occurring between 1980 and 1981, and splitting the series in two stable intervals. Our new proxy relations bring a strong improvement and show the importance of temporal scale for choosing the appropriate proxy and the F 10.7 quiet-Sun background level. From historical evidence, we conclude that the 1981 jump is most likely due to a unique change in the F 10.7 scientific team and the data processing, and that the newly re-calibrated sunspot number (version 2) will probably provide the only possible reference to correct this inhomogeneity.

An elementary proof of the Eichler–Selberg trace formula

We give a purely algebraic proof of the trace formula for Hecke operators on modular forms for the full modular group{\mathrm{SL}_{2}(\mathbb{Z})}, using the action of Hecke operators on the space of period polynomials. This approach, which can also be applied for congruence subgroups, is more elementary than the classical ones using kernel functions, and avoids the analytic difficulties inherent in the latter (especially in weight two). Our main result is an algebraic property of a special Hecke element that involves neither period polynomials nor modular forms, yet immediately implies both the trace formula and the classical Kronecker–Hurwitz class number relation. This key property can be seen as providing a bridge between the conjugacy classes and the right cosets contained in a given double coset of the modular group.

Turbulence decay in the density-stratified intracluster medium

Turbulence evolution in a density-stratified medium differs from that of homogeneous isotropic turbulence described by the Kolmogorov picture. We evaluate the degree of this effect in the intracluster medium (ICM) with hydrodynamical simulations. We find that the buoyancy effect induced by ICM density stratification introduces qualitative changes to the turbulence energy evolution, morphology, and the density fluctuation–turbulence Mach number relation, and likely explains the radial dependence of the ICM turbulence amplitude as found previously in cosmological simulations. A new channel of energy flow between the kinetic and the potential energy is opened up by buoyancy. When the gravitational potential is kept constant with time, this energy flow leaves oscillations to the energy evolution, and leads to a balanced state of the two energies where both asymptote to power-law time evolution with slopes shallower than that for the turbulence kinetic energy of homogeneous isotropic turbulence. We discuss that the energy evolution can differ more significantly from that of homogeneous isotropic turbulence when there is a time variation of the gravitational potential. Morphologically, ICM turbulence can show a layered vertical structure and large horizontal vortical eddies in the central regions with the greatest density stratification. In addition, we find that the coefficient in the linear density fluctuation–turbulence Mach number relation caused by density stratification is in general a variable with position and time.

PENANAMAN PEMAHAMAN HUBUNGAN ANTARA BILANGAN PADA SISWA BARU SEKOLAH DASAR MELALUI PEMBELAJARAN PMRI

Students who have a good number sense can think and reason flexibly use numbers to solve problems, find answers that don't make sense, understand how numbers can be separated and put together in different ways, see connections between number operations, do mental calculations, and make reasonable estimates. Whereas, on the other hand, students with a poor number sense tend to depend on procedures rather than reasons, often not paying attention when answers or estimates don't make sense and have limited numerical reasoning. Therefore planting an understanding of numbers with the right method must be done early. This study aims to know how can PMRI approach context fabel construct early elementary school students understanding in finding number relation. Design research method used to reach this goal and tested in SDN Angkasa Penfui with 36 early grade 1 elementary school students as the subject. The results show that the PMRI approach using fable as a context construct students understanding in finding number relation flexibly.

Thermal transfer in Rayleigh–Bénard cell with smooth or rough boundaries

Several Rayleigh–Bénard experiments in water are performed with smooth or rough boundaries. We present new thermal transfer measurements obtained with large roughness elements arranged in a square lattice. The data are compared to previous data obtained with smaller elements in the same cell (Tisserand et al., Phys. Fluids, vol. 23, 2011). Experiments in the same apparatus without roughness are presented, as reference results, to allow for comparison. In the rough case, several regimes of heat transfer are identified: one similar to the smooth case, an enhanced heat transfer regime characterized by a modification of the Nusselt versus Rayleigh number relation and a third part where the relation can be similar to a smooth one with a corrected prefactor.

Solutions of the cubic Fermat equation in ring class fields of imaginary quadratic fields (as periodic points of a 3-adic algebraic function)

Explicit solutions of the cubic Fermat equation are constructed in ring class fields [Formula: see text], with conductor [Formula: see text] prime to [Formula: see text], of any imaginary quadratic field [Formula: see text] whose discriminant satisfies [Formula: see text] (mod [Formula: see text]), in terms of the Dedekind [Formula: see text]-function. As [Formula: see text] and [Formula: see text] vary, the set of coordinates of all solutions is shown to be the exact set of periodic points of a single algebraic function and its inverse defined on natural subsets of the maximal unramified, algebraic extension [Formula: see text] of the [Formula: see text]-adic field [Formula: see text]. This is used to give a dynamical proof of a class number relation of Deuring. These solutions are then used to give an unconditional proof of part of Aigner’s conjecture: the cubic Fermat equation has a nontrivial solution in [Formula: see text] if [Formula: see text] (mod [Formula: see text]) and the class number [Formula: see text] is not divisible by [Formula: see text]. If [Formula: see text], congruence conditions for the trace of specific elements of [Formula: see text] are exhibited which imply the existence of a point of infinite order in [Formula: see text].